Eguchi-Hanson singularities in U(2)-invariant Ricci flow

We show that a Ricci flow in four dimensions can develop singularities modeled on the Eguchi-Hanson space. In particular, we prove that starting from a class of asymptotically cylindrical $U(2)$-invariant initial metrics on $TS^2$, a Type II singularity modeled on the Eguchi-Hanson space develops in finite time. Furthermore, we show that for these Ricci flows the only possible blow-up limits are (i) the Eguchi-Hanson space, (ii) the flat $\mathbb{R}^4 /\mathbb{Z}_2$ orbifold, (iii) the 4d Bryant soliton quotiented by $\mathbb{Z}_2$, and (iv) the shrinking cylinder $\mathbb{R} \times \mathbb{R} P^3$. As a byproduct of our work, we also prove the existence of a new family of Type II singularities caused by the collapse of a two-sphere of self-intersection $|k| \geq 3$.

The main result of this paper is to show A Ricci flow on a four dimensional non-compact manifold may develop a Type II singularity modeled on the Eguchi-Hanson space in finite time.
The Eguchi-Hanson space is diffeomorphic to the cotangent bundle of the two-sphere and asymptotic to the flat cone R 4 /Z 2 . It is the simplest example of a Ricci flat asymptotically locally euclidean (ALE) manifold and in the physics literature known as a gravitational instanton. The Eguchi-Hanson singularities constructed in this paper are the first examples of orbifold singularities in Ricci flow, and are also the first examples of singularities with Ricci flat blow-up limits. As a byproduct of our work we also show that A Ricci flow on a four dimensional non-compact manifold may collapse an embedded two-dimensional sphere with self-intersection k ∈ Z to a point in finite time and thereby produce a singularity.
The singularities we construct when |k| ≥ 3 are of Type II and the author conjectures that their blow-up limits are homothetic to the steady Ricci solitons found in [A17].
1.1. Background. A family of time-dependent metrics g(t) on a manifold M is called a Ricci flow if it solves the equation Here Ric g(t) is the Ricci tensor of the metric g(t). In local coordinates the Ricci flow equation can be written as a coupled system of second order non-linear parabolic equations. Heuristically speaking, the Ricci flow smoothens the metric g(t), while simultaneously shrinking positively curved and expanding negatively curved directions at each point of the manifold. Ricci flow was introduced by Hamilton [Ham82] in 1982 to prove that a closed three dimensional manifold admitting a metric of positive Ricci curvature also admits a metric of constant positive sectional curvature. This success demonstrated the power of Ricci flow and ignited much research in this area, culminating in Perelman's proof of the Poincaré and Geometrization Conjectures for three dimensional manifolds.
Even though every complete Riemannian manifold (M, g) of bounded curvature admits a short-time Ricci flow starting from g, singularities may develop in finite time. Understanding their geometry is central to the study of Ricci flow and has topological implications. For instance, Perelman proved the Geometrization Conjecture by analyzing the singularity formation in three dimensional Ricci flow and showing that a Ricci flow nearing its singular time exhibits one of the following two behaviors: • Extinction: The manifold becomes asymptotically round before shrinking to a point • (Degenerate or non-degenerate) Neckpinch: A region of the shape of a small cylinder R × S 2 develops Based on this knowledge Perelman was able to construct a Ricci flow with surgery, which performs the decomposition of a three manifold into pieces corresponding to the eight Thurston geometries, yielding a proof of the Geometrization Conjecture. In order to understand the formation of singularities in Ricci flow it is very useful to take blow-up limits. Roughly speaking one zooms into the region in which the singularity is forming by parabolically rescaling space and time. The resulting blow-up limit is an ancient Ricci flow called the singularity model. It encapsulates most of the geometric information of the singularity. Note that a Ricci flow is called ancient if it can be extended infinitely into the past. To date all known singularity models are either shrinking or steady Ricci solitons. These are self-similar solutions to the Ricci flow equation that, up to diffeomorphism, homothetically shrink or remain steady, and can be understood as a generalization of Einstein manifolds of positive or zero scalar curvature, respectively. Hamilton distinguishes between Type I and Type II singularities, depending on the rate at which the curvature blows up to infinity as one approaches the singular time. It has been proven that Type I singularities are modeled on shrinking Ricci solitons [EMT11], however it is unknown whether all Type II singularity models are steady Ricci solitons. In three dimensions the only Type I singularity models are S 3 , R × S 2 and their quotients.
As three dimensional singularity formation is now well understood the next step is to analyze the four dimensional case, where currently very little is known other than that the possibilities are far more numerous. Below we list all Type I singularity models known in four dimensions: (1) S 3 × R and its quotients (2) S 2 × R 2 and its quotients (3) Einstein manifolds of positive scalar curvature (e.g. S 4 , CP 2 , etc.) (4) Compact gradient shrinking Ricci solitons that are not Einstein (5) The FIK shrinker [FIK03] Note that (1) and (2) are just products of a three dimensional Type I singularity model with the real line. As Einstein manifolds in four dimensions remain to be classified, list item (3) may contain a very large set of manifolds. As for (4), to date only few examples of compact shrinking Ricci solitons are known and a list of these can be found in Cao's survey [Cao10]. The FIK shrinker is a non-compact U (2)invariant shrinking Kähler-Ricci soliton, which is diffeomorphic to the blow-up of C 2 at the origin. It is an open question whether there are other non-flat one-ended shrinking Ricci solitons in four dimensions. Maximo proved that Type I singularities modeled on the FIK shrinker may occur in U (2)-invariant Kähler-Ricci flow [M14].
The FIK shrinker models an interesting singularity in four dimensional Ricci flow -namely the collapse of an embedded two-dimensional sphere with non-trivial normal bundle. Topologically, real rank 2 vector bundles over the two-dimensional sphere are classified by their Euler class, which is an integer multiple of the generator of H 2 (S 2 , Z). We call this multiple the twisting number and denote it by k.
Recall that the self-intersection of an embedded two-dimensional sphere in a fourdimensional manifold is equal to the twisting number of its normal bundle. Unlike in Kähler geometry, where there is a canonical choice for the generator of H 2 (S 2 , Z) and the sign of the self-intersection number is crucial, in the Riemannian case only its absolute value affects the geometry and behavior of embedded two-spheres under Ricci flow. Heuristically speaking, the larger |k|, the more negative curvature there is in the vicinity of the sphere and the less likely it collapses to a point. In the list above S 2 × R 2 and the FIK shrinker model the collapse of two dimensional spheres with self-intersection equal to 0 and −1, respectively. The main goal of this paper is to show that embedded spheres of self-intersection number |k| ≥ 2 may also collapse in finite time. To explain this in greater detail we give an overview of our setup below.
1.2. Overview of setup. Let M k , k ≥ 1, be diffeomorphic to the blow-up of C 2 /Z k at the origin, and denote by S 2 o the two-sphere stemming from the blow-up. Alternatively one can also view M k as a plane bundle over S 2 o . Fix an arbitrary point o, for 'origin', on S 2 o . Note that S 2 o , with respect to the orientation inherited from C 2 , has self-intersection −k. Then equip M k with an U (2)-invariant metric g.
It turns out that with help of the Hopf fibration π : S 3 /Z k → S 2 such U (2)-invariant metrics can be conveniently written as a warped product metric of the form (1.2) g = ds 2 + a 2 (s)ω ⊗ ω + b 2 (s)π * g S 2 ( 1 2 ) , on the open dense subset R >0 × S 3 /Z k ⊂ M k . ω is the 1-form dual to the vertical directions of the Hopf fibration and s is a parametrisation of the R >0 factor. Note that g pulls back to a Berger sphere metric on the cross-sections S 3 /Z k . One can complete g to a smooth metric on all of M k by requiring and that a(s) and b(s) can be extended to an odd and even function around s = 0, respectively. Via the boundary condition a s (0) = k is how topology enters the analysis of the Ricci flow equation. We would like to mention here that throughout this paper we often take the warping functions a and b to be functions of points (p, t) in spacetime rather than of (s, t). This will always be clear from context. An upshot of writing the metric g in the form (1.2) is that the Ricci flow equation (1.1) reduces to a (1 + 1)-dimensional system of parabolic equations for the warping functions a and b, which simplifies the analysis greatly. In addition to this, both the FIK shrinker, which is diffeomorphic to M 1 , and the Eguchi-Hanson space, which is diffeomorphic to M 2 , are U (2)-invariant, and therefore their metrics can be written in the form (1.2). In this paper we only study Riemannian manifolds diffeomorphic to M k , k ∈ N, equipped with a U (2)-invariant metric of the form (1.2).
We will consider numerous scale-invariant quantities, the most fundamental and important of which we introduce here: The quantity Q measures the 'roundness' of the cross-sectional S 3 /Z k . That is, when Q = 1 the metric on the cross-section is round. The quantities x and y are more interesting, as they measure the deviation of the metric g from being Kähler. In particular, when y = 0 the manifold (M k , g), k ≥ 1, is Kähler with respect to the standard complex structure induced from C 2 . Moreover, when x = y = 0 the manifold (M k , g) is hyperkähler, and as we show in section 4, homothetic to the Eguchi-Hanson space.
1.3. Overview of results. The first main result of this paper is to show that For a large class of U (2)-invariant asymptotically cylindrical initial metrics on M k , k ≥ 2, the Ricci flow develops a Type II singularity in finite time, as the area of S 2 o decreases to zero. When k = 2 the blow-up limit of the singularity is homothetic to the Eguchi-Hanson space.
We define the class of metrics for which this result holds in subsection 1.4. Note that in the k = 2 case the Eguchi-Hanson space is the first example of a Ricci flat singularity model. Based on numerical simulations the author believes that the Type II singularities in the k ≥ 3 case are modeled on the steady solitons found in [A17]. A paper on the numerical results is in preparation.
The above result should be contrasted with the behavior of a Ricci flow starting from a Kähler metric. It is well known that the Kähler condition is preserved by Ricci flow, and that for such a flow the area of a complex submanifold evolves in a fixed manner. In particular, if (M, g) is a Kähler manifold with Kähler form ω, then under Ricci flow the Kähler class evolves by . This shows that for a Kähler-Ricci flow (M k , g(t)), k ∈ N the two sphere S 2 o can only collapse to a point when k = 1. In fact, when k = 2 the area of S 2 o is stationary and for k ≥ 3 increasing. Maximo in [M14] uses the Kähler condition and (1.4) to show that an embedded sphere of self-intersection −1 may collapse to a point in finite time under Ricci flow. Note that in our construction the metrics are not assumed to be Kähler and hence we cannot rely on (1.4).
The second main result of this paper is the classification of all possible blow-up limits in the k = 2 case, including those at larger distance scales from the tip of M 2 . In particular, we show that For a large class of U (2)-invariant asymptotically cylindrical initial metrics on M 2 any sequence of blow-ups subsequentially converges to one of the following spaces: (i) The Eguchi-Hanson space (ii) The flat R 4 /Z 2 orbifold (iii) The 4d Bryant soliton quotiented by Z 2 (iv) The shrinking cylinder R × RP 3 The blow-up limits (ii) and (iii) show that the Eguchi-Hanson singularity results in the formation of an orbifold point, which to our knowledge the first concrete example of such in four dimensional Ricci flow.
We expect that many of our methods generalize to the analysis of Ricci flow on other cohomogeneity one manifolds. These are manifolds that admit an action by isometries of a compact Lie group G for which the quotient is one dimensional. The author believes that this work could contribute towards a complete picture of Ricci solitons and ancient Ricci flows on cohomogeneity one manifolds in four dimensions.
1.4. Precise statement of results. Before presenting the main theorems of this paper, we list the definition of a class I of metrics needed to state our results.
Definition 7.2. For K > 0 let I K be the set of all complete bounded curvature metrics of the form (1.2) on M k , k ≥ 1, with positive injectivity radius that satisfy the following scale-invariant inequalities: Q ≤ 1 a s , b s ≥ 0 y ≤ 0 sup a s < K sup |bb ss | < K Denote by I the set of metrics g such that for sufficiently large K > 0 we have g ∈ I K .
For any k ∈ N the set I of metrics on M k is non-empty, as for example the metric on M k defined by a(s) = Q = tanh(ks) b(s) = 1 is contained in I. Moreover, as we prove in Lemma 7.9, the class I K of metrics is preserved by the Ricci flow for sufficiently large K > 0. In our paper we will mainly consider Ricci flows (M k , g(t)), t ∈ [0, T ), starting from an initial metric g(0) ∈ I. Now we list the precise statements of the main results of this paper.
Then g(t) encounters a Type II curvature singularity in finite time T sing > 0 and sup 0≤t<T sing Furthermore, there exists a sequence of times t i → T sing such that the following holds: Consider the rescaled metrics Then (M k , g i (t), o) subsequentially converges, in the pointed Gromov-Cheeger sense, to an eternal Ricci flow (M k , g ∞ (t), o), t ∈ (−∞, ∞). When k = 2 the metric g ∞ (t) is stationary and homothetic to the Eguchi-Hanson metric.
Remark 1.1. We would like to make the following remarks: (1) In Theorem 9.1, case k = 2, we only prove that there exists a blow-up sequence which converges to the Eguchi-Hanson space. In Theorem 12.1 below we extend this result and show that in fact any blow-up around the tip of M 2 is homothetic to the Eguchi-Hanson space.
(2) The initial metric g(0) ∈ I with sup p∈M b(p, 0) < ∞ is asymptotic to R × S 3 /Z k , where S 3 /Z k is equipped with a squashed Berger sphere metric. This is because metrics in I satisfy a s , b s ≥ 0 and Q ≤ 1.
The second main result of our paper is the classification of all possible blow-up limits in the k = 2 case: Theorem 12.1 (Blow-up limits). Let (M 2 , g(t)), t ∈ [0, T sing ), be a Ricci flow starting from an initial metric g(0) ∈ I (see Definition 7.2) with sup p∈M 2 b(p, 0) < ∞. Let (p i , t i ) be a sequence of points in spacetime with b(p i , t i ) → 0. Passing to a subsequence, we may assume that we are in one of the four cases listed below.
Consider the dilated Ricci flows Then (M 2 , g i (t), p i ), t ∈ [−b(p i , t i ) −2 t i , 0], subsequentially converges, in the Cheeger-Gromov sense, to an ancient Ricci flow (M ∞ , g ∞ (t), p ∞ ), t ∈ (−∞, 0]. Depending on the limiting property of the sequence (p i , t i ) we have: (i) M ∞ ∼ = M 2 and g ∞ (t) is stationary and homothetic to the Eguchi-Hanson metric (ii) M ∞ ∼ = R 4 \ {0}/Z 2 and g ∞ (t) can be extended to a smooth orbifold Ricci flow on R 4 /Z 2 that is stationary and isometric to the flat orbifold R 4 /Z 2 (iii) M ∞ ∼ = R 4 \ {0}/Z 2 and g ∞ (t) can be extended to a smooth orbifold Ricci flow on R 4 /Z 2 that is homothetic to the 4d Bryant soliton quotiented by Z 2 (iv) M ∞ ∼ = R × RP 3 and g ∞ (t) is homothetic to a shrinking cylinder Remark 1.2. Note that in Theorem 12.1 we do not prove that all blow-up limits (i)-(iv) occur. In fact, it may turn out that the Eguchi-Hanson singularity is isolated, in which case we would only see blow-up limits (i) and (ii).
As a byproduct of our work we also prove the following two theorems, which are of independent interest. Firstly, we exclude shrinking Ricci solitons in a large class of metrics.
Theorem 6.1 (No shrinker). On M k , k ≥ 2, there does not exists a complete U (2)invariant shrinking Ricci soliton of bounded curvature satisfying the conditions (1) sup p∈M k |b s | < ∞ (2) T 1 = a s + 2Q 2 − 2 > 0 for s > 0 (3) Q = a b ≤ 1 As we show in section 5 the inequalities T 1 > 0 for s > 0 and Q ≤ 1 are preserved by a Ricci flow (M k , g(t)), k ≥ 2, with g(t) ∈ I. For this reason Theorem 6.1 can be used to exclude Type I singularities for such flows.
Secondly, we prove a uniqueness result for ancient Ricci flows on M 2 .
We rely heavily on this result when we investigate all possible blow-up limits of a Ricci flow (M 2 , g(t)) encountering a singularity at S 2 o .
1.5. Outline of paper and proofs. Our paper is organized by sections. Section 2 is preliminary and its goal is to set up in more detail the manifolds and metrics considered in this paper. Here we also derive the full curvature tensor and Ricci flow equation for U (2)-invariant metrics. In section 3 we prove a maximum principle for degenerate parabolic differential equations on M k . Beginning from section 4 we present new results. Below we outline the main results of those sections and their proofs.
Outline of section 4. A key ingredient in our paper are the scale-invariant quantities x = a s + Q 2 − 2 and y = b s − Q, that measure the deviation of a U (2)-invariant metric from being Kähler with respect to two fixed complex structures J 1 and J 2 on M k , k ≥ 1 (see section 4 for the precise definition of J 1 and J 2 ). In particular, a metric is Kähler with respect to J 1 whenever y = 0 and with respect to J 2 whenever x = y = 0.
Interestingly, a U (2)-invariant metric of the form (1.2) is Kähler with respect to J 2 if, and only if, the underlying manifold is diffeomorphic to M 2 and the metric is homothetic to the Eguchi-Hanson metric, as we show in Lemma 4.1. Therefore the quantities x and y can be used to measure how much a metric on M 2 deviates from the Eguchi-Hanson metric -a tool that is indispensable to our analysis. In the later sections we develop methods to control the behavior of x and y under the Ricci flow. This will allow us to prove that certain singularities of Ricci flows (M 2 , g(t)) are modeled on the Eguchi-Hanson space.
In Lemma 4.2 of this section we also derive various properties of the Eguchi-Hanson metric. These are frequently used throughout the paper.
Outline of section 5. The goal of this section is to derive various scale-invariant inequalities that are conserved by Ricci flow. We say that on a Riemannian manifold (M, g) a geometric quantity T g : M → R is scale-invariant if for every point p ∈ M , we have T g (p) = T λg (p) for all λ > 0. The scale-invariance of the inequalities derived is crucial, as it ensures that they pass to blow-up limits and thus also constrain their geometry.
We construct these inequalities from the scale-invariant quantities a s , b s and Q := a b , where a and b are the warping functions of the metric g of the form (1.2). Note that subscript s denotes the derivative with respect to s. The key observation is that the evolution equation of the scale-invariant quantity can be written in the form where C (α,β,γ) is a function of a s , b s and Q. For certain choices of α, β, γ and δ ∈ R one can determine the sign of C (α,β,γ) at a local extremum at which T (α,β,γ) = δ.
Depending on the sign, this allows one to prove, via the maximum principle, that either T (α,β,γ) ≥ δ or T (α,β,γ) ≤ δ is a conserved inequality. One of the conserved inequalities of this form is x ≤ 0, however we derive many others.
In this section we also find conserved inequalities not of the above form. For instance, we show that each of the inequalities listed below are conserved by the Ricci flow: The proof is carried out by applying the maximum principle to their evolution equations or, in the case of a s , b s ≥ 0, to their system of evolution equations. The conserved inequalities Q ≤ 1, y ≤ 0 and a s , b s ≥ 0 are especially important, as they are part of the definition of the class of metrics I mentioned above, and constitute the first step in showing that I is preserved by the Ricci flow.
Outline of section 6. The main result of section 6 is Theorem 6.1, which rules out shrinking solitons on M k , k ≥ 2, within a large class of U (2)-invariant metrics. Before we outline the proof, note that from the evolution equation (2.12) of b under Ricci flow it follows by an application of L'Hôpital's rule that at s = 0 This formula is a generalization of (1.4) to the non-Kähler case, as the area of S 2 o at time t equals b(o, t) 2 π. Hence a shrinking soliton must satisfy which for k ≥ 2 implies that y s < 0 at s = 0. For the proof of Theorem 6.1 we have to rely on the inequality which by Lemma 5.8 is conserved by the Ricci flow. In particular, we show that amongst metrics of the form (1.2) on M k , k ≥ 2, satisfying Q ≤ 1, T 1 > 0 when s > 0, and sup p∈M k |b s | < ∞ there are no shrinking solitons. We briefly sketch the proof here: First we show in Lemma 6.4 that Q s ≥ 0 for shrinking solitons. This follows from the Ricci soliton equation, which for metrics of the form (1.2) reduces to a system of ordinary differential equations. Then we consider the evolution equation (1.6) ∂ t y = y ss + a s a y − y a 2 G, of y, where G is a function of a s , b s and Q. In Lemma 6.5 we show that whenever Q s , T 1 > 0 we have G > 0. This shows that under Ricci flow satisfying these inequalities a negative minimum of y is strictly increasing and a positive maximum is strictly decreasing. However, since y is a scale-invariant quantity, and a shrinking Ricci soliton, up to diffeomorphism, homothetically shrinks under Ricci flow, we see that the maximum or minimum of y must remain constant throughout the flow. We conclude that y = 0 everywhere, excluding a shrinking soliton. In the proof of Theorem 6.1, rather than working with the evolution equation (1.6) of y, we use the corresponding ordinary differential equation on a Ricci soliton background.
Outline of section 7. The goal of this section is to prove Theorem 7.5, which states that for a Ricci flow (M k , g(t)), k ≥ 1, starting from an initial metric g(0) ∈ I with sup p∈M k b(p, 0) < ∞ there exists a C 1 > 0 such that the curvature bound The proof is carried out by a contradiction/blow-up argument: Assume there exists a sequence of numbers D i → ∞ and points (p i , t i ) in spacetime such that Consider the rescaled metrics Then Perelman's no-local-collapsing theorem shows that (M k , g i (t), p i ) subconverges to an ancient non-collapsed Ricci flow (M ∞ , g ∞ (t), p ∞ ), t ≤ 0. As D i → ∞ the warping functions b i corresponding to the metrics g i (t) satisfy b i (p i , 0) → ∞. Recalling that the warping function b describes the size of the base manifold S 2 in the Hopf fibration of the S 3 /Z k crosssections, one can see that (M ∞ , g ∞ (t)) splits as M ∞ = R 2 ×N , where R 2 is equipped with the flat euclidean metric and the restriction of g ∞ (t) to N is a 2d non-compact κ-solution. However, the only κ-solutions in 2d are either the shrinking sphere or its Z 2 quotient, both of which are compact. This is a contradiction and the proof of the curvature bound follows. In Corollary 7.6 we show that ancient Ricci flows in I, which are κ-non-collapsed at all scales, also satisfy the curvature bound This curvature bound will be important in section 11.
Outline of section 8. In this section we prove various local and global compactness results for U (2)-invariant Ricci flows in the class of metrics I. To state the results we need to first introduce the following notation for a U (2)-invariant Riemannian manifold (M, g): • Let Σ p ⊂ M denote the orbit of p under the U (2)-action.
• Let C g (p, r) := q ∈ M d g (q, Σ p ) < r One sees that C g (p, r) is the tubular neighborhood of 'radial width' r of the orbit Σ p of p under the U (2)-action. See Definition 2.1 for more details.
The main result of this section is Theorem 8.1, which states under which conditions a sequence of U (2)-invariant Ricci flows of the form (C g i (0) (p i , r), g i (t), p i ), t ∈ [−∆t, 0], ∆t > 0, r > 0, subsequentially converges, in the Cheeger-Gromov sense, to a limiting U (2)-invariant Ricci flow ( We also show that after choosing suitable coordinates the warping functions of the metrics g i (t) subsequentially converge to the corresponding warping functions of g ∞ (t). The compactness result of Theorem 8.1 is used frequently throughout the paper, especially its variation, stated in Proposition 8.3.
Outline of section 9. The main goal of this section is to constrain the geometry of ancient Ricci flows (M k , g(t)), k ∈ N, t ∈ (−∞, 0] in the class of metrics I that are κ-non-collapsed at all scales. This is achieved by proving that various scale-invariant inequalities hold. For instance, in Theorem 9.1 we prove that three inequalities of the form T (α,β,γ) ≥ 0, as in introduced in the outline of section 5 above, hold on such ancient flows. Furthermore, we prove in Theorem 9.2 that an ancient Ricci flow on M 2 in I which is Kähler with respect to J 1 , i.e. y = 0 everywhere, is stationary and homothetic to the Eguchi-Hanson space. This result will be used in section 10, where we construct an eternal blow-up limit of a Ricci flow on M 2 that is homothetic to the Eguchi-Hanson space.
The proof of these theorems is via a contradiction/compactness argument frequently employed throughout the paper. We briefly sketch the method here: Assume we want to prove that a scale-invariant inequality T ≥ 0 holds on M k × (−∞, 0]. We argue by contradiction and assume that We then take a sequence of points (p i , t i ) in spacetime such that T (p i , t i ) → ι as i → ∞, and consider the dilated metrics 2 ) (see Definition 2.1) for some small ∆t > 0. By the compactness results of section 8, in particular Proposition 8.3, the Ricci flows (C g i (0) (p i , 1 2 ), g i (t), p i ), [−∆t, 0], subsequentially converges to a Ricci flow by the scale invariance of T . If, however, the evolution equation of T precludes a negative infimum from being attained, we have arrived at a contradiction and proven the desired result.
Outline of section 10. The goal of this section is to prove Theorem 10.1, which states that a Ricci flow (M k , g(t)), k ≥ 2, starting from an initial metric g(0) ∈ I with sup p∈M k b(p, 0) < ∞ encounters a Type II singularity in finite time at the tip of M k as the area of S 2 o decreases to zero. In the k = 2 case we show that such a singularity possesses a blow-up limit that is stationary and homothetic to the Eguchi-Hanson space. We do not further investigate the k ≥ 3 case, however the author conjectures that their blow-up limits are homothetic to the steady Ricci solitons found in [A17].
The proof is carried out in multiple steps. First we show in Lemma 10.5 that g(t) encounters a singularity in finite time T sing ∈ (0, ∞) and b(o, t) → 0 as t → T sing . This shows that the two-sphere S 2 o at the tip of M k collapses to a point in finite time and thereby produces a singularity.
In the second step, we rely on the results of section 6 to show that a blow-up limit around o ∈ S 2 o cannot be a shrinking Ricci soliton when k ≥ 2. As all Type I singularities are modeled on shrinking Ricci solitons we deduce that the singularity is of Type II.
In the third step we borrow a trick due to Hamilton to pick a sequence of times t i → T sing such that the following holds: Take the rescaled metrics In the final step we analyze the geometry of M ∞ when k = 2. It turns out that for the choice of times t i it follows that Applying a strong maximum principle we deduce that y = 0 everywhere. By the results of section 9 it then follows that g ∞ (t) is stationary and homothetic to the Eguchi-Hanson metric.
We mention here that the k = 2 case of Theorem 10.1 is superseded by Corollary 11.2 of Theorem 11.1. However, since the proof of Theorem 10.1 is simpler we present it here.
Outline of section 11. The goal of this section is to show that an ancient Ricci flow (M 2 , g(t)), t ∈ (−∞, 0], which is κ-non-collapsed at all scales and satisfies g(t) ∈ I, is stationary and homothetic to the Eguchi-Hanson space. The most important consequence of this is that in Theorem 10.1 in fact any blow-up of the singularity forming at the tip of M 2 is homothetic to the Eguchi-Hanson space, whereas we had previously only proven that there exists a blow-up sequence that converges to the Eguchi-Hanson space.
The proof idea, which we call successive constraining, is to find a continuously varying family of preserved inequalities Z θ ≥ 0, θ ∈ [0, 1], for which Z 0 ≥ 0 on M 2 × (−∞, 0] implies that g(t) is homothetic to the Eguchi-Hanson metric. For our choice of conserved inequalities Z θ ≥ 0, θ ∈ [0, 1], it follows from the work of section 9 that Z 1 ≥ 0 on M 2 × (−∞, 0]. Then we deform the inequality Z 1 ≥ 0 along the path Z θ ≥ 0, θ ∈ [0, 1], to the inequality Z 0 ≥ 0 with help of the strong maximum principle applied to the evolution equation of Z θ . This allows us to deduce that g(t) is stationary and homothetic to the Eguchi-Hanson space. In subsection 11.1 we give a more detailed outline of the proof of Theorem 11.1.
Outline of section 12. The main result of this section is Theorem 12.1, which characterizes all the possible blow-up limits of a Ricci flow (M 2 , g(t)) starting from an initial metric g(0) ∈ I with sup p∈M 2 b(p, 0) < ∞. We show that the only possible blow-up limits are (i) the Eguchi-Hanson space, (ii) the flat orbifold R 4 /Z 2 , (iii) the 4d Bryant soliton quotiented by Z 2 and (iv) the shrinking cylinder R × RP 3 .
Below we give a brief outline of the proof of Theorem 12.1: Assume we are given a sequence of points (p i , t i ) in spacetime with b(p i , t i ) → 0. Consider the rescaled metrics By passing to a subsequence we may assume that either By section 11 we already know that in case (I) we converge to the Eguchi-Hanson space. Therefore we only need to investigate the behavior in case (II), i.e. at scales larger than the forming Eguchi-Hanson singularity. For this we need to divide case (II) into three subcases: By passing to a subsequence we may assume that For (II.a) and (II.c) we show in Lemma 12.9 and Lemma 12.6 that (M 2 , g i (t), p i ) subsequentially converges to the flat orbifold R 4 /Z 2 and the shrinking cylinder R × RP 3 , respectively. The proof of these lemmas is relatively easy. Proving in Lemma 12.8 that the blow-up limit in case (II.b) is homothetic to the 4d Bryant soliton quotiented by Z 2 is trickier. Here we rely on Lemma 12.3, which characterizes the geometry of the high curvature regions of g(t) at distance scales larger than the Eguchi-Hanson singularity away from the tip of M 2 . In subsection 12.1 we give a more detailed outline of the proof of Theorem 12.1.
1.6. Further questions and conjectures. In this section we collect some conjectures and further questions that arise from our results. The central open question remaining in this paper is whether or not the Eguchi-Hanson singularity of Theorem 12.1 is isolated. By isolated we mean that the only blow-up limits are the Eguchi-Hanson space and its asymptotic cone, the flat orbifold R 4 /Z 2 . We conjecture that Conjecture 1. The Eguchi-Hanson singularity of Theorem 12.1 is not isolated and all four blow-up limits (i)-(iv) occur. In particular, it is accompanied by a Type I singularity modeled on the shrinking cylinder R × S 3 /Z 2 .
An affirmative answer to this conjecture would provide evidence for a longstanding conjecture in Ricci flow stating that a Type II singularity is always accompanied by a Type I singularity in its vicinity. The author has an argument showing that if the Eguchi-Hanson singularity were isolated, the curvature would blow up at a rate faster than (T sing − t) −λ , where λ is any positive constant.
Although we have not analyzed the blow-up limits of a U (2)-invariant Ricci flow (M k , g(t)), t ∈ [0, T sing ), in the k = 2 case, we believe that for each k ∈ N there exists a unique blow-up limit of the singularity arising from the collapse of the two sphere S 2 o at the tip of M k . In collaboration with Jon Wilkening the author has already conducted numerical simulations confirming this, and a paper is in preparation [AW19]. In particular, we conjecture that Conjecture 2. Let (M k , g(t)) be a U (2)-invariant Ricci flow encountering a singularity at the tip of M k , as the area of S 2 o decreases to zero. Then the following picture holds: By isolated we mean that the singularity is not accompanied by a Type I singularity in its vicinity. For instance, in the case k ≥ 2 we expect a singularity caused by the collapse of the two-sphere S 2 o at the tip of M k to always be accompanied by a Type I singularity modeled on the shrinking cylinder R × S 3 /Z k and therefore not to be isolated. If for each k ≥ 3 the corresponding steady Ricci soliton of [A17] is in fact the unique blow-up limit at the tip of M k , then these singularities are necessarily accompanied by a Type I singularity modeled on S 3 /Z k , because these solitons are asymptotically cylindrical.
Another interesting question is the following: Question 1. Can the Eguchi-Hanson singularity occur on a closed four dimensional manifold?
The author conjectures that the answer is yes, however only non-generically. The simplest model on which to investigate this question is M = M 2 # RP 3 M 2 ∼ = S 2 × S 2 equipped with an U (2)-invariant metric. One could carry out a construction as follows: Vary between an initial metric that encounters a R × RP 3 neckpinch singularity and an initial metric that leads to the collapse of one of the S 2 factors of M . On the path between these two metrics there should be a metric whose Ricci flow evolution forms an Eguchi-Hanson singularity in finite time.
This paper has not touched upon the behavior of a general non-U (2)-invariant metric on T S 2 . A first question would be: Question 2. Does the picture of Theorem 12.1 also hold for Ricci flows starting from non-U (2)-invariant perturbations of asymptotically cylindrical U (2)-invariant metrics on T S 2 ?
And a final big question mark is the following:  [KronI89], [KronII89]. These spaces have one end that is asymptotic to the cone R 4 /Γ, where Γ ⊂ U (2) is a certain finite group -a binary dihedral, tetrahedral, octahedral or icosahedral group. In the case that Γ = Z k is cyclic, Gibbons and Hawking [GH78], [GH79] discovered a closed form (3k − 6)parameter family of such metrics. In the physics literature these metrics are known as multi-center Eguchi Hanson spaces. It would be interesting to see whether the results of this paper can be generalized to prove the existence of singularities modeled on these spaces. 1.7. Acknowledgments. The author would like to thank his PhD advisors Richard Bamler and Jon Wilkening for their constant support and encouragement. This work was supported by a GSR fellowship, which was funded by NSF grant DMS-1344991.

Preliminaries
2.1. Notation. Here we collect some of the notation used throughout the paper.
• M k , k ∈ N: a manifold diffeomorphic to the blow-up of C 2 /Z k , k ≥ 1, at the origin. • S 2 o : the two-sphere added during the blow-up of C 2 /Z k . • ξ: the radial coordinate on M k or the parametrization of the R >0 factor in the product • s: the geodesic distance from S 2 o , and often considered as a function of ξ and t.
• origin: refers to the point o.
• g: a metric of the form (2.2) or (2.3) unless otherwise stated • d g : the metric distance induced by g.
• g(t): a time dependent family of metrics of the form (2.2) or (2.3). • u, a, b: the warping functions of the metric (2.2). Depending on context these will be considered as functions of (ξ, t), (s, t) or (p, t), where p is a point on M k . • Q := a b . • B g (p, r): the ball centered at p of radius r with respect to the metric g.
• C g (p, r), r > 0: the subset of a cohomogeneity one U (2)-invariant Riemannian manifold (M, g) defined by The set C g (p, r) is diffeomorphic to either M k or R × S 3 /Z k . • C g (p, r): the closure of C g (p, r). • T sing : the singular time of a Ricci flow.
2.2. The manifold and metric. For k ∈ N let M k be diffeomorphic to the blowup of C 2 /Z k at the origin. Denote by S 2 o the embedded two-sphere in M k stemming from the blow-up, and fix some point o for 'origin' on S 2 o . We now describe the U (2)-invariant metrics on M k , k ≥ 1, studied in this paper. Let z 1 , z 2 be the standard coordinates on C 2 and let U (2) act on C 2 by left multiplication. This action descends to M k , k ∈ N. Note that M k can be seen as the total space of the complex line bundle O(−k) via Then U (2) ∼ = U (1) × SU (2) acts on M k in the following way: The action of U (1) rotates the fibres of π and SU (2) acts on the base S 2 o via rotations. Now introduce the Hopf coordinates where ξ > 0, η ∈ [0, π/2] and ψ, φ ∈ [0, 2π). These coordinates descend to M k . In particular, this allows us to endow M k with the radial coordinate ξ : M k → R ≥0 , by continuously extending ξ to S 2 o by taking ξ = 0 on S 2 o . Note that the coordinate ξ is only smooth on M k \ S 2 o . A computation shows that the standard euclidean metric g eucl = dx 2 1 + dy 2 1 + dx 2 2 + dy 2 2 may be written as in Hopf coordinates. The 1-form ω := dψ − cos(2η)dφ is dual to the Hopf fibre directions, or equivalently dual to the vector field generated by the U (1) action. Furthermore (2.1) dη 2 + sin 2 (2η)dφ 2 is the pull-back of the Fubini-Study metric g F S on CP 1 , normalized to have constant sectional curvature equal to 1 4 . From the above we see that the warped-product metric is the most general U (2)-invariant metric on C 2 * and descends to a U (2)-invariant metric on the open dense set C 2 * /Z k ⊂ M k . It will be useful to introduce the change of coordinates defined by ds = u(ξ)dξ and s = 0 at ξ = 0. Then for p ∈ M k the quantity s(p) = d g (p, S 2 o ) describes the radial distance of p from S 2 o . In these coordinates the metric becomes (2.3) where in a slight abuse of notation we consider a and b as functions of s. Depending on the context we will consider a and b either as functions of s or ξ.
The metric g can be extended to a metric on all of M k by taking a(0) = 0 and b(0) > 0. In other words we shrink the Hopf fibre directions to zero as s → 0 or equivalently as we approach S 2 o . Note that for every p ∈ S 2 0 ds 2 + a(s) 2 ω ⊗ ω is the pull-back of g onto the fibre π −1 (p). As U (1) acts on the fibre π −1 (p), we see that π −1 (p) is a union of S 1 orbits and p. Furthermore, such a S 1 orbit in π −1 (p) ⊂ M k is parametrized by 0 ≤ ψ < 2π k and, by the form of the metric (2.3), such an S 1 orbit at radial distance s from S 2 o has a circumference of length 2π k a(s).
we must require that a s (0) = k in order to avoid a conical singularity at S 2 o . This is how the topology of the manifold enters the analysis of the Ricci flow equation. Additionally requiring that a(s) and b(s) can be extended to an odd and even function, respectively, around s = 0 is a sufficient condition for the metric g to be smoothly extendable to all of M k [VZ18]. In the rest of the paper all metrics considered will be of the form (2.2) or equivalently (2.3). In Figure 1 the manifold M k and its metric close to the two sphere S 2 o is schematically depicted. (M k , g), k ∈ N, are cohomogeneity one manifolds, meaning that the generic orbits of the U (2) action are of codimension 1. The generic orbit is also called the principal orbit. The non-generic orbits are called non-principal orbits. In the case of M k the principal orbits are diffeomorphic to S 3 /Z k and the single non-principal orbit is S 2 Figure 1. Diagram of the manifold M k close to the tip and of codimension 2. Below we introduce some notation that we frequently employ throughout the paper: Definition 2.1. Assume (M, g) is a U (2)-invariant cohomogeneity one manifold with principal orbit S 3 /Z k for some fixed k ∈ N and g is a metric of the form (2.3). Let p ∈ M and r > 0. Then • Let Σ p ⊂ M denote the orbit of p under the U (2)-action.
• Let Σ + p be the set of all points q ∈ M that can be joined via path τ to p with g τ , ∂ ∂s ≥ 0.
Note that we have Σ p ∼ = S 3 /Z k if p lies on a principal orbit and Σ p ∼ = S 2 if p lies on a non-principal orbit.
2.3. The connection, Laplacian and curvature tensor. We now compute the connection, Laplacian and curvature tensor for metrics of the form (2.3). To obtain the corresponding expressions for the metric (2.2) use the relation Take the orthonormal basis of T * M . Let e i , i = 1, 2, 3, 4, be the corresponding dual basis of T * M . Define the connection 1-forms θ j i by ∇e i = θ j i e j and the curvature 2-forms Ω j i by R(·, ·)e i = Ω j i e j . With help of Cartan's structure equations one can compute the connection 1-forms and curvature 2-forms. First note that Hence we obtain the connection 1-forms θ i j : from which we can derive the expression for the Laplacian of a U (2)-symmetric function f (s) on M k .
Finally, we may compute the components R ijkl = g (R(e k , e l )e j , e i ) .
of the curvature tensor. Below we list its non-zero components All other components are either determined by the standard symmetries of the curvature tensor or are zero.
Define the time dependent radial distance function s = s(ξ, t) by ds = u(ξ, t)dξ Furthermore the commutation relation holds. In terms of s we can use (2.9) to rewrite the Ricci flow equation in a slightly simpler form Note that the dependence of the right hand side of this system of equations on ξ is hidden in the variable s = s(ξ, t). However we can write the equations in terms of (s, t) by introducing the functionsã and noting that ∂ t a ξ = ∂ tã s + ∂ sã t ∂s ∂t ξ ∂ t b ξ = ∂ tb s + ∂ sb t ∂s ∂t ξ . By slight abuse of notation, however, we will drop the tilde and consider the warping functions a,b and u as functions of either (p, t), p ∈ M k or (ξ, t) or (s, t), depending on context. In (s, t) coordinates the Ricci flow equation reads Whenever we differentiate a function f : M k × [0, T ] with respect to time, unless stated otherwise, assume that the point on the manifold M k is held fixed. If we differentiate with respect to time while holding s fixed we will denote the partial derivative by ∂ t f | s to avoid confusion. Because s is a function of (ξ, t), in general for fixed s 0 > 0 the set {s = s 0 } ⊂ M k is dependent on time. Therefore holding s or ξ fixed during partial differentiation produces very different results.
This following property of the warping functions a and b will be used throughout the paper.
Lemma 2.2. Let (M k , g(t)), t ∈ [0, T ), be a smooth Ricci flow solution. Then for all t ∈ [0, T ] the warping functions a(s, t) and b(s, t) can be extended to an odd and even function, respectively, on R.
Proof. Note that a necessary condition for a metric g of the form (2.3) to be smooth is that its corresponding warping functions a and b are extendable to odd and even functions, respectively, on R. Therefore the desired result follows. Alternatively, notice that if the warping functions of the initial data a(s, 0) and b(s, 0) can be extended to an odd and even function, respectively, on R, we can also extend the equations (2.13), (2.14) and (2.15) to all of R. An inspection of these equations shows that the parity of a and b is preserved under the flow.
2.5. Recap of blow-up limits of singularities. As mentioned above, every complete Riemannian manifold (M, g) of bounded curvature admits a short-time Ricci flow starting from g, however singularities may develop in finite time. Similar to the study of other nonlinear equations, it is very useful to consider blow-up limits of singularities. We briefly sketch the idea here: Assume (M, g(t)), t ∈ [0, T sing ), is a Ricci flow encountering a curvature singularity as t → T sing . Let (p i , t i ) with p i ∈ M and t i → T sing be a sequence of points in spacetime such that and K i → ∞ as i → ∞. Take the rescaled metrics Then (M k , g i (t), p i ) subsequentially converge, in the Cheeger-Gromov sense, to a pointed ancient Ricci flow solution (M ∞ , g ∞ (t), p ∞ ), t ∈ (−∞, 0] (see [ChI,Theorem 6.68] for more details). Note that in general M ∞ = M . A Ricci flow is called ancient if it can be extended to a time interval of the form (−∞, T ), T ∈ R. The blow-up limit (M ∞ , g ∞ (t), p ∞ ) is called the singularity model and yields important geometrical information on the shape of the singularity. Hamilton [Ham95] distinguishes between Type I and II singularities, depending on the rate of curvature blow-up, i.e. for Type I and for Type II sup By the work of Naber [N10], and Enders, Müller and Topping [EMT11] it is known that Type I singularities are modeled on shrinking Ricci solitons. One hopesalthough it has not been proven -that all Type II singularities are modeled on steady solitons, as to date all known examples are.

The maximum principle
Assume we are given a Ricci flow (M k , g(t)), t ∈ [0, T ], k ≥ 1. We make the following definition: In this section we prove a maximum principle for operators ) that in (ξ, t) coordinates and away from the non-principal orbit S 2 o of M k can be written in the form It is useful to work in (s, t) coordinates, in which case the operator P [u] can be expressed as where we recall the expression (2.15) for ∂s ∂t . Note P [u] is degenerate at the origin However, in (s, t) coordinates the smoothness of u and c is equivalent to saying that u(s, t), c(s, t) can be extended to smooth even functions around s = 0 by defining u(s, t) = u(−s, t) and v(s, t) = v(−s, t) for s ≤ 0. Hence we see that u s = c s = 0 at s = 0 and via L'Hôpital's Rule we obtain the following representation of P [u] on the principal orbit S 2 0 : The maximum principle derived for P below depends on the sign of (m + 1): Theorem 3.2. Let (M k , g(t)), k ∈ N, t ∈ [0, T ] be a Ricci flow with bounded curvature. Let P be an operator of the form (3.1) and and there exist constants M, σ > 0 such that the growth conditions u(s, t) ≥ −M exp(−σs 2 ) c(s, t) ≤ M |s| 2 + 1 are satisfied, then the following holds true: Before proving Theorem 3.2 we need to derive some bounds on as a , bs b and 1 b 2 for metrics g of bounded curvature. This will allow us to bound the coefficients appearing in the expression (3.2) of the operator P [u].
Lemma 3.3. Let (M k , g), k ∈ N, and K > 0 such that |Rm g | g ≤ K on M k . Then everywhere on M k we have From the curvature components derived in subsection 2.3 we see that At a local minimum b s = 0 we thus have Now we argue by contradiction. Assume that there exists a s * > 0 and δ > 0 such that at s = s * we have b 2 < 1−δ K . From above it follows that b s < 0 for s ≥ s * and hence, because b > 0 everywhere, lim s→∞ b s = 0. Equation (3.3) then shows that Dividing by bs Q shows that a s → −∞ as s → ∞ contradicting a ≥ 0. This proves the first bound. To prove the second bound note , where the last inequality follows from (1). For the third bound we have Lemma 3.4. Let (M k , g), k ∈ N, and K > 0 such that |Rm g | g ≤ K on M k . Then everywhere on M k we have Proof. The quantity φ = as a s satisfies the ODE dφ ds = s a ss a + φ(1 − φ) s and by L'Hôpital's rule we have φ(0) = 1. Note that the function φ can be extended to an even function on R. Therefore dφ ds (0) = 0 and there exists a small > 0 such that φ ≤ 1 + √ Ks for 0 ≤ s ≤ . Actually the inequality holds for all s ≥ 0, since whenever φ(s) = 1 + √ Ks we have dφ ds = s a ss a − K − √ K < 0, since | ass a | = |R 0101 | ≤ K. this proves the the upper bound of (3.5). To prove the lower bound assume that a s (s 0 ) < 0. For every s 1 > s 0 there exists a s * ∈ (s 0 , s 1 ) such that by the mean value theorem. It follows that This concludes the proof.
Now we may proceed to proving the maximum principle of Theorem 3.2.
Proof of Case 1 of Theorem 3.2. Let K > 0 such that Introduce the new variable r := r(s, t) defined by r(r + 2) = s 2 and let u(r, t) = u( r(r + 2), t). Note that 2r ∼ s 2 for s 1 and r ∼ s for s 1. We make this substitution to remove the apparent singularity at s = 0 in the (s, t) coordinate representation (3.2) of the operator P [u]. The function u is smooth, because u is extendable to an even function around the origin (see [W43]). Rewriting (3.2) in terms of (r, t) coordinates we see that u satisfies the inequality where A, B and C are smooth functions defined by Note that above we regard s as a function of r. Recall that by Lemma 2.2 the functions a and b can be extended to an odd and even function, respectively, around the origin. Therefore the quantity considered as a function of s, can be extended to an even function around the origin by [W43]. Hence this expression depends smoothly on r, showing that B(r, t) is smooth. Similarly, we see that C(r, t) is smooth. From the expression (2.15) for ∂s ∂t and the curvature components listed in subsection 2.3 it follows that By Lemma 3.3 and Lemma 3.4 we hence see that for some some positive constant M depending on K. Finally, noting that A(r) is bounded and positive for r > 0, we can apply [F64, Theorem 9, p.43] to deduce that the weak maximum principle holds. Note that for any compact we may assume that c < 0 on U by performing the transformation u ← ue −γt , for γ = γ(U ) chosen sufficiently large. Therefore the strong maximum principle follows from a slight adaptation of [F64, Theorem 1, p.34].
Proof of Case 2 of Theorem 3.2. We first prove the weak maximum principle. Taking u = ue −γt we see that u satisfies As c is a smooth function of (s, t), we can choose γ sufficiently large such that in a neighborhood of which is a contradiction. The weak maximum principle now follows by the proof of [F64, Theorem 9, p.43].
In this paper we only apply the strong maximum principle for m ∈ N and therefore only prove this case here. For the general case refer to [Fee13,Theorem 5.17]. Given a Ricci flow (M k , g(t)), t ∈ [0, T ], define the corresponding family of rotationally symmetric spaces (R m+1 , h(t)), t ∈ [0, T ], by is the round metric on S m of sectional curvature k 2 . A sufficient condition for h to be smooth at s = 0 is that a is extendable to an odd function around the origin and a s (0) = k. Both these conditions are satisfied and we conclude that h is a smooth metric. The Laplacian of a rotationally symmetric function f on (R m+1 , h) is given by a s a f s and thus the condition P [u] ≤ 0 may be written as Note that for any bounded U ⊂ M k × [0, T ] we may assume that c < 0 on U by performing the transformation u ← ue −γt , for γ = γ(U ) chosen sufficiently large. Hence the desired result follows from [ChII, Theorem 12.40].
Remark 3.5. It was crucial in our analysis that u is extendable to an even function around the origin, as the following example demonstrates: Consider the degenerate parabolic equation a computation shows that the above PDE corresponds to however the solution v to the heat equation becomes positive at some later time t > 0 and x > 0. This shows that u ≤ 0 is not necessarily preserved by (3.8).
In this paper we also rely on a maximum principle for a system of parabolic inequalities on u 1 , We prove the following Lemma: T ]) satisfy the above system (3.10)-(3.11) of parabolic inequalities and for some constants M, σ > 0 Proof. Writing the equation in terms of (r, t) as in the proof of Theorem 3.2 we obtain where A(r), B(r, t) are as in the proof of Theorem 3.2 and After constructing a barrier function of the form

Kähler quantities and the Eguchi-Hanson space
Recall that a complex structure J on a Riemannian manifold (M, g) satisfying (2) ∇J = 0 defines a Kähler structure. On the manifolds M k , k ≥ 1, we define two complex structures J 1 and J 2 by J 1 e 1 = e 0 J 1 e 2 = e 3 and J 2 e 0 = e 2 J 2 e 1 = e 3 . Note that being Kähler with respect to J 2 automatically implies Kählerity with respect to J 1 . This motivates the definition of the following scale-invariant quantities to measure the deviation of a metric from being Kähler with respect to the complex structures J 1 and J 2 . For example, the FIK shrinker [FIK03] is Kähler with respect to the complex structure J 1 and in our notation satisfies y = 0. The Eguchi-Hanson space is the unique Kähler manifold with respect to J 2 as the following lemma shows.
Lemma 4.1. Amongst all Riemannian manifolds (M k , g),k ≥ 1, equipped with U (2)invariant metric g of the form (2.3), up to scaling the Eguchi-Hanson space is the unique Kähler manifold with respect to the complex structure J 2 . Furthermore being Kähler with respect to J 2 is equivalent to x = y = 0.
Proof. By the above discussion being Kähler with respect to J 2 is equivalent to Notice at s = 0 we have forcing the underlying manifold to be diffeomorphic to M 2 by the boundary conditions (1.3). Then in terms of a and b the condition x = y = 0 is equivalent to the first order system of equations Let a E and b E be a solution to this system of equations satisfying the initial conditions Then by the scale-invariance of condition (4.1), for every λ > 0 the metric given by the warping functions λa E (λs) and λb E (λs) also satisfies (4.1). Hence up to rescaling there is a unique Kähler manifold with respect to the complex structure J 2 . From [EH79] or [Cal79] we see that the metric given by a E and b E is homothetic to the Eguchi-Hanson metric.
In the rest of the paper we denote by g E the Eguchi-Hanson metric with warping functions a E and b E normalized such that b E = 1 on S 2 o . Note that the normalization condition is equivalent to saying that the area of the exceptional divisor S 2 o is equal to 2π.
Lemma 4.2. The warping functions a E and b E of the Eguchi-Hanson metric satisfy the following properties Proof. For brevity write a and b for a E and b E , respectively. Note that on the Eguchi-Hanson background we have where the last equality follows from (4.2) and (4.3). As Q = 0 at s = 0 it follows that Q < 1 for ≥ 0 and hence Q s > 0 for s ≥ 0. As both exist. From the system of differential equations (4.2) and (4.3) we then see that a ∞ = b ∞ = 1. This concludes the proof.

Some preserved conditions
In this section we derive various scale-invariant inequalities that are preserved by a Ricci flow (M k , g(t)), t ∈ [0, T ], k ∈ N. The scale-invariance is crucial, as it ensures that the inequalities pass to blow-up limits and therefore also constrain their geometry. The preserved inequalities we list in this section will play an important role in all subsequent sections.

Section Outline. A central quantity in our analysis is
In geometric terms, Q measures the deviation of the cross-sectional S 3 /Z k in M k from being round. That is, when Q = 1 the cross-section is round and as Q → 0 the cross-sectional S 3 /Z k collapses along the S 1 Hopf fibres to a two-sphere. A computation shows that the evolution equation of Q is Therefore one expects that the inequality Q ≤ 1 is preserved by Ricci flow, which in Lemma 5.2 we prove to be the case. Apart from Q, the Kähler quantities x and y introduced in section 4 are used throughout this paper and are one of the key ingredients in showing that certain Ricci flows on M 2 develop singularities modeled on the Eguchi-Hanson space. We show in Lemma 5.3 and Lemma 5.5 that the inequalities are both preserved. Furthermore, using the maximum principle for systems of weakly coupled parabolic equations of Lemma 3.6, we show in Lemma 5.6 that a s , b s ≥ 0 is preserved. In Lemma 5.7 we show that on a Ricci flow background satisfying Q ≤ 1 and y ≤ 0, for any C > 2 the inequality a s ≤ C is preserved. In the following sections we will mainly consider Ricci flows satisfying a s , b s ≥ 0, y ≤ 0, Q ≤ 1 and a s ≤ C. This gives us enough control on a and b to prove many interesting results. Finally, we show that whenever a subset of the inequalities Q ≤ 1, y ≤ 0 and a s , b s ≥ 0 hold, the details of which are discussed below, the following inequalities are preserved by the Ricci flow. The precise statements and proofs of these preserved inequalities can be found in Lemmas 5.8, 5.9, 5.10 and 5.11 below.
The main idea in constructing the above inequalities is to study the evolution equation of the scale-invariant quantities For this we need to compute the evolution equations of a s , Qb s and Q 2 . Recall To simplify the formulae slightly, define the linear operator away from the non-principal orbit S 2 o . As in section 3 we may use L'Hôptital's rule to find a representation of L on the non-principal orbit S 2 o . Then, as we show in the Appendix A, the evolution equations of a s , Qb s and Q 2 can be written as Since the operator L is linear, one sees that T (α,β,γ) satisfies an evolution equation of the form is a function of a s , b s and Q. This evolution equation is very useful, as it allows us to systematically search for preserved inequalities. In particular, if we can find α, β, γ, δ ∈ R for which we can determine the sign of C (α,β,γ) at a local extrema of T (α,β,γ) at which T (α,β,γ) = δ, it follows from the maximum principle of Theorem 3.2 that, depending on the sign, either is a preserved inequality. We searched for real numbers α, β, γ and δ leading to preserved conditions that yield the most useful control of the geometry of the flow. This is how we found the quantities T 1 , T 2 , T 3 and T 4 . In later sections we will make heavy use of each of their respective inequalities. For instance, we use the preserved inequalities T 1 ≥ 0 to exclude shrinking solitons on M k , k ≥ 2, in the next section. Finally, in section 11 we generalize the above idea to find a continuously varying family of conserved inequalities.
Statement and proof of results. In this subsection we list the precise statements and proofs of the results stated in the section outline. Before we begin, we prove the following technical lemma, which we need for verifying the growth conditions of the maximum principle of Theorem 7.5. Ks)).
Proof. By the curvature components listed in section 2.3 we see that from which we conclude that Similarly, Lemma 3.3 shows |b s | ≤ √ 5Kb from which we deduce that ). This concludes the proof. Now we begin proving the conserved inequalities listed above.
Lemma 5.2. Let (M k , g(t)), t ∈ [0, T ], k ≥ 1, be a Ricci flow with bounded curvature. Then the inequality Q ≤ 1 is preserved by the Ricci flow.
Proof. Define the quantityQ = 1 − Q. From the evolution equation (5.1) of Q we see is non-positive. Furthermore, by Lemma 5.1 we have |Q| = o(exp(s 2 )). Therefore we may apply the maximum principle of Theorem 3.2 to deduce thatQ ≥ 0 on M k × [0, T ]. The desired result thus follows.
Proof. The evolution equation of x, as derived in the Appendix A, is Note that |x| = o(exp(s 2 )) by Lemma 5.1. Therefore applying the maximum principle of Theorem 3.2 yields the desired result.
Remark 5.4. Note that x = 2 − k at s = 0 by the boundary conditions (1.3). Therefore the result can only hold for k = 1, 2.
Lemma 5.5. Let (M k , g(t)), t ∈ [0, T ], k ≥ 1, be a Ricci flow with bounded curvature. Then the inequality y ≤ 0 is preserved by the Ricci flow.
Since y is an odd quantity, we consider the quantity Qy = Qb s − Q 2 instead. Its evolution equation is where M 1 is one of the curvature components listed in section 2.3. By Lemma 3.3 we see that for some C > 0 Furthermore |Qy| = o(exp(s 2 )) by Lemma 5.1. Now the result follows from applying the maximum principle of Theorem 3.2.
Proof. The evolution equations (5.3) and (5.4) of a s and Qb s can be written as a system of weakly coupled parabolic equations By Lemma 3.3 and Lemma 3.4 the zeroth order coefficients of a s and Qb s are bounded. Lemma 5.1 shows that |a s |, |b s | = o(exp(s 2 )). Finally, note that the off-diagonal coefficients 8 Q 3 b 2 and 4 Q 2 b 2 are non-negative. Thus the desired result follows by the maximum principle for weakly coupled parabolic equations of Lemma 3.6.
Lemma 5.7. Let (M k , g(t)), t ∈ [0, T ], k ≥ 1, be a Ricci flow with bounded curvature satisfying y ≤ 0, Q ≤ 1 and a s , b s ≥ 0. Then for C ≥ 2 the inequality a s ≤ C is preserved by the Ricci flow.
Proof. Define the quantity A := a s − C. Then from the evolution equation (5.9) of a s it follows that where we used Q ≤ 1 and y = b s − Q ≤ 0. As |A| = o(exp(s 2 )) by Lemma 5.1 it follows from the maximum principle of Theorem 7.5 that the inequality A ≤ 0 is preserved by the Ricci flow. This proves the desired result.
Lemma 5.8. Let (M k , g(t)), t ∈ [0, T ], k ≥ 1, be a Ricci flow with bounded curvature satisfying y ≤ 0, b s ≥ 0 and Q ≤ 1. Then the inequality which can be derived from the evolution equations (5.3), (5.4) and (5.5) for a s , Qb s and Q 2 listed above. Inspecting the quadratic expression we see that when y = 0 it is equal to by the assumptions y ≤ 0, b s ≥ 0 and Q ≤ 1, and furthermore the quadratic expression (5.12) is concave in y, we conclude that ) by Lemma 5.1. Hence the result follows from applying the maximum principle of Theorem 3.2.
Below we prove some further preserved conditions. These can be skipped on the first reading of the paper.
Proof. Note that T 2 = 2−k when s = 0 by the boundary conditions (1.3). Therefore the result can only hold true for k = 1, 2. The evolution equations of T 2 is The coefficients are bounded by Lemma 3.3. Furthermore |T 2 | = o(exp(s 2 )) by Lemma 5.1. Therefore applying the maximum principle of Theorem 3.2 yields the desired result.
Lemma 5.10. Let (M k , g(t)), t ∈ [0, T ], k ≥ 1, be a Ricci flow with bounded curvature satisfying Q ≤ 1. Then the inequality Proof. The evolution equations of T 3 is Note that the coefficients are bounded by Lemma 3.3. Furthermore |T 3 | = o(exp(s 2 )) by Lemma 5.1. Applying the maximum principle of Theorem 3.2 yields the desired result.
Lemma 5.11. Let (M k , g(t)), t ∈ [0, T ], k ≥ 1, be a Ricci flow with bounded curvature satisfying y ≤ 0, b s ≥ 0 and Q ≤ 1. Then the inequality is preserved by the Ricci flow. Here Proof. By Lemma 5.8 we already know that the inequality By the assumption y ≤ 0 we have which implies that Note that the zeroth order coefficient of T 4 is bounded by Lemma 3.3. Furthermore |T 4 | = o(exp(s 2 )) by Lemma 5.1. Applying the maximum principle of Theorem 3.2 yields the desired result.

Exclusion of shrinking solitons
In this section we rule out U (2)-invariant shrinking solitons on M k , k ≥ 2, within a large class of metrics. In particular, we show Theorem 6.1 (No shrinker). On M k , k ≥ 2, there does not exists a complete U (2)invariant shrinking Ricci soliton of bounded curvature satisfying the conditions This theorem is the key ingredient in section 10, where we show that certain Ricci flows on M k , k ≥ 3, develop Type II singularities in finite time.
Soliton equations. Recall that a shrinking Ricci soliton (M, g(t)) is a solution to the Ricci flow equation that up to diffeomorphism homothetically shrinks. Such a soliton solution may be written as , where σ(t) = 1 − 2ρt for some ρ > 0 and Φ t is a family of diffeomorphisms. The reader may consult [Top06] for more details. Hence for a U (2)-invariant shrinking Ricci soliton (M k , g(t)), k ≥ 1, the corresponding warping functions can be written as The above formulae are with respect to the radial coordinate s, which is equivalent to fixing a gauge. For this reason the family of diffeomorphisms Φ t does not appear explicitly. Differentiating with respect to t at time 0 yields where f : M k → R is the potential function satisfying f ss = ρ + a ss a + 2 b ss b and we used the expression (2.15) for ∂s ∂t derived in section 2.4. Similarly we obtain Substituting the expressions ∂ t a and ∂ t b from the Ricci flow equations (2.11) and (2.12), respectively, we see that the soliton equations for the warping functions a and b at time In a slight abuse of notation we will denote a and b as functions of s only when we are considering Ricci solitons. In that case a and b should be interpreted as the initial data a(s, 0) and b(s, 0) at time zero that leads to a Ricci soliton solution, via the correspondence (6.1) and (6.2).
Remark 6.2. The above shows that all U (2)-invariant Ricci solitons on M k are automatically gradient Ricci solitons with potential function f .
Evolution of x, y and Q on soliton background. Since x, y and Q are scaleinvariant quantities, their evolution on a Ricci soliton background can be expressed as follows: , 0 Differentiating, we therefore obtain With help of the evolution equations (5.7), (12.7) and (5.1) for x, y and Q, this yields the following ordinary differential equations for x, y and Q at time zero on a soliton background Alternatively these equations can be derived from the soliton equations (6.3)-(6.5). In a slight abuse of notation we will often denote x, y and Q as functions of s only when we are considering Ricci solitons.
Exclusion of shrinking solitons. By [CZ10] we know that the potential function of a non-compact complete shrinking Ricci soliton grows quadratically with the distance to some fixed point. In our setting this translates into the following lemma: Lemma 6.3. Let (M k , g), k ≥ 1, be a complete non-compact shrinking Ricci soliton of bounded curvature. Then Proof. See Theorem 1.1, equation (2.3) and equation (2.8) of [CZ10].
This allows us to prove the following lemma: Proof. First notice that for a complete shrinking Ricci soliton with Q ≤ 1, the strong maximum principle applied to the evolution equation (5.1) of Q forces as otherwise we would have Q = 1 everywhere, which cannot be the case. Similarly, unless we are at the origin s = 0. By equation (6.8) we have We now argue by contradiction. Assume there exists an s * > 0 such that Q s (s * ) < 0. Then Q s (s) < 0 for all s > s * , because at any extremum of Q we have Q s = 0 and shows that bs b is bounded and from Lemma 6.3 it follows that f s → ∞ as s → ∞.
Therefore eventually from which it follows by equation (6.9) that Q ss < 0 for sufficiently large s. This, however, contradicts that Q > 0 unless s = 0.
In the lemma below we bound the term which appears in the evolution equation (6.7) of y, away from zero.
Now we prove the non-existence of shrinking solitons.
Proof of Theorem 6.1. We argue by contradiction. Assume such a shrinking Ricci soliton exists. Applying L'Hôpital's Rule to the evolution equation (2.12) of b shows that at s = 0 Clearly, every shrinking soliton satisfies The boundary conditions (1.3) of a and b at s = 0 imply that and thus we deduce from the above that as k ≥ 2 by assumption. The ordinary differential equation (6.7) for y can be written as (6.10) y ss = f s − a s a y s + y a 2 G. Lemma 6.4 and Lemma 6.5 imply that G > 0 for s > 0, which in turn shows that y s ≤ 0 everywhere, as at a negative local minimum of y we would have y ss = y a 2 G < 0. The asymptotic properties of f listed in Lemma 6.3 and the bounds on as a proven in Lemma 3.4 show that eventually and hence from the equation (6.10) it follows that y ss < 0 for s sufficiently large. From this it follows that which contradicts our assumptions on b s and Q.

Curvature bound
The aim of this section is to prove that a Ricci flow (M k , g(t)), k ∈ N, t ∈ [0, T ), starting from an initial metric g(0) ∈ I -where I is a class of metrics to be discussed below -with sup p∈M k b(p, 0) < ∞ satisfies the curvature bound where C 1 > 0 is some constant. This allows us to control the geometry via the warping function b, which will be crucial for constructing blow-up limits in the following parts of the paper. Note that this bound was already derived in the compact case in [IKS17] and we will follow their strategy to prove it in our noncompact setting.
Recall the following definition (see also [ChI][Definition 8.23]): , be a Ricci flow and κ > 0. We say that the Ricci flow is κ-non-collapsed at a point (p 0 , t 0 ) in spacetime at scale ρ if the following two conditions hold for all r ≤ ρ: We now define the class of metrics I.
Definition 7.2. For K > 0 let I K be the set of all complete bounded curvature metrics of the form (2.3) on M k , k ≥ 1, with positive injectivity radius that satisfy the following scale-invariant inequalities: Denote by I the set of metrics g such that for sufficiently large K > 0 we have g ∈ I K .
Note that for any k ∈ N the set I of metrics on M k is non-empty, as for example the metric on M k defined by is contained in I. In Lemma 7.9 below we show that if g(0) ∈ I K 0 for some K 0 > 0 then there exists a K > K 0 such that g(t) ∈ I K for t ∈ [0, T ). Note that conditions (7.1)-(7.5) are scale-invariant, and therefore pass to blow-up limits.
An adaptation of [ChI,Theorem 8.26] to our setting yields the following result: be a Ricci flow starting from an initial metric g(0) ∈ I. Then there exists a κ > 0 depending on T , inj(g(0)) and sup M ×[0,T /2] Ric g(t) such that g(t) is κ-non-collapsed at every (p, t) ∈ M × ( T 2 , T ) at every scale ρ < T /2. Remark 7.4. Recall that if a Ricci flow g(t) is κ-non-collapsed at scale ρ, then the parabolically dilated Ricci flow α 2 g(α −2 t) is κ-non-collapsed at scale αρ. As the κ-non-collapsedness property is preserved under Cheeger-Gromov limits, a blow-up limit of a Ricci flow (M k , g(t)), [0, T sing ) is κ-non-collapsed at all scales.
Having set up the necessary terminology, we may now state the main theorem of this section: Then there exists a constant C 1 > 0 such that A useful variant of Theorem 7.5 is: Corollary 7.6. Let (M k , g(t)) with g(t) ∈ I (see Definition 7.2) for t ∈ (−∞, 0] be an ancient Ricci flow solution which is κ-non-collapsed at all scales. Then there exists a constant C 1 > 0 such that Remark 7.7. Corollary 7.6 follows immediately from Theorem 7.5 for ancient κnon-collapsed Ricci flows that arise as blow up limits of Ricci flows (M k , g(t)), t ∈ [0, T sing ), g(0) ∈ I, as the curvature bound is scale-invariant. Nevertheless, we give a proof of the general case.
Let us now prove the assertions made above. We begin with the following lemma: , is a Ricci flow starting from an initial metric g(0) ∈ I K 0 . Then there exists a constant K ≥ 0, depending only on the initial metric g(0), such that Proof. We follow the proof strategy of [IKS17, Lemma 7]. Consider the quantities where C > 0 is a constant to be determined later. The goal is to show that the inequalities H + ≥ 0 and H − ≤ 0 are preserved by the Ricci flow for sufficiently large C > 0. The quantities H ± satisfy the evolution equations In the Appendix A we carry out the derivation of the evolution equation. We now show that H − ≤ 0 is preserved. Using Young's inequality to bound the terms involving a ss and then disregarding non-positive terms not involving C, we obtain and a s ≤ C for some C > 0 by Lemma 5.5, Lemma 5.2, Lemma 5.6 and Lemma 5.7, respectively. Therefore we obtain the following bounds away from the non-principal orbit S 2 o : Hence for a sufficiently large C > 0 it follows that away from the non-principal orbit S 2 o . Switching to coordinates (s, t) we see that for s > 0 On the non-principal orbit S 2 o , or equivalently when s = 0, we have where we used the expression for the curvature component R 0202 derived in section 2.3. This shows that for each time t < T the function H − (s, t) grows subexponentially. Note that by Lemma 3.3 and Lemma 3.4 the coefficient Similarly, we see from the bound (3.7) on | ∂s ∂t | presented in the proof of the maximum principle of Theorem 3.2, Case 1, that the coefficient As T ∈ [0, T ) was arbitrary it follows that H − ≤ 0 is preserved by the Ricci flow.
We repeat the same process to prove that H + ≥ 0 is preserved. Applying Young's inequality to bound terms involving a ss and then disregarding non-negative terms not involving C, we see that Bounding the zeroth order terms via Young's inequality as above, we see that for C > 0 sufficiently large away from the non-principal orbit S 2 o . On the non-principal orbit S 2 o we have where we used that b s ≥ 0 with equality at s = 0 to deduce that b ss ≥ 0 at s = 0. From here the above proof that H − ≤ 0 is preserved carries over and we may conclude that H + ≥ 0 is preserved as well. Recalling the bounds on a s and b s , the desired result now follows. Now we can prove that I (see Definition 7.2) is preserved by Ricci flow: Lemma 7.9. Let K 0 > 0. Then there exists a K ≥ K 0 such that the following holds: Let (M k , g(t)), k ≥ 1, t ∈ [0, T ), be a Ricci flow solution starting from an initial metric g(0) ∈ I K 0 . Then g(t) ∈ I K for every t ∈ [0, T ).
Proof. By Lemma 5.2, Lemma 5.6, Lemma 5.5, Lemma 5.7 we see that for K > 2 the conditions (7.1) -(7.4) are preserved. By Lemma 7.8 we see that there exists a K ≥ K 0 such that inequality (7.5) holds for t ∈ [0, T ). Now we only need to prove that for every time t ∈ [0, T ) the metric g(t) has bounded curvature and positive injectivity radius. As the curvature of g(0) is bounded by the assumption that g(0) ∈ I K , it follows by Shi's Theorem [Shi89] that for every time T ∈ [0, T ) the Ricci flow g(t) has bounded curvature on the time interval [0, T ]. As inj g(0) > 0 it follows that the metric g(0) is non-collapsed. By standard volume distortion estimates it follows that for each t ∈ [0, T /2] the metric g(t) is non-collapsed, and hence inj g(t) > 0. By Theorem 7.3 there exists a κ > 0 and ρ > 0 such that for each t ∈ [0, T ) the metric g(t) is κ-non-collapsed at scale ρ < T /2. This shows that inj g(t) > 0 for all t ∈ [T /2, T ).
Before proving Theorem 7.5, we need to prove the following two lemmas in preparation: Lemma 7.10. Let (M k , g(t)), k ≥ 1, t ∈ [0, T ), be a Ricci flow starting from an initial metric g(0) ∈ I. Then there exists a constant C 0 ≥ 0, depending only on the initial metric g(0), such that Proof. By Lemma 7.9 there exists a K > 0 such that g(t) ∈ I K for t ∈ [0, T ). From the evolution equation (2.12) of b and Definition 7.2 of I K it follows that This concludes the proof.
Proof. From the evolution equation (2.12) of b and expression (2.4) for the Laplacian with respect to the background metric g(t) it follows Applying the maximum principle [ChII, Theorem 12.14] yields the desired result.
We now proceed to proving Theorem 7.5.
Proof of Theorem 7.5. We argue by contradiction. Assume there exists a sequence of points (p i , t i ) in spacetime and constants D i → ∞ as i → ∞ such that By the assumption that g(0) ∈ I the initial metric g(0) has bounded curvature. Hence by Shi's theorem [Shi89] we have that for every T ∈ [0, T ) the metric g(t) has bounded curvature on M k × [0, T ]. As by Lemma 7.11 the warping function b is uniformly bounded on M k × [0, T ), we thus see that D i → ∞ forces K i → ∞ and therefore t i → T . Consider the rescaled Ricci flows , where ∆t i > 0 is to be determined below. As K i → ∞ we see that g i (t) are blow-ups rather than blow-downs, which is important for the following reason: By Theorem 7.3 there exists a κ > 0 such that g(t) is κ-non-collapsed at every scale p ≤ T /2 at every spacetime point (p, t) ∈ M k × [T /2, T ). As K i → ∞ we see that g i (t) are κ-non-collapsed at scales tending to infinity as i → ∞. By Lemma 7.9 there exists a K > 0 such that g(t) ∈ I K for all t ∈ [0, T ). Furthermore, by Lemma 7.10 there exists a C 0 such that Recall the Definition 2.1 of C g (p, r). Set and consider the parabolic neighborhoods As Thus we deduce that and hence the curvatures of the rescaled metrics g i (t) satisfy on the parabolic neighborhoods Ω i , subsequentially converges, in the Cheeger-Gromov sense, to an ancient pointed Ricci flow (M ∞ , g ∞ (t), p ∞ ), t ∈ (−∞, 0]. Claim 1: The Ricci flow (M ∞ , g ∞ (t), p ∞ ), t ∈ (−∞, 0], splits as (R 2 × N, g eucl + g N (t)), t ∈ (−∞, 0], where g eucl is the flat euclidean metric, and (N, g N (t)) is a non-compact ancient Ricci flow.
Proof of Claim: Denote by a i and b i the warping functions of the rescaled metrics g i (t). Then by (7.9) we see that As D i → ∞, the warping functions b i tend to infinity uniformly. As b i describes the size of the base S 2 in the Hopf fibration,intuitively one can see that this claim is true. Nevertheless, we provide a formal proof below: Inspecting the curvature components listed in section 2.3, we see that all the curvature components of g i (t), apart from R 0101 , tend to zero on Ω i . Hence the curvature operator of g ∞ (t) is of rank 1. Furthermore, as g(0) has bounded curvature by the assumption that g(0) ∈ I we see that the scalar curvature R g(t) is pointwise bounded below by inf p∈M k R g(0) (p) > −∞. Hence the blow-up limit g ∞ (t) has non-negative scalar curvature, which in turn implies that the curvature operator is non-negative. By [Ham86,8.3. Theorem & p. 178] we conclude that (M ∞ , g ∞ (t)) splits as a product (R 2 × N, g eucl + g N (t)). Note also that N is diffeomorphic to the leafs of the distribution spanned by e 0 and e 1 , as these are the only planes with non-flat sectional curvature. Recalling that e 0 = ∂ ∂s we see that the integral curves of e 0 are non-compact and therefore N is non-compact as well.
As (M ∞ , g ∞ (t)) is κ-non-collapsed at all scales, the above claim implies that (N, g N (t)) is a 2d κ-solution. However, by Hamilton's work a two dimensional κsolution is either the shrinking round sphere S 2 or its Z 2 quotient [CLN06, §1 of Chapter 9]. Since N is non-compact we have arrived at a contradiction. Therefore the desired result follows.
Proof of Corollary 7.6. The proof is the same as for Theorem 7.5. Since the Ricci flow is assumed to be κ-non-collapsed at all scales, we may also take blow-down limits and do not need to assume that b is uniformly bounded. Furthermore, since ancient Ricci flows have non-negative scalar curvature, Claim 1 of the proof of Theorem 7.5 also carries over.

Compactness properties
In this section we prove some compactness properties of U (2)-invariant cohomogeneity one Ricci flows. For general Ricci flows the compactness properties are well-known [ChI,Chapter 3]. Therefore the main technical difficulty is to show that the U (2)-symmetry passes to the limit.
The main theorem of this section is Theorem 8.1 which roughly states the following compactness property: Let (U i , g i (t), p i ), [−∆t, 0], be a sequence of U (2)-invariant cohomogeneity one manifolds in the class I of metrics. Here the U i are open manifolds and assumed to compactly contain the sets C g i (0) (p i , r) (see Definition 2.1) for some fixed r > 0. This condition can be understood as requiring U i to have 'radial diameter' of at least r. Furthermore the metrics g i (t) are normalized such that b = 1 at the points (p i , 0) in spacetime. We show that if the flows g i (t) are κ-non-collapsed and of uniformly bounded curvature, then (U i , g i (t), p i ), [−∆t, 0], subsequentially converges to a limiting U (2)-invariant Ricci flow (C ∞ , g ∞ (t), p ∞ ). Moreover, if we correctly pick/normalize the coordinate ξ, the warping functions u i (ξ, t), a i (ξ, t) and b i (ξ, t) of the metrics g i (t) converge on compact parabolic sets in C ∞ to the warping functions u ∞ (ξ, t), u ∞ (ξ, t) and b ∞ (ξ, t) of g ∞ (t). This in essence shows that when taking limits of U (2)-invariant Ricci flows, we may work with the warping functions only, without having to concern ourselves with the underlying manifold.
Theorem 8.1 has two important applications: Firstly, it implies the corresponding compactness result for complete Ricci flows. In particular, a sequence of uniformly bounded and non-collapsed U (2)-invariant cohomogeneity one Ricci flows (M k , g i (t), p i ), t ∈ [−t i , 0], t i → ∞, normalized such that b(p i , 0) = 1, subsequentially converges, in the Cheeger-Gromov sense, to a limiting Ricci flow (M ∞ , g ∞ (t), p ∞ ), t ∈ [−∞, 0], that is also U (2)-invariant and cohomogeneity one. Secondly, we prove a variant of Theorem 8.1 in Proposition 8.3, where we specialize to the case in which the 'radial diameter' of the U i is equal to 1 2 . This will allow us to alter one assumption of Theorem 8.1 and yield a very useful tool for proving certain scale-invariant inequalities via a contradiction/compactness argument, as introduced in the outline of section 9 in section 1.5 of this paper.
Below we state the main results of this section. For this recall Definition 2.1 of C g (p, r), C + g (p, r) and Σ p .
Theorem 8.1 (Local compactness). Let k ∈ N and κ, ρ, K, r, ∆t > 0. Assume that is a sequence of pointed cohomogeneity one U (2)-invariant Ricci flows satisfying the following properties: (1) U i is an open U (2)-invariant manifold with principal orbit S 3 /Z k .
(2) For t ∈ [−∆t, 0] we have g i (t) ∈ I (see Definition 7.2). Denote by u i , a i and b i the warping functions of g i (t).
, subsequentially converges, in the Cheeger-Gromov sense, to a pointed Ricci flow satisfying the following properties: (a) C ∞ is a cohomogeneity one U (2)-invariant manifold such that either (i) All orbits are principal: In this case C ∞ is diffeomorphic to the cylinder R × S 3 /Z k and we equip C ∞ with a radial coordinate ξ : C ∞ → R defined by ξ(p) = d g∞(0) (p, Σ p∞ ).
(ii) There is exactly one non-principal orbit: In this case C ∞ is diffeomorphic to M k and we equip C ∞ with the radial coordinate ξ : such that the metric g ∞ (t), t ∈ [−∆t, 0], is of the form (2.2) and in the class I (c) Choosing the coordinate ξ on (U i , g i (t), p i ) corresponding to whether we are in case (i) or (ii) above, the warping functions u i (ξ, t), a i (ξ, t) and b i (ξ, t) converge on compact sets to u ∞ (ξ, t), a ∞ (ξ, t) and b ∞ (ξ, t). (d) For every r < r the closed set C g∞(0) (p ∞ , r ) ⊂ C ∞ is compact.
From Theorem 8.1 the following corollary follows immediately: Corollary 8.2 (Compactness of complete Ricci flows). Let k ∈ N, κ, K > 0 and , is a sequence of pointed U (2)-invariant cohomogeneity one Ricci flows satisfying: (1) For t ∈ [−t i , 0] we have g i (t) ∈ I. Denote by u i , a i and b i the warping functions of g i (t). Proof. This follows from Theorem 8.1 by a diagonal argument.
The following proposition is a variant of Theorem 8.1 in the case we take r = 1 2 . Proposition 8.3. Let k ∈ N, κ, ρ, C 1 > 0, r = 1 2 and ∆t ∈ (0, 1 is a sequence of pointed U (2)-invariant cohomogeneity one Ricci flows satisfying conditions (1)-(5) of Theorem 8.3. If, instead of condition (6) of Theorem 8.1, we require subsequentially converges, in the Cheeger-Gromov sense, to a pointed Ricci flow Proof of Proposition 8.3. For brevity we write Ω i = C g i (0) p i , 1 2 × [−∆t, 0]. Let a i and b i denote the warping functions of g i (t). As g i (t) ∈ I we have as b i (p i , 0) = 1 by assumption. By the Ricci flow equation we have as ∆t ≤ 1 48C 1 by assumption. This yields the uniform curvature bound The result now follows from Theorem 8.1.
The main proof idea of Theorem 8.1 is to construct a set of four Killing vector fields X j , j = 1, 2, 3, 4, generated by the U (2)-action on each (U i , g i (t)), and show that these Killing vector fields pass to the limit (C ∞ , g ∞ (t)). This allows us to reconstruct the U (2)-action on C ∞ , proving the desired result. The main difficulty, however, is to show that the orbits corresponding to the flows of the Killing vector fields do not degenerate in the limit and thereby ensure that the full U (2) symmetry group is preserved. For this we will rely on Lemma 8.4 below, where we prove that κ-non-collapsedness implies a lower positive bound on Q away from a non-principal orbit.
Lemma 8.4. Let k ∈ N, r 0 ∈ (0, 1] and κ, C 1 , c > 0. Assume that (M, g) is a U (2)invariant cohomogeneity one manifold with principal orbit S 3 /Z k equipped with a metric g ∈ I. Take p ∈ M . If g is κ-non-collapsed at scale cb(p): If for r ≤ cb(p) the ball B g (p, r) is compactly contained in M and |Rm g | g < r −2 on B g (p, r) then vol(B g (p, r)) ≥ κr 4 then there exists an > 0 depending on k, κ, C 1 , c and r 0 for which the following holds: If for q ∈ C + g (p, b(p)r 0 ) the set C g (q, b(p) r 0 4 ) is compactly contained in C + g (p, b(p)r 0 ) then Q(q) ≥ .
Proof. By rescaling we may assume without loss of generality b(p) = 1 and that the metric g is κ-non-collapsed at scale c > 0. The latter follows from the fact that if g is κ-non-collapsed at scale ρ then α 2 g is κ-non-collapsed at scale αρ. Fix a q ∈ C + g (p, b(p)r 0 ) such that the assumptions of the lemma hold. Take U := C g (q, b(p) r 0 4 ). Note that U is a union of orbits of the U (2)-action. Recall that non-principal orbits are non-generic and characterized by a = 0. As a s ≥ 0 we see that all the orbits of U are principal and therefore diffeomorphic to S 3 /Z k . Because 0 ≤ b s ≤ Q ≤ 1 for metrics in I we see that 1 ≤ b ≤ 2 in C + g (p, r 0 ) and hence |Rm g | g ≤ C 1 in C + g (p, r 0 ) by assumption (2). From the expression we have Q ≤ 2Q(q) on C g (q, r 1 ).
Proof of Claim: Let q ∈ C g (q, r), r < r 2 . Then Σ q is isometric to S 3 /Z k equipped with a squashed Berger metric. In particular, if we denote by ι : Σ q → M the inclusion, then ι * g = a(q ) 2 ω ⊗ ω + b(q ) 2 π * (g F S ), where g F S is the Fubini-Study metric on S 2 normalized to have curvature equal to 1 4 and π : S 3 /Z k → S 2 is the Hopf fibration. Note that Furthermore, as the Hopf fibers of Σ q ∼ = S 3 /Z k have length 2π k a(q ), we see that vol(π −1 (B g F S (π(q ), r))) = 2π k a(q )vol(B g F S (π(q ), r)) ≤ Ca(q )r 2 , for some constant C > 0 depending on k only. Since Q = a b , Q ≤ 2Q(q) and b ∈ [1, 2] in C g (q, r 1 ) it follows that Integrating this inequality proves the claim.
As |Rm g | g ≤ C 1 on C + g (p, r 0 ), the ball B g (q, r 0 4 ) is compactly contained in M , and g is κ-non-collapsed at scale c, we see that for r ≤ r 3 := min 1 √ C 2 , c, r 0 4 vol(B g (q, r)) ≥ κr 4 .
Before proving the compactness theorems listed above, we construct a set of four Killing vector fields on a general U (2)-invariant cohomogeneity one manifold M with principal orbit S 3 /Z k . By passing to the universal cover we may assume without loss of generality that k = 1. Pick the basis for the Lie algebra of U (2). Then X i , i = 1, 2, 3, 4, satisfy the commutation relations and [X 0 , X i ] = 0 for i = 1, 2, 3. Extend X i , i = 1, 2, 3, 4, to left-invariant vector fields on U (2). Note that the integral curves generated by X i , i = 1, 2, 3, 4, have period 2π. The U (2)-action generates four corresponding Killing vector fields X i , i = 1, 2, 3, 4, on M k by taking We now prove the following: Lemma 8.5. For i = 1, 2, 3, 4 we have Proof. By the form (2.3) of the metric we see that |X 0 | g = a. Hence we only need to prove the result for i = 1, 2, 3. First note that the vector fields X i , i = 1, 2, 3, are orthogonal to ∂ ∂s and therefore parallel to the orbits of the U (2) action on M k . Hence it suffices to study the metric g restricted to these directions. Here we see that a 2 ω ⊗ ω + b 2 π * (g F S ) ≤ max(a, b) 2 g S 3 , where g S 3 is the round metric on S 3 with sectional curvatures equal to 1. Thus it suffices to show that |X i | g S 3 ≤ 1.
If we identify S 3 with SU (2), the vectors X i correspond to right-invariant vector fields on SU (2). Moreover, one can check that these vector fields are orthonormal with respect to the metric g S 3 . Hence the desired result follows.
Remark 8.6. In fact one can show that min(a, b) ≤ |X i | g ≤ max(a, b). Recalling that the isometry generated by the Killing vector field X i descends to a rotation of the base S 2 in the Hopf fibration π : S 3 → S 2 , one can see that the upper bound is attained on π −1 ({Equator of S 2 }) and the lower bound is attained on π −1 ({N, S}), where N , S denote the north and south pole with respect to the rotation induced by X i . Now we proceed to proving the main theorem of this section: Proof of Theorem 8.1. As g i (t) is κ-non-collapsed at (p i , 0) at scale min(ρ, r, √ ∆t) it follows from [ChI, Lemma 6.54] that there exists a uniform δ > 0 such that By assumption g i (t) has bounded curvature on the parabolic neighborhood By an adaptation of [ChI, Theorem 3.16] we therefore deduce that after passing to a subsequence C g i (0) (p i , r) , g i (t), p i , t ∈ [−∆t, 0], converges, in the Cheeger-Gromov sense, to a pointed Ricci flow where C ∞ is an open manifold.
Proof of Claim: Recall the construction of the Killing vector fields X j , j = 1, 2, 3, 4 for a general U (2)-invariant manifold M explained above. Let X ij , i ∈ N, j = 1, 2, 3, 4, denote the corresponding Killing vector fields on the manifolds C g i (0) (p i , r). Recall that g i (t) ∈ I implies that 0 ≤ b s ≤ Q ≤ 1. Therefore b ≤ r + 1 on C g i (0) (p i , r). Note that from the evolution equation (2.12) of b it follows that where c > 0 is some universal constant. As |Rm g i (t) | g i (t) ≤ C 1 on Ω i by assumption, we see that there exists a C > 0, depending on r and C 1 only, such that b ≤ C on Ω i . From Lemma 8.5 it hence follows that for i ∈ N, j = 1, 2, 3, 4, Recall that in general a Killing vector field X a on a manifold satisfies the relation Therefore we see that the Killing vector fields X ij are uniformly bounded in C 2 (Ω i ), and converge to C 1 Killing vector fields X ∞,j , j = 1, 2, 3, 4, on (C ∞ , g ∞ (t)) after passing to a subsequence. However, since the group of isometries of a smooth manifold is a smooth Lie group, the vector fields X ∞,j , j = 1, 2, 3, 4 are in fact smooth. As the Killing vector fields X ij , i ∈ N, j = 1, 2, 3, 4, are complete, so are X ∞,j , j = 1, 2, 3, 4. Integrating the Killing vector fields X ∞,j , j = 1, 2, 3, 4, then yields the desired U (2)-action on (C ∞ , g ∞ (t)).
It remains to be shown that this action is faithful by proving that the Killing vector fields are non-zero at times t ∈ [−∆t, 0].
Fix t ∈ [−∆t, 0] and choose points q i ∈ Σ + p i (see Definition 2.1) and q ∞ ∈ C ∞ with d g i (t ) (q i , Σ + p i ) = 1 2 r 1 and q i → q ∞ . Checking the conditions of Lemma 8.4, we see that there exists an > 0, independent of i, such that Q(q i , t ) ≥ .
As g ∈ I and therefore 0 ≤ b s ≤ Q ≤ 1, we see that Therefore the geometry of the orbit Σ q i ∼ = S 3 /Z k is controlled -the curvature and diameter are uniformly bounded from above, and its volume and Hopf fiber lengths are uniformly bounded away from zero. Hence the norms of the Killing vector fields X ij , j = 1, 2, 3, 4, at the points (q i , t ) in spacetime are uniformly bounded away from zero, proving that on the limiting space (C ∞ , g ∞ (t )) the Killing vector fields X ∞,j , j = 1, 2, 3, 4, are non-zero. As t ∈ [∆t, 0] was arbitrary, the desired result follows.
By the slice theorem we we see that either (i) all orbits of C ∞ are principal and diffeomorphic to S 3 /Z k or (ii) there exists exactly one non-principal orbit, which is diffeomorphic to S 2 and as usual we denote by S 2 o . Below it will become clear why C ∞ cannot possess two non-principal orbits. In case (i) C ∞ is diffeomorphic to the manifold R × S 3 /Z k and in case (ii) it is diffeomorphic to M k . In both cases there is a dense open set of the form R × S 3 /Z k ⊂ C ∞ .
We now show that the metrics g ∞ (t) can be expressed in the form (2.3). Denote the warping functions of the metrics g i (t) by a i and b i . In case (i) we define the radial coordinates ξ i (p) = ±d g i (0) (p, Σ p i ), and ξ ∞ (p) = ±d g∞(0) (p, Σ p∞ ) on C g i (0) (p i , r) and C ∞ , respectively. We choose the sign of ξ i (p) depending on which side of the hypersurface Σ p i the point p lies, and in such a way that The sign of ξ ∞ (p) is chosen such that ξ i → ξ ∞ as i → ∞. In case (ii) we may assume without loss of generality that for all i ∈ N the open manifolds Then define radial coordinates on C g i (0) (p i , r) and C ∞ . Note that the coordinates ξ i and ξ ∞ are smooth away from a non-principal orbit and furthermore that ξ i → ξ ∞ in C ∞ away from a non-principal orbit. Hence we obtain the one-forms dξ i and dξ ∞ away from a non-principal orbit, which are orthogonal to all orbits of C g i (0) (p i , r) and C ∞ , respectively. For brevity we drop the subscript and write ξ for the coordinates ξ i or ξ ∞ . Since the metric g ∞ (t) is U (2)-invariant, as shown above, there exists warping functions u ∞ , a ∞ , b ∞ : C ∞ × [−∆t, 0] → R ≥0 such that the metric can be expressed as g ∞ (t) = u 2 ∞ (ξ, t)dξ 2 + a 2 ∞ (ξ, t)ω ⊗ ω + b 2 ∞ (ξ, t)π * (g F S ), where at time 0 we have u = 1 on C ∞ .
As a ∞ (p, t) = |X ∞,0 | g∞(t) (p) and X i,o → X ∞,0 as i → ∞ by above, we see that away from a non-principal orbit a i → a ∞ smoothly. Similarly, one can show with help of the remaining Killing vector fields X i,j , j = 1, 2, 3, that away from a non-principal orbit b i → b ∞ smoothly. Hence away from a non-principal orbit,a i (ξ, t), b i (ξ, t) → a ∞ (ξ, t), b ∞ (ξ, t) in C ∞ as i → ∞. Furthermore, from the curvature bounds on Ω i and the boundary conditions on a i , b i , a ∞ , b ∞ at a non-principal orbit (see section 2.2 for the smoothness conditions on the warping functions at the non-principal orbit), one can show that in fact a i (ξ, t), b i (ξ, t) → a ∞ (ξ, t), b ∞ (ξ, t) smoothly everywhere. Hence the metric g ∞ (t), t ∈ [−∆t, 0], is in the class I. As a s ≥ 0 for metrics in I we see that C ∞ can possess at most one non-principal orbit. Finally, we note that by [ChI,Theorem 3.16] the closed set C g ∞(0) (p ∞ , r ) ⊂ C ∞ is compact for every r < r.

Ancient Ricci flows Part I
In this section we prove some properties of ancient Ricci flows g(t) ∈ I, −∞ < t ≤ 0, that are non-collapsed at all scales. This yields important geometric information on the blow-up limits of singular Ricci flows, which we exploit and refine in later chapters. The main goal is to prove the following theorem: Theorem 9.1. Let κ > 0 and (M k , g(t)), k ≥ 2, t ∈ (−∞, 0], be an ancient Ricci flow, which satisfies the following properties: (i) κ-non-collapsed at all scales (ii) g(t) ∈ I for t ∈ (−∞, 0].
Proof strategy. In both of these theorems we are given an ancient Ricci flow (M, g(t)), t ≤ 0, and want to show that a scale invariant quantity T satisfies We prove such statements by a contradiction/compactness argument. First we assume that ι := inf and take a sequence of points (p i , t i ) in spacetime such that Then we consider the rescaled Ricci flows where ∆t > 0 is chosen such that the conditions of Proposition 8.3 are met. Then (C g i (0) p i , 1 2 , g i (t), p i ) subsequentially converges to a Ricci flow (C ∞ , g ∞ (t), p ∞ ), t ∈ [−∆t, 0]. By construction However, if the evolution equation of T precludes the possibility of a negative infimum being attained, we have arrived at a contradiction and proven the desired result.
Proof of main theorems of this section. Before proving Theorem 9.1 we need to state a technical lemma in preparation: Lemma 9.3. Let (M k , g(t)), k ∈ N, t ≤ 0, be an ancient Ricci flow satisfying the conditions of Theorem 9.1. Then for every > 0 there exists a δ > 0 such that whenever at a point (p, t) in spacetime one of the following inequalities holds (i) T 1 (p, t) ≤ − and k ≥ 2 (ii) T 2 (p, t) ≤ − and k ≤ 2 (iii) T 3 (p, t) ≤ − and k ∈ N (iv) |x(p, t)| ≥ and k = 2 then s(p, t) ≥ δb(p, t).
Proof. Recall that by Corollary 7.6 there exists a C 1 > 0 such that |Rm g(t) | g(t) ≤ C 1 0 on M k × (−∞, 0]. We first prove (i). We fix > 0 and argue by contradiction. Assume there exists a sequence of points (p i , t i ) in spacetime such that Define the rescaled metrics For sufficiently small ∆t > 0 the conditions of Proposition 8.3 are satisfied and hence (C g i (0) p i , 1 2 , g i (t), p i ) subsequentially converges to a Ricci flow (C ∞ , g ∞ (t), p ∞ ). By (9.1) one sees that p ∞ lies on the non-principal orbit S 2 o of C ∞ . By construction we have T 1 (p ∞ , 0) ≤ − as T 1 is a scale invariant quantity. This however contradicts the fact that T 1 = a s + 2 (Q 2 − 1) = k − 2 ≥ 0 on S 2 o . Note that T 2 = 2 − k, T 3 = k + 1 and x = k − 2 on S 2 o . Therefore by the same argument applied to T 2 , T 3 and x the desired result holds true.
Next we prove Theorem 9.1. Proof of Theorem 9.1. Recall that by Corollary 7.6 there exists a C 1 > 0 such that |Rm g(t) | g(t) ≤ C 1 0 on M k × (−∞, 0]. We first show that T 1 ≥ 0 in M k × (−∞, 0]. We argue by contradiction. Assume that As g(t) ∈ I we know that ι > −∞. Take a sequence of points (p i , t i ) in spacetime such that T 1 (p i , t i ) → ι as i → ∞. From Lemma 9.3 it follows that for sufficiently large i for some δ > 0. Define the rescaled metrics For sufficiently small ∆t > 0 the conditions of Proposition 8.3 are satisfied and hence as T 1 is a scale invariant quantity. Since T s (p ∞ , 0) = 0, we see from the evolution equation (5.11) of T 1 that showing that either case a) or case b) must hold. We now show that both of these cases are impossible, thereby arriving at a contradiction. First note that by (9.2) we know that p ∞ does not lie on the non-principal orbit S 2 o . Therefore the strong maximum principle applied to the evolution equation (5.1) of Q shows that in case a) Q = 0 everywhere on C ∞ × [−∆t, 0]. This, however, contradicts the non-collapsedness of C ∞ and therefore case a) cannot occur. In case b) the same argument shows that Q = 1 everywhere on C ∞ × [−∆t, 0]. Then applying the strong maximum principle to the evolution equation (5.8) of Qy, which simplifies when Q = 1, shows that y = 0 everywhere on C ∞ × [−∆t, 0]. This, however, implies that T 1 = 1 > 0 on C ∞ × [−∆t, 0] contradicting our assumption that ι < 0.
It remains to be shown that T 1 (p, t) = 0 if, and only if, k = 2 and p lies on the non-principal orbit S 2 o . We argue by contradiction. Assume there exists a point (p, t) in spacetime such that p / ∈ S 2 o and T 1 (p, t) = 0.
Then arguing as above, we see that either case a) or case b) must hold true, both of which lead to the same contradiction.
By the same method we may prove that T 2 ≥ 0 and T 3 ≥ 0 on M k × (−∞, 0]. Note that the evolution equations (5.13) and (5.14) show that T 2 and T 3 cannot attain a negative infimum, leading to the desired contradiction.
Next we prove Theorem 9.2.
Proof of Theorem 9.2. Recall that by Corollary 7.6 there exists a C 1 > 0 such that |Rm g(t) | g(t) ≤ C 1 0 on M 2 × (−∞, 0]. Also recall Lemma 4.1, which states that (M 2 , g(t), t ∈ (−∞, 0], is homothetic to the Eguchi-Hanson space if, and only if, Therefore it suffices to show that x = 0. We follow the proof strategy of Theorem 9.1 and argue by contradiction. Assume x < 0 and take a sequence of points (p i , t i ) in spacetime such that Note that ι > −∞ as g(t) ∈ I for t ∈ (−∞, 0]. From Lemma 9.3 it follows that for some δ > 0. Define the rescaled metrics where ∆t > 0 is chosen such that the conditions of Proposition 8.3 are satisfied. Then (C g i (0) p i , 1 2 , g i (t), p i ) subsequentially converges to a Ricci flow (C ∞ , g ∞ (t), p ∞ ), t ∈ [−∆t, 0], on which by construction x(p ∞ , 0) = ι < 0, since x is a scale-invariant quantity. Furthermore, we see by (9.4) that p ∞ does not lie on the non-principal orbit S 2 o . The evolution equation (5.7) for x in the Kähler case y = 0 simplifies to This, however, cannot happen, as otherwise the strong maximum principle applied to the evolution equation (5.1) of Q would imply that Q = 0 on C ∞ × [−∆t, 0]. Hence we have arrived at a contradiction and conclude By the same argument one shows that as well, which concludes the proof.

Eguchi-Hanson and a family of Type II singularities
In this section we show that Ricci flow solutions (M k , g(t)), k ≥ 2, starting from a large class of initial metrics encounter a Type II singularity in finite time at the origin. In the case k = 2 we show that the Eguchi-Hanson metric can occur as a blow-up limit. Below we state the precise result: Theorem 10.1 (Type II singularities). Let (M k , g(t)), k ≥ 2, be a Ricci flow starting from an initial metric g(0) ∈ I (see Definition 7.2) with (10.1) sup Then g(t) encounters a Type II curvature singularity in finite time T sing > 0 and sup 0≤t<T sing Furthermore, there exists a sequence of times t i → T sing such that the following holds: Consider the rescaled metrics Then (M k , g i (t), o) subsequentially converges, in the pointed Gromov-Cheeger sense, to an eternal Ricci flow (M k , g ∞ (t), o), t ∈ (−∞, ∞). When k = 2 the metric g ∞ (t) is stationary and homothetic to the Eguchi-Hanson metric.
In this paper we do not study the detailed geometry of the singularity models of the Type II singularities arising in the k ≥ 3 case, however, as stated in Conjecture 2 in the introduction, the author believes that these singularities are modeled on the non-collapsed steady Ricci solitons found in [A17]. The author, in collaboration with Jon Wilkening, has carried out numerical simulations supporting this conjecture. A paper summarizing the results is in preparation [AW19].
Outline of proof. Here we sketch the proof of Theorem 10.1. First we show in Lemma 10.5 that the condition (10.1) forces a Ricci flow solution (M k , g(t)), k ≥ 2, to develop a singularity in finite time T sing > 0 at the origin. Then we take a sequence of times t i → T sing and define the rescaled metrics These metrics subsequentially converge to a singularity model (M k , g ∞ (t)), −∞ < t ≤ 0 -an ancient solution of the Ricci flow. Now recall the dichotomy between Type I and Type II singularities and that every Type I singularity is modeled on a shrinking Ricci soliton [EMT11]. Therefore we can prove that the singularity is of Type II by showing that (M k , g ∞ (t)) is not a shrinking Ricci soliton. For this we apply Theorem 6.1, which excludes shrinking solitons whenever (i) sup |b s | < ∞, (ii) T 1 > 0 for s > 0 and (iii) Q ≤ 1 hold. By definition, every metric in I satisfies conditions (i) and (iii). As these conditions are scale-invariant, they pass to the blow-up limit (M k , g ∞ (t)). From Theorem 9.1 it follows that condition (iii) holds true as well, allowing us to conclude that (M k , g ∞ (t)) is not a shrinking soliton and that the singularity is of Type II. By the work of Hamilton we can then choose a sequence of times t i → T sing , possibly different from the sequence t i , such that the corresponding blow-ups around the origin converge to an eternal Ricci flow (M k , g ∞ (t)), −∞ < t < ∞.
In the k = 2 case we show that (M 2 , g ∞ (t)) is stationary under Ricci flow and homothetic to the Eguchi-Hanson space. What makes k = 2 special is that the second term of the right hand side of is zero and therefore as y ≤ 0 with equality at S 2 0 for metrics in I. It turns out that for the specific choice of t i → T sing from Hamilton's trick we have that on (M 2 , g ∞ (t)) at S 2 o at time 0 we have ∂ t b(0, t) = 2y s (0, t) = 0.
An application of L'Hôpital's Rule shows that on S 2 o we have y Q = y s k .
Therefore we can apply the strong maximum principle of Theorem 3.2, Case 2, to the evolution equation (12.8) of y Q to show that y = 0 everywhere. From Theorem 9.2 it then follows that (M 2 , g ∞ (t)) is homothetic to the Eguchi-Hanson space. In fact one can show with help of the evolution equation of |Rm g(t) | 2 g(t) that Hamilton [Ham95] introduced the notion of Type I and Type II Ricci flows, which are defined by the rate at which the curvature blows up as t T . In particular, In the case that such a constant C > 0 does not exists, that is we say the singularity is of Type II. By the work of Naber [N10] and Enders, Müller and Topping [EMT11] every Type I singularity model is a non-flat Ricci shrinking soliton. Hamilton A point p ∈ M in a Type I Ricci flow is called a (general) Type I singular point if there exists an essential blow-up sequence with p i → p on M . Now we state the main result of [EMT11], asserting that Type I singularities are modeled on shrinking Ricci solitons.
Theorem 10.4 (see [EMT11,Theorem 1.4] ). Let (M, g(t)) be a Type I Ricci flow on [0, T ) and suppose p is a Type I singular point as in Definition 10.3. Then for every sequence λ j → ∞, the rescaled Ricci flows (M, g j (t), p) defined on [λ j T, 0) by g j (t) := λ j g T + t λ j subconverge to a non-flat gradient shrinking soliton.
We use Theorem 10.4 to exclude Type I singularities for Ricci flows satisfying the assumptions of Theorem 10.1.
Proof of the main theorem. First we show that a singularity must occur in finite time: Lemma 10.5. The maximal extension of a Ricci flow (M k , g(t)), k ≥ 1, starting from an initial metric g(0) ∈ I with where we used expression (2.4) of the Laplacian. By the maximum principle (see for instance [ChII,Theorem 12.14]) we see that there exists a T < ∞ such that inf p∈M k b 2 (p, t) → 0 as t → T.
As b s ≥ 0 we conclude that lim . Moreover, by Theorem 7.3 there exist constants κ, ρ > 0 such that g(t) is κ-noncollapsed at scales less or equal to ρ. Now take a sequence of times t i T sing such that Claim 1: The sequence of points (o, t i ) in spacetime is an essential blow-up sequence.
Proof of Claim: We argue by contradiction. Assume, after passing to a subsequence, that Then by Lemma 3.3 and the fact that b s ≥ 0 for metrics in I we have where we used the expression for the curvature component R 2323 derived in section 2.3. This shows that which contradicts (10.3). This proves the claim.
Define the rescaled metrics By property (2) above and the fact that b s ≥ 0 for metrics in I it follows that Note also that the rescaled metrics g i (t) are κ-non-collapsed at scales tending to infinity. Corollary 8.2 then implies that (M k , g i (t), o) subsequentially converges, in the Cheeger-Gromov sense, to an ancient Ricci flow (M ∞ , g ∞ (t), o), t ∈ (−∞, 0], where M ∞ ∼ = M k . By Theorem 9.1 we have 0] on the blow-up limit g ∞ (t). Theorem 6.1 shows that g ∞ (t) cannot be a shrinking soliton, which by the contrapositive of Theorem 10.4 proves that the singularity is of Type II. Therefore Now we mimic the proof of [Ham95,Theorem 16.4, Type II(a)] to construct an eternal blow-up limit. Pick a sequence of times T i < T sing satisfying as the latter goes to zero as t → T i . Consider the rescaled Ricci flow solutions If we write a i , b i for the warping functions of the rescaled metric g i (t) we obtain from equation (10.4) the following inequality Note that here t is the time variable of the rescaled Ricci flow g i (t). Therefore for any fixed t we have and (10.6) b −2 i (o, 0) = 1. From this, the fact that b s ≥ 0 and the curvature bound of Theorem 7.5, we see that on bounded time intervals the curvatures of g i (t) eventually become bounded by 2C 1 . In addition to this the metrics g i (t) are κ-non-collapsed at larger and larger scales. Therefore Corollary 8.2 implies that (M 2 , g i (t), o) subsequentially converges to an eternal Ricci flow (M 2 , g ∞ (t), o). Furthermore (10.5) and (10.6) show that that where we write a ∞ and b ∞ for the warping functions of the metric g ∞ (t). Notice that (10.7) and (10.8) imply that at time 0 on S 2 o we have where y ∞ = (b ∞ ) s − a∞ b∞ corresponds to the Kähler quantity y on the g ∞ background.
Now it only remains to be shown that in the k = 2 case g ∞ (t) is stationary and homothetic to the Eguchi-Hanson metric. In the following we drop the ∞ subscript and let a, b, Q, y be with respect to the metric g ∞ (t). Note that equation (10.9) and an application of L'Hôpital's Rule show that at time 0 on S 2 o we have The evolution equation for y Q derived in the Appendix A is Because g ∞ (t) ∈ I is of bounded curvature we see that is bounded. Note that we applied Lemma 3.3 to show that 1 b 2 is bounded. Therefore we may apply the strong maximum principle of Theorem 3.2, Case 2, to deduce that yielding that g ∞ (t) is Kähler in the the k = 2 case. By Theorem 9.2 we then deduce that g ∞ (t) is homothetic to the Eguchi-Hanson metric, which proves the desired result.
11. Ancient Ricci flows Part II: k = 2 case In this section we prove that every non-collapsed ancient Ricci flow in the class of metrics I is isometric to the Eguchi-Hanson metric: Theorem 11.1 (Unique ancient flow). Let κ > 0 and (M 2 , g(t)), t ∈ (−∞, 0], be an ancient Ricci flow that is κ-non-collapsed at all scales and g(t) ∈ I, t ∈ (−∞, 0] (see Definition 7.2). Then g(t) is stationary and homothetic to the Eguchi-Hanson metric.
An immediate consequence of this theorem is that for every sequence of times t i → T sing in Theorem 10.1, the rescaled Ricci flows subsequentially converge to the Eguchi-Hanson space. In other words, the Eguchi-Hanson space is the unique limit of blow-ups around the origin. With a little extra work one can show the slightly more general result, asserting that blow-up limits centered at points close to, but not necessarily on the tip of M 2 , subsequentially converge to the Eguchi-Hanson space: Corollary 11.2. Let (M 2 , g(t)), t ∈ [0, T sing ), be a Ricci flow starting from an initial metric g(0) ∈ I (see Definition 7.2) that develops a singularity at time T sing . Let (p i , t i ) be a sequence of points in spacetime with t i → T sing satisfying and consider the rescaled metrics Then (M 2 , g i (t), p i ) subsequentially converges, in the Gromov-Cheeger sense, to a blow-up limit (M 2 , g ∞ (t), p ∞ ), t ≤ 0, which is homothetic to the Eguchi-Hanson space.
We defer the proof of Corollary 11.2 to the end of subsection 11.5.
11.1. Outline of Proof. Here we outline the proof of Theorem 11.1. Below we take (M 2 , g(t)), t ∈ (−∞, 0], to be a non-collapsed ancient Ricci flow with g(t) ∈ I, t ∈ (−∞, 0]. We construct a continuously varying one-parameter family of functions satisfying the following five requirements: (1) For every θ ∈ (0, 1] the condition is preserved on the (M 2 , g(t)) background.
Remark 11.3. We briefly remark on some of the properties of f θ : • In the expression for Z θ of requirement (1) we take x, Q and a s to be functions of spacetime. For brevity we do not express the dependence explicitly.
• The term x Q 2 can be extended smoothly to the non-principal orbit S 2 o , as x = x s = 0 at s = 0. Therefore Z θ is well-defined on M 2 . • When θ = 1 we already know that , as from Theorem 9.1 it follows that • The family f θ (Q), θ ∈ (0, 1], we construct below is smooth everywhere apart from when Q = Q θ . It will become clear later that this does not pose a problem. In subsection 11.2 we show that at points (p, t) in spacetime at which Q(p, t) = Q θ , or equivalently at points where f is smooth, the evolution equation of the corresponding Z θ can locally be written as where W θ andD θ are bounded and scale-invariant expressions involving b s , Q, f θ (Q), f θ (Q) and f θ (Q). Again, all quantities in the evolution equation of Z θ should be interpreted as functions of spacetime. In subsection 11.3 we construct a family of functions f θ , θ ∈ (0, 1], by solving an initial value problem for a second order non-linear ordinary differential equation. Subsequently we show that the family satisfies requirements (1)-(5) listed above. In particular, we show in subsection 11.4 that for the constructed family -on a non-collapsed ancient Ricci flow background -the following property holds true: For all points (p, t) in spacetime such that Q(p, t) < Q θ , we have W θ (p, t) ≥ 0, θ ∈ (0, 1]. This fact, in conjunction with the fourth bullet point of Remark 11.3, essentially shows that for each θ ∈ (0, 1] the inequality Z θ ≥ 0 is preserved on the g(t) background.
Once we have shown that our family f θ , θ ∈ (0, 1], satisfies requirements (1)-(5), we will use a blow-up argument in conjunction with the strong maximum principle to show that if for some θ 0 ∈ (0, 1] the inequality Z θ ≥ 0 holds for all θ ∈ [θ 0 , 1], then there must exists an θ 1 < θ 0 such that the inequality also holds for all θ ∈ [θ 1 , 1]. This shows that the set is open. As E is defined by a closed condition and therefore closed, it follows that E = (0, 1] and therefore Z θ ≥ 0 for all θ ∈ (0, 1]. By property (2) of f θ we deduce that at all points (p, t) in spacetime such that Q(p, t) < 1 we have x(p, t) ≥ 0. Note that by the strong maximum principle applied to the evolution equation (5.1) of Q it follows that Q < 1 and hence x ≤ 0 everywhere. Now recall Theorem 9.1 which states that x ≤ Qy ≤ 0 on M 2 × (−∞, 0]. Therefore x = y = 0 on M 2 × (−∞, 0] and we conclude that the metric g(t) is homothetic to the Eguchi-Hanson metric by Lemma 4.1.
11.2. Evolution equations. The main difficulty in carrying out the proof is to find a family of functions f θ , θ ∈ (0, 1], for which requirement (1) is satisfied. Our strategy is to first derive the evolution equation of Z θ for a general f θ and then reduce the problem to solving a second order ordinary differential equation in f θ . For this, first note that the evolution equation of f θ (Q) away from the non-principal orbit S 2 o can be written as The computation is carried out in the Appendix A.
Remark 11.4. Some remarks on the evolution equation of f θ (Q): • We often omit the dependence of our quantities on spacetime, i.e. by f θ (Q) we mean f θ (Q(p, t)). • For brevity we often omit the dependence of f on θ and Q, as in the expression for C f above. For instance, we write f for f θ (Q) and f for f θ (Q). • Note that by Lemma 2.2 the quantity Q = a b as a function of s can be extended to an odd function around the origin. Therefore as long as f θ : [0, 1] → [0, 1] is extendable to an even function around the origin, the term f Q and hence C f can be smoothly extended to all of M 2 . From equation (11.1) and the evolution equations (5.3) and (5.5) of a s and Q 2 , respectively, we see that the evolution equation of Z θ can be written as after having eliminated any occurring a s by substituting A computation carried out in the Appendix A shows that Similarly, we compute the expressions for C Z,1 and C Z,2 in the Appendix A, however their exact forms are not important for our analysis. It is only important to note that when f is extendable to an even function around 0, the quantities C Z,i , A i , i = 0, 1, 2, are scale-invariant, bounded, and can be extended smoothly to S 2 o . For reasons that will become clear below, we rewrite the equation (11.2) in the form and D θ = C Z,1 + ZC Z,2 + A 1 − A 2 Z θ − 2 b s Q Sometimes it will be useful to regard W θ as a quadratic polynomial. Therefore we define In the proof of Theorem 11.1 we also need the evolution equation of Z 1 = x Q 2 + 1, which can be written as where C Z 1 ,0 = 1 Q 2 −4 1 + Q 2 y 2 + 8Q 1 − 2Q 2 y + 16Q 2 1 − Q 2 and C Z 1 ,1 , C Z 1 ,2 are a bounded scale-invariant functions of a s , b s and Q. The derivation of this evolution equation is carried out in the Appendix A. Note the following lemma: Lemma 11.5. Let (M 2 , g(t)), t ∈ (−∞, 0], be an ancient Ricci flow as in Theorem 11.1. Then Proof. By Theorem 9.1 we know that Z 1 = T 1 Q 2 ≥ 0 in M 2 × (−∞, 0]. Moreover, notice the similarity of C Z 1 ,0 to the zeroth order term in the evolution equation (5.11) of T 1 . Therefore we see by the proof of Lemma 5.8 that C Z 1 ,0 ≥ 4 (1 − Q 2 ) for metrics in I.
In the proof of Theorem 11.1 we deform the inequality Z 1 ≥ 0 along a path of conserved inequalities Z θ ≥ 0, θ ∈ (0, 1]. Thus Z 1 ≥ 0 is the starting point for successively constraining the ancient Ricci flow towards the Eguchi-Hanson space.
Note that we are now regarding Q as an independent variable and not as a function of spacetime. Before we explain how we arrived at this differential equation, we list some of its properties below. For clarity of exposition we defer their proofs to subsection 11.6.
Lemma 11.6. For every f 0 ∈ R the ordinary differential equation (11.6) possesses an even analytic solution around the origin with initial condition Furthermore, f varies smoothly with f 0 .
Recalling that by definition the above Lemma 11.12 and concavity of w θ (z) show that to prove Proposition 11.10 it suffices to check that for θ ∈ (0, 1] and 0 ≤ Q < Q θ we have where f θ is as defined in Proposition 11.9. Note that γ is only defined for Q 2 > 0. This however does not pose a problem as implies that Q 2 ≥ 3 4 > 0. Recall that by the properties of f θ (Q) summarized in Proposition 11.9 we have w θ (−f θ (Q) + 1) = 0 for 0 ≤ Q < Q θ , and therefore only need to investigate the sign of β and γ in their respective regimes. This explains why we chose to define f θ (Q) via the ordinary differential equation (11.6). In the following technical lemma, the proof of which we defer to subsection 11.6, we show that for our choice of f θ the functions β and γ are in fact positive in their respective regimes: Lemma 11.13. Fix θ ∈ (0, 1] and let f θ (Q) as defined in Proposition 11.9. Then for 0 ≤ Q < Q θ we have Now we can prove Proposition 11.10.
Proof of Proposition 11.10. By Lemma 11.11 and Lemma 11.13 we know that W θ ≥ 0 whenever 0 ≤ Q < Q θ , with equality if and only if b which by the definition of Z θ is equivalent to T 2 = 0.
Remark 11.14. The proof of Proposition 11.10 essentially implies that for every θ ∈ (0, 1] the inequality Z θ ≥ 0 is preserved on Ricci flow backgrounds in I satisfying T 1 ≥ 0, T 2 ≥ 0 and T 3 ≥ 0. We do not prove this here, as our proof of Theorem 11.1 does not rely on this fact.
11.5. Proof of main theorem. Next, we prove that the Eguchi-Hanson space is the unique ancient Ricci flow in the class I.
Proof of Theorem 11.1. Recall that by Corollary 7.6 there exists a C 1 > 0 such that and by Lemma 11.5 Z 1 ≥ 0 on M 2 × (∞, 0]. Hence we may assume that there exists a θ 0 ∈ (0, 1] such that for all θ ∈ [θ 0 , 1] Claim 1: For every 0 ≤ Q * < 1 we have Proof of Claim: We argue by contradiction. Assume there exists a sequence of points (p i , t i ) in spacetime such that

By construction
Now we need to distinguish two cases: Then there exists an r ∈ (0, 1 2 ) and ∆t ∈ (0, ∆t) such that on the parabolic set we have Q < Q θ 0 . By the strong maximum principle of Theorem 3.2 applied to the evolution equation (11.3) of Z θ 0 we have Z θ 0 = 0 on Ω and therefore (Z θ 0 ) s = (Z θ 0 ) ss = 0 on Ω . By the evolution equation (11.3) of Z θ 0 we see that that which by Proposition 11.10 implies However, the evolution equation (5.13) of T 2 then implies y = 0 on Ω . and thus also x = 0 on Ω . That in turn implies which is a contradiction.
Case 2: Q(p ∞ , 0) ≥ Q θ 0 . Recall that at points (p, t) in spacetime satisfying Q(p, t) ≥ Q θ 0 we have Z θ 0 (p, t) = Z 1 (p, t). In this case we therefore have By the strong maximum principle applied to the evolution equation (11.5) of Z 1 and Lemma 11.5 we deduce Z 1 = 0 on Ω. Furthermore, we see that this is only possible when Q = 1 on Ω.
Thus for every Q * ∈ (0, 1) there exists a δ > 0 such that for all points (p, t) in spacetime satisfying 0 ≤ Q(p, t) ≤ Q * we have By the continuous dependence of Z θ and Q θ on θ, and the fact that Z θ = Z θ at points (p, t) in spacetime at which Q(p, t) ≥ max(Q θ , Q θ ), there exists an θ 1 < θ 0 such that for θ ∈ [θ 1 , 1] Z θ ≥ 0 on M 2 × (−∞, 0]. Now consider the set The above argument shows that E is an open subset of (0, 1]. As the condition Z θ ≥ 0 is closed and f θ depends continuously on θ, it follows that E is also a closed subset of (0, 1]. Hence by connectedness of (0, 1] it follows that E = (0, 1] and thus for all θ ∈ (0, 1] Z θ ≥ 0 on M 2 × (−∞, 0]. Note that by the strong maximum principle applied to the evolution equation (5.1) of Q Q < 1 on M 2 × (−∞, 0], as otherwise Q = 1 everywhere, which is not true. As Z θ = x Q 2 + f θ (Q) and by Proposition 11.9 for every 0 ≤ Q < 1 we have f θ (Q) → 0 as θ → 0 it follows that
Proof of Corollary 11.2. By Theorem 7.5 and the fact that b s ≤ 0 for metrics in I there exists a C 1 > 0 such that This shows that b(o, t) → 0 as t → T sing . As by assumption and y ≤ 0 by the fact that g(t) ∈ I it follows that Consider the rescaled metrics These satisfy the curvature bound By Theorem 7.3 the rescaled metrics g i (t) are κ-non-collapsed at larger and larger scales. Hence by Corollary 8.2 the Ricci flows (M 2 , g i (t), p i ) subsequentially converge to a pointed ancient Ricci flow (M 2 , g ∞ (t), p ∞ ), −∞ < t < 0, with g ∞ (t) ∈ I. By Theorem 11.1 it follows that g ∞ (t) is stationary and homothetic to the Eguchi-Hanson metric.
11.6. Proof of technical lemmas. In this subsection we collect the proofs of the technical lemmas we relied on above.
Proof of Lemma 11.6. We apply [A17, Theorem 9.2] to prove this lemma. Define where denotes the derivative with respect to Q and subscript r denotes the derivative with respect to r. Rewriting the differential equation (11.6) with respect to the independent variable r, we obtain (11.11) rf rr = 1 At r = 0 the right hand side must equal zero, which can be ensured by requiring Then (11.11) can be written as a system of equations of the form is an analytic vector-valued function of several variables satisfying P ( 0, 0, f 0 ) = 0 for all f 0 ∈ (−1, 1). A computation shows that ∂P ∂u ( 0,0,f 0 ) = 0 0 1 − 3 2 f 0 −2 This matrix has no positive integer eigenvalues and furthermore where I 2 is the 2 × 2 identity matrix. By [A17, Theorem 9.2] the desired result follows.
Hence for Q 2 > 1 − θ 4 and 0 < f < 1 we obtain the following inequality Furthermore we observe that If l < 1, then there would exists a Q * < 1 such that for Q ≥ Q * we have f ≥ 1 10 Here 1 10 can be replaced by any positive number less that 1 8 . However, integrating this differential inequality shows that in this case f would reach 1 before Q = 1, leading to a contradiction of our assumption. Therefore we may assume that l = 1. Defining we obtain the differential inequality By Lemma 11.7 we know that g(Q) > 0 and g (Q) < 0 on Q ∈ (0, 1).
Claim 1: The function g(Q) reaches zero before Q = 1.
Proof of Claim: By our assumption that l = 1 we know that there exists a Q * < 1 such that for Q > Q * g(Q) < θ.
Furthermore, by choosing Q * < 1 sufficiently close to 1, we may assume that for Q ≥ Q * (11.16) g (Q) ≤ 3 7 as 3 7 < 1 2 and 5 49 < 1 8 . Now take the substitution g(Q) = u (r) for r = − ln(1 − Q). Then the (11.16) becomes The corresponding ordinary differential equation is of oscillatory type, which motivates the substitution u(r) = e − 2 7 r v(r) yielding the inequality Hence v reaches 0 in finite r, which tracing back the substitutions, shows that g must reach zero before Q = 1.
Assertion (2) follows from the fact that for the initial condition f (0) = 0 the corresponding solution to the ODE (11.6) is f (Q) = 0. By the continuous dependence of f on f (0) = 0 we deduce that Q θ → 1 as θ → 0.

Finally, assertion (3) follows by definition.
Proof of Lemma 11.11. First note that by Lemma 11.7 we know that f, f ≥ 0 on [0, Q θ ]. Solving the ODE (11.6) for f , we obtain Substituting expression (11.17) into (11.9) yields Defining we then only need to check that with equality only at F = 0, Q = 1. Therefore the minimum of p 2 is attained when Q = 1, in which case we have we havẽ As 0 ≤ Q ≤ Q θ < 1, we actually have p 2 (f, Q 2 ) > 0 on (f, Q) ∈ [0, 1] × [0, Q θ ] and the result follows.
Proof of Lemma 11.13. A computation shows that where we omitted the dependence of f on θ and Q for brevity. We first show that β > 0 in the region of the f -Q-plane. Plugging the expression (11.17) of f into the expression (11.18) for β, we obtain An application of L'Hôpital's Rule shows that β = 12f 2 > 0 at Q = 0 and therefore we may assume that Q > 0. Recall that f, f > 0 on (0, Q θ ) by Lemma 11.7. Hence it suffices to show that the polynomials and and hence for every (f, Q) ∈ R 1 we have To show that p 4 > 0 on R 1 is more complicated. For this we introduce the variable F = f Q 2 and polynomialp 4 (F, Q 2 ) = p 4 (f, Q 2 ) Thenp which gives As this expression is concave in F one can easily check that in the region of the Q 2 -F -plane we have ∂ Q 2p 4 ≤ 0 and thusp 4 (F, Q 2 ) ≥p 4 F, From this we conclude that p 4 > 0 on R 1 ∩ {Q > 0} and hence β > 0 on R 1 . We adopt the same procedure to show that γ > 0 in the region Substituting the expression (11.17) for f into the expression (11.19) for γ we obtain First notice that for any point (f, Q) ∈ R 2 Then from . Therefore the first term in the expression for γ is positive and we only need to prove non-negativity of the second term. For this define the polynomials where we again took F = f Q 2 . Computing the partial derivatives We deduce that at an local extrema ∂ Q 2p 3 = ∂ Fp4 = 0 F = −17Q 4 + 8Q 2 + 8 5Q 2 + 2 and 80 + 144Q 2 − 188Q 4 − 200Q 6 + 307Q 8 = 0.
In Lemma 11.15 below we show that the equation for Q 2 has no zeros in the interval Q 2 ∈ [ 3 4 , 1]. Therefore p 5 (F, Q 2 ) has no local extrema in the region R 3 of the (F, Q 2 )plane enclosed by the curves As the set of the (F, Q 2 )-plane corresponding to R 2 is a subset of R 3 it suffices to check non-negativity ofp 5 on the boundary of the region R 3 . There we havẽ This concludes the proof. ≥ 24 + 50 1 + 3r − 4r 2 − 4r 3 + 6r 4 .
12. Discussion of blow-up limits in k = 2 case In this section we investigate the possible blow-up limits of a Ricci flow (M 2 , g(t)), t ∈ [0, T sing ), starting from an initial metric g(0) ∈ I with sup p∈M 2 b(p, 0) < ∞. By Lemma 10.5 and Corollary 11.2 we know that such flows develop a Type II singularity modeled on the Eguchi-Hanson space as the area of the non-principal orbit S 2 o shrinks to zero. One expects, however, that at larger distance scales from S 2 o one could also see other blow-up limits. The goal of this section is to show that these are in fact limited to the following four possibilities: (i) the Eguchi-Hanson space, (ii) the flat R 4 /Z 2 orbifold, (iii) the 4d Bryant soliton quotiented by Z 2 and (iv) the shrinking cylinder R × RP 3 .
Next, we state the main theorem of this section: Theorem 12.1 (Blow-up limits). Let (M 2 , g(t)), t ∈ [0, T sing ), be a Ricci flow starting from an initial metric g(0) ∈ I (see Definition 7.2) with sup p∈M 2 b(p, 0) < ∞. Let (p i , t i ) be a sequence of points in spacetime with b(p i , t i ) → 0. Passing to a subsequence, we may assume that we are in one of the four cases listed below. on R 4 /Z 2 that is stationary and isometric to the flat orbifold R 4 /Z 2 (iii) M ∞ ∼ = R 4 \ {0}/Z 2 and g ∞ (t) can be extended to a smooth orbifold Ricci flow on R 4 /Z 2 that is homothetic to the 4d Bryant soliton quotiented by Z 2 (iv) M ∞ ∼ = R × RP 3 and g ∞ (t) is homothetic to a shrinking cylinder Remark 12.2. Notice that in Theorem 12.1 we do not claim that all blow-up limits (i)-(iv) actually occur. If the Eguchi-Hanson singularity is isolated one would only see (i) and (ii).
12.1. Outline of proof. Assume we are given a sequence of points (p i , t i ) in spacetime with b(p i , t i ) → 0. Consider the rescaled metrics normalized such that b(p i , 0) = 1. By passing to a subsequence we may assume that either In case (I) we know by Corollary 11.2 that (M 2 , g i (t), p i ) subsequentially converges to the Eguchi-Hanson space, which is the blow-up limit (i) from above. Therefore we only need to investigate the behavior in case (II), i.e. at scales larger than the forming Eguchi-Hanson singularity. At these scales Lemma 12.3 yields very important geometric information. In particular, we show that for every > 0 there exist constants C, δ > 0 such that the following holds: For all points (p, t) in are the only non-zero components of the curvature tensor of a rotationally symmetric metric, we see that blow-up limits of case (II) satisfy the curvature bound We now briefly explain some of the geometric intuition behind the quantities T F 1 and T F 2 for rotationally symmetric metrics. When T F 1 = 0 the underlying space is of constant curvature and therefore isometric to a sphere, the flat plane or hyperbolic space, depending on the sign of the scalar curvature. On the other hand solving the ODE T F 2 = 0 one can show that b s → 0 as s → ∞ and the underlying space is asymptotically cylindrical. Thus blow-up limits in case (II) are rotationally symmetric spaces that are 'sandwiched' between a sphere and an asymptotically cylindrical space.
We need to divide case (II) into three subcases in order to investigate the possible blow-up limits: By passing to a subsequence we may assume that For (II.a) and (II.c) we show in Lemma 12.9 and Lemma 12.6 that (M 2 , g i (t), p i ) subsequentially converges to the flat orbifold R 4 /Z 2 and the shrinking cylinder R × RP 3 , respectively. The main idea is that by the strong maximum principle applied to the evolution equation (12.10) of b s when Q = 1 a minimum b s = 0 or a maximum b s = 1 can only be attained if b s is constant everywhere.
Proving that the blow-up limit in case (II.b) is an ancient orbifold Ricci flow, which is homothetic to the 4d Bryant soliton quotiented by Z 2 , is trickier. The construction is carried out in Lemma 12.8, the proof of which we sketch here: Fix a T > 0 and define Then consider the rescaled metrics g i (t), defined as above, on the parabolic neighborhoods Ω i,n := E p i ,t i ,n × [−T − 1, 0] in spacetime. By Lemma 12.3 we know that ∂ t b 2 → 0 uniformly as b → 0. Hence from the curvature bound (12.1) |Rm g(t) | g(t) ≤ C 1 b 2 of Theorem 7.5, we see that the curvature of g i (t) is bounded by Cn 2 on Ω i,n for i sufficiently large and C > 0 some constant. The difficulty in constructing the limiting orbifold flow arises from the fact that the curvature bound (12.1) degenerates as n → ∞. We get around this by exploiting the inequalities on T F 1 and T F 2 derived in Lemma 12.3, to find a uniform curvature bound independent of n. From here it is then easy to construct the orbifold Ricci flow g ∞ (t), t ∈ [−T, 0], on R 4 \ {0}/Z 2 by taking the limit n → ∞. Via Lemma 12.12, and Theorem 12.11 in the Appendix B, we show that g ∞ (t) can be extended to a smooth orbifold Ricci flow on R 4 /Z 2 . Apriori the curvature bound of g ∞ (t) on R 4 /Z 2 × [0, T ] may deteriorate as T → ∞. Nevertheless we can use a diagonal argument to construct an ancient orbifold Ricci flow on R 4 /Z 2 . Hamilton's trace Harnack inequality then implies that g ∞ (t) has bounded curvature on R 4 /Z 2 × (−∞, 0]. Finally, we apply the result [LZ18] by Xiaolong Li and Yongjia Zhang to deduce that g ∞ (t) is homothetic to the 4d Bryant soliton quotiented by Z 2 .
12.2. Proof of main theorem. We begin by proving the central lemma of this section, which yields important geometric information on the high curvature regions of a Ricci flow (M 2 , g(t)) as in the Theorem 12.1.
Proof. Fix ∈ (0, 1). Recall the following facts of the Eguchi-Hanson space (M 2 , g E ) derived in section 4: (a) x = y = 0 on M 2 (b) Q → 1 as s → ∞ (c) g E is normalized such that its warping function b E satisfies b E (0) = 1. Using (a) a computation shows that bb ss = 2 1 − Q 2 Pick C > 10 such that on (M 2 , g E ) we have Q > 1 − , bb ss < , T F 1 > − and T F 2 < whenever s > C. This is possible by property (b).
Take a path γ : [0, T sing ) → M 2 such that where we recall that s(p, t) is the distance of a point p ∈ M 2 from the non-principal orbit S 2 o at the tip of M 2 . By Corollary 11.2 we know that at distance scales comparable to b(o, t) away from S 2 o we converge to the Eguchi-Hanson space as t → T sing . From the scale-invariance of Q, bb ss , T F 1 , T F 2 and ∂ t b 2 it follows that at spacetime points (γ(t), t) inequalities (i)-(v) eventually hold as t → T sing .
Let A be the set of all sequences of points {(p i , t i )} i∈N in spacetime satisfying the following two properties: ( (2) implies that for such sequences t i → T sing as i → ∞.
We first prove inequality (i), arguing by contradiction. Assume that Then there exists a sequence {A n } n∈N of sequences A n = {(p n,i , t n,i )} i∈N of points in spacetime satisfying properties (1) and (2) above, and For each n ∈ N take N (n) ∈ N such that for m ≥ N (n) we have For each n ∈ N take I(n) ∈ N such that Q(p n,I(n) , t n,I(n) ) − lim inf i→∞ Q(p n,i , t n,i ) ≤ 1 n and b(p n,I(n) , t n,I(n) ) ≤ 1 n .
Let (p n , t n ) = (p N (n),I(N (n)) , t N (n),I(N (n)) ) for n ∈ N. Then we see that and both properties (1) and (2) from above hold. Before we carry on recall that by Theorem 7.5 there exists a C 1 > 0 such that Recall also Theorem 7.3, from which it follows that there exist constants κ, ρ > 0 such that g(t) is κ-non-collapsed at scale ρ. Next, consider the rescaled metrics where ∆t > 0 is chosen such that Proposition 8.3 holds. Then (C g i (0) p i , 1 2 , g i (t), p i ) subsequentially converges to a pointed Ricci flow ( We argue by contradiction. Assume there exists a C > 0 such that after passing to a subsequence of (p i , Consider the rescaled metrics By Corollary 11.2 we see that (M 2 , g i (t), p i ) subsequentially converges to (M 2 , g ∞ (t), p ∞ ), where g ∞ (t) is stationary and homothetic to the Eguchi-Hanson metric. By con- 0), where the first inequality follows from the fact that 1 ≥ Q ≥ b s ≥ 0 for metrics in I and the second inequality follows from the definition of C. Thus We prove (ii)-(v) by the same strategy. Below we first prove inequality (ii) by contradiction. Assume that As before we can construct a sequence of points (p i , t i ) in spacetime satisfying properties (1) and (2), and such that lim i→∞ bb ss (p i ,t i ) = ι.
The derivation is carried out in the Appendix A, in the subsection on the evolution equations when Q = 1. From this we see that at the point (p ∞ , 0) in spacetime we have ∂ t (bb ss ) which is a contradiction. This proves (ii). We prove inequality (iii) similarly. Assume that As before, (C g i (0) p i , 1 2 , g i (t), p i ) subsequentially converges to a pointed Ricci flow (C ∞ , g ∞ (t), p ∞ ). By construction By inequality (i) of this lemma Q = 1 on C ∞ × [−∆t, 0]. For Q = 1 the evolution equation of T F 1 can be written as The derivation is carried out in the Appendix A. From this we see that at the point (p ∞ , 0) in spacetime we have with equality if, and only if, b s (p∞,0) = 0. Therefore we conclude that b s = 0 at (p ∞ , 0). Applying the strong maximum principle to the evolution equation (12.10) of b s when Q = 1, it then follows that b s = 0, and hence b ss = 0, everywhere in C ∞ × [−∆t, 0]. This, however, implies T F 1 = 1 at (p ∞ , 0), which is a contradiction and thus proves (ii). We proceed to prove (iv) in the same fashion. Assume that As before, (C g i (0) p i , 1 2 , g i (t), p i ) subsequentially converges to a pointed Ricci flow (C ∞ , g ∞ (t), p ∞ ). By construction By statement (i) and (ii) of this lemma we have Q = 1 and bb ss ≤ 0 on C ∞ ×[−∆t, 0]. When Q = 1 the evolution equation of T F 2 can be written as where C F 2 is a polynomial expression in bb ss and 1 − b 2 s . The derivation is carried out in the Appendix A. By Lemma 12.10 in the Appendix A, C F 2 < 0 whenever T F 2 > 0 and bb ss ≤ 0. This, however, implies which is a contradiction and thus proves (iv).
Finally we prove (v), also by contradiction. Assume that As before, (C g i (0) p i , 1 2 , g i (t), p i ) subsequentially converges to a pointed Ricci flow (C ∞ , g ∞ (t), p ∞ ). By construction as ∂ t b 2 is a scale-invariant quantity. By (i) we have Q = 1 on C ∞ × [−∆t, 0] and the evolution equation (2.12) of b simplifies to , as b s ∈ [0, 1] for metrics in I. This, however, implies that which is a contradiction and thus proves (v).
Lemma 12.5. Let (M 2 , g(t)), t ∈ [0, T sing ), be a Ricci flow starting from an initial metric g(0) ∈ I with sup p∈M 2 b(p, 0) < ∞. Then for every ∈ (0, 1) there exists a δ > 0 such that at all points (p, t) in spacetime at which b(p, t) ≤ δ we have Proof. Fix > 0. By Lemma 12.3 we only need to prove that there exists a δ > 0 such that the result holds when b(p, t) ≤ Cb(o, t) ≤ δ, where C > 0 is as in Lemma 12.3. Note that ∂ t b 2 = 0 on the Eguchi-Hanson space background. By Corollary 11.2 we know that on the scale b(p, t) ≤ Cb(o, t) we converge to the Eguchi-Hanson space as t → T sing . As b(o, t) → 0 as t → T sing we see that there exists a δ > 0 such that at all points (p, t) in spacetime at which b(p, t) ≤ Cb(o, t) ≤ δ. This completes the proof.
Below we prove the simplest case of Theorem 12.1.
Lemma 12.6. Let (M 2 , g(t)), t ∈ [0, T sing ), be a Ricci flow starting from an initial metric g(0) ∈ I with sup p∈M 2 b(p, 0) < ∞. Let (p i , t i ) be a sequence of points in spacetime satisfying Consider the rescaled metrics That is, the metric g ∞ (t) is cylindrical. From here one can inductively show that for every r > 0 the Ricci flows (C g i (0) (p i , r), g i (t), p i ), t ∈ [−T, 0] subsequentially converge to a limiting cylindrical Ricci flow. Hence (M 2 , g i (t), p i ), t ∈ [−T, 0], subsequentially converges to the shrinking cylinder (R×RP 3 , g ∞ (t), p ∞ ), t ∈ [−T, 0]. As T > 0 is arbitrary the desired result follows by a diagonal argument.
Before we carry on constructing the blow-up limit (iii) of Theorem 12.1, we need to state two technical lemmas. Their proofs can be skipped on the first reading.
Write B = b s . Then the ODI becomes Since b s > 0 we may treat b as the independent variable, yielding the following ODI Note that as B = b s ∈ (0, 1) we may rearrange the inequality and integrate to obtain where we denote by b 0 and B 0 the values of b and B at s 0 , respectively. Evaluating the integrals and rearranging we deduce By the initial conditions (12.3) and (12.4) we have Kη 2 , which can be rearranged to Choosing K sufficiently large the desired result follows. Now we may construct the orbifold Ricci flow blow-up: Lemma 12.8. Let η ∈ (0, 1) and (M 2 , g(t)), t ∈ [0, T sing ), be a Ricci flow starting from an initial metric g(0) ∈ I with sup p∈M 2 b(p, 0) < ∞. Assume that (p i , t i ) is a sequence of points in spacetime satisfying

Consider the rescaled Ricci flows
, subsequentially converges, in the Cheeger-Gromov sense, to an ancient Ricci flow can be extended to a smooth orbifold Ricci flow on R 4 /Z 2 that is homothetic to the 4d Bryant soliton quotiented by Z 2 .
Proof. Fix T > 0. By Lemma 12.5 we have that for every > 0 there exists a δ > 0 such that at points (p, t) in spacetime at which b(p, t) < δ we have ∂ t b 2 ≤ . This shows that for every N > 0 there exists a δ = δ (N ) > 0 such that whenever b(p, t) < δ then Consider the rescaled metrics For n ∈ N ≥2 and (p, Furthermore, define the parabolic neighborhoods Recall that by Theorem 7.5 there exists a C 1 > 0 such that . Hence for fixed n and sufficiently large i the curvatures of g i (t) on Ω i,n are uniformly bounded: |Rm g i (t) | g i (t) ≤ 4n 2 C 1 on Ω i,n . This follows from (12.5), b(p i , t i ) → 0 and the fact that b s ≥ 0 for metrics in I. By Theorem 7.3 the Ricci flows g i (t) are κ-non-collapsed at larger and larger scales. By a slight adaptation of the local compactness Theorem 8.1 we see that for each n ∈ N the Ricci flows (E p i ,t i ,n , g i (t), p i ), t ∈ [−T − 1, 0], subsequentially converge to a Ricci flow (E ∞,n , g ∞,n (t), p ∞,n ), t ∈ [−T − 1, 0]. The manifolds E ∞,n are diffeomorphic to R 4 \ {0}/Z 2 and therefore incomplete. By a diagonal argument we may assume that E ∞,n ⊂ E ∞,n+1 and g ∞,n (t) = g ∞,n+1 (t) on E ∞,n . This allows us to drop the dependence on n and write g ∞ (t) and p ∞ for brevity. By Lemma 12.3 we have Q = 1, bb ss ≤ 0, T F 1 ≤ 0, T F 2 ≥ 0 and ∂ t b 2 ≤ 0 on E ∞,n .
Proof of Claim: We argue by contradiction. Assume that t ∈ [−T, 0] is such that for t ∈ [t , 0] we have b s (p ∞ , t) ≥ 0 with equality if, and only if, t = t . Applying the strong maximum principle to the evolution equation (12.10) of b s when Q = 1, we obtain that b s = 0 on E ∞,n × [−T − 1, t ]. That is, the metric g ∞ (t) is cylindrical for times t ∈ [−T − 1, t ]. We now show that this leads to a contradiction. Take times Because b s (p i , t i ) → 0 as i → ∞, Lemma 12.6 implies that after passing to a subsequence (M 2 , g i (t), p i ) converges to the shrinking cylinder R × RP 3 . For every n ∈ N take N n ∈ N such that for i ≥ N n the region C g i (0) (p i , n) ⊂ M 2 is close, in the Cheeger-Gromov sense, to a cylinder R × RP 3 of length 2n and radius 1. By Perelman's pseudolocality theorem there exists a K > 0 and τ > 0 such that g i (t) has bounded curvature on , subsequentially converges to a limiting Ricci flow (M ∞ , g ∞ (t), p ∞ ), t ∈ (−∞, τ ], where M ∞ ∼ = R × RP 3 and g ∞ (0) is cylindrical. By the uniqueness of Ricci flow solutions [CZ06], we see that g ∞ (t) remains cylindrical for t ∈ [0, τ ] and therefore b s = 0 on M ∞ × (−∞, τ ). Now we have arrived at a contradiction, as this implies that t is not the earliest time at which b s = 0 on the wordline through the point (p ∞ , 0) in the spacetime E ∞,n × [−T − 1, 0]. This proves the claim.
and hence for some universal constant c > 0, as 1−b 2 s b 2 and − bss b are the only non-zero curvature components of a rotationally symmetric metric.
Claim 2: There exists a K = K(η) > 1, independent of n, such that Proof of Claim: Fix n ≥ 2. As bb ss ≤ 0 and b s ≥ 0 it follows from Claim 1 that b s ≥η > 0 in the region As ∂ t b 2 ≤ 0 we have b(p ∞ , t) ≥ b(p ∞ , 0) = 1 for t ∈ [−T, 0]. Now choose a K = K(η) > 1 such that Lemma 12.7 holds true. If at some point (p , t ) ∈ E ∞,n × [−T, 0] we had 1 − b 2 s b 2 ≥ K then on the time slice {t = t } ⊂ E ∞,n the result of Lemma 12.7 would imply that b s <η when b ≥ 1. This cannot be true, as by Lemma 12.5 we have that ∂ t b 2 ≤ 0 on E ∞,n × [−T, 0] and therefore b(p ∞ , t) ≥ 1 for t ∈ [−T, 0]. Hence we deduce by (12.6) that the curvature is bounded by cK on the region R. As on E ∞,n × [−T, 0] \ R we have b > 1, it follows by (12.6) and the fact that b s ∈ [0, 1] for metrics in I that the curvature is uniformly bounded by c there.
Claim 2 shows that as n → ∞ we may extract a limiting Ricci flow (M ∞ , g ∞ (t), p ∞ ), t ∈ [−T, 0], with curvature bounded by cK. By construction M ∞ is diffeomorphic to (R 4 \ {0})/Z 2 . Define the radial coordinate ξ : M ∞ → R by Note that by the Ricci flow equation (2.12) for b we have Working in (ξ, t) coordinates we see that Hence for all t ∈ [−T, 0] we have b(ξ, t) → 0 as ξ → 0. As M ∞ has bounded curvature, we see that 1−b 2 s b 2 is bounded as well and hence b s (ξ, t) → 1 as ξ → 0. From Theorem 12.11 in Appendix B it then follows that g ∞ (t), t ∈ (−T, 0], can be extended to a smooth orbifold Ricci flow on R 4 × Z 2 . Since T was arbitrary, a diagonal argument produces an ancient orbifold Ricci flow (R 4 \ /Z 2 , g ∞ (t), p ∞ ), t ∈ (−∞, 0]. Note that apriori g ∞ (t) might have unbounded curvature as t → −∞.
As Q = 1, b s ∈ [0, 1] and b ss ≤ 0 we see that g ∞ (t) is rotationally symmetric and has positive sectional curvature. Furthermore, for each t ∈ (−∞, 0] the metric g ∞ (t) is asymptotically cylindrical, as the following argument shows: Either b is bounded, in which case bb ss ≤ 0 and b s ≥ 0 show that lim s→∞ b s = 0, or b is unbounded, in which case the inequality T F 2 ≤ 0 and the proof of Lemma 12.7 show that on each time slice b s → 0 as b → ∞.
By the Hamilton's trace Harnack inequality (see for instance [ChII,Theorem D.26]) and the fact that for any T > 0 the metric g ∞ (t) has bounded curvature on Therefore g ∞ (t) has bounded curvature on R 4 /Z 2 × (∞, 0]. By the result of Li and Zhang [LZ18] we conclude that g ∞ (t) is homothetic to the four dimensional Bryant soliton quotiented by Z 2 .
Lemma 12.9. Let (M 2 , g(t)), t ∈ [0, T sing ), be a Ricci flow starting from an initial metric g(0) ∈ I with sup p∈M 2 b(p, 0) < ∞. Let (p i , t i ) be a sequence of points in spacetime satisfying .
, subsequentially converges, in the Cheeger-Gromov sense, to an ancient Ricci flow can be extended to a smooth orbifold Ricci flow on R 4 /Z 2 that is stationary and isometric to the flat orbifold R 4 /Z 2 .
Proof. First note that Proof of Claim: We argue by contradiction. Assume there exists a C > 0 such that after passing to a subsequence of (p i , t i ) we have Consider the rescaled metrics Then by by Corollary 11.2 the sequence (M 2 , g i (t), p i ) subsequentially converges to a blow-up limit (M 2 , g ∞ (t), p ∞ ), which is homothetic to the Eguchi-Hanson space. By construction b(p ∞ , 0) = 1 and b s (p ∞ , 0) = 1.
The latter follows from the assumption that b s (p i , t i ) → 1 as i → ∞. However, by Lemma 4.2 we have b s < 1 everywhere on the Eguchi-Hanson space. This is a contradiction and the claim follows.
Fix T > 0 and consider the rescaled metrics g i (t) on the parabolic sets E (p i ,t i ,n) × [−T, 0] as in the proof of Lemma 12.8. By the same reasoning, we see that for all n ∈ N ≥2 the flows (E p i ,t i ,n , g i (t), p i ) subsequentially converges to a Ricci flow (E ∞,n , g ∞,n (t), p ∞,n ), t ∈ [−T, 0]. As in the proof of Lemma 12.8, we may assume that E ∞,n ⊂ E ∞,n+1 and g ∞,n = g ∞,n+1 on E ∞,n . Therefore we drop the dependence on n and write p ∞ and g ∞ (t). By construction we have b(p ∞ , 0) = 1 where the latter follows from the assumption that b s (p i , t i ) → 1 as i → ∞.
Inserting the expressions for ∂ t a and ∂ t b from the evolution equations (2.11) and (2.12) for a and b we obtain Evolution equations of a s , b s , Qb s , x, y and y Q . By the commutation relations above we have Hence plugging in the expressions for ∂ t a and ∂ t b from the evolution equations (2.11) and (2.12) for a and b we obtain From here we can compute the evolution equation of Qb s : Now we may compute the evolution equations of the Kähler quantities x and y: where in the last step we made the substitutions a s = x − Q 2 + 2, b s = y + Q and a = Qb. Similarly, = y ss + a s a y − y a 2 (x + 2) 2 + Q 2 2x + y 2 Then we can compute where we substituted a s = x − Q 2 + 2, b s = y + Q and a = Qb in the last step.
Evolution equation of H ± . In section 7 we define the quantities H ± := bb ss ∓ a 2 s − b 2 s ± C for some constant C > 0. Below we derive its evolution equation.
First note that we have Substituting the evolution equation for b s derived above we obtain Hence we can compute the evolution equation of H via and substituting the expressions for ∂ t b, ∂ t b ss , ∂ t a s and ∂ t b s derived above. Noting that ss a longer computation shows that Evolution equation of f θ (Q).
by the evolution equation (5.1) of Q. Note that we omitted the dependence the quantities on spacetime (ξ, t) and and the dependence of f on Q and θ. For example we wrote f for f θ (Q(ξ, t)). Noting that Evolution equation of Z θ . We have We computed the evolution equation for f θ (Q) above. Therefore it remains to compute ∂ t x Q 2 . For this recall the evolution equation (5.7) of x Differentiation shows that Therefore we get In the last step, we used the expressions for x, y and Q s in terms of a s , b s and Q to eliminate x, y and Q s from the expression for C x Q 2 . Hence we have by definition, we can solve for a s to obtain Using this substitution to eliminate all occurring a s from the expression C Z we obtain Evolution equation of Z 1 . The evolution equation for Z 1 = x Q 2 +1 follows quickly from the evolution equations for Z θ by setting f = 1. One obtains Note that we wrote C Z 1 ,0 in terms of y = b s − Q instead of b s in order to see the similarity with the zeroth order term of the evolution equation of T 1 presented in the proof of Lemma 5.8.
Evolution equations when Q = 1. When Q = 1 we have a = b and the Ricci flow equations simplify. In particular, we obtain Using the commutation relation of ∂ t and ∂ s we can also compute the evolution equation of b s and b ss : 2 (b 2 s − 1) b ss b 2 Let us introduce the notation We need the evolution equations of scale-invariant quantities of the form where F : [0, 1] → R is a smooth function. In particular, we see that s ∂ t b s Expanding this expression, we obtain In this paper we make use of three different choices of F : Plugging these into the expression for C F above we compute We also prove the following lemma here: Lemma 12.10. Let X ∈ [0, 1]. Then whenever 0 ≤ Y < X − X 2 we have P (X, Y ) := (2 − 3X)Y 2 + 2X 2 − 4X + 2 Y − 2(X − 1) 2 X < 0.
The last inequality follows by the fact that the polynomial 3X 3 + X 2 + 2X − 2 is increasing on [0, 1] and evaluates to 2 3 at X = 2 3 . Hence the claim follows by concavity of ∂ X P (X, Y ) in Y .
By above we know that P (X, X 2 − X) < 0 for X ∈ (0, 1). Hence using the result of the claim, we see that P (X, Y ) < 0 on R ∩ {X ≥ 2 3 }.
Taking ξ ∈ (0, ∞) to be a radial coordinate on R 4 \ {0} the metric g(t) may be written as g(t) = u 2 (ξ, t)dξ 2 + b 2 (ξ, t)g S 3 , where u, b : (0, ∞) → R are smooth warping functions, and g S 3 is the round metric on S 3 with sectional curvatures equal to one. If for all t ∈ [0, T ] the warping function b(ξ, t) → 0 as ξ → 0, then g(t) can be extended to a smooth Ricci flow on R 4 ×(0, T ]. Below we assume (R 4 \ {0}, g(t)), t ∈ [0, T ], is a Ricci flow as in Theorem 12.11. The proof strategy will be as follows: First we prove in Lemma 12.12 that for every t 0 ∈ [0, T ] there exist coordinates x i , i = 1, 2, 3, 4, of R 4 for which the metric g(t 0 ) can be extended to a C 1,1 metric on R 4 . Note, however, without control on the derivative of the curvature tensor the metric g(t) at times t = t 0 may not to be C 1,1 with respect to the coordinates x i . To get around this issue we show in Lemma 12.14 and Lemma 12.16 that in fact all derivatives ∇ m Rm, m ∈ N, of the curvature tensor are bounded on R 4 \{0}×(δ, T ] for any δ > 0. The proof utilizes Shi's interior derivative estimates and is based on a De Giorgi-Nash-Moser iteration argument. With these results in place, we use harmonic coordinates to prove Theorem 12.11. Let us begin by proving Lemma 12.12. Let g = ds 2 + b(s) 2 g S 3 be a smooth, rotationally symmetric metric with bounded curvature on R 4 \ {0}. Here g S 3 is the round metric of curvature one on S 3 and b : (0, ∞) → R is a smooth positive function. If b → 0 as s → 0 then g can be extended to a C 1,1 metric on R 4 . Furthermore, if we take standard Euclidean coordinates x i , i = 1, 2, 3, 4, on R 4 we have g ij = δ ij and ∂ k g ij = 0 at the origin, and ∂ k ∂ l g ij locally bounded on R 4 \ {0}.
Proof. As g has bounded curvature there exists a K > 0 such that because these are the only non-zero curvature components of a rotationally symmetric metric. In particular, this shows that b s → 1 as s → 0 + and b ss → 0 as s → 0 + . Let x i , i = 1, 2, 3, 4, be Euclidean coordinates of R 4 , normalized such that i (x i ) 2 = s 2 . In these coordinates  Next, extend g to the origin by setting g = δ ij there. As it follows by Claim 1 that this defines a continuous extension of g to the origin.
As (2x k δ ij − δ ik x j − x i δ jk ) = O(s) and it follows that we may continuously extend ∂ k g ij to the origin by setting ∂ k g ij = 0 there. Finally, note that Hence ∂ k ∂ l g ij is bounded in a neighborhood around, but excluding the origin. This shows that ∂ k g ij is Lipshitz.
Next, we prove the boundedness of the gradient of the curvature tensor. For this we recall some interior curvature estimates. Note the following differential inequalities for the evolution of the curvature tensor and its derivatives under Ricci flow (See for instance [BC04, Chapter 7]): ∂ t − ∆ |Rm| 2 ≤ −2|∇Rm| 2 + 16|Rm| 3 (12.12) ∂ t − ∆ |∇ m Rm| 2 ≤ −2|∇ m+1 Rm| 2 (12.13) + m j=0 c mj |∇ j Rm| · |∇ m−j Rm| · |∇ m Rm| Here c mj are positive constants depending on j, m and the dimension of the manifold only. Note also that the laplacian is with respect to the evolving metric g(t). Using these inequalities one can show the following interior derivative estimate (See for instance [CZ06, Theorem 1.4.2]). If for p ∈ U and r > 0 the closed set B g(0) (p, r) is contained in U then Next, we prove that for all τ > 0 the gradient |∇Rm| is bounded on R 4 \{0}×[τ, T ]. First note that due to Shi's estimates of Theorem 12.13 |∇Rm g(t) | g(t) (p) = O 1 d g(t) (p, 0) on R 4 \ {0} × [τ, T ].
Hence when curvature is bounded by K, the function u := e −CKt |∇Rm| is a subsolution to the heat equation, i.e.
With help of a De Giorgi-Nash-Moser iteration argument, this is enough to prove that u is bounded for t > τ . We carry this out in the lemma below: Lemma 12.14. Let (R 4 \ {0}, g(t)), t ∈ [0, T ], be a Ricci flow as in Theorem 12.11. Then for any τ > 0 there exists a C = C(K, τ ) > 0 such that Proof. As shown above, the function u = e −CKt |∇Rm| is a subsolution to the heat equation, i.e. (∂ t − ∆) u ≤ 0.
By Lemma (12.12) we may choose Euclidean coordinates x i , i = 1, 2, 3, 4, on R 4 for which g(0) is C 1,1 . Take s 2 = i (x i ) 2 and write B R (x) for the ball centered at x with radius R with respect to g(0). Hence it suffices to show that for some R > 0 the function u is bounded on B R (0) × [τ, T ]. We achieve this via a De Giorgi-Nash-Moser iteration argument. In the Claim below we derive the crucial estimate.
Claim 1: Let δ > 0, p ≥ 2, R 0 ∈ [1, 10] and t 0 ∈ [δ, T ). Then there exists a constant C = C(K, δ, T ) > 0 such that the following holds: If u ∈ L p (B R 0 × [t 0 , T ]), then for R 1 ∈ [ 1 2 , R 0 ) and t 1 ∈ (t 0 , T ] Proof of Claim: In the following a constant C is assumed to only depend on K, δ and T and might vary from line to line. Fix a number A > 1 that we later take to ∞. Then choose a C 2 function F : R ≥0 → R with the following properties: (1) F (s) = s p for s ≤ A (2) F is linear for s ≥ A + 1 with slope pA p−1 + 1 (3) On [A, A + 1] take F (s) to be defined such that F ≥ 0 We see that these properties imply that F (s) ≤ ps p−1 . Next, define the cut-off functions η , > 0, and φ : R 4 → R. For this take a smooth function h : R → R with h = 0 on (−∞, 1 2 ] and h = 1 on [1, ∞). Then define That is, φ = 1 on B R 1 (0) and φ = 0 on R 4 \ B R 0 (0). Clearly, |∇φ| g(t) ≤ c R 0 −R 1 and |∇η | g(t) ≤ c for some universal constant c depending on h and λ only.
Since u ∈ L p (B R 0 × [t 0 , T ]) is a positive function there exists a t ∈ [t 0 , t 1 ] such that (12.14) Next, we compute via integration by parts Integrating with respect to time from t to T and noting that η F (u)∇u∇φ 2 dx dt := I 1 − I 2 − I 3 .
Next we estimate each of these integrals I 1 , I 2 and I 3 separately, in order to analyze their behaviors as → 0. For the first integral we have For the second integral I 2 , note that Shi's estimates and Kato's inequality yield |∇u| = |∇ e −CKt |∇Rm| | ≤ |∇ 2 Rm| = O 1 s 2 as s → 0.
Taking B = 32 and assuming without loss of generality that K > 1, we obtain ∂ t S ≤ ∆S − S 2 CK 4 + CK 6 From here we may follow the proof of [CZ06, Theorem 1.4.2] to deduce the desired result.
With help of Theorem 12.15 we inductively prove that the higher derivatives of the curvature tensor are bounded.
As in the proof of Lemma 12.14, choose coordinates x i , i = 1, 2, 3, 4, such that g(0) can be extended to a C 1,1 metric on R 4 , and write s 2 = i (x i ) 2 . As the curvature of g(t), t ∈ [0, T ], is bounded there exists a λ > 0 such that By the modified Shi's estimates of Theorem 12.15 we see that for some C that depends on τ, K, and C l , l = 1, 2, · · · , N , only. In particular, this implies that for all R > 0 the function u ∈ L 2 (B R (0)) × [0, T ]. By the differential inequality (12.13) for the evolution of the curvature derivatives we see that (∂ t − ∆) |∇ N +1 Rm| 2 ≤ −2|∇ N +2 Rm| 2 + CK 2 |∇ N +1 Rm| + CK|∇ N +1 Rm| 2 and hence (∂ t − ∆) |∇ N +1 Rm| ≤ CK K + |∇ N +1 Rm| . Thus defining u = e −CKt |∇ N +1 Rm| + K we deduce that (∂ t − ∆) u ≤ 0. Now we are in the same setup as in the proof of Lemma 12.14. Therefore we may use the same De Giorgi-Nash-Moser iteration argument to show that u and hence |∇ N +1 Rm| are bounded in R 4 × [τ, T ]. This proves the desired result. Now we may prove the main Theorem 12.11: Proof of Theorem 12.11. By Lemma 12.12 we can choose coordinates x i , i = 1, 2, 3, 4, for R 4 such that g(T ) can be extended to a C 1,1 metric on all of R 4 . Below we write g = g(T ) for brevity. By [DK81, Lemma 1.2] there exist C 2,α harmonic coordinates y i : U → R, i = 1, 2, 3, 4, in an open neighborhood U of R 4 containing the origin and satisfying (1) y i = 0 (2) ∂y i ∂x j = δ i j at the origin. Furthermore, as g is smooth on U \ {0}, it follows from interior elliptic regularity that y i are smooth on U \ {0}. Write g ij = g ∂ ∂y i , ∂ ∂y j and Ric ij = Ric g ∂ ∂y i , ∂ ∂y j .
We have that g ij ∈ C 1,α (U ) with respect to the y i coordinates. Furthermore, g ij is smooth on U \ {0}. where Q(g, ∂g) is some universal analytic expression that is polynomial in the matrix g, quadratic in ∂g ∂y i , and has a denominator term depending on det g ij . The equation (12.16) makes sense on all of U if we interprete it in the weak sense.

Proof of Claim: Write
Since covariant differentiation commutes with contractions, we can use the product rule to express the above derivative as a sum of terms, which only involve ∇ m Ric, m = 1, 2, · · · , k, and ∇ m Y i l , m, l = 1, 2, 3, · · · , k, contracted with Y i , Y j and Y i l , l = 1, 2, · · · , k. As by Lemma 12.16 all the derivatives of the curvature tensor are bounded and g ij (y) ∈ C k (U ) we see that ∂ k ∂y i 1 ∂y i 2 · · · ∂y i k Ric ij is bounded as well. Hence the k-th spatial derivatives ∂ k Ric ij are bounded, which implies that ∂ k−1 Ric ij is a Lipshitz function and can be continuously extended to a all of U . Similarly, the lower order derivatives ∂ m Ric ij , m = 0, 1, 2, · · · , k − 2, can be continuously extended to the origin.
First note that g is a C 1,1 (U ) weak solution of the elliptic equation (12.16). Furthermore Q(g, ∂g) ∈ C 0,1 (U ) and by Claim 1 we have Ric ij ∈ C 0,1 (U ) as well. Since such weak solutions are unique, and there exists a C 2,α (U ) solution that agrees on the boundary ∂U , we see that g is in fact C 2,α (U ). Bootstrapping standard Schauder estimates and the result of Claim 1, we conclude that g ij is smooth with respect to the harmonic coordinates y i , i = 1, 2, 3, 4.
It remains to be shown that g(t) can be extended to a smooth Ricci flow on R 4 × (0, T ]. Recall that by Lemma 12.16, for all τ > 0 the derivatives of the curvature tensor are bounded on U × [τ, T ]. Moreover g(T ) is bi-lipshitz to the euclidean metric δ ij on U and by the previous paragraph the covariant derivatives of g(T ) with respect to δ ij are all bounded. Therefore we may follow the proof of [ChI,Lemma 3.11] with t 0 = T to deduce that ∂ m ∂t m ∂ n ∂y i 1 · · · ∂y in (g(t)) ij ≤ K m,n on U \ {0} × [τ, T ], for some constants K m,n > 0. This shows that g(t) can be smoothly extended to U × [τ, T ]. As τ > 0 was arbitrary the desired result follows.