报告人：韩青（University of Notre Dame）

时间：2024年04月18日 16：00-

地点：理科楼LA106

摘要：A surface in the 3-dimensional Euclidean space can be viewed as the image of a map from a planar domain to the 3-dimensional Euclidean space, at least locally. The standard metric in Euclidean space induces a metric on the surface, which allows us to compute the lengths of curves on the surface and to compute the distance of any two points on the surface. The induced metric on the surface can be transformed into an abstract metric by the abovementioned map. Now, we consider the converse question. Given an abstract metric on a planar domain or on a closed surface, can we find a surface in the 3-dimensional Euclidean space whose induced metric is the given abstract metric? This is the isometric embedding problem. It started with a conjecture by Schlaefli in 1873 that this can always be achieved near any given point. This conjecture is widely open and there are only a few results under various conditions. The question can be reformulated in terms of partial differential equations. Despite the technical description, the underlying equation has a simple form but is hard to solve. In this talk, I will give a historical account of isometric embedding and present several open problems. If time permits, I will also discuss applications of isometric embedding in general relativity. The talk is aimed at a general audience.

简介：韩青，美国圣母大学数学系终身教授。美国纽约大学库朗数学研究所博士，美国芝加哥大学博士后。获美国Sloan Research Fellowship. 韩青教授长期致力于非线性偏微分方程和几何分析的研究，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。文章发表于CPAM, Duke Math. J., GAFA, JDG, Adv. Math., Crelle’s Journal, Math. Ann.等国际知名期刊中。

邀请人：穆春来、黄小军、张林、陈天聪

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