Global Kahler-Ricci Flow on Complete Non-Compact Manifolds

TL;DR

This paper investigates the long-term behavior of the K"ahler-Ricci flow on complete non-compact K"ahler manifolds with specific potential function conditions, establishing conditions for finite-time blow-up or convergence to Ricci-flat metrics.

Contribution

It provides new results on the global existence and convergence of the K"ahler-Ricci flow under potential function and Sobolev inequality assumptions on non-compact manifolds.

Findings

Flow either blows up in finite time or converges to Ricci-flat metric

Conditions for global existence and convergence established

Results extend understanding of K"ahler-Ricci flow on non-compact manifolds

Abstract

In this paper, we study the global K\"ahler-Ricci flow on a complete non-compact K\"ahler manifold. We prove the following result. Assume that $(M,g_0)$ is a complete non-compact K\"ahler manifold such that there is a potential function $f$ of the Ricci tensor, i.e., $$ R_{i\bar{j}}(g_0)=f_{i\bar{j}}. $$ Assume that the quantity $|f|_{C^0}+|\nabla_{g_0}f|_{C^0}$ is finite and the L2 Sobolev inequality holds true on $(M,g_0)$. Then the Kahler-Ricci flow with the initial metric $g_0$ either blows up at finite time or infinite time to Ricci flat metric or exists globally with Ricci-flat limit at infinite time. A related is also discussed.