Bessel Functions, Heat Kernel and the Conical K\"ahler-Ricci Flow

TL;DR

This paper establishes heat kernel estimates for conical metrics, enabling the proof of short-time existence of the conical K"ahler-Ricci flow, using Bessel functions and Schauder estimates.

Contribution

It introduces new heat kernel estimates for conical metrics and proves short-time existence of the conical K"ahler-Ricci flow, extending Donaldson's openness theorem.

Findings

Established heat kernel estimates for conical metrics

Proved short-time existence of the conical K"ahler-Ricci flow

Applied Bessel functions and Schauder estimates in the analysis

Abstract

Following Donaldson's oppenness theorem on deforming a conical K\"ahler-Einstein metric, we prove a parabolic Schauder-type estimate with respect to conical metrics. As a corollary, we show that the conical K\"ahler-Ricci Flow exists for short time. The key is to establish the relevant heat kernel estimates, where we use the Weber's formula on Bessel function of the second kind and Carslaw's heat kernel representation in \cite{Car}.