The conical K\"ahler-Ricci flow on Fano manifolds

TL;DR

This paper investigates the long-term evolution of the conical K"ahler-Ricci flow on Fano manifolds, establishing existence, uniform estimates, and convergence to conical K"ahler-Einstein metrics.

Contribution

It constructs long-time solutions via twisted flows, derives uniform Perelman's estimates, and proves convergence to conical K"ahler-Einstein metrics under existence conditions.

Findings

Established long-time existence of the flow

Derived uniform Perelman's estimates

Proved convergence to conical K"ahler-Einstein metrics

Abstract

In this paper, we study the long-term behavior of the conical K\"ahler-Ricci flow on Fano manifold $M$. First, based on our work of locally uniform regularity for the twisted K\"ahler-Ricci flows, we obtain a long-time solution to the conical K\"ahler-Ricci flow by limiting a sequence of these twisted flows. Second, we study the uniform Perelman's estimates of the twisted K\"ahler-Ricci flows. After that, we prove that the conical K\"ahler-Ricci flow must converge to a conical K\"ahler-Einstein metric if there exists one.