Mixed-type Partial Differential Equations and the Isometric Immersion Problem

：李思然（上海交通大学）
：2026-04-16 10:00
：海韵园实验楼S307

报告人：李思然（上海交通大学）

时  间：2026年4月16日10:00

地  点：海韵园实验楼S307

内容摘要:

This talk is about a classical problem in differential geometry and global analysis: the isometric immersions of Riemannian manifolds into Euclidean spaces. We focus on the PDE approach to isometric immersions, i.e., the analysis of Gauss--Codazzi--Ricci equations, especially in the regime of low Sobolev regularity. Such equations are not purely elliptic, parabolic, or hyperbolic in general, hence calling for analytical tools for PDEs of mixed types. We discuss various recent contributions -- in line with the pioneering works by G.-Q. Chen, M. Slemrod, and D. Wang (2010) -- on the weak continuity of Gauss--Codazzi--Ricci equations, the interior regularity of flat isometric immersions, and the fundamental theorem of submanifold theory with low regularity. Two mixed-type PDE techniques are emphasised throughout these developments: the method of compensated compactness and the theory of Coulomb--Uhlenbeck gauges. (Joint with Reza Pakzad (Toulon), Armin Schikorra (Pittsburgh), and Xiangxiang Su (Shanghai Jiao Tong).)

个人简介：

李思然，上海交通大学副教授。2017年英国牛津大学博士毕业。入选中国科协第九届青年人才托举工程项目（2024--2027）；主持国家自然科学基金青年项目（2023--2025）；主持国家重点研发计划青年科学家项目（2024--2029）；获加拿大Applied Mathematics, Modeling and Computational Science协会Kolmogorov–Wiener Prize for Young Researchers（2019）。研究领域涉及微分几何，流体力学，随机分析。主要工作包括：子流形的低正则嵌入问题；流体力学的几何化理论；具有热传导、动态燃烧等效应的Navier-Stokes方程的整体适定性问题等等。在Arch. Ration. Mech. Anal., J. Math. Pures Appl., J. Funct. Anal., Ann. IHP Probab. Statist.等国际期刊发表多篇学术论文。

 

联系人：王金花