Ancient solution to Kahler-Ricci flow

TL;DR

This paper investigates ancient solutions to the K"ahler-Ricci flow, establishing their asymptotic volume behavior and classifying certain solitons, thereby advancing understanding of K"ahler geometry and Ricci flow dynamics.

Contribution

It proves that non-flat ancient solutions with nonnegative bisectional curvature have zero asymptotic volume ratio and classifies gradient shrinking solitons with positive bisectional curvature as compact, generalizing earlier results.

Findings

Ancient solutions have asymptotic volume ratio zero.

Gradient shrinking solitons with positive bisectional curvature are compact.

Results lead to a compactness theorem for ancient solutions.

Abstract

In this paper, we prove that any non-flat ancient solution to K\"ahler-Ricci flow with bounded nonnegative bisectional curvature has asymptotic volume ratio zero. We also prove that any gradient shrinking solitons with positive bisectional curvature must be compact. Both results generalize the corresponding earlier results of Perelman in \cite{P1} and \cite{P2}. The results can be applied to study the geometry and function theory of complete K\"ahler manifolds with nonnegative bisectional curvature via K\"ahler-Ricci flow. It also implies a compactness theorem on ancient solutions to K\"ahler-Ricci flow.