Instantaneously complete Chern-Ricci flow and K\"ahler-Einstein metrics

TL;DR

This paper establishes existence results for Chern-Ricci flows on complex manifolds with incomplete initial data, explores their behavior near initial time, and provides conditions for the existence of K"ahler-Einstein metrics on noncompact Hermitian manifolds.

Contribution

It generalizes previous Ricci flow existence results to higher dimensions and noncompact Hermitian manifolds, including cases with unbounded curvature.

Findings

Existence of Chern-Ricci flows with incomplete initial data.

Behavior analysis of solutions as time approaches zero.

Conditions for the existence of K"ahler-Einstein metrics on noncompact manifolds.

Abstract

In this work, we obtain some existence results of Chern-Ricci Flows and the corresponding Potential Flows on complex manifolds with possibly incomplete initial data. We discuss the behaviour of the solution as $t\rightarrow 0$. These results can be viewed as generalization of an existence result by Giesen and Topping for surfaces of hyperbolic type of Ricci flow to higher dimensions in certain sense. On the other hand, we also discuss the long time behaviour of the solution and obtain some sufficient conditions for the existence of K\"ahler-Einstein metric on complete noncompact Hermitian manifolds, which generalizes the work of Lott-Zhang and Tosatti-Weinkove to complete noncompact Hermitian manifolds with possibly unbounded curvature.