Ricci flow does not preserve positive sectional curvature in dimension four

We find examples of cohomogeneity one metrics on $S^4$ and $\mathbb C P^2$ with positive sectional curvature that lose this property when evolved via Ricci flow. These metrics are arbitrarily small perturbations of Grove--Ziller metrics with flat planes that become instantly negatively curved under Ricci flow.


Introduction
The Ricci flow ∂ ∂t g(t) = −2 Ric g(t) of Riemannian metrics on a smooth manifold is an evolution equation that continues to drive a wide range of breakthroughs in Geometric Analysis, see e.g. [Bam21] for a survey. One of the keys to using Ricci flow is to control how the curvature of g(t) evolves; in particular, which curvature conditions of the original metric g(0) are preserved. Our main result establishes that, in dimension n = 4, positive sectional curvature (sec > 0) is not among them: Theorem A. There exist smooth Riemannian metrics with sec > 0 on S 4 and CP 2 that evolve under the Ricci flow to metrics with sectional curvatures of mixed sign.
In contrast, sec > 0 is preserved on closed manifolds of dimension n ≤ 3, by the seminal work of Hamilton [Ham82]. Moreover, it was previously known [Máx14] that Ric > 0 is not preserved in dimension n = 4, even among Kähler metrics, but these examples do not have sec > 0. Although Theorem A does not readily extend to all n > 4, there are examples of homogeneous metrics on flag manifolds of dimensions 6, 12, and 24 with sec > 0 that lose that property when evolved via Ricci flow, see [BW07,CW15,AN16]. A state-of-the-art discussion of Ricci flow invariant curvature conditions can be found in [BCRW19], see also Remark 5.1.
Theorem A builds on our earlier result [BK19] that certain metrics with sec ≥ 0, introduced by Grove and Ziller [GZ00] in a much broader context (see Section 2.1.1), immediately acquire negatively curved planes on S 4 and CP 2 , when evolved under Ricci flow. In light of the appropriate continuous dependence of Ricci flow on its initial data [BGI20], the metrics in Theorem A are obtained by means of: Theorem B. Every Grove-Ziller metric on S 4 or CP 2 is the limit (in C ∞ -topology) of cohomogeneity one metrics with sec > 0.
In full generality, the problem of perturbing sec ≥ 0 to sec > 0 is notoriously difficult, see e.g. [Wil07,Prob. 2]. Aside from clearly being unobstructed on S 4 and CP 2 , the deformation problem is facilitated here by the presence of natural directions for perturbation, given by the round metric and the Fubini-Study metric, respectively. Indeed, we deform sec ≥ 0 into sec > 0 in Theorem B by linearly interpolating lengths of Killing vector fields for the SO(3)-action which is isometric for both the Grove-Ziller metric g 0 and the standard metric g 1 on these spaces. The resulting SO(3)-invariant metrics g s , s ∈ [0, 1], are smooth and have sec > 0 for all sufficiently small s > 0. For a lower-dimensional illustration, consider the T 2 -action on S 3 ⊂ C 2 via (e iθ1 , e iθ2 ) · (z, w) = e iθ1 z, e iθ2 w , and invariant metrics g = dr 2 + ϕ(r) 2 dθ 2 1 + ξ(r) 2 dθ 2 2 , 0 < r < π 2 , written along the geodesic segment γ(r) = (sin r, cos r). The functions ϕ and ξ encode the g-lengths of the Killing fields ∂ ∂θ1 and ∂ ∂θ2 respectively, and must satisfy certain smoothness conditions at the endpoints r = 0 and r = π 2 . The unit round metric g 1 is given by setting ϕ and ξ to be ϕ 1 (r) = sin r and ξ 1 (r) = cos r, while a Grove-Ziller metric g 0 corresponds to concave monotone functions ϕ 0 and ξ 0 that plateau at a constant value b > 0 for at least half of 0, π 2 . The curvature operator of g is easily seen to be diagonal, with eigenvalues −ϕ ′′ /ϕ, −ξ ′′ /ξ, and −ϕ ′ ξ ′ /ϕξ, see e.g. [Pet16,Sec. 4 give rise to metrics g s deforming g 0 to have sec > 0 for s > 0. It turns out that a similar approach works for proving Theorem B, with the addition of a third (nowhere vanishing) function ψ, to deal with SO(3)-invariant metrics on 4-manifolds. The biggest challenge is verifying that these metrics have sec > 0, since that is no longer equivalent to positive-definiteness of the curvature operator if n ≥ 4. To overcome this difficulty, we use a much simpler algebraic characterization of sec > 0 in dimension n = 4, given by the Finsler-Thorpe trick (Proposition 2.2).
Motivated by the above, it is natural to ask whether the set of cohomogeneity one metrics with sec ≥ 0 on a given closed manifold coincides with the closure (say, in C 2 -topology) of the set of such metrics with sec > 0, if the latter is nonempty. In contrast to Theorem B, there is some evidence to suggest that Grove-Ziller metrics on certain 7-manifolds cannot be perturbed to have sec > 0, see [Zil09,Sec. 4].
This paper is organized as follows. Background material on cohomogeneity one manifolds and the Finsler-Thorpe trick in dimension 4 is presented in Section 2. The smoothness conditions and curvature operator of SO(3)-invariant metrics on S 4 and CP 2 are discussed in Section 3. Section 4 contains the proof of Theorem B, focusing mainly on the case of S 4 , since the proof for CP 2 is mostly analogous. Finally, Theorem A is proved in Section 5.
A cohomogeneity one manifold is a Riemannian manifold (M, g) endowed with an isometric action by a Lie group G, such that the orbit space M/G is one-dimensional. Let π : M → M/G be the projection map. Throughout, we assume M/G = [0, L] is a closed interval, and the nonprincipal orbits B − = π −1 (0) and B + = π −1 (L) are singular orbits. In other words, B ± are smooth submanifolds of dimension strictly smaller than the principal orbits π −1 (r), r ∈ (0, L), which are smooth hypersurfaces of M . Fix x − ∈ B − , and consider a minimal geodesic γ(r) in M joining x − to B + , meeting it at x + = γ(L); that is, γ is a horizontal lift of [0, L] to M . Denote by K ± the isotropy group at x ± , and by H the isotropy at γ(r), for r ∈ (0, L). By the Slice Theorem, given r ± max > 0 so that r + max + r − max = L, the tubular neighborhoods D(B − ) = π −1 ([0, r − max ]) and D(B + ) = π −1 ([L − r + max , L]) of the singular orbits are disk bundles over B − and B + . Let D l±+1 be the normal disks to B ± at x ± . Then K ± acts transitively on the boundary ∂D l±+1 , with isotropy H, so ∂D l±+1 = S l± = K ± /H, and the K ± -action on ∂D l±+1 extends to a K ± -action on all of D l±+1 . Moreover, there are equivariant diffeomorphisms D( where the latter is given by gluing these disk bundles along their common boundary ∂D(B ± ) ∼ = G × K± S l± ∼ = G/H. In this situation, one associates to M the group diagram Conversely, given a group diagram as above, where K ± /H are spheres, there exists a cohomogeneity one manifold M given as the union of the above disk bundles.
Fix a bi-invariant metric Q on the Lie algebra g of G, and set n = h ⊥ , where h ⊂ g is the Lie algebra of H. Identifying n ∼ = T γ(r) (G/H) for each 0 < r < L via action fields X → X * γ(r) , any G-invariant metric on M can be written as (2.1) g = dr 2 + g r , 0 < r < L, along the geodesic γ(r), where g r is a 1-parameter family of left-invariant metrics on G/H, i.e., of Ad(H)-invariant metrics on n. As r ց 0 and r ր L, the metrics g r degenerate, according to how G(γ(r)) ∼ = G/H collapse to B ± = G/K ± . Namely, they satisfy smoothness conditions that characterize when a tensor defined by means of (2.1) on M \ (B − ∪ B + ) ∼ = (0, L) × G/H extends smoothly to all of M , see [VZ20].
2.1.1. Grove-Ziller metrics. If both singular orbits B ± of a cohomogeneity one manifold M have codimension two, then M can be endowed with a new G-invariant metric g GZ with sec ≥ 0, as shown in the celebrated work of Grove and Ziller [GZ00, Thm. 2.6]. We now describe this construction, building metrics with sec ≥ 0 on each disk bundle D(B ± ) that restrict to a fixed product metric dr 2 + b 2 Q| n near ∂D(B ± ) ∼ = G/H, so that these two pieces can be isometrically glued together. Consider one such disk bundle D(B) at a time, say over a singular orbit B = G/K, and let k be the Lie algebra of K. Set m = k ⊥ and p = h ⊥ ∩k, so that g = m ⊕ p ⊕ h is a Q-orthogonal direct sum. Since p is 1-dimensional, the metric Q a,b on G, given by has sec ≥ 0 whenever 0 < a ≤ 4 3 and b > 0, see [GZ00, Prop. 2.4] or [BM17, Lemma 3.2]. Fix 1 < a ≤ 4 3 , and let r max > 0 be such that (2.2) y := ρ √ a √ a−1 satisfies y < r max , where ρ = ρ(b) is the radius of the circle(s) K/H endowed with the metric b 2 Q| p . Then, we can find a smooth nondecreasing function f : [0, r max ] → R and some 0 < r 0 < r max , with f (0) = 0, f ′ (0) = 1, f (2n) (0) = 0 for all n ∈ N, f ′′ (r) ≤ 0 for all r ∈ [0, r max ], f (3) (r) > 0 for all r ∈ [0, r 0 ), and f (r) ≡ y for all r ∈ [r 0 , r max ]. The rotationally symmetric metric g D 2 = dr 2 + f (r) 2 dθ 2 , 0 < r ≤ r max , on the punctured disk D 2 \ {0} extends to a smooth metric g D 2 on D 2 with sec ≥ 0 that, near ∂D 2 = {r = r max }, is isometric to a round cylinder [r 0 , r max ] × S 1 (y) of radius y. Thus, the product manifold (G × D 2 , Q a,b + g D 2 ) has sec ≥ 0, and so does the orbit space D(B) ∼ = G × K D 2 of the K-action on G × D 2 , when endowed with the metric g GZ that makes the projection map Π : (G×D 2 , Q a,b +g D 2 ) → (G× K D 2 , g GZ ) a Riemannian submersion. Writing this metric g GZ in the form (2.1), we have [GZ00, Lemma 2.1, Rem. 2.7] or [BM17, Lemma 3.1 (ii)]. In particular, g GZ = dr 2 + b 2 Q| n for all r ∈ [r 0 , r max ], since f (r) 2 a f (r) 2 +aρ 2 ≡ 1 for all such r; hence (D(B), g GZ ) is isometric to the prescribed product metric near ∂D(B) ∼ = G/H. This construction can be performed on each disk bundle D(B ± ) with the same b > 0, provided r ± max > 0 are chosen sufficiently large so that (2.2) holds for the corresponding radii ρ ± (b) of the circles K ± /H endowed with the metric b 2 Q| p± . Gluing these two disk bundles together, we obtain the desired G-invariant metric g GZ with sec ≥ 0 on M ∼ = D(B − )∪D(B + ) and M/G = [0, L], where L = r + max +r − max . Although it is natural to pick the same (largest) value for r ± max , so that the gluing occurs at r = L 2 , it is convenient to not impose this restriction. Note that Remark 2.1. Although this is not a requirement in the original Grove-Ziller construction, we assume that f (3) (r) > 0 on [0, r 0 ), hence the curvature of (D 2 , g D 2 ) is monotonically decreasing for r ∈ [0, r 0 ). As a consequence, for each 0 < r * < r 0 , there is a constant c > 0, depending on r * , so that sec g D 2 ≥ c for all r ∈ [0, r * ].
2.2. Finsler-Thorpe trick. In order to verify sec > 0 on Riemannian 4-manifolds, we shall use a result that became known in the Geometric Analysis community as Thorpe's trick, attributed to Thorpe [Tho72], but that actually follows from much earlier work of Finsler [Fin36], and is often referred to as Finsler's Lemma in Convex Algebraic Geometry. This rather multifaceted result is also known as the S-lemma, or S-procedure, in the mathematical optimization and control literature, see e.g. [PT07]. Details and other geometric perspectives can be found in [BKM21].
Let Sym 2 b (∧ 2 R n ) ⊂ Sym 2 (∧ 2 R n ) be the subspace of symmetric endomorphisms R : ∧ 2 R n → ∧ 2 R n that satisfy the first Bianchi identity. These objects are called algebraic curvature operators, and serve as pointwise models for the curvature operators of Riemannian n-manifolds. For instance, R ∈ Sym 2 b (∧ 2 R n ) is said to have sec ≥ 0, respectively sec > 0, if the restriction of the quadratic form R(σ), σ to the oriented Grassmannian Gr + 2 (R n ) ⊂ ∧ 2 R n of 2-planes is nonnegative, respectively positive. A Riemannian manifold (M n , g) has sec ≥ 0, or sec > 0, if and only if its curvature operator R p ∈ Sym 2 b (∧ 2 T p M ) has sec ≥ 0, or sec > 0, for all p ∈ M . The orthogonal complement to Sym 2 b (∧ 2 R n ) is identified with ∧ 4 R n ; so, if n = 4, it is 1-dimensional, and spanned by the Hodge star operator * . Since σ ∈ ∧ 2 R 4 satisfies σ ∧ σ = 0 if and only if * σ, σ = 0, the quadric defined by * in ∧ 2 R 4 is precisely the Plücker embedding Gr + 2 (R 4 ) ⊂ ∧ 2 R 4 . As shown by Finsler [Fin36], a quadratic form R(σ), σ is nonnegative when restricted to the quadric * σ, σ = 0 if and only if some linear combination of R and * is positive-semidefinite, yielding: be an algebraic curvature operator. Then R has sec ≥ 0, respectively sec > 0, if and only if there exists τ ∈ R such that R + τ * 0, respectively R + τ * ≻ 0. 3. Cohomogeneity one structure of S 4 and CP 2 Both S 4 and CP 2 admit a cohomogeneity one action by G = SO(3) as we now recall, see [BK19, Sec. 3] and [Zil09, Sec. 2] for details. The G-action on S 4 is the restriction to the unit sphere of the SO(3)-action by conjugation on the space of symmetric traceless 3 × 3 real matrices, while the G-action on CP 2 is a subaction of the transitive SU(3)-action. The corresponding orbit spaces are S 4 /G = 0, π 3 and CP 2 /G = 0, π 4 , endowing S 4 with the round metric with sec ≡ 1, and CP 2 with the Fubini-Study metric with 1 ≤ sec ≤ 4. Their group diagrams are as follows: according to an appropriate choice of minimal geodesic γ(r), r ∈ [0, L], see [BK19, Sec. 3]. In both cases, since H is discrete, n ∼ = g = so(3). We henceforth fix Q to be the bi-invariant metric such that {E 23 , E 31 , E 12 } is a Q-orthonormal basis of so (3), where E ij is the skew-symmetric 3 × 3 matrix with a +1 in the (i, j) entry, a −1 in the (j, i) entry, and zeros in the remaining entries. The 1-dimensional subspaces n k = span(E ij ), where (i, j, k) is a cyclic permutation of (1, 2, 3), are pairwise inequivalent for the adjoint action of H in the case of S 4 , while n 1 and n 2 are equivalent in the case of CP 2 , but neither is equivalent to n 3 . Collectively denoting S 4 and CP 2 with the above cohomogeneity one structures by M 4 , we consider diagonal G-invariant metrics g on M 4 , i.e., metrics of the form where L = π 3 or L = π 4 according to whether M 4 = S 4 or M 4 = CP 2 , cf. (2.1). Note that every G-invariant metric on S 4 is of the above form, i.e., n k are pairwise orthogonal, but n 1 and n 2 need not be orthogonal for all G-invariant metrics on CP 2 , i.e., the off-diagonal term g(E 23 , E 31 ) need not vanish identically. The standard metric on M 4 , with curvatures normalized as above, is obtained setting ϕ, ψ, ξ to (3.2) S 4 : ϕ 1 (r) = 2 sin r, ψ 1 (r) = √ 3 cos r + sin r, ξ 1 (r) = √ 3 cos r − sin r, CP 2 : ϕ 1 (r) = sin r, ψ 1 (r) = cos r, ξ 1 (r) = cos 2r, see Figure 1 below for their graphs.
Remark 3.2. Since the isotropy groups K ± for the G-action on S 4 are conjugate, the smoothness conditions at the endpoints r = 0 and r = L can be obtained from one another by interchanging the roles of ϕ and ξ. Furthermore, just as the round metric (3.2), all metrics we consider on S 4 have the following additional symmetries: However, metrics on CP 2 do not have any of these features or extra symmetries, as K ± are not conjugate, and, in general ϕ(r) = ξ (L − r) and ψ(r) = ψ (L − r). Figure 1. Graphs of ϕ 1 , ψ 1 , ξ 1 , for S 4 (left) and CP 2 (right).
Remark 3.4. Diagonal entries in R i are sectional curvatures sec(e i ∧ e j ) = R ijij of coordinate planes, while off-diagonal entries are R ijkl , with i, j, k, l all distinct, so the Finsler-Thorpe trick states that sec ≥ 0 and sec > 0 are respectively equivalent to the existence of τ such that all R ijij R klkl − (R ijkl + τ ) 2 are ≥ 0 and > 0.
To illustrate the above, note that setting ϕ, ψ, ξ to be the functions in (3.2) that correspond to the standard metrics in S 4 and CP 2 , the blocks R i become constant: (3.5) In particular, τ can be chosen constant, and R + τ * 0 if and only if τ ∈ [−1, 1] for S 4 , and τ ∈ [0, 2] for CP 2 , and R + τ * ≻ 0 if and only if τ is in the open intervals. Similarly, the curvature of a Grove-Ziller metric with gluing interface ∂D(B ± ) isometric to (G/H, b 2 Q| n ) and L = r + max + r − max can be computed by setting ϕ, ψ, ξ instead to be the functions that make (3.1) match with (2.3), namely (see Figure 2) as m = n 2 ⊕ n 3 and p = n 1 for the disk bundle D(B − ), but ϕ and ξ switch roles on the disk bundle D(B + ), in which m = n 1 ⊕ n 2 and p = n 3 . Recall that f (r) ≡ √ a ρ √ a−1 for r 0 ≤ r ≤ r max on each of D(B ± ), so, in a neighborhood of the gluing interface r = r − max = L − r + max , the functions ϕ = ψ = ξ are all constant and equal to b . In what follows, to simplify the exposition, we shall work with ϕ, ψ, ξ only on the interval (0, r − max ], which, at least on S 4 , determines their values for all 0 < r < L by setting r + max = r − max and imposing the additional symmetries (3.3), see Remark 3.2. Straightforward computations using Proposition 3.3 imply the following: Proposition 3.5. The curvature operator of the Grove-Ziller metric (2.3); i.e., the metric (3.1) with ϕ, ψ, ξ as in (3.6), for r ∈ (0, r − max ], is R = diag(R 1 , R 2 , R 3 ), with: .
It is a straightforward consequence of Proposition 3.1 that g s are smooth metrics: where φ k , k = 1, 2, are smooth functions, hence (ii) and (iii) are also satisfied.
4.2. Regularity of perturbation. By (4.3), Lemma 4.1, and Proposition 3.3, each entry of the curvature operator matrix R s of g s along γ(r) is a smooth function where P is a polynomial. Note that the g s -orthonormal basis on which the matrix R s is being written varies smoothly with s. The singularities in (4.5) at r = 0 and r = L, due to ϕ s (0) = 0 and ξ s (L) = 0, are removable as a consequence of Lemma 4.1. This corresponds to the fact that also P vanishes to the appropriate order because ϕ s , ψ s , ξ s satisfy the required smoothness conditions. Moreover, these smoothness conditions imply that (4.5) equals where Q is continuous. Furthermore, by (4.4), the numerator above can be written as a polynomial P in the parameter s, the functions ϕ 0 , ψ 0 , ξ 0 and their first and second derivatives, and the functions ∆ ϕ , ∆ ψ , ∆ ξ and their first and second derivatives (indicated as . . . below). Thus, (4.6) and hence (4.5) are equal to (4.7) P (s, ϕ 0 , ψ 0 , ξ 0 , . . . , ∆ ϕ , ∆ ψ , ∆ ξ , . . . ) ϕ 2 0 ψ 2 0 ξ 2 0 + Q(s, r) s.
In particular, the dependence of the above on s is polynomial in the first term, and smooth on the second. Expanding in s, we have where P n are polynomials. Each coefficient in this sum is a smooth function of r that vanishes at r = 0 and r = L in such way that the limits of (4.7) as r ց 0 and r ր L are both finite, so the corresponding coefficients in (4.7) extend to smooth (hence bounded) functions on [0, L]. Thus, P (s, ϕ 0 , ψ 0 , ξ 0 , . . . , ∆ ϕ , ∆ ψ , ∆ ξ , . . . )/ϕ 2 0 ψ 2 0 ξ 2 0 can be regarded as a polynomial in the variable s whose coefficients are continuous functions of r. We will implicitly (and repeatedly) use this fact in what follows. 4.3. Positive curvature on S 4 . To simplify the exposition, we shall focus primarily on the case M 4 = S 4 , in which r ± max = L 2 = π 6 and it suffices to verify sec > 0 along the geodesic segment γ(r) with r ∈ [0, r − max ] due to the additional additional symmetries (3.3), cf. Remark 3.2.
Let R s = diag (R s ) 1 , (R s ) 2 , (R s ) 3 be the curvature operator of (S 4 , g s ) along γ(r), given by Proposition 3.3, where ϕ, ψ, ξ are set to be ϕ s , ψ s , ξ s defined in (4.2). As discussed above, R s , s ∈ [0, 1], extends smoothly to r = 0, and this extension (as well as its entries) will be denoted by the same symbol(s). Clearly, R 0 is the curvature operator of the Grove-Ziller metric g 0 , so R 0 +τ 0 * 0 for all r ∈ [0, r − max ], where τ 0 := − ϕ ′ 0 2b 2 , see Proposition 3.5. The proof of Theorem B hinges on the next: Claim 4.2. If s > 0 is sufficiently small, then R s +τ s * ≻ 0 for all r ∈ [0, r − max ], with We begin the journey towards Claim 4.2 observing that certain diagonal entries of R s , which are sectional curvatures with respect to g s , are positive for all s ∈ (0, 1].
Proposition 4.6. For all r ∈ [0, r − max ], the entries of (R s ) i , for i = 2, 3, satisfy: . Proof. First, let us consider η 2 . From Proposition 3.3, We analyze these three terms separately using (4.4). The first two satisfy  Second, the proof that η 3 has the same expansion as η 2 is similar. Namely, where the first term was already considered above, and the second term satisfies by similar considerations involving L'Hôpital's rule and Proposition 3.1 (i). Thus, , cf. (4.9). Next, consider µ 2 . From Proposition 3.3, The first term above satisfies , cf. (4.9). The proof that µ 3 has the same expansion as µ 2 is analogous, and left to the reader.

Positive turns negative
In this section, we prove Theorem A, using the fact that Grove-Ziller metrics on S 4 and CP 2 immediately acquire negatively curved planes under Ricci flow [BK19], together with Theorem B, and continuous dependence on initial data [BGI20].
Proof of Theorem A. Let M 4 be either S 4 or CP 2 , and consider the 1-parameter family of metrics g s on M 4 , defined in (4.3), such that g 0 is a Grove-Ziller metric and g 1 is either the round metric or the Fubini-Study metric, accordingly. From Lemma 4.1, the metrics g s are smooth, and it is evident from (4.2) and (4.3) that, for all k ≥ 0 and 0 < α < 1, there exists a constant λ k,α > 0 such that (5.1) g s − g 0 C k,α ≤ λ k,α s, for all 0 ≤ s ≤ 1, where · C k,α denotes the Hölder norm on sections of the bundle E = Sym 2 T M 4 with respect to a fixed background metric. For 0 ≤ s ≤ 1, let g s (t), 0 ≤ t < T (g s ), be the maximal solution to Ricci flow starting at g s (0) = g s , where 0 < T (g s ) ≤ +∞ denotes the maximal (smooth) existence time of the flow. For all 0 ≤ s ≤ 1 and 0 ≤ t < T (g s ), we have that g s (t) ∈ C ∞ (E), so g s (t) is in the proper closed subspace h k,α (E) ⊂ C k,α (E) for all k ≥ 0 and 0 < α < 1, in the notation of [BGI20]. From the main theorem in [BK19], there exist a 2-plane σ tangent to M 4 and t 0 > 0 such that sec g0 (σ) = 0 and sec g0(t) (σ) < 0 for all 0 < t < t 0 . Fix 0 < t * < t 0 , and let δ > 0 be such that sec g (σ) < 0 for all metrics g with g − g 0 (t * ) C 2,α < δ. By the continuous dependence of Ricci flow on initial data [BGI20, Thm A], there exist constants r > 0 and C > 0, depending only on t * and g 0 , such that, if g s − g 0 C 4,α ≤ r, then T (g s ) ≥ t 0 and g s (t) − g 0 (t) C 2,α ≤ C g s − g 0 C 4,α for all t ∈ [0, t 0 ]. Together with (5.1), this yields that if 0 ≤ s ≤ r/λ 4,α , then g s (t) − g 0 (t) C 2,α ≤ C g s − g 0 C 4,α ≤ C λ 4,α s, for all 0 ≤ t ≤ t 0 .
Remark 5.1. The curvature operators R(t) : ∧ 2 T M → ∧ 2 T M of metrics g(t) on M n evolving under Ricci flow satisfy the PDE ∂ ∂t R = ∆R + 2Q(R), where Q(R) depends quadratically on R. By Hamilton's Maximum Principle, if an O(n)invariant cone C ⊂ Sym 2 b (∧ 2 T M ) is preserved by the ODE d dt R = 2Q(R), then it is also preserved by the above PDE. It was previously known that the cone C sec>0 of curvature operators with sec > 0 is not preserved under the above ODE on R in dimensions n ≥ 4, since it is easy to find R 0 ∈ ∂C sec>0 with Q(R 0 ) pointing outside of C sec>0 . Nevertheless, this observation alone does not imply the existence of metrics realizing such a family of curvature operators on some closed n-manifold, thus evolving under Ricci flow and losing sec > 0, as the above metrics g s (t) do.