Ricci Flow under Kato-type curvature lower bound

In this work, we extend the existence theory of non-collapsed Ricci flows from point-wise curvature lower bound to Kato-type lower bound. As an application, we prove that compact three dimensional non-collapsed strong Kato limit space is homeomorphic to a smooth manifold. The result also holds in higher dimension under uniform strong Kato lower bound of 1-isotropic curvature. We also use the Ricci flow smoothing to study stability problem in scalar curvature geometry.


Introduction
There is a long history on the study of compactness of sets of manifolds with uniform lower bound of Ricci curvature.For instance, the Gromov compactness Theorem [20] states that sequence of compact manifolds with uniform Ricci lower bound and diameter upper bound admits a convergent subsequence in the Gromov-Hausdorff topology.Since then, there have been many studies toward understanding the structure of the metric space arising as limits of smooth manifolds, see [10,11,12,13,14,17] and the reference therein.The analytic properties such as comparison geometry and heat kernel estimates on manifolds with Ricci lower bound play an important role there.
On the other hand, there are however many interesting scenarios in which uniform Ricci lower bound is missing, especially in the study of Ricci flow.It is then natural and important to consider the case when a uniform Ricci lower bound is further relaxed to bound in a weaker sense.In [35], Petersen and Wei generalized the classical fundamental Laplacian and volume comparison for uniform Ricci curvature lower bounds to smallness assumptions of L p norm of Ric − for p > n/2 where Ric − (x) = inf{α ≥ 0 : Ric(g(x)) + αg(x) ≥ 0}.This opens the door to understanding the structure of limit spaces under small ||Ric − || p assumption for p > n/2, see also [18,30].
More recently, it was further generalized by Carron [5] and Rose [36] that in the compact case, a Dynkin-type lower bound of Ricci curvature also suffices to obtain the Li-Yau estimates for the heat kernel, building on the idea of Zhang-Zhu [46].For a complete Riemannian manifold (M n , g 0 ) of dimension n ≥ 2, we denote κ t (M n , g 0 ) = sup x∈M ˆt 0 ˆM H g 0 (x, y, s)(Ric) − (y) dvol g 0 (y)ds where H g 0 is the heat kernel, that is H(•, •, t) is the kernel of the operator e −t∆g 0 for any t > 0.
Definition 1.1.Let {(M i , g i )} ∞ i=1 be a sequence of compact manifolds 1 .We say that {(M i , g i )} ∞ i=1 satisfies • a uniform Dynkin bound on Ric − if there exists T > 0 such that for all i ∈ N, ; • a uniform Kato bound on Ric − if there exist a non-decreasing function f : (0, T ] → (0, +∞), T > 0 such that f (t) → 0 as t → 0 + and for all t ∈ (0, T ] and i ∈ N, • a strong uniform Kato bound on Ric − if there exist a non-decreasing function f : (0, T ] → (0, +∞), T, Λ > 0 such that f (t) → 0 as t → 0 + and for all t ∈ (0, T ] and i ∈ N, When Ric ≥ −K, it satisfies the uniform Dynkin for T = (16nK) −1 .In the compact case, it was also shown that a strong uniform Kato bound can be achieved for some small T under suitable L p bound of Ric − [38] or suitable Morrey bound [9].Perhaps more importantly, it was proved by Carron [5] that the set of compact manifolds satisfying a uniform Dynkin bound on Ric − is pre-compact in the Gromov-Hausdorff topology.When it is strengthened to strong uniform Kato sense, the structure theory of the corresponding volume non-collapsed Gromov-Hausdorff limit has been developed by Carron-Mondello-Tewodrose [6].This generalizes the earlier works of Cheeger-Colding on Ricci limit spaces.We refer readers to [7,8,37,39] for more interesting and important developments.
In this work, we are interested in the regularity of the Gromov-Hausdorff limits of volume non-collapsed compact manifolds (without boundary) under the strong Kato lower bound on 1-isotropic curvature.To clarify the notion, denote the space of algebraic curvature tensors on R n by C B (R n ).For any given R ∈ C B (R n ), we extend it complex linearly to C n .We say that R ∈ C PIC2 if for each two complex dimensional subspace Σ of C n and orthonormal basis v, w ∈ C n of Σ, we have R(v, w, v, w) ≥ 0. If one instead asks for nonnegativity of complex sectional curvature only for PIC1 sections, defined to be those Σ that contain some non-zero vector v whose conjugate v is orthogonal to the entire section Σ.The algebraic curvature tensors R with non-negative complex sectional curvature for each such restricted Σ form a cone we denote by C PIC1 .When n = 3, it is known that C PIC1 is also the cone of curvature tensors R with Ric(R) ≥ 0.
For n ≥ 4, one can equivalently describe C PIC1 as follows (see [3]): C PIC1 is the cone consisting of curvature tensors R such that for any orthonormal four-frame {e i } 4 i=1 ⊂ R n and λ ∈ [0, 1], we have We are interested in studying manifolds where the curvature is bounded from below with respect to C PIC1 .For a Riemannian manifold (M n , g) with n ≥ 3 and x ∈ M, we define where I is the curvature tensor defined by I ijkl = δ ik δ jl − δ il δ jk .This is the negative part of the lowest eigenvalue of Rm with respect to C PIC1 .This generalizes the notion of Ric − in the sense that when n We follow the spirit of κ t and define, for all t > 0, κ t,IC1 (M n , g 0 ) = sup x∈M ˆt 0 ˆM H g 0 (x, y, s)(IC 1 ) − (y) dvol g 0 (y)ds.
We say that (M n , g) ∈ K IC1 (n, f ) if for all t ∈ (0, T ] we have We will mainly focus on the case when M is compact and the Kato-bound is uniformly strong.Given a sequence of compact manifolds When n = 3, it coincides with the concept of strong Kato bound of Ric − considered in [6], modulus scaling2 . In contrast with the work [6,7], we intend to strengthen the curvature lower bound from Ricci curvature to 1-isotropic curvature (which two notions coincide when n = 3).This is largely motivated by the works of Bamler-Cabezas-Wilking [2], Simon-Topping [32], Lai [25] and Lee-Tam [26] on obtaining the regularity of the Gromov-Hausdorff limit using Ricci flow.
The following is the main technical result which says that such manifolds can be regularized by the Ricci flow uniformly.This generalizes the earlier work of Bamler-Cabezas-Wilking [2] on the theory of Ricci flows with curvature bounded from below in point-wise sense initially.
Theorem 1.1.Suppose (M n , g 0 , x 0 ), n ≥ 3 is a pointed compact manifold which is inside K IC1 (n, f, v) for some non-decreasing function f , T and Λ satisfying (1.2) and diam(M, g 0 ) ≤ D for some D > 0, then there exist S, α, ṽ depending only on n, f, v, D, T, Λ and a solution g(t), t ∈ [0, S] to the Ricci flow such that for all (x, t) ∈ M × (0, S], (a A similar result also holds in the Kähler case under strong uniform Kato lower bound on bisectional curvature, see Theorem 5.1. Remark 1.1.Using the partial Ricci flow approach in [25] (see also [32,26]), the result also holds in the complete non-compact case without bounded curvature assumption if suitable static heat kernel estimates hold.In the presence of uniform Ricci lower bound, this usually follows from Li-Yau estimates.In the compact case, it has been obtained by Carron [5] and Rose [36] under the Kato-type lower bound of Ricci curvature.The comparison geometry in the complete non-compact case seems out of reach at the moment due to the absence of exhaustion function with good control.We will therefore only focus on the compact case in this work.
As an application, we prove that the Gromov-Hausdorff limit of sequence of compact manifolds satisfying the strong uniform IC1-Kato bound and a uniform diameter upper bound is homeomorphic to a smooth manifold.
) is a sequence of compact manifold such that (a) (M i , g i,0 , x i ) ∈ K IC1 (n, f, v) for some x i ∈ M i , v > 0 and a nondecreasing function f satisfying (1.2); (b) diam(M i , g i,0 ) ≤ D for some D > 0. Then there exist a smooth compact manifold M ∞ and a distance function d ∞ on M ∞ so that (M i , g i,0 ) sub-converges to (M ∞ , d ∞ ) in the measured Gromov-Hausdorff sense.Moreover, there exists a Ricci flow g ∞ (t), t ∈ (0, S] on M ∞ such that lim The final part of the result also holds if C PIC1 is replaced by other curvature cone in Lemma 3.1.Moreover, it particularly infers that if a compact manifold supports a metric with sufficiently small negative lower bound of IC 1 in the strong Kato sense relative to n, D, v, then it also admits a smooth metric whose curvature operator lies in C PIC1 via a contradiction argument. Another application is to understand the stability of metrics on torus with almost non-negative scalar curvature.When M = T n , the celebrated work of Schoen and Yau [41,42], Gromov and Lawson [19] stated that metrics with non-negative scalar curvature R ≥ 0 must be flat.In [21], Gromov asked if a sequence of metrics g i with R(g i ) ≥ −i −1 on T n will sub-converge to a flat metric in some appropriate weak sense.For related works, we refer interested readers to the survey paper of Sormani [43] for a comprehensive discussion.In this regard, we show that if in addition the 1-isotropic curvature is bounded from below in the strong Kato sense, then the sequence will sub-converge to the flat torus in the measured Gromov-Hausdorff sense.
Acknowledgement: The author would like to thank C. Rose for his interest and D. Tewodrose for comments on improving the manuscript.The work was partially supported by Hong Kong RGC grant (Early Career Scheme) of Hong Kong No. 24304222, a direct grant of CUHK and a NSFC grant.

the strong kato bound and implications
In the following, compact manifolds are referred those without boundary.In this section, we will collect some important properties from [6] for compact manifolds (M n , g) satisfying the strong Kato bound.The properties will play crucial role in obtaining estimates of Ricci flows.
We say that (M n , g) Similarly, we define K(n, f, v) to be the class of pointed compact manifolds (M, g, x 0 ) such that the volume Vol g B g (x 0 , √ T ) ≥ vT n/2 .The following Proposition says that the heat kernel will behave in a similar way as the Euclidean one under the Kato control.This is originated in [5] Proposition 2.1.[7, Proposition 2.6] There exists γ Moreover, in this case, it is known that (M n , g) ∈ K(n, f ) is a volume doubling space: for any .
If in addition f satisfies a stronger integrability condition: for some Λ > 0. As pointed out in [6], f satisfying (2.2) automatically implies that f satisfies (2.1) by shrinking T .It was first shown by Carron [5] that under the stronger assumption (2.2), we can strength the conclusion to Ahlfors n-regular.
2) for some Λ > 0. Then there exists C n > 0 such that for all x ∈ M and 0 < s < r ≤ √ T , Moreover, for any x, y ∈ M and 0 ≤ r ≤ √ T , We are primarily interested in the non-collapsed case in this work.We end this section by observing that the non-collapsed strong Kato assumption can be re-phased as an estimate in an integral form.

Ricci flow smoothing and estimates
The idea of this work is to mollify the metric g 0 using the Ricci flow.The Ricci flow is a one parameter family of metrics g(t) satisfying In the coming few sections, we will focus on obtaining estimates of Ricci flows.If (M n , g 0 ) is a compact manifold, it was shown by Hamilton [40] that the Ricci flow admits a short-time solution g(t).It is by-now known to be extremely powerful in regularizing metrics even in case without bounded curvature, for instances see the works [22,33,25,32,24] and the reference therein.Although we are primarily interested in the compact case, we will state our result in this section using a localized form so that it also applies to the noncompact case.
It is a general philosophy that the Ricci flow will tend to be positively curved as it evolves, for instances see the list in Lemma 3.1 and the recent work of Brendle [4].The following Proposition illustrates its relation to curvature estimates in the non-collapsed case.This is a slight generalization of the curvature estimates in [25,26,31].Proposition 3.1.For any n ≥ 3, v 0 , L > 0, there exist S(n, v 0 , L), a(n, v 0 , L) > 0 such that the following holds.Suppose (M n , g(t)), t ∈ [0, S] is a smooth solution to the Ricci flow (not necessarily complete) such that for some x 0 ∈ M, we have (c) for all x ∈ B g(t) (x 0 , 1) and t ∈ (0, S], we have either Proof.The proof of the curvature estimate uses the Perelman-inspired pointpicking argument from [31, Lemma 2.1].We illustrate the case of (ii) and will point out the modification needed for (i) and (iii).
Suppose the conclusion is false for some n, v 0 , L > 0. Then for any a k → +∞, we can find a sequence of manifold M n k , Ricci flows g k (t), t ∈ [0, S k ] and p k ∈ M k satisfying the hypotheses but so the curvature estimate fails with a = a k in an arbitrarily short time.We might assume a k S k → 0. Using the smoothness of the flow, we can choose 2 ).By (v) and the fact that a k t k → 0, [31, Lemma 5.1] implies that for sufficiently large k ∈ N, we can find β(n) > 0, tk ∈ (0, t k ] and xk ∈ B g k ( tk ) (x k , 34 − We note here that from the proof of [31, Lemma 5.1], for (x, t) we have d g k (t) (x, x k ) < 1 so that the conditions applied.
We now consider the parabolic rescaled Ricci flows gk (t By (b), (d) and the result of Cheeger, Gromov and Taylor [15], the injectivity radius of gk (0) at xk is bounded from below uniformly.Together with the curvature estimates from (b), we may apply Hamilton's compactness theorem [23] to conclude that, (M k , gk (t), xk ) sub-converges in the Cheeger-Gromov sense to (M ∞ , g∞ (t), x∞ ) which is a complete non-flat ancient solution to the Ricci flow with bounded curvature.By (e), Rm(g ∞ (t)) ∈ C PIC1 for all t ≤ 0.Moreover, (d) implies that it is of Euclidean volume growth.But this contradicts with [2, Lemma 4.2].This completes the proof.
In case of (iii), each g k (t) is in addition Kähler and the conclusion (e) is replaced by OB(g k (t)) ≥ − L t+Q k tk instead so that the limiting Ricci flow g∞ (t) is Kähler, non-flat with Euclidean volume growth and has non-negative bounded holomorphic orthogonal bisectional curvature.This contradicts with [29, Proposition 6.1].
In case of (i), the Ricci curvature of gk (t) is not almost non-negative but we might instead apply Hamilton-Ivey estimates (for instances see [16, Proposition 9.8]) on the limiting Ricci flow g∞ (t) to conclude that g∞ (t) has nonnegative sectional curvature and Euclidean volume growth.The non-flatness contradicts with Perelman's Theorem [34,Section 11].

Volume non-collapsing along flow
In view of Proposition 3.1, the volume non-collapsing along the flow plays an important role to avoid curvature blowup.In this section, we will show that under the mildly integrable curvature lower bound, if the initial metric is volume non-collapsed, then so does the flow for a uniform short time.We rely heavily the idea of Simon-Topping [31] which considered the case when the Ricci curvature is uniformly bounded from below along the flow.
Lemma 4.1.Suppose (M n , g(t)), t ∈ [0, S] is a smooth solution to the Ricci flow (not necessarily complete) such that for some x 0 ∈ M, we have B g(t) (x 0 , 5) ⋐ M for all t ∈ [0, S].Then the following holds.
Proof.The proof is almost identical to that of [31,Lemma 11.1].We only give a sketch and point out the necessary modifications.We first note that since φ ∈ L 1 ([0, S]), for any ε > 0, there exists S ε such that if 0 < t < S ε , Hence, as t → 0 + , the Ricci lower bound behaves as This serves as the crucial replacement of the uniform Ricci lower bound assumption in [31,Lemma 11.1].With this replacement in mind, we can carry out the exact same argument in the proof of [31,Lemma 11.1] to obtain the following.If the conclusion fails, then for any sufficiently small ε0 , there exist a sequence of n-manifold M i , a sequence of point x i ∈ M i , a sequence of Ricci flows gi (t) on M i × [0, ti ] with ti decreases to 0, t i ∈ (0, ti ] and a sequence of points ) for all t ∈ [0, t i ]; (iv) v −1 0 r n ≥ Vol gi (0) (y, r) ≥ v 0 r n for all r ∈ (0, 1) and y ∈ B gi (0) (y i , 1); We remark here that since t i → 0, [31, (11.15)] now follows from (4.3) when i is sufficiently large.Now consider the rescaled Ricci flow Here the property (I) follows from (4.3) while (V) follows from the initial Ahlfors n-regularity assumption.Moreover, using the same proof where [31,Lemma 3.4] is now replaced by Lemma 4.1, the Claim 1 in [31, page 513] also holds for g ∞ (1) in the sense that if we have a ball B g∞(1) (y, L) for y ∈ M ∞ and L > 0, that is covered by N balls of radius r ≥ R(n, α), then we must have . Now we are in position to apply [31,Lemma 11.2] as in the proof of [31,Lemma 11.1] to derive contradiction when ε0 is chosen to be too small.Finally, we show that the volume non-collapsing is indeed preserved in a large scale for a uniform short time.
Proof.Since B g(t) (x 0 , 1) is compactly contained inside M, we might take finitely many {B g(t) It remains to estimate N from below.By Lemma 4.1, we might restrict S so that By measuring the set using dvol g 0 and volume growth assumption of g 0 , we deduce that This completes the proof by substituting it back to (4.4).

Ricci flow smoothing on Compact manifolds
In this section, we establish a existence theory of Ricci flow and use it to prove the stability.
In higher dimension, when g 0 is in addition Kähler, one usually consider the holomorphic curvature which is more complex geometric.We say that the holomorphic bisectional curvature BK of a Kähler curvature tensor R is non-negative (denoted by R ∈ C BK ) if for any X, Y ∈ T 1,0 M, we have R(X, X, Y, Ȳ ) ≥ 0.
If one instead ask for the non-negativity only for X, Y ∈ T 1,0 M such that g(X, Ȳ ) = 0, then we say that the orthogonal bisectional curvature OB is nonnegative (denoted by R ∈ C OB ).One might then analogously define the class of compact Kähler manifolds satisfying the strong Kato-type lower bound on BK (or OB).Define for t > 0, where BK − (x) = inf{α ≥ 0 : Rm g(x) + αB ∈ C BK } and B is the Kähler curvature tensor given by B(X, Ȳ , Z, W ) = g(X, Ȳ )g(Z, W )+g(X, W )g(Z, Ȳ ).In this case, the same proof of Theorem 1.1 yields the following.
Theorem 5.1.Suppose (M n , g 0 , x 0 ) is a pointed compact Kähler manifold with for some non-decreasing function f , T and Λ satisfying (1.2) and diam(M, g 0 ) ≤ D for some D > 0, then there exist S, α, ṽ depending only on n, f, v, D, T, Λ and a solution g(t), t ∈ [0, S] to the Kähler-Ricci flow such that for all (x, t) ∈ M × (0, S], When a sequence of compact manifold (M n i , g i ) satisfy a uniform Dynkintype lower bound of Ricci, it is already shown by Carron [5] that the sequence admits a Gromov-Hausdorff limit after passing to subsequence.The structure theory was further investigated by Carron-Mondello-Tewodrose [6,7] in the non-collapsed case when the Dynkin-type lower bound of Ricci is strengthened to be a strong Kato-type lower bound.We now shows that in dimension three, the limit will be Gromov-Hausdorff close and homeomorphic to a smooth manifold.Moreover, the result also holds in higher dimension if we require the strong Kato bound on 1-isotropic curvature.
Proof of Corollary 1.1.The proof is almost identical to that of [2, Corollary 4].We include it for reader's convenience.By Theorem 1.1, there exist α, S > 0 and φ ∈ L 1 ([0, S]) ∩ C 0 loc ((0, S)) such that for each i ∈ N, we can find a Ricci flow g i (t), t ∈ [0, S] such that g i (0) = g i,0 and for all t ∈ (0, S], It follows from Hamilton's compactness [23] that after passing to subsequence, there exists a smooth Ricci flow (M ∞ , g ∞ (t), x ∞ ), t ∈ (0, S] such that (M i , g i (t), x i ) converges to (M ∞ , g ∞ (t), x ∞ ) in the C ∞ Cheeger-Gromov sense.If we first let i → +∞ and followed by s → 0, we see that M ∞ is compact since diam(M i , g i,0 ) ≤ D and d g∞(s) → d ∞ for some distance function d ∞ on M ∞ .By interchanging the order of taking limit, it is easy to see that d g i,0 converges to d ∞ modulus diffeomorphism.This proved the Gromov-Hausdorff convergence.It follows from [6, Theorem A] that the convergence is also in the measured Gromov-Hausdorff sense.
Suppose now (M i , g i,0 , x i ) ∈ K IC1 (n, f i , v) so that f i (T ) decreases to 0. We want to show that g ∞ (t) satisfies a stronger curvature estimate.
Fix any x i ∈ M. If r ≥ D where diam(M i , g i,0 ) ≤ D, then for all x ∈ B g i,0 (x i , r) = M i and t ∈ (0, r 2 ∧ T ], ˆBg i,0 (x i ,2r) ≤ C 1 e C 1 D 2 (T −1 −t −1 ) f i (T ) T . (5.6) For t ∈ [T, r 2 ], we extend f i (t) by defining Since f i (T ) → 0 as i → +∞, f i satisfies the assumption (iii) in Proposition 3.2 for r → +∞ in the sense that for any r ≥ D, there exists N ∈ N so that for all i > N, Proposition 3.2 applies on B g 0 (x i , r) so that (IC 1 ) − (g i (x i , t)) ≤ Λ f i ( Λt) t + 1 r 2 for some uniform Λ > 0. Since x i is arbitrary in M i , by letting i → +∞ for t ∈ (0, S] and followed by letting r → +∞, we see that Rm(g ∞ (t)) ∈ C PIC1 on M ∞ for t ∈ (0, S].This completes the proof. As another application, we obtain a stability in view of Geroch conjecture.This is motivated by the work of [1] where compactness under uniform L p curvature bound for p = 3 is studied when n = 3.Since a L p , p > 3  2 lower bound on Ric implies the uniform strong Kato bound of Ric [38], together with the regularity theory of Ricci flow (for instances, see [44,45]), we can view Corollary 1.2 as a generalization.
Proof of Corollary 1.2.By Corollary 1.1 it suffices to identify the Gromov-Hausdorff limit.Recall that there exists (M ∞ , g ∞ (t), x ∞ ), t ∈ (0, S] such that the Ricci flow (M n i , g i (t), x i ) with g i (0) = g i,0 converges to (M ∞ , g ∞ (t), x ∞ ) in the smooth Cheeger-Gromov sense after passing to subsequent.By maximum principle, we have R(g i (t)) ≥ −i −1 for all i and hence R(g ∞ (t)) ≥ 0. Since M ∞ is compact, the smooth convergence implies that M ∞ is diffeomorphic to M i for i sufficiently large.In particular, σ(M ∞ ) ≤ 0 and hence g ∞ (t) is Ricci flat for t ∈ (0, S] by strong maximum principle.Thus, g ∞ = g ∞ (S) is desired limit.
If b 1 (M i ) = n for all i, we also have b 1 (M ∞ ) = b 1 (M i ) = n for i large.Since g ∞ (t) is of Ricci flat, it is the standard torus by the classical Bochner theorem.This completes the proof.