几何分析学术讲座 （二）

题目 : Generalized Kaehler-Einstein metrics and the Kaehler-Ricci flow

报告人： 张雅山（北京大学）

时间：4月12日（周五） 9:30-10：30

地点：工商楼105

摘要: Recent years have seen important progresses on geometric analysis aspect of semi-ample canonical line bundles. In this talk, we shall first recall Song-Tian's (possibly singular) generalized Kaehler-Einstein metrics on semi-ample canonical line bundles and determine their metric asymptotics near singular points in Kodaira dimension one case. Then we shall move to long-time Riemann curvature behaviors of the Kaehler-Ricci flow on compact Kaehler manifolds with semiample canonical line bundle and explain that these results provide an analytic viewpoint to classify the complex structures on the underlying manifolds. Part of this talk is based on joint works with Frederick Fong.

题目: Geometric structures and new collapsing models of Einstein spaces

报告人：张若冰(Stony Brook University)

时间：4月12日（周五） 10:30-11：30

地点：工商楼105

摘要: An Einstein manifold, by definition, has a Riemannian metric with constant Ricci curvature. Roughly, with some fixed gauge, the metric solves certain system of highly degenerate nonlinear elliptic PDEs. In both metric Riemannian geometry and geometric analysis, it is always a central topic to study the degeneration behaviors of a family of Einstein metrics and geometric evolutions of the underlying spaces. This talk centers on the geometric analysis of a family of collapsing Einstein manifolds with sufficiently wild analytic properties, for instance, the uniform Sobolev inequality never holds in any collapsing sequence. We will explain some entirely new tools from both metric-geometric and algebro-geometric sides in analyzing Einstein equations. Our specific concerns of the talk include the following:

(1) the sharp topological condition for the "existence" of the epsilon-regularity of collapsed Einstein metrics,

(2) geometric structures of collapsing Einstein limits,

(3) new constructions of collapsed Einstein spaces: in both low and high dimensional cases.

联系人：王枫（wfmath@zju.edu.cn）