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 C0 and C2-estimates for potential functions, we finally obtain a C2,α-estimate based on Tian’s method in his PKU thesis.