odern Mathematics Lecture Series | Asymptotic analysis for harmonic maps with prescribed singularities

Abstract

Motivated by studies of axially symmetric stationary solutions of the Einstein vacuum equations in general relativity, we study singular harmonic maps from the 3-dimensional Euclidean space to the hyperbolic plane, with prescribed singularities. We prove that every such harmonic map has a unique tangent map at the black hole horizon and the harmonic map depends on the location of the black hole smoothly. The harmonic map equation restricted to the unit sphere has a singularity at the north and south poles. The collection of parameters representing the conical singularities in the tangent map determines a flow along which the reduced energy of the harmonic map is decreasing. This leads to an explicit and optimal lower bound for the ADM mass in terms of the total angular momentum, in asymptotically flat, axially symmetric, and maximal initial data sets for the Einstein equations with multiple black holes. The talk is based on joint work with Marcus Khuri, Gilbert Weinstein, and Jingang Xiong.

About the speaker

韩青教授本科毕业于北京大学数学系，后保送至美国Courant研究所，师从林芳华教授。1993年博士毕业后进入芝加哥大学从事博士后研究工作。现为美国Notre Dame大学终身教授。在美国获得Sloan Fellowship，长期致力于偏微分方程和几何分析的研究工作，在等距嵌入、Monge-Ampere方程、调和函数的零点集和奇异集、退化方程等方面做出了一系列原创性的重要研究成果。

Personal Homepage：

https://math.nd.edu/people/faculty/qing-han/