Interior derivative estimates for the Kähler–Ricci flow

Abstract

We give a maximum principle proof of interior derivative estimates for the Kähler-Ricci flow, assuming local uniform bounds on the metric.

Regularizing properties of the twisted Kähler–Ricci flow

“…Thanks to the complex parabolic Evans-Krylov theory and Schauder estimates (see [32] for a recent account in the Kähler-Ricci flow context), these bounds allow us to show that ' t;j ! ' t in C 1 .X 0; T /, as j !…”

A local version of Bando's theorem on the real-analyticity of solutions to the Ricci flow

“…The estimates (1.2) are variants of the well-known local estimates of Shi [10] and Hamilton [5] (see also [8,9]), which likewise take the form…”

The Kähler–Ricci flow around complete bounded curvature Kähler metrics

“…The next lemma shows that we have a local curvature estimate provided g(t) stays uniformly equivalent to a good reference metric. In [27], Sherman-Weinkove showed essentially the same estimate but with less detail on the dependence of the various constants in the Lemma. Lemma 2.1.…”

“…By [4, Lemma 3.3], we also have local uniform upper bound for h k (t) with respect to h on [0, ǫ n K −1 ]. In particular, by the Evans-Krylov theory [14,18] or Kähler-Ricci flow local estimates [27], we may conclude subequence convergence h k i (t) → g(t) in C ∞ loc (M × (0, ǫ n K −1 ] where g(t) solves (1.1). By applying Proposition 2.1 on each h k (t), it is easy to see that…”

“…Thus there is a sequence {h k } ⊂ S(1, c, h) such that h k → g 0 in C 0 loc (M), and by Theorem 1.1, there is a sequence of solutions h k (t) to (1.1) on M × [0, T h ) each satisyfing the conclusions in Theorem 1.1. In particular, by the uniform estimates in Theorem 1.1, there exists T (n, c, K) such that given any T < T (n, c, K) we may have C −1 h ≤ h k (t) ≤ Ch on M × [0, T ) for some some C independent of k. Thus by the Evans-Krylov theory [14,18] or Kähler-Ricci flow local estimates [27], we may conclude subequence convergence of h k i (t) → g(t) in C ∞ loc (M × (0, T (n, c, K)) where g(t) solves (1.1). The fact that we may extend this solution to M × (0, T h ) follows from the same argument as in Theorem 1.1.…”