Smoothing conic Kahler metrics and the conical Kahler-Ricci flow

Abstract

In this thesis, we first give two different smoothing methods for conic K¨ahler metrics.

One is constructing an approximating sequence of smooth K¨ahler metrics with

the same lower Ricci curvature bound with the original conic K¨ahler metric with

some lower Ricci curvature bound based on Tian’s approximation method for the

conic K¨ahler-Einstein metric. Another one is constructing a smooth approximating

sequence of K¨ahler metrics with uniformly bisectional curvature upper bound, based

on C. Li and Y. Rubinstein’s bisectional curvature upper bound estimate for a standard

conic metric. Then we will use approximation method to construct solutions to

the conical K¨ahler-Ricci flow which preserves the conic structure. After $C^{0}$ and $C^{2}$-estimates for potential functions, we finally obtain a $C^{2,\alpha}$-estimate based on Tian’s

method in his PKU thesis.