Diameter and Ricci curvature estimates for long-time solutions of the Kahler-Ricci flow

Abstract

It is well known that the Kähler-Ricci flow on a Kähler manifold X admits a long-time solution if and only if X is a minimal model, i.e., the canonical line bundle KX is nef. The abundance conjecture in algebraic geometry predicts that KX must be semi-ample when X is a projective minimal model. We prove that if KX is semi-ample, then the diameter is uniformly bounded for long-time solutions of the normalized Kähler-Ricci flow. Our diameter estimate combined with the scalar curvature estimate in [34] for long-time solutions of the Kähler-Ricci flow are natural extensions of Perelman's diameter and scalar curvature estimates for short-time solutions on Fano manifolds. We further prove that along the normalized Kähler-Ricci flow, the Ricci curvature is uniformly bounded away from singular fibres of X over its unique algebraic canonical model Xcan if the Kodaira dimension of X is one. As an application, the normalized Kähler-Ricci flow on a minimal threefold X always converges sequentially in Gromov-Hausdorff topology to a compact metric space homeomorphic to its canonical model Xcan, with uniformly bounded Ricci curvature away from the critical set of the pluricanonical map from X to Xcan.

On the collapsing of Calabi-Yau manifolds and Kähler-Ricci flows

“…Next, using the results in [3], in [38,Theorem 1.1] it was very recently proved that (4.5) diam(M, g(t)) C, uniformly for all t 0. Also, in [38, Corollary 1.1] it is shown that there is a uniform C such that for all x ∈ M, 0 < r < diam(M, g(t)), t 0 we have…”

“…After earlier work in [51,22,60,35,21], it was recently shown in [13] that ω(t) → f * ω can in C α loc (M • ) as t → ∞, for any 0 < α < 1. Furthermore, in [38] it is shown that diam(M, ω(t)) C, for all t 0, and [53] shows that the metric completion (Z, d Z ) of (N • , ω can ) is a compact metric space, which is homeomorphic to N when this is smooth.…”

On the Hölder estimate of Kähler-Ricci flow

“…In [3], the second named author and Fong considered the case when the generic fibres are biholomorphic to each other and developed a sharp parabolic Schauder estimate on cylinder using the idea of Hein-Tosatti in [10], and thus confirmed the above conjecture in the locally product case. More recently, Jian and Song [13] considered the case when m+n = 3 and proved that the Ricci curvature is uniformly locally bounded. For further discussions, we refer interested readers to [1,3,4,6,7,9,13,20,22,25] and the references therein.…”

“…More recently, Jian and Song [13] considered the case when m+n = 3 and proved that the Ricci curvature is uniformly locally bounded. For further discussions, we refer interested readers to [1,3,4,6,7,9,13,20,22,25] and the references therein.…”

Second-order estimates for collapsed limits of Ricci-flat Kähler metrics

“…The partial second-order estimate is motivated by the study of collapsing problems in Kähler geometry (see, e.g., [1,6,14,22,23,28,29,32,33,34,35,41,42,47,59,53,54,55,56,58,63,64,66,67,68,69,71,74,80]), as well as the study of canonical metrics in Kähler geometry and the behavior of the Kähler-Ricci flow. More specifically, the estimate in Theorem A plays a crucial role in establishing the following conjectural picture for collapsed Gromov-Hausdorff limits of Ricci-flat Kähler metrics which were originally proposed in [66,67], inspired by [24,38,39] and has been intensively studied since:…”