Heat kernel and curvature bounds in Ricci flows with bounded scalar curvature

Abstract

Abstract.In this paper we analyze Ricci flows on which the scalar curvature is globally or locally bounded from above by a uniform or time-dependent constant. On such Ricci flows we establish a new time-derivative bound for solutions to the heat equation. Based on this bound, we solve several open problems: 1. distance distortion estimates, 2. the existence of a cutoff function, 3. Gaussian bounds for heat kernels, and, 4. a backward pseudolocality theorem, which states that a curvature bound at a later time implies a curvature bound at a slightly earlier time.Using the backward pseudolocality theorem, we next establish a uniform L 2 curvature bound in dimension 4 and we show that the flow in dimension 4 converges to an orbifold at a singularity. We also obtain a stronger ε-regularity theorem for Ricci flows. This result is particularly useful in the study of Kähler Ricci flows on Fano manifolds, where it can be used to derive certain convergence results.

The extension problem of the mean curvature flow (I)

“…Let T 0 = N + β where N ∈ N, N ≥ 2 and β ∈ [0, 1). For the interval I 0 := (−2, 0), we have a subsequence of {t i }, which we denote by {i (1) k }, such that {Σ i (1) k ,s , s ∈ I 0 } converges in smooth topology, possibly with multiplicity at most N 0 , to a limit minimal surfaceΣ ∞ away from a singular setS = {q 1 , · · · , q l }. Consider the interval I ′ 0 := (−1 − λ, 1 − λ) with λ ∈ (0, 1).…”

“…Consider the interval I ′ 0 := (−1 − λ, 1 − λ) with λ ∈ (0, 1). Since I ′ 0 ∩ I 0 = ∅, we can take a further subsequence of {i (1) k }, denoted by {i (2) k }, such that {Σ i (2) k ,s , s ∈ I ′ 0 } converges to the same limit surfaceΣ ∞ away from the same singular setS. Similarly, we consider I 2 = (−2 − β, −β).…”

“…follows from Chen-Wang [7](See also [1] for a different approach). The limit of the above flow is then a Ricci-flat gradient shrinking soliton with finite singular points, which is nothing but a flat metric cone over S 3 /Γ for some finite subgroup Γ of O(4).…”

“…Theorem 4.9 of[26], or Theorem B.1 in Appendix B). Thus, we can take T → ∞ in Lemma 4.15 and obtain a function, still denoted by w, defined on Ann(3ǫ,1 3 ǫ −1 ) × [t ǫ + 1, ∞) satisfying (4.70), (4.71) and (4.72). Replacing ǫ by1 3 ǫ and t by t − t ǫ 3 − 1, we know that the function w is defined on Ann(ǫ, ǫ −1 ) × [0, ∞) and satisfies (4.16), (4.17) and(4.18).…”

Local Sobolev constant estimate for integral Ricci curvature bounds

“…Moreover, integral curvature bounds have recently been discovered in various geometric situations, such as the L 2 bound of the curvature tensor for noncollapsed manifolds with bounded Ricci curvature, and the (almost) L 4 bound of the Ricci curvature for the Kähler-Ricci flow as well as the (real) Ricci flow (under certain conditions) [9,17,29,5,25,30,4]. In [23], the important Laplacian comparison and volume comparison are generalized to integral Ricci lower bound.…”