日程表

学术报告题目与摘要

Regularity of minimal submanifolds

丁琪（上海数学中心）

Abstract：We will discuss the regularity of minimal graphs of high codimensions in Euclidean space including special Lagrangian graphs under some conditions as well as several typical related examples.

Riemannian plane without conjugate points

葛剑（北京大学）

Abstract: In this talk, we discuss geometric properties and open problems of Riemannian plan without conjugate points.

Gromov-Hausdorff convergence of Kahler manifolds

刘钢（华东师范大学）

Abstract: we discuss some recent development of Gromov-Hausdorff convergence of Kahler manifolds with geometric applications.

Regularity of transition layers in Allen-Cahn equation

王克磊（武汉大学）

Abstract: In this talk I will survey the regularity theory of transition layers of solutions to singularly perturbed Allen-Cahn equation, from zeroth order regularity to second order one. Some applications of this regularity theory will also be discussed, including De Giorgi conjecture, classification of finite Morse index solutions and construction of minimal hypersurfaces by Allen-Cahn approximation.

On the fill-ins of nonnegative scalar curvature metrics

魏国栋（中山大学（珠海））

Abstract: In this talk,we first show the extensibility of an arbitrary boundary metric to a positive scalar curvature metric inside for a compact manifold with boundary, which solves an open problem due to Gromov. Then we introduce a fill-in invariant and discuss its relationship with the positive mass theorems for asymptotically flat (AF) and asymptotically hyperbolic (AH) manifolds. In particular, we prove that the positive mass theorem for AH manifolds implies that for AF manifolds. In the end, we give some estimates for the fill-in invariant, which provide some partially affirmative answers to two conjectures by Gromov. This is a joint work with Prof. Yuguang Shi and Dr. Wenlong Wang.

The Weyl law revisited

徐国义（清华大学）

Abstract: H. Weyl proved the Weyl law about the limit behavior of eigenvalues for 2-dimensional domains. He claimed that his method also works in higher dimensional case. For Dirichlet eigenvalues, his claim can be verified directly. The case of Neumann eigenvalues is not trivial due to the lack of monotonicity comparison results in this case. We will sketch our proof of the Weyl law for all dimensions, following the Weyl’s original method of ‘cutting-pasting'. And we will point out the difficulty in higher dimensions, and the way to overcome them.The key technical idea is linear approximation of any domain and its related comparison results for eigenvalues. This a joint work with Weiwei Wang and Zuoqin Wang.

Recognizing shape of a hypersurface via 1st eigenvalue, mean curvature

胥世成（首都师范大学）

Abstract: In this talk, we prove the following almost rigidity result: Any closed Riemannian manifold M of dimension at least 2 is diffeomorphic and almost isometric to a round sphere in the Euclidean space, if M is isometrically immersed in a Hadamard manifold N with sectional curvature -1< ="K_N<=" 0, such that

1) the 1st eigenvalue and maximum norm of the mean curvature of M are close to the round sphere;

2) the volume of M, the extrinsic diameter of M in N, and the L^{2n}-norm of 2nd fundamental form admit a uniform bounded.

On the non-existence of solutions to the Dirichlet problem for minimal surface system

张永胜（同济大学）

Abstract: In this talk, we will show how to generalize the non-existence result by Lawson-Osserman.