报告人：Professor Harold Rosenberg

                巴西国家纯粹和应用数学研究所

                National Institute for Pure and Applied Mathematics (IMPA)

几何分析系列讲座：三维流形中的极小曲面

时间&地点：2016年06月06日上午 9:30-11:30数学物理大楼661

2016年06月07日上午 9:30-11:30数学物理大楼661

2016年06月08日上午 9:30-11:30数学实验楼108

2016年06月09日上午 9:30-11:30数学物理大楼661

2016年06月10日上午 9:30-11:30数学实验楼108

学院联系人:教授

报告人简介：Rosenberg教授,巴西国家纯粹和应用数学研究所。于1963年获得加州大学伯克利分校博士学位。曾任巴黎七大教授，Jussieu数学研究所所长。1987年获巴黎科学院Marie Guido-Triossi奖。巴西国家科学院院士，国际著名微分几何专家。

报告简介：Lecture 1. The starting point is a theorem of Calabi concerning geodesics on convex surfaces. Suppose S is a topological 2-sphere with a metric whose curvature K is greater than 0 and less than or equal to one. Let C be a simple closed geodesic in S. Then the length of C is at least 2π.

If the length of C equals 2π, then S is isometric to the sphere of radius one of R^3, (denoted S(2)).

This inspired the following theorem by Laurent Mazet and me.

Suppose the sectional curvatures of M are less than or equal to one, and the Ricci curvature is non-negative. Let S be a minimal embedded sphere in M. Then the area of S (denoted |S|) is at least 4π. If |S| equals 4π, then M is an isometric quotient of S(3) or a quotient of S(2) x R.

Lecture 2: We discuss area minimizing surfaces S when the scalar curvature R of M has a lower bound R(0). We will prove.

Theorem. Suppose M has scalar curvature at least R(0), and S is an area minimizing surface in its homotopy class. Then R(0) |S| is at most 4π times the Euler characteristic of S. Equality implies M is an isometric quotient of S x R.

This theorem was proved by Cao-Galloway when R(0) = 0, Bray-Brendle-Neves when S is a sphere, and by Micallef-Morin in all cases.

Lecture 3. We will discuss lower area bounds for minimal S in hyperbolic M. When the Heegard genus of M is at least 6, Mazet and I have proved the area of S is at least 2π. The proof uses the min-max theory of Algren-Pitts. We will discuss the ideas.

Lectures 4 and 5. I will discuss the global theory of properly immersed minimal surfaces in complete hyperbolic 3-manifolds of finite volume.

When S has finite topology, Collin, Hauswirth and I proved the ends of S are smoothly asymptotic to totally geodesic cusp ends. The total curvature of S equals 2π times the Euler characteristic of S. I will discuss the ideas of the proof and several other structure theorems of minimal surfaces in these hyperbolic manifolds.

my best regards, Harold

欢迎广大师生参加！