Ricci flow of warped Berger metrics on R4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{4}$$\end{document}

We study the Ricci flow on R4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{4}$$\end{document} starting at an SU(2)-cohomogeneity 1 metric g0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{0}$$\end{document} whose restriction to any hypersphere is a Berger metric. We prove that if g0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{0}$$\end{document} has no necks and is bounded by a cylinder, then the solution develops a global Type-II singularity and converges to the Bryant soliton when suitably dilated at the origin. This is the first example in dimension n>3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n > 3$$\end{document} of a non-rotationally symmetric Type-II flow converging to a rotationally symmetric singularity model. Next, we show that if instead g0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{0}$$\end{document} has no necks, its curvature decays and the Hopf fibres are not collapsed, then the solution is immortal. Finally, we prove that if the flow is Type-I, then there exist minimal 3-spheres for times close to the maximal time.

Shi proved that if (M, g 0 ) is complete and has bounded curvature, then the Ricci flow problem admits a solution [41]. Moreover, such solution is unique in the class of complete solutions with bounded curvature by the work of Chen and Zhu [18]. A solution to the Ricci flow encounters a finite-time singularity at some T < ∞ if and only if [29,41] lim sup Finite time singularities of the Ricci flow are classified as follows [29]: According to results of Naber [36] and Enders-Müller-Topping [20] any parabolic dilation of a Type-I Ricci flow at a singular point converges to a non-flat gradient shrinking soliton. On the other hand, far less is known about Type-II singularities.
The first examples of Type-II singularities in dimension n ≥ 3 were found in [24] by Gu and Zhu, who considered a family of rotationally invariant Ricci flows on S n . Angenent, Isenberg and Knopf later discovered Type-II spherically symmetric Ricci flows on S n that are modelled on degenerate neckpinches [1]. Type-II singularities were also derived for rotationally invariant Ricci flows on R n by Wu in [44] and later, for a larger set of initial data, by the author in [19].
Only very recently the first explicit examples of non rotationally symmetric Type-II Ricci flows in dimension higher than three have been analysed by Appleton in [4] and by Stolarski in [43], where they both obtained Ricci flat singularity models.
In our first result we show that a large family of 4-dimensional cohomogeneity 1 Ricci flows develop Type-II singularities modelled on the Bryant soliton [10]. The Ricci flow on 4-dimensional cohomogeneity 1 manifolds has been recently studied on various topologies [4,6,31,32]. In [31] Isenberg, Knopf and Šešum showed that the Ricci flow starting at a family of generalized warped Berger metrics on S 1 × S 3 is Type-I and becomes rotationally symmetric around any singularity. This behaviour is regarded as a Type-I example of symmetry enhancement along the Ricci flow.
In this work we study the Ricci flow evolving from a generalized warped Berger metric on R 4 . Namely, consider a metric g 0 invariant under the cohomogeneity 1 left-action of SU(2) on R 4 = C 2 . We can then write g 0 in Bianchi IX form as [22] g 0 = (ds) 2 + a 2 (s) σ 1 i=1 is a coframe dual to some Milnor frame {X i } 3 i=1 on SU (2), with X 3 tangent to the Hopf fibres. We further assume that g 0 is invariant under rotations of the Hopf fibres. The last condition means that the left-invariant vector field X 3 is Killing thus extending the Lie algebra of g 0 -Killing vectors to u (2). In particular, we can write g 0 as g 0 = (ds) 2 + b 2 (s) (σ 1 ⊗ σ 1 + σ 2 ⊗ σ 2 ) + c 2 (s) σ 3 ⊗ σ 3 .
In analogy with [31] we finally assume that c ≤ b so that each non-degenerate fiber {s} × S 3 is a Berger sphere. We call such metric a warped Berger metric on R 4 .
We first focus on initial data with linear volume growth.

Definition 1
We let G be the set of complete bounded curvature warped Berger metrics g 0 on R 4 satisfying the following conditions: is the mean curvature of the centred Euclidean sphere of Euclidean radius r with respect to g 0 .
The control on the sign of H amounts to ruling out the existence of necks [2]. We also note that the condition in (i) is weaker than asking for both b and c to be monotone. We prove that any Ricci flow starting in G converges to the Bryant soliton once suitably rescaled. This provides Type-II examples of symmetry enhancement and constitutes the first explicit case in dimension higher than three of a non-conformally flat Type-II Ricci flow converging to a rotationally symmetric singularity model. 4 , g(t)) 0≤t<T be the maximal complete, bounded curvature solution to the Ricci flow starting at some g 0 ∈ G. The solution develops a Type-II singularity at some T < ∞ which is modelled on the Bryant soliton once suitably dilated.
Theorem 1 resembles an analogous result recently derived in [4], where Appleton studied the Ricci flow on the blow-up of C 2 /Z k starting from a subclass of U(2)-invariant metrics as above. In particular, he showed that when k = 2 if the initial metric is bounded by a cylinder at infinity and both b and c are increasing and satisfy a differential inequality, then the flow is Type-II and converges to the Eguchi-Hanson metric once suitably dilated.
Theorem 1 characterizes the Type-II singularity only partially. Indeed, while the Type-II singularity is not isolated, being the Bryant soliton asymptotically cylindrical (see, e.g., [8]), in general there is no control on the blow-up sequence giving rise to a family of shrinking cylinders. Moreover, the symmetries and the lack of necks suggest that the curvature ought to become large locally around the singular orbit (see also [4]). Equivalently, one should detect the Bryant soliton when dilating the flow at the origin o by suitable factors. In our next result we address these issues hence providing a much clearer picture of the Type-II singularity developed by Ricci flows in G. In the following statement R g (t) represents the scalar curvature of the Ricci flow solution.
Theorem 2 Let (R 4 , g(t)) 0≤t<T be the maximal complete, bounded curvature solution to the Ricci flow starting at some g 0 ∈ G. Then the following conditions hold: (i) (The Bryant soliton appears at the origin.) There exists t j T such that the rescaled Ricci flows (R 4 , g j (t), o) defined by g j (t) .
= R g(t j ) (o)g(t j + (R g(t j ) (o)) −1 t) converge to the Bryant soliton in the Cheeger-Gromov sense.
(ii) (The singularity is global.) For any p ∈ R 4 we have (iii) (Type-I blow-up at infinity.) For any t j T there exist a sequence { p j } and α > 0 such that d g 0 (o, p j ) → ∞, (T − t j )R g(t j ) ( p j ) ≤ α, and the rescaled Ricci flows (R 4 , g j (t), p j ) defined by g j (t) .
= R g(t j ) ( p j )g(t j + (R g(t j ) ( p j )) −1 t) converge to the self-similar shrinking cylinder in the Cheeger-Gromov sense. (iv) (Classification of singularity models.) Any non-trivial singularity model is isometric to either the self-similar shrinking cylinder or the Bryant soliton.
We note that as an immediate consequence of (i) the scalar curvature and the full curvature are comparable in certain regions up to the singular time. We also point out that the phenomenon of Type-II enhancement of symmetries along the Ricci flow is intrinsic to the classification of 3-dimensional κ-solutions obtained by Brendle in [9]. We also note that item (iv) in Theorem 2 relies on the recent extension of Brendle's work to higher dimensions by Li and Zhang [35].
Next, we show that the long-time property is satisfied by a class of Berger metrics whose curvature decays at infinity. General long-time existence results on non-compact manifolds usually rely on controlling the sign of the curvature and the volume growth [16]. From a different perspective, similar conclusions may be achieved when the analysis is restricted to families of homogeneous Riemannian metrics [34]. In this case the behaviour of the flow for long times is also understood [7]. Instead of assuming a transitive action of a Lie group, one may study cohomogeneity 1 manifolds. In this direction, Oliynyk and Woolgar proved that the Ricci flow on R n starting at an asymptotically flat spherically symmetric metric without necks is immortal [38]. The author improved this result by allowing any decay of the curvature [19].
In our setting we consider the following set, whose intersection with G is empty.

Definition 2
We let G ∞ be the set of complete warped Berger metrics g on R 4 with positive injectivity radius and satisfying the following conditions: (ii) |Rm g | g (s) → 0 as s → ∞ and there exist μ > 0 and s 0 > 0 such that c(s) ≥ μ for any s ≥ s 0 .
We prove the following: ) 0≤t<T be the maximal complete, bounded curvature solution to the Ricci flow evolving from some g 0 ∈ G ∞ . Then the solution is immortal.
The long-time property may in general fail if we omit the requirement on the monotonicity of b and H . Indeed by the adaptation of [2] to R n+1 obtained in [19] we deduce that if g 0 is asymptotically flat with b = c and (R 4 , g 0 ) contains a neck which is sufficiently pinched (in a precise way), then the Ricci flow is Type-I. It therefore remains to address the relation between Type-I singularities and existence of minimal hyperspheres for Berger Ricci flows. We recall that Angenent and Knopf constructed the first examples of nondegenerate neckpinches by evolving rotationally invariant metrics on S n containing minimal (stable) hyperspheres [2]. Later the link between Type-I singularities and minimal spheres has been explored for Ricci flows on closed 3-manifolds by Song in [42]. In our setting we prove the following: 0≤t<T be the maximal complete, bounded curvature solution to the Ricci flow evolving from a complete warped Berger metric g 0 with positive injectivity radius and curvature decaying at infinity. If g(t) develops a Type-I singularity at T < ∞, then there exists δ > 0 such that (R 4 , g(t)) contains minimal embedded 3-spheres for any t ∈ [T − δ, T ).
We may also apply Theorem 1 and Theorem 3 to derive two simple corollaries. First we immediately deduce that neither G nor G ∞ contain shrinking solitons.

Corollary 1
There are no Taub-NUT like shrinking Ricci solitons on R 4 .
In the second application we classify warped Berger Ricci flows with bounded nonnegative curvature. In particular we show that for positively curved warped Berger Ricci flows bounded by a cylinder at infinity, parabolic dilations at the origin along any sequence of times give rise to the Bryant soliton.
Corollary 2 Let (R 4 , g(t)) 0≤t<T be the maximal complete, bounded curvature Ricci flow starting at some complete warped Berger metric g 0 with bounded nonnegative curvature. Then T is finite if and only if b(·, 0) is bounded. If T is finite, then for any t j T the rescaled Ricci flows

Outline
We briefly describe the organization of the paper.
In Sect. 2 we discuss Berger metrics on R 4 and we comment on the main assumptions. In Sect. 3 we show that the condition on the lack of necks persists along the Ricci flow. The main step consists in adapting the analogous argument adopted in [4], which relies on the application of a general maximum principle for systems of parabolic equations. In Sect. 4 we study warped Berger Ricci flows evolving from initial data either in G or in G ∞ . Similarly to [31] we show that the curvature is controlled by the size of the principal orbits and that the solution becomes rotationally symmetric around any singularity at some rate that breaks scale-invariance. An important ingredient, for the case of G ∞ , is also given by the application of the Pseudolocality formula in [17] by Chau, Tam and Yu. In Sect. 5 we prove that any singularity model is rotationally symmetric by showing that the left-invariant Milnor frame diagonalizing the metric generates a copy of su (2) in the Lie algebra of Killing fields acting on the singularity model. We then apply the rigidity result obtained by Zhang in [45] to classify these singularity models. In Sect. 6 we prove the main results. Theorem 1 heavily relies on the characterization of Type-I singularities in [36] and [20]. The appearance of the Bryant soliton follows from a result by Hamilton [27] once we know that the singularity is Type-II. The localization of the Bryant soliton in (i) of Theorem 2 is a direct consequence of the convergence of left-invariant vector fields obtained in Sect. 5. The property that the singularity is global depends on the monotonicity assumption (b s ≥ 0, H ≥ 0), which allows us to control the space-time region where the flow stays smooth. The Type-I blow-up at infinity follows once we know that the solution becomes singular everywhere at some finite time T . We then obtain the classification of singularity models by combining the characterization of singularity models in Sect. 5 with the analysis in [35]. The proof of Theorem 3 follows from a contradiction argument. We show that if a Ricci flow in G ∞ develops a finite-time singularity, then any singularity model is a non-compact κ-solution with Euclidean volume growth. However, in [39] Perelman showed that this is not possible. We finally address the proof of Theorem 4, which again depends on the characterization of Type-I Ricci flows obtained in [20]. The last section is devoted to deriving some easy applications of the main results.

Warped Berger metrics on R 4
A compact Lie group G acting on a Riemannian manifold (M, g) via isometries is said to act with cohomogeneity 1 if the orbit space M/G is 1-dimensional. If M is a non-compact manifold, then the orbit space is either homeomorphic to [0, 1) or to R, depending on whether there exists a singular orbit of codimension greater than one (see, e.g., [23]). We analyse the first case, with G and M being SU(2) and R 4 respectively.
Let now g be a Riemannian metric on R 4 = C 2 invariant under the cohomogeneity 1 left-action of SU (2). The action admits one singular orbit consisting of the origin o of R 4 . All the geometric information can then be obtained by restricting g along a radial geodesic starting at o and meeting the 3-hyperspheres orthogonally. Namely, on R 4 \ {o} we have (see also [22]) where ξ, a, b, c : (0, +∞) → (0, +∞) are smooth radial functions. Since we are interested in SU(2)-invariant metrics on R 4 whose restrictions to any hypersphere are Berger metrics we further require the metric g to be invariant under rotations of the Hopf fibres. Equivalently, we assume that a ≡ b in (2). Moreover, we also restrict the analysis to those metrics satisfying the ordering constraint c ≤ b. For any radial coordinate x > 0 the metric is then a left-invariant metric on the Euclidean hypersphere S(o, x) with the S 1 -fiber squashed by a factor c(x)/b(x) ∈ (0, 1]. If we denote the g-distance from the origin by s, then we can write g as We note that given a radial map f on R 4 , then we interpret f = f (s) = f (s(x)) as a function of x unless otherwise stated. We also have the relation It is a general fact that g in (3) extends smoothly at the origin o ∈ R 4 if and only if b and c extend to smooth odd functions at x = 0 and the following is satisfied: We note that the underlying topology plays a role in the analysis of the Ricci flow dynamics via the boundary conditions above. We point out that, as previously observed, the Lie algebra of Killing vectors for g contains a copy of u(2). Indeed, any U(2)-invariant metric on R 4 = C 2 can be written as in (3), up to choosing a suitable Milnor frame (see, e.g., the case k = 1 in Sect. 2.1 of [4]). In analogy with [31] we refer to any (U(2)-invariant) metric on R 4 of the form (3) and satisfying c ≤ b as a (generalized) warped Berger metric.

Curvature terms
Given a warped Berger metric g 0 a simple application of the Koszul formula (see also [31, Appendix A]) allows to compute the sectional curvatures of the vertical planes and of the mixed ones We also note that we can write the scalar curvature as R g = 2(k 01 + k 02 + k 03 + k 12 + k 13 + k 23 ).

Initial data for the Ricci flow
In this work we study the Ricci flow problem on R 4 with initial condition given by a warped Berger metric g 0 . We first assume that g 0 is bounded by a cylinder at infinity so that the Ricci flow evolving from g 0 always encounters a finite-time singularity. According to [2,19] if (R 4 , g 0 ) contains necks, then the Ricci flow solution may be Type-I and converge to a shrinking cylinder once rescaled. In order to construct Type-II singularities we thus need to exclude these initial geometries. A generalization of the notion of neck discussed in [2] to the SU(2)-invariant setting consists in considering whether the mean curvature of embedded hyperspheres changes sign. Namely, we introduce the quantity H : representing the mean curvature of the centred Euclidean sphere of Euclidean radius x with respect to g 0 . We say that g 0 does not have necks when the mean curvature H is nonnegative on R 4 \ {o}.
While in the rotationally symmetric setting a Sturmian type of argument guarantees that minimal hyperspheres cannot appear along the flow, one might expect that in the SU(2)-case the mean curvature could generally change sign along the flow. In order to prevent the latter phenomenon from happening, we require the spatial derivative b s to be nonnegative as well.
Definition 2. 1 We let G be the set of complete bounded curvature warped Berger metrics on R 4 satisfying the following conditions:

Remark 2.2
From the formula for the mean curvature of the embedded hyperspheres we immediately derive that the assumption (i) in Definition 2.1 is weaker than asking for the monotonicity of both b and c.
In the second class of initial data for the Ricci flow we consider warped Berger metrics without necks but whose behaviour at infinity is not controlled by that of a cylinder. Namely, we require the curvature to decay to zero and the Hopf fibres to be not collapsed.

Definition 2.3
We let G ∞ be the set of complete warped Berger metrics g on R 4 with positive injectivity radius and satisfying the following conditions: Remark 2. 4 We point out that the sets G and for some m > 0, and hence by (6) we find Since the curvature is decaying to zero at infinity and c ≤ b we see that |b s | ≥ 1/2 outside some Euclidean ball B(o, r ), for r large enough. Therefore b(s) → ∞ and this is a contradiction. We conclude that if Remark 2. 5 We observe that the well known Taub-NUT metric on R 4 [30] is a hyperkähler metric belonging to G ∞ since the curvature decays to zero at cubic rate while both b and c are increasing.

The Ricci flow equations
If g 0 is a complete bounded curvature warped Berger metric on R 4 , then by [41] there exists a smooth complete solution to the Ricci flow problem. Such solution is unique among those complete solutions with bounded curvature [18]. Therefore we have a well-defined notion of maximal time of existence for the Ricci flow solution. In the following we always let (R 4 , g(t)) 0≤t<T be the maximal complete, bounded curvature solution to the Ricci flow starting at some complete bounded curvature Berger metric g 0 . The Ricci flow diffeomorphism invariance and the uniqueness property ensure that the symmetries persist. Moreover, since the Ricci tensor is diagonal along the global frame {∂ x , X 1 , X 2 , X 3 }, the maximal Ricci flow solution starting at g 0 must be of the form where s = s(x, t) is the g(t)-distance from the origin. In terms of the variables s and t the Ricci flow equations can be written as The choice of a meaningful geometric coordinate s provides us with a parabolic form of the Ricci flow equations. However, we get a non vanishing commutator between ∂ t and ∂ s given by We also report the formula for the (time-dependent) Laplacian along the Ricci flow. For any We dedicate the end of this subsection to proving that the Ricci flow solution g(t) starting at g 0 remains a warped Berger metric until its maximal time of existence T ≤ ∞. More precisely, we show the following: 0≤t<T be the maximal solution to the Ricci flow starting at some complete bounded curvature warped Berger metric g 0 and let ε .
Proof We first verify that the ordering c ≤ b is preserved along the flow for any t ∈ [0, T ). By [41] the curvature is bounded at any time slice R 4 × {t}, with t ∈ [0, T ); thus from the Ricci flow equations we find that there exists some time dependent positive constant α(t) such that c/b(·, t) ≤ α(t) < ∞ on R 4 . As long as a (smooth) solution exists the boundary conditions (5) are satisfied, which then imply that the function f . = log(c/b) is smoothly defined on R 4 and equal to zero at the origin for any time. From the evolution equations (12), (13) and the formula for the Laplacian (15) we get Therefore whenever c/b > 1 we find We can then apply the maximum principle [15,Corollary 7.45] and conclude that since c/b(·, 0) ≤ 1, the same ordering persists along the flow. In fact, once we know that c ≤ b is preserved in time, a standard application of the strong maximum principle shows that if in a space-time neighbourhood of the point and thus c = b everywhere for all earlier times by real analyticity of solutions to the Ricci flow [5].
We now let ε ∈ [0, 1) be defined as in the statement. If ε = 0 there is nothing to show; we can then take ε > 0. Again from [41] We can apply the maximum principle and conclude that c(·, t) ≥ εb(·, t) for any t ∈ [0, T ).

Ricci flow without necks
In this section we show that the monotonicity assumptions b s ≥ 0 and H ≥ 0 are preserved along the Ricci flow solution. The main ingredient is given by a maximum principle for systems of parabolic equations [40,Theorem 13,p. 190] that recently Appleton used to derive similar conclusions for a family of U (2)-invariant Ricci flows with cylindrical asymptotics [4]. In the following we mainly adapt the argument in [4] to the topology of R 4 , i.e. to the boundary conditions given in (5).

Basic estimates
Let g 0 be a complete bounded curvature Berger metric on R 4 . We collect a few preliminary bounds that are necessary to apply the maximum principle for systems to the evolution equations of cb s /b and cH.
for any x ≥ x 0 which contradicts the fact that b is bounded. Therefore there exists a sequence of points p j → ∞ such that b( p j ) > δ, with δ given above. Assume that there exists a sequence q j → ∞ such that b(q j ) ≤ δ. It follows that there exists a sequence of minimaq j → ∞ such that b(q j ) ≤ δ.
which is not possible. The proof is then complete.
A simple consequence of the previous Lemma is the following Given x 0 > 0 we may apply Lemma 3.1 and conclude the proof.
We also need to check that both b s and c s are exponentially bounded at spatial infinity. Proof According to (8) and the uniform bound on the curvature we see that b and hence b ss are exponentially bounded; thus the same holds for b s by integrating b ss . Similar conclusions are satisfied by c s .
Finally a bound similar to Corollary 3.2 is satisfied by c s /c as well.

Maximum principle for systems
We consider the maximal Ricci flow solution (R 4 , g(t)) 0≤t<T evolving from a complete bounded curvature warped Berger metric g 0 . We note that given t 0 < T then the estimates above hold uniformly for any t ∈ [0, t 0 ] being the curvature uniformly bounded in the spacetime region R 4 × [0, t 0 ]. From the evolution equations (12), (13), the commutator formula (14) and the expression for the mean curvature of embedded hyperspheres H : We may now prove the main result of this section.
Using the commutator formula (14) we may rewrite the evolution equations (17) and (18) in and From Lemma 3.1 and Corollary 3.2 we derive that the zero order coefficients are uniformly bounded in ( . Moreover, by Lemma 2.6 we know that the ordering c ≤ b is preserved along the flow, therefore the coupling coefficients 4c 2 /b 4 and 16/b 2 (1 − c 2 /b 2 ) are both nonnegative. Similarly to [4] we can introduce a barrier function for t ≤ min{t 0 , (2β) −1 } and compute the evolution equations of cb s /b + W and cH + W for any > 0. Using Corollary 3.2, Lemma 3.4 and standard distortion estimates of the distance function it is straightforward to check that there exist β = β(t 0 ) and λ = λ(t 0 ) such that and similarly for the evolution equation of cH + W . By assumption

Analysis of the Ricci flow
In this section we derive the main curvature estimates for the Ricci flow solution (R 4 , g(t)) evolving from a warped Berger metric metric g 0 . In the first part we focus on the case g 0 ∈ G.
Similarly to the analyses in [31] and [32] (which are performed on S 1 × S 3 and S 2× S 2 respectively) we prove that away from the origin the Ricci flow is controlled by the size of the principal orbits. In particular, we show that the formation of a singularity at some positive x (i.e. along the Euclidean hypersphere of radius x) is equivalent to b(x, t) converging to zero as t → T . We also describe the behaviour of the flow as the time approaches T . Analogously to [31], we prove that around any singularity the solution becomes rotationally symmetric at some rate that breaks scale-invariance.
In the second part we extend the previous estimates to Ricci flows starting at some g 0 ∈ G ∞ . Moreover, for this class of solutions we also prove that (a scale-invariant version of) the mean curvature of minimal hyperspheres admits a uniform positive lower bound in the compact region where singularities may form.
For notational reasons we always let α denote a positive constant only depending on g 0 that may change from line to line.

Curvature estimates in G
Throughout this section we let (R 4 , g(t)) 0≤t<T be the maximal complete, bounded curvature Ricci flow solution evolving from some g 0 ∈ G. Since b(·, 0) is bounded from above and uniformly in the space-time R 4 × [0, T ). We observe that (19) is not available for the topologies analysed in [32] and [4]. Next, we show that the maximal time of existence T is finite. g(t)) 0≤t<T be the Ricci flow solution starting at g 0 ∈ G. Then Proof From the boundary conditions we deduce that b 2 (·, t) is a smooth function on R 4 as long as the solution exists. By (12) and (15) we get The conclusions then follow from the maximum principle [12, Theorem 12.14].

Remark 4.2
From Lemmas 3.5 and 4.1 we derive that the set G is preserved along the Ricci flow.
Next, we prove that b s and c s are uniformly bounded in the space-time. The evolution equations of the first order spatial derivative are given by and Proof From Lemma 3.5 and Lemma 4.1 we derive that b s (·, t) is integrable for any t ∈ [0, T ). Moreover, by [41] we see that for any t ∈ [0, T ) there exists α(t) < ∞ such that |b ss /b| ≤ α(t). Since by Lemma 4.1 b is uniformly bounded from above we deduce that b s (x, t) → 0 as x → ∞ for any t ∈ [0, T ). Therefore, if b s becomes unbounded (from above) as t T , then there exists a critical point p 0 for b s (·, t 0 ) where b s =ᾱ for the first time, for someᾱ large to be chosen below and for some t 0 > 0. Evaluating (20) By choosingᾱ > max{sup|b s |(·, 0), 2} one easily checks that the c s − quadratic polynomial in the brackets does not admit roots, thus proving (b s ) t ( p 0 , t 0 ) < 0. The exact same argument works for the lower bound of b s . In fact, the lower bound for b s also follows from Lemma 3.5.
We now adapt the argument for c s . Since b 2 cH = (b 2 c) s and b 2 c is bounded from above, we see that b 2 cH(·, t) is integrable. By differentiating we find (b 2 c) ss = 2bcb ss + 2b 2 s c + 4bb s c s + b 2 c ss . From (7) we derive |b s c s |(·, t) ≤ α(t)bc(·, t) + c 3 /b 3 (·, t) ≤ α(t) being the curvature bounded at any time slice R 4 × {t} for t ∈ [0, T ). Similarly |c ss |(·, t) ≤ α(t). Therefore, since b s is uniformly bounded in the space-time we conclude that |(b 2 c) ss |(·, t) ≤ α(t), which implies b 2 cH(x, t) → 0 as x → ∞ for any t ∈ [0, T ). In particular c s (x, t) → 0 as x → ∞ because b (and hence c) is uniformly bounded from above and b s (x, t) → 0. One can then argue as above that if c s becomes unbounded as t T , then there exists a first maximum p 0 where c s attains a sufficiently large valueᾱ at some t 0 > 0 for the first time. It follows that By Lemma 2.6 we know that the ordering c ≤ b is preserved. Therefore forᾱ large enough the right hand side is strictly negative. The same conclusion holds for the lower bound.
From the previous Lemma and the condition c ≤ b we immediately derive the following bounds for the vertical sectional curvatures. From now on any estimate is satisfied in the space-time R 4 × [0, T ) unless otherwise stated.

Corollary 4.4
There exists α > 0 such that The following estimate is a necessary step to prove that the solution to the Ricci flow becomes spherically symmetric at any singularity forming away from the origin.

Lemma 4.5
The following holds as long as the solution exists: Proof Let us denote the quantity (b 2 s − 4)/b by ϕ. By the boundary conditions ϕ is uniformly bounded from above as x → 0. Moreover, as we have already argued in the proof of Lemma 4.3, we find that ϕ(x, t) becomes negative for x large enough. We may then let ( p 0 , t 0 ) be the maximum point among prior times where ϕ attains some positive value α. A direct computation gives Evaluating the evolution equation at ( p 0 , t 0 ) we get We now regard the term in the brackets as a quadratic polynomial in c s . Chosen α > 0, we can find > 0 such that b 2 s = 4 + . The discriminant of the polynomial is given by where we have again used that the ordering c ≤ b is preserved by Lemma 2.6. Therefore, the quantity sup R 4 ϕ + (·, t) is non-increasing along the solution.
In the next Lemma we prove that if c(x, t) converges to zero as t → T (along some sequence of times) for some x > 0, then the metric becomes rotationally symmetric at x.

Lemma 4.6
There exists α > 0 such that Proof We first prove a useful characterization of the behaviour of the second order spatial derivatives at infinity.

Proof of Claim 4.7
From the proof of Lemma 4.3 we see that b s → 0 at infinity, which implies that the integral of b ss (·, t) has a finite limit for any t ≥ T /2. By Shi's derivative estimates and the Koszul formula we find that for any t ∈ [T /2, T ) there exists α(t) > 0 such that which then proves that |b sss |(·, t) ≤ α(t) being both b and b s uniformly bounded. Therefore b ss (x, t) → 0 as x → ∞ for any t ∈ [T /2, T ).
Again from the proof of Lemma 4.3 we derive that the integral of (b 2 c) ss (·, t) is convergent for any t ∈ [T /2, T ). By computing the derivative (b 2 c) sss and using again Shi's derivative estimates we obtain that (b 2 c) ss → 0, which also implies c ss (x, t) → 0 as x → ∞ for any t ∈ [T /2, T ).
By the boundary conditions the function ϕ = 1/c − 1/b is continuously defined at the origin and identically zero. From (19) we see that ϕ is bounded along any time slice R 4 × {t}, for t ∈ [0, T ). The evolution equation of ϕ is given by By Lemma 4.3 and Claim 4.7 we see that for any δ > 0 and t ∈ [T /2, T ) there exists x 0 = x 0 (δ, t) such that the time derivative of ϕ can be bounded for x larger than x 0 as for some η > 0, where we have used Lemma 2.6 and Lemma 4.1. Therefore, if ϕ does not stay bounded as t T , then there exists a sequence of maxima diverging as the solution approaches its maximal time of existence.
We introduce the quantityᾱ .
) and ε is chosen to satisfy Lemma 2.6. Suppose that ( p 0 , t 0 ) is a space-time maximum point among prior times where ϕ attains some value greater thanᾱ. By evaluating (22) at ( p 0 , t 0 ) we see that Using Lemma 4.5, we can estimate the time derivative from above as which then yields An analogous bound holds for the first order spatial derivatives. Namely, we have the following Lemma 4.8 There exists α > 0 such that The function ψ extends to zero at the origin due to the boundary conditions. As argued before ψ(x, t) → 0 as x → ∞ for any t ∈ [0, T ). Consider the upper bound. Suppose that there exists a large valueᾱ which ψ attains for the first time at some maximum space-time point ( p 0 , t 0 ). The evolution equation of ψ is We can then evaluate both sides at ( p 0 , t 0 ) and use Lemma 4.6 to get forᾱ sufficiently large, with ε satisfying Lemma 2.6. The same argument shows the existence of a uniform lower bound.
We may now improve Lemma 4.5.

Lemma 4.9
There exists α > 0 such that Since ϕ is uniformly bounded from above at the origin and at spatial infinity we take ( p 0 , t 0 ) to be the maximum space-time point where ϕ =ᾱ for the first time, for some positiveᾱ to be chosen below. We have According to Lemma 4.8 we can bound c s in terms of b s . It follows that there exists some positive constant α > 0 such that We finally use Lemmas 4.3, 4.6 and the fact that ϕ > 0 implies b 2 s > 1 to derive forᾱ large enough.
Next, we extend the previous arguments to the second spatial derivatives. We show that away from the origin the mixed sectional curvatures are controlled by c and hence by b as in and Lemma 4.10 There exists α > 0 such that Proof By [41] there exists α > 0 such that |k 01 | ≤ α in R 4 × [0, T /2]. Since by Lemma 4.1 b is uniformly bounded from above, we deduce that b 2 |k 01 | ≤ α and similarly for b 2 |k 03 | using Lemma 2.6. We may hence consider the case t ∈ [T /2, T ). Define the map ψ .
= bb ss − μb 2 s − νc 2 s in R 4 × [T /2, T ), for some μ and ν positive constants we will determine below. According to Claim 4.7 ψ(x, t) → 0 as x → ∞ for any t ∈ [T /2, T ). We then adapt the argument in [32,Lemma 7] to show that ψ is uniformly bounded in the space-time region. Explicitly, at any stationary point of ψ(·, t) we have Suppose ψ attains some negative value −ᾱ for the first time at t 0 ∈ [T /2, T ). By Lemma 4.3 we can chooseᾱ sufficiently large such that bb ss ≤ −ᾱ/4. Therefore we get −bb ss Evaluating the evolution equation of ψ at ( p 0 , t 0 ), we have (provided we set μ ≥ 1) One can then estimate the remaining terms exactly as in [32] by using the uniform boundedness of the first spatial derivatives and Lemma 2.6. Namely, we can use the weighted Cauchy-Schwarz inequality to get 2bb s c s c ss c 2 ≤ c 2 and similarly for the others. Therefore there exists a uniform constant α such that the time derivative at ( p 0 , t 0 ) is bounded from below as once we choose μ = 1, ν = 2 andᾱ sufficiently large. The existence of a uniform upper bound follows from a similar argument by consideringψ . = bb ss + μb 2 s + νb 2 s . The very same proof applies for k 03 .
We can finally show that both c and b admit limits as the flow approaches its maximal time of existence T < ∞.

Corollary 4.11 For any x ≥ 0 the limits lim t T c(x, t) and lim t T b(x, t) exist and are finite.
Proof By applying Lemmas 4.3 and 4.10 we get The same argument works for b as well.
The curvature is hence uniformly controlled along any Euclidean hypersphere where the components b and c do not converge to zero as t T . Namely, from Corollary 4.4 and Lemma 4.10 it follows that there exists a positive constant α such that Next, we show that around a singularity rotational symmetry type of bounds hold for the second spatial derivatives as well.

Lemma 4.12
There exists α > 0 such that Proof We adapt the proof in [31], whose argument works for a compact Type-I Ricci flow setting. Once we define ψ .
The boundary conditions (5) Evaluating ϕ at the maximum point ( p 0 , t 0 ) we get From (25) it easily follows that there exists some uniform constant α > 0 such that |G| ≤ α/b 3 . Being ψ uniformly bounded (Lemma 4.8), we have According to Lemmas 4.6 and 4.8 an analogous estimate can be found for |Hb 2 ψ s |. Then forᾱ sufficiently large. Therefore ϕ is uniformly bounded and we get where we have used Lemmas 4.8 and 2.6.
We finally discuss the existence of lower bounds for the mixed sectional curvatures.

Lemma 4.13
There exists α > 0 such that Proof We adapt the analogous argument in [31]. Consider the map f . = c ss c log c, which is smooth in R 4 \ {o} × [0, T ). Moreover f extends continuously to the origin and f (o, t) = 0 as long as a solution exists. By Claim 4.7 and the fact that c is uniformly bounded from above, we deduce that f (x, t) → 0 at spatial infinity for any t ∈ [T /2, T ). Suppose that there exist ( p 0 , t 0 ) ∈ (R 4 \ {o}) × [T /2, T ) andᾱ large to be chosen below such that f ( p 0 , t 0 ) = −ᾱ for the first time. From (9) and (25) it follows Since by Lemma 4.1 c is uniformly bounded from above the last inequality implies log c( p 0 , t 0 ) < 0 and forᾱ large enough. By direct computation we get In the following the signs of b s and c s are not relevant. In particular we assume without loss of generality that c s ≥ 0. By Lemma 4.8 we have where we have used Lemmas 4.6 and 4.9 to derive the last inequality. According to Lemma 4.12 it holds By choosingᾱ large enough (and hence c( p 0 , t 0 ) small) it follows that 2(c s c ss /c)(2 + 1/log c)( p 0 , t 0 ) ≥ 0. Finally, Lemmas 4.6, 4.8, and 4.12 yield the bounds Evaluating the evolution equation of f at ( p 0 , t 0 ) and using (26) we get the lower bound forᾱ sufficiently large (i.e. c small enough). The case of b ss b log b does not require modifications.

Curvature estimates in G ∞
We consider (R 4 , g(t)) 0≤t<T a maximal complete, bounded curvature Ricci flow solution evolving from some g 0 ∈ G ∞ . If the solution develops a finite-time singularity at some T < ∞, then we can apply [17, Theorem 1.1] and conclude that there exists ρ > 0 such that Remark 4.14 We note that the set G ∞ is preserved along the Ricci flow. Consider the maximal Ricci flow evolving from some g 0 ∈ G ∞ . By Lemma 3.5 condition (i) in Definition 2.3 persists in time. From [29] we also derive that |Rm|(s, t) → 0 as s → ∞ for all t ∈ [0, T ). If T < ∞, then given ρ as in (27) we can find μ > 0 such that c( If instead T = ∞, then c is uniformly bounded from below away from the origin in any compact interval of existence being the curvature bounded. An immediate consequence of (27) is that for any radial coordinate x 1 > ρ the spatial derivatives (up to second order) of b and c are uniformly bounded in time along the hypersphere of radius x 1 . In particular, for any for any t ∈ [0, T ). For the function c/b is uniformly bounded from below at the origin and along the hypersphere of radius x 1 and the evolution equation (16) prevents c/b from attaining interior minima approaching zero.
By inspection one can check that given x 1 > ρ any bound derived in Subsection 4.1 extends to the space-time region B(o, x 1 ) × [0, T ). For any argument relies on a maximum principle which still applies to this setting once we know that any relevant quantity is uniformly bounded along the parabolic boundary of the region B(o, x 1 ) × [0, T ). Explicitly, we have the following Lemma 4.15 Let (R 4 , g(t)) 0≤t<T , with T < ∞, be the maximal Ricci flow solution evolving from some g 0 ∈ G ∞ and let ρ > 0 satisfy Then for any Remark 4. 16 One can verify that Lemma 4.15 holds for a larger class of Ricci flows than G ∞ . Indeed, it suffices to control the flow uniformly along the parabolic boundary of some spacetime region and then apply maximum principle arguments without relying on the quantities b s and H being nonnegative.
Next, we prove that b s and H remain positive along a hypersphere of sufficiently large radius. In the following ρ still denotes the radius defined by (27).

Lemma 4.17
Let (R 4 , g(t)) 0≤t<T , with T < ∞, be the maximal Ricci flow solution evolving from a warped Berger metric g 0 ∈ G ∞ . There existx 2 ≥x 1 > ρ, δ > 0 andt ∈ [0, T ) such that Proof We have already shown that b(x, 0) → ∞ as x → ∞. Since the curvature is uniformly bounded in the complement of the Euclidean ball B(o, ρ), we can pick ρ < for some ς > 0 and for any t ∈ [0, T ). We can use the Koszul formula to write the evolution equation of b s as Therefore given x > ρ, by (27) and Shi's derivative estimates there exists α(x) > 0 such that |(b s ) t (x, t)| ≤ α uniformly in time. Since T < ∞ the last property implies that b s (x, ·) is Lipschitz and hence admits a finite limit as t T , which we know to be nonnegative according to Lemma 3.5. Let us assume for a contradiction that any such limit is zero.
Since b ss is bounded in the annular region (x 0 , x 1 ) × S 3 uniformly in time, we deduce that On the other hand, being the curvature controlled in the annular region (x 0 , x 1 ) × S 3 , we get for any t ∈ [0, T ), which gives a contradiction. Therefore, there existsx 1 ∈ [x 0 , x 1 ] as in the statement. The proof for H is similar. Indeed we can write H = (log(b 2 c)) s and then adapt the argument above noting that by Definition 2.3 log(b 2 c))(x, 0) → ∞ as x → ∞. In particular we can always pickx 2 ≥x 1 .
Next we show that cH stays away from zero in the compact region B(o, ρ) for times close to the maximal time of existence T . In the following we letx 2 andt be defined as in the previous Lemma.

Corollary 4.18 There exists
Proof By the boundary conditions we have cH(o, t) = 3 as long as the solution exists. According to Lemmas 3.5 and 4.17 there exists δ > 0 such that cH(x 2 , t) ≥ δ for any t ∈ [t, T ) and cH(x,t) ≥ δ for any 0 ≤ x ≤x 2 . Suppose that cH gets smaller than min{δ, 3} in B(o,x 2 ) × [t, T ). Then there exists a minimum point ( p 0 , t 0 ) and from (18), (28) and Lemma 3.5 we get

From Lemma 4.8 it follows that
for some α = α(x 2 ) > 0. Thus we can find a uniform constant α only depending onx 2 such that Therefore, whenever cH ≤δ, for someδ only depending onx 2 andt, the function t → min B(o,x 2 ) (cH)(·, t) is Lipschitz and satisfies We conclude that cH cannot approach zero in the interior of B(o,x 2 ) as t T .
Definition 5.1 A complete bounded curvature ancient solution to the Ricci flow (M ∞ , g ∞ (t)) −∞<t≤0 is a singularity model for a warped Berger Ricci flow (R 4 , g(t)) 0≤t<T if T < ∞ and there exists a sequence of space-time points ( p j , t j ) with t j T such that λ j . = |Rm g(t j ) | g(t j ) ( p j ) → ∞ and the rescaled Ricci flows (R 4 , g j (t), p j ) defined by

Remark 5.2
We note that by the Cheeger-Gromov convergence any singularity model (M ∞ , g ∞ (t)) of a warped Berger Ricci flow is non-compact and non-flat.
The main goal of this section consists in classifying the singularity models of warped Berger Ricci flows. Namely, we show the following result g(t)) 0≤t<T , with T < ∞, be the maximal Ricci flow solution evolving from a warped Berger metric g 0 belonging to either G or G ∞ . Then any singularity model is either the self-similar shrinking soliton on the cylinder or a positively curved rotationally symmetric κ-solution.
We prove the characterization of singularity models by showing that the symmetries of warped Berger Ricci flows are enhanced when dilating the flow around a singularity. More precisely, we show that the left-invariant vector fields in (1) become Killing vectors when passing to the limit hence forcing the singularity model to be rotationally symmetric. Given g 0 ∈ G there exists ε > 0 such that c/b(·, 0) ≥ ε. Therefore g 0 is bounded between two round cylinders outside some compact region and there exists α > 0 such that Vol g 0 (B g 0 ( p, 1)) ≥ α for any p ∈ R 4 . The latter condition is satisfied by any g 0 ∈ G ∞ being the injectivity radius positive and the curvature bounded. Thus if (R 4 , g(t)) 0≤t<T , with T < ∞, is the maximal Ricci flow solution evolving from some g 0 which belongs to either G or G ∞ , then by [11,Theorem 8.26] there exists κ > 0 such that g(t) is (weakly) κ−noncollapsed in R 4 × (T /2, T ) at any scale r ∈ (0, √ T /2). Accordingly, there exist blow-up sequences satisfying Definition 5.1 and hence any warped Berger Ricci flow evolving from either G or G ∞ admits singularity models ( [29,Sect. 16]). In particular, any singularity model of a warped Berger Ricci flow is (weakly) κ-non-collapsed at all scales.
We first consider a maximal Ricci flow solution (R 4 , g(t)) 0≤t<T , with T < ∞, starting at some warped Berger metric g 0 ∈ G. Later we check that the same conclusions are satisfied by Ricci flows in G ∞ . We let ( p j , t j ) be a blow-up sequence of space-time points giving rise to a singularity model (M ∞ , g ∞ (t), p ∞ ) as in Definition 5.1 and we denote the rescaling factors |Rm g(t j ) | g(t j ) ( p j ) by λ j . Due to the SU(2)-symmetry we may fixθ ∈ S 3 and we may set p j = (x j ,θ). We also let ( j ) be the diffeomorphisms given by the Cheeger-Gromov-Hamilton convergence (see [11,Chapter 4]).
We first provide a topological characterization of the limit manifold.
Proof The proof follows from adapting the argument in [19,Lemma 4.1] which extends to the SU(2)-invariant case due to (25).
Next, we prove that the symmetries of the flow are enhanced when dilating. To this aim, we first show that the Milnor frame passes to the singularity model. Since the proof of that relies on an Ascoli-Arzelà argument, we need C 3 -bounds with respect to the rescaled solutions.

Lemma 5.5 There exists a continuous function f
for any t ∈ (−∞, 0] and ν > 0 and for i = 1, 2, 3. Furthermore, there exists α > 0 such that for i = 1, 2, 3 up to passing to a subsequence. Proof We fix t = 0 and ν > 0 and we let q ∈ B g j (0) ( p j , ν). In the following we only analyse the case of X 1 since the others are proved similarly. We deal with the bounds for |∇ k g j (0) X 1 | g j (0) with k = 0, 1, 2, 3 separately. Case k = 0. We consider a g(t j )-unit speed geodesic from p j to q. From Lemma 4.3 we get The desired estimate then follows from (25) which gives λ j b 2 ( p j , t j ) ≤ α. Case k = 1. By direct computation we get for any t ∈ [0, T ). Lemmas 2.6 and 4.3 imply that |∇ g(t) X 1 |(·, t) is uniformly bounded and that the estimate (30) is satisfied. Case k = 2. We analyse in detail only one exemplificative instance. One of the terms appearing in the computation of the norm where we have used (8). The last term is then bounded because it coincides with the case k = 0 we have already discussed. Case k = 3. One of the terms appearing in the computation of (λ j ) −1 |∇ 3 According to Shi's first derivative estimate the covariant derivatives of the curvature are bounded on the singularity models, therefore there exists a uniform constant α such that Thus we have We can then bound the right hand side of (31) as where the last inequality follows again from the case k = 0. The other terms are dealt with similarly. Let now t ∈ (−∞, 0]. By Corollary 4.11 we get We may then extend the proof of the bound for the case k = 0 for any t ∈ (−∞, 0]. The cases k = 1, 2, 3 generalize easily. Since the rescaled Ricci flows converge to the limit ancient flow in the pointed Cheeger-Gromov sense, from Lemma 5.5 it follows that the sequence ( −1 j ) * X 1 is uniformly C 3bounded in B g ∞ (0) ( p ∞ , 1) with respect to g ∞ (0). We can then apply the Ascoli-Arzelà theorem and obtain the following Corollary 5.6 There exists a subsequence ( −1 j ) * X 1 that converges in C 2 to a vector field X 1,∞ on B g ∞ (0) ( p ∞ , 1).
From now on we re-index the subsequence given by the previous Corollary. In order to prove that X 1,∞ is actually a Killing vector field for g ∞ (0) we need a preliminary result. The following shows that the singularity model cannot be Ricci flat.

Lemma 5.7
For any q ∈ M ∞ and for any t ∈ (−∞, 0] the following is satisfied: Proof Suppose for a contradiction that there exist q ∈ M ∞ , t ∈ (−∞, 0], a subsequence (which we still denote by j) and μ > 0 such that b( j (q), t j + (λ j ) −1 t) → μ. By (25) we immediately derive that R g ∞ (t) (q) = 0. Since any complete ancient solution to the Ricci flow has nonnegative scalar curvature [13], a standard application of the maximum principle and the uniqueness of the flow among complete and bounded curvature solutions yield Ric ∞ ≡ 0 everywhere in the space-time. We then fix the time to be 0 and assume that g ∞ (0) is not flat. By the uniform C 1 -lower bound in (30) there exists an open subset U ⊂ B g ∞ (0) ( p ∞ , 1) where |X 1,∞ | g ∞ (0) | U > 0, with X 1,∞ given by Corollary 5.6. From the real analyticity of the ancient limit flow [5] it follows that there existsq ∈ U such that |Rm g ∞ (0) | g ∞ (0) (q) > 0, otherwise the limit would be flat. Moreover, b( j (q), t j ) → 0 as j → ∞. For if such condition did not hold, then by (25) the Riemann tensor would vanish atq. Since g ∞ (0) is Ricci flat we get = lim j→∞ 1 λ 2 j b 4 b 4 (2k 01 + k 03 ) 2 + 2(k 01 + k 12 + k 13 ) 2 + (k 03 + 2k 13 ) 2 ( j (q), t j ). By the Cheeger-Gromov convergence we get j (q) ∈ B g j (0) ( p j , 2) for j large enough. Therefore, from Lemma 5.5 (the case of k = 0) it follows that λ j b 2 ( j (q), t j ) ≤ α for j large and for some positive α. From the estimate in Lemma 4.12 we finally derive that the limit above is zero if and only if lim j→∞ b 2 |sec g(t j ) | ( j (q), t j ) = 0, with sec g(t j ) the maximal sectional curvature of g(t j ). Therefore by Corollary 5.6 and the choice ofq, up to passing to a diagonal subsequence, we conclude that which is a contradiction because we have just proved that the right hand side must vanish.
We can now show that X 1,∞ is a Killing vector field on the limit manifold for any time.

Lemma 5.8
There exists a unique smooth extension of X 1,∞ to the limit manifold M ∞ such that ( −1 j ) * X 1 converges in C 2 to X 1,∞ on compact sets. Moreover X 1,∞ is a g ∞ (t)-Killing vector field for any t ∈ (−∞, 0]. for any (q, t) ∈ M ∞ × (−∞, 0]. We now show that Lemma 5.9 actually extends to any singularity model of a warped Berger Ricci flow in G ∞ . Indeed, given a blow-up sequence ( p j , t j ) as above and a radial coordinate x 1 > ρ, with ρ satisfying (27), then by Lemma 4.15 it suffices to prove that any rescaled geodesic ball B g j (t) ( p j , ν) lies in B(o, x 1 ) for j sufficiently large. Lemma 5.10 Let (M ∞ , g ∞ (t), p ∞ ) −∞<t≤0 be a singularity model for a warped Berger Ricci flow (R 4 , g(t)) 0≤t<T , with T < ∞, starting at some g 0 ∈ G ∞ . For any t ≤ 0, for any ν > 0 and for any x 1 > ρ, with ρ satisfying (27), there exists j 0 = j 0 (t, ν, x 1 ) such that for all j ≥ j 0 we have Proof Given a blow-up sequence ( p j , t j ) with p j = (x j , θ) for some θ ∈ S 3 , we observe that up to a finite number of indices we have x j < ρ otherwise |Rm g(t j ) | g(t j ) ( p j ) would be bounded. Suppose for a contradiction that there exist a time t, a radius ν, a coordinate x 1 > ρ and a subsequence q j k = (y j k , θ j k ) ⊂ B g j (t) ( p j , ν) such that y j k > x 1 . Then by (27) and standard distortion estimates of the Riemannian distance we get which then gives us a contradiction for k large enough.
We may then adapt all the arguments above and conclude that any singularity model of a warped Berger Ricci flow in G ∞ is rotationally symmetric. We finally address the proof of the classification result in Proposition 5.3. Lemma 5.9 implies that any singularity model is in particular conformally flat. Thus by [45] we derive that any singularity model has nonnegative curvature operator. Since we have shown that singularity models are weakly κ-non-collapsed at all scales, we find that any singularity model is a κ-solution to the Ricci flow. If the curvature operator is not strictly positive at some point in the space-time, then the same argument in [19,Lemma 4.3] shows that (M ∞ , g ∞ (t)) splits off a line and must hence be isometric to the self-similar shrinking soliton on the cylinder R × S 3 . Conversely, if the curvature operator is strictly positive at a point, then by the strong maximum principle we conclude that the singularity model is positively curved.

Remark 5.11
We point out that Proposition 5.3 extends to warped Berger Ricci flows for which both the estimate (25) and the rotational symmetry type of bounds in Lemmas 4.6, 4.8 and 4.12 are satisfied.

Bryant soliton singularities
In this subsection we show that any Ricci flow in G encounters a Type-II singularity and that Theorem 2 is satisfied.
Proof of Theorem 1 According to Lemma 4.1 the Ricci flow develops a finite-time singularity at some T < ∞. Suppose that the Ricci flow is Type-I and let be the singular set defined as in [20,Definiton 1.5].
If the origin o does not belong to , then the flow stays smooth on B(o, 2r ) for some r > 0. Thus there exists δ > 0 such that b(r , t) ≥ δ > 0 for any t ∈ [0, T ). Lemma 3.5 then implies b(x, t) ≥ δ for any x ≥ r and for all t ∈ [0, T ). From the estimate (25) we finally deduce that the curvature stays uniformly bounded outside B(o, r ) and hence on R 4 . The latter condition contradicts that the flow develops a singularity at T [41].
If o ∈ , then we can apply [20, Theorem 1.1] and derive that any parabolic dilation of the flow at o (sub)converges to a non-flat shrinking soliton. By the classification in Proposition 5.3 we get that any such singularity model is a shrinking cylinder [33]. However by the SU(2) symmetry and the Cheeger-Gromov convergence we have just shown that the cylinder R × S 3 is exhausted by open sets diffeomorphic to R 4 , which is not possible. Therefore, the singularity is Type-II.
Since the flow is Type-II and non-collapsed we can choose a blow-up sequence giving rise to a singularity model which consists of an eternal solution [29,Sect. 16]). By the classification in Proposition 5.3 we deduce that such eternal solution is rotationally symmetric and positively curved 1 . Therefore the scalar curvature and the Riemann curvature are comparable up to the singular time and we can hence adapt the argument in [29] to extract a space-time sequence ( p j , t j ), with t j T , such that if we set λ j . = R g(t j ) ( p j ), then the rescaled Ricci flows (R 4 , g j (t), p j ) defined by g j (t) .
= λ j g(t j + (λ j ) −1 t) (sub)converge in the pointed Cheeger-Gromov sense to a κ-solution whose scalar curvature attains its supremum in the space-time. According to [27] the singularity model is then a gradient steady soliton and must hence be isometric to the Bryant soliton by the classification in Proposition 5.3.

Proof of Theorem 2
We prove the four points in Theorem 2 separately.
(i) The Bryant soliton appears at the origin. Let ( p j , t j ) and λ j be defined as in the proof of Theorem 1, let j be the family of diffeomorphisms given by the Cheeger-Gromov convergence and let (R 4 , g ∞ (t), p ∞ ) be the Bryant soliton arising as limit singularity model. By the SU(2) symmetry we may choose p j of the form (x j , θ) for some θ ∈ S 3 . Suppose for a contradiction that there exists δ > 0 such that for j sufficiently large we have d g j (0) (o, p j ) ≥ 2δ > 0. We may then find points q j ≡ (x j , θ), withx j < x j , such that, up to passing to a subsequence, −1 j (q j ) → q ∞ for some q ∞ satisfying d g ∞ (0) ( p ∞ , q ∞ ) = δ. Since the scalar curvature of the Bryant soliton g ∞ (t) attains its maximum at the centre of symmetry, i.e. at the origin of R 4 , we deduce that p ∞ = o ∈ R 4 and therefore that the Killing vectors {X i,∞ } constructed above need to vanish at p ∞ . Equivalently, from the argument in Lemma 5.9 we derive that Since the warping coefficient b is monotone in space andx j < x j we have which is not possible because the Killing fields generating the rotational symmetry cannot vanish along a principal orbit, i.e. away from the origin. Therefore, up to choosing a subsequence, we have d g j (0) (o, p j ) → 0. In particular, we may pick a subsequence such that R g(t j ) (o) ≥ (1 − δ j )λ j for some δ j → 0. If we then dilate the Ricci flow by factors R g(t j ) (o) we still obtain the Bryant soliton as pointed Cheeger-Gromov limit.
(ii) The singularity is global. Consider the set of points where the flow becomes singular as t T : We note that the previous definition makes sense due to Corollary 4.11 and the estimate (25). Part (ii) in the statement of Theorem 2 is equivalent to showing = R 4 . Indeed we have proved above that the curvature cannot stay uniformly bounded at the origin, while away from the origin the estimate (25) implies that both b and c need to converge to zero as t T for the curvature to blow-up. We assume for a contradiction that = R 4 . By Lemma 3.5 there existsx ≥ 0 satisfying = B(o,x). We may always take the Euclidean ball B(o,x) to be closed because by Corollary 4.11 there exists a uniform constant α > 0 such that b 2 (x, t) ≤ α(T − t) for all x <x. Claim 6.1 Let (R 4 , g(t)) 0≤t<T be the Ricci flow starting at some g 0 ∈ G. Then lim t T cH(x, t) = 0 for any x >x.
where we have used (25). Analogously, given ν > 0, t ≤ 0 and p ∈ B g(t j ) ( p j , ν(λ j ) −1/2 ) we see that λ j b 2 ( p, t j + (λ j ) −1 t) ≥ λ j b 2 ( p, t j ) + α|t|, for j large enough. From (36) we also derive the following spatial control: We may finally estimate the curvature of the rescaled Ricci flows as Since the flow is weakly κ-non-collapsed for some κ > 0 we may apply Hamilton's compactness theorem and conclude that the sequence of rescaled Ricci flows converge in the pointed Cheeger-Gromov sense to a singularity model (M ∞ , g ∞ , p ∞ ) −∞<t≤0 to which the classification in Proposition 5.3 applies. In particular, g ∞ (t) is of the form (33). Arguing as in the proof of Claim 6.1 and using (36) we see that for ε small enough, where we have used the fact that b is uniformly bounded from above. We finally conclude that |∂ y φ ∞ | < 1/2 which by the boundary conditions (5) and the classification in Proposition 5.3 implies that (M ∞ , g ∞ (t)) is the self-similar shrinking soliton on R × S 3 .
(iv) Classification of singularity models. According to Proposition 5.3 if the singularity model is not a family of shrinking cylinders, then it must be a positively curved rotationally symmetric κ-solution. By the recent classification in [35] we conclude that in this case the singularity model is isometric to the Bryant soliton (up to scaling).

Immortal warped Berger Ricci flows
Let (R 4 , g(t)) 0≤t<T be the maximal Ricci flow starting at some g 0 ∈ G ∞ and suppose that T < ∞.
Proof of Theorem 3 Letx 2 ,t and μ be given by Corollary 4.18 and consider a blow-up sequence giving rise to a singularity model (M ∞ , g ∞ (t), p ∞ ). Since by Lemma 5.10 the rescaled geodesic balls are included in B(o,x 2 ) for j large enough, we can argue exactly as in the proof of Claim 6.1 and deduce that any singularity model for the flow is in fact flat. This shows that the maximal time of existence cannot be finite.

Type-I Berger Ricci flows contain minimal 3-spheres
The existence of sufficiently pinched minimal embedded hyperspheres gives rise to Type-I singularities for (asymptotically flat) rotationally symmetric Ricci flows on R n [19]. Thus, in general we cannot extend the conclusions of Theorem 1 and Theorem 3 to include initial data containing minimal 3-spheres.
While in the SO(n)-invariant case no minimal spheres can appear along the flow, in the SU(2)-cohomogeneity 1 setting an analogous property might fail. On the other hand, minimal spheres can disappear in finite time [19,Proposition 1.7].
In the following we consider a Type-I warped Berger Ricci flow whose curvature is controlled at spatial infinity uniformly in time. A priori one might expect that there exist examples of Type-I singularities where both b and c have local minima while the mean curvature of the embedded hyperspheres remains positive. The next result rules out this possibility. We prove that for times close to the maximal time of existence a Type-I warped Berger Ricci flow solution (R 4 , g(t)) must contain minimal 3-spheres.