On the long time behaviour of the Conical K\"ahler- Ricci flows

TL;DR

This paper proves the long-term existence and convergence of conical K"ahler-Ricci flows to K"ahler-Einstein metrics under certain conditions, using a Liouville type theorem for conical singularities.

Contribution

It establishes the global existence and exponential convergence of conical K"ahler-Ricci flows, extending previous results to the conical setting with new regularity and Liouville theorems.

Findings

Flows exist for all time with maximal regularity.

Converge exponentially to K"ahler-Einstein metrics when the twisted first Chern class is non-positive.

Proves a Liouville type theorem for conical K"ahler-Ricci flat metrics.

Abstract

We prove that the conical K\"ahler-Ricci flows introduced in \cite{CYW} exist for all time $t\in [0,+\infty)$. These immortal flows possess maximal regularity in the conical category. As an application, we show if the twisted first Chern class $C_{1,\beta}$ is negative or zero, the corresponding conical K\"ahler-Ricci flows converge to K\"ahler-Einstein metrics with conical singularities exponentially fast. To establish these results, one of our key steps is to prove a Liouville type theorem for K\"ahler-Ricci flat metrics (which are defined over $\mathbb{C}^{n}$) with conical singularities.