Schedule
Schedule
09:30 - 10:00
Opening Remarks
Speaker: Shing-Tung Yau
10:00 - 10:45
Minimal surfaces defined by extremal eigenvalue problems
Speaker: Richard Schoen
Abstract: Minimal surfaces in spheres are characterized by the condition that their embedding functions are eigenfunctions on the surface with its induced metric. The metric on the surface turns out to be an extremal for the eigenvalue among metrics on the surface with the same area. In recent decades, this extremal property has been used to construct new minimal surfaces by eigenvalue maximization. There is an analogous theory for minimal surfaces in the euclidean ball with a free boundary condition. In this talk we will describe new work that generalizes this idea to products of balls. We will describe the general theory and apply it in a specific case to explain and generalize the Schwarz p-surface, which is a free boundary minimal surface in the three dimensional cube with one boundary component on each face of the cube. We will show how the method can be used to construct such surfaces in rectangular prisms with arbitrary side lengths.
10:45 - 11:00
Tea Time
11:00 - 11:25
General relativistic constraint equations and their properties
Speaker: Piotr Chrusciel
Abstract: I will review some old and new results on the topic.
11:30 - 11:55
The existence of minimal Lagrangians
Speaker: Yng-Ing Lee
Abstract: A minimal Lagrangian is minimal with respect to Riemannian structure and Lagrangian with respect to symplectic structure. It admits better properties than a general minimal submanifold and can be a good representative for middle dimensional submanifolds. However, its existence is still wildly open. I will first survey possible methods in studying the problem and then report my ongoing projects. It aims to generalize a result of mine in 1998 to noncompact settings and to the asymptotic Lagrangian Plateau problem in complex hyperbolic surfaces. The theorem in the compact setting is: if there exists a compact branched minimal Lagrangian surface with respect to one Kahler-Einstein metric with , there also exists a compact branched minimal Lagrangian in the same homotopic class with respect to any Kahler-Einstein metrics in the same connected component of the original Kahler-Einstein metric.
12:00 - 14:00
Lunch
14:00 - 14:25
Topological classification of 5D stationary biaxial Einstein spacetime, and its geometric applications
Speaker: Sumio Yamada
Abstract: In this talk, I introduce the topological classification of 5D stationary Einstein spacetime with biaxial symmetry, utilizing the so-called plumbing construction of 4D toric varieties. As its applications, constructions of nontrivial 5D soliton solutions to the vacuum Einstein equation, and also new set of 4D gravitational instantons. The classification theorem is a collaborative work with Marcus Khuri, Yukio Matsumoto and Gilbert Weinstein.
14:30 - 14:55
The Positive Mass Theorem and Beyond
Speaker: Lan-Hsuan Huang
Abstract: The celebrated Positive Mass Theorem, pioneered by Schoen and Yau, has been a cornerstone in the development of geometric analysis and mathematical general relativity. In recent years, the field has seen a burst of new activity and exciting progress. In this talk, I will discuss some of these recent developments and their broader significance.
15:00 - 16:00
Tea Time
16:00 - 16:25
The geometry and analysis of free boundary minimal surfaces
Speaker: Martin Li
Abstract: In this talk, I will survey on some of the recent results and open questions on the geometric and analytic properties of free boundary minimal surfaces, which has become a very active area of research in the past decade due to the groundbreaking discovery of Fraser and Schoen.
16:30 - 16:55
Minimal spheres and tori in 3-spheres with bumpy metrics
Speaker: Zhichao Wang
Abstract: In this talk, we show the strong Morse inequalities for the area functional in the space of embedded tori and spheres in the three sphere. As a consequence, we prove that in the 3-sphere with bumpy metrics, there are least 4 embedded minimal spheres. If in addition the metric has positive Ricci curvature, there are 9 distinct embedded minimal tori. This is based on the joint work with X. Zhou and X. Li.
17:30 -