Albert Chau

PSEUDOLOCALITY FOR THE RICCI FLOW AND APPLICATIONS

In \cite{P1}, Perelman established a differential Li-Yau-Hamilton (LYH) type inequality for fundamental solutions of the conjugate heat equation corresponding to the Ricci flow on compact manifolds (also see \cite{N2}). As an application of the LYH inequality, Perelman proved a pseudolocality result for the Ricci flow on compact manifolds. In this article we provide the details for the proofs of these results in the case of a complete non-compact Riemannian manifold. Using these results we prove that under certain conditions, a finite time singularity of the Ricci flow must form within a compact set. We also prove a long time existence result for the \KRF flow on complete non-negatively curved \K manifolds.

Rigidity of entire self-shrinking solutions to curvature flows

Abstract We show (a) that any entire graphic self-shrinking solution to the Lagrangian mean curvature flow in ℂn with the Euclidean metric is flat; (b) that any space-like entire graphic self-shrinking solution to the Lagrangian mean curvature flow in ℂn with the pseudo-Euclidean metric is flat if the Hessian of the potential is bounded below quadratically; and (c) the Hermitian counterpart of (b) for the Kähler Ricci flow.

A C 0 -estimate for the parabolic Monge–Ampère equation on complete non-compact Kähler manifolds

In this article we study the Kahler–Ricci flow, the corresponding parabolic Monge–Ampere equation and complete non-compact Kahler–Ricci flat manifolds. Our main result states that if ( M , g ) is sufficiently close to being Kahler–Ricci flat in a suitable sense, then the Kahler–Ricci flow has a long time smooth solution g ( t ) converging smoothly uniformly on compact sets to a complete Kahler–Ricci flat metric on M . The main step is to obtain a uniform C 0 -estimate for the corresponding parabolic Monge–Ampere equation. Our results on this can be viewed as parabolic versions of the main results of Tian and Yau [ Complete Kahler manifolds with zero Ricci curvature. II , Invent. Math. 106 (1990), 27–60] on the elliptic Monge–Ampere equation.