In this talk we will show that there is a natural Willmore deformation between minimal surfaces in H^n and S^n. The deformation of the Veronese two sphere and its generalization provide examples of complete minimal surfaces in H^4 with varying Willmore energy.

本报告系统介绍千禧问题之一: 不可压缩Navier-Stokes方程相关性质. 首先介绍该方程的来源及背景, 弱解存在性、正则性等的研究历史和现状; 其次, 介绍解的长时间渐近行为性质; 最后列出一些目前尚未解决的重要问题.

The hyperbolic metric plays an important role in complex analysis and geometric function theory. In this talk, we will establish several interesting properties for the hyperbolic metric of a simply connected domain of the complex plane, which can be applied to discuss the Roper-Suffridge extension operator and its modification. This hyperbolic metric can be viewed as providing an alternative approach to study biholomorphic mappings in several complex variables. This is joint work with Prof. Taishun Liu.

In collaboration with G. Edwards we produce (local) Calabi-Yau metrics, in two complex dimensions, with cone singularities along intersecting complex lines, for cone angles that strictly violate the Troyanov condition. We identify the tangent cone at the origin as a product of two 2-cones. In the tangent cone limit, the line with the smallest cone angle remains apart while the other lines collide into a single cone factor.

We apply Menke’s JSJ decomposition for symplectic fillings to several families of contact 3-manifolds. Among other results, we complete the classification up to diffeomorphism of strong symplectic fillings of lens spaces. We reduce the classification of the strong symplectic fillings of large families of Seifert fibered spaces to that of lens spaces. We show that fillings of contact manifolds obtained by Legendrian surgery on certain Legendrian knots are the result of attaching a symplectic 2-handle to a filling of a lens space. This is joint work with Austin Christian.

In mathematics it is usually a routine way to decompose the mathematical object under consideration into some simpler ones which are easier to study. By analyzing how the pieces assemble the information is obtained about the original object. In homotopy theory, we usually need to decompose a topological space as the product or wedge of some smaller spaces. This method helps us to easily investigate the homotopy groups or homology groups of the original space via the ones of the smaller spaces. In this talk, I will simply introduce the method of homotopy decomposition and then give some our recent results on the homotopy decompositon of some looped Co-H spaces.

We recently improved our result on Kahler-Ricci type flow whose stationary points are cscK metrics, introduced by Yuan Yuan, Zhang YuGuang and the speaker.

In this talk, we will present some recent progress on the geometry of strictly nef vector bundles, with focus on the characterization of projective spaces.