Cusp K\"ahler-Ricci flow on compact K\"ahler manifold

TL;DR

This paper establishes the long-term existence, uniqueness, and convergence of the cusp K"ahler-Ricci flow on certain compact K"ahler manifolds, leading to a unique cusp K"ahler-Einstein metric.

Contribution

It introduces a new approach by limiting twisted conical K"ahler-Ricci flows to study cusp flows on manifolds with ample twisted canonical bundle.

Findings

Proves long-time existence and uniqueness of cusp K"ahler-Ricci flow.

Shows convergence of the flow to a cusp K"ahler-Einstein metric.

Abstract

In this paper, by limiting twisted conical K\"ahler-Ricci flows, we prove the long-time existence and uniqueness of cusp K\"ahler-Ricci flow on compact K\"ahler manifold $M$ which carries a smooth hypersurface $D$ such that the twisted canonical bundle $K_M+D$ is ample. Furthermore, we prove that this flow converge to a unique cusp K\"ahler-Einstein metric.