中国科学院数学与系统科学研究院
数学研究所
数学科学全国重点实验室
非线性泛函分析研讨班
Speaker: Prof.Yannick Sire (Johns Hopkins University)
Inviter: 孙黎明
Language: English
Title：Heat flow of harmonic maps and singular spaces I
Time&Venue：2026年8月11日（星期二）10:00-11:00 & 南楼N202
Abstract: The heat flow of harmonic maps from a smooth, compact Riemannian manifold without boundary, (M,g) into another smooth, compact Riemannian manifold without boundary (N,h) was first studied in the seminal work of Eells and Sampson when the target manifold (N,h) has non-positive curvature. Gromov-Schoen studied harmonic maps from M into a singular Cat(0) space which was used to understand the p-adic superrigidity of lattices in groups of rank one. A key analytical property of such harmonic maps is the Lipschitz continuity, from which one derives Bochner type estimates and vanishing theorems. As for Eells-Sampson theorem, it is rather natural to study the associated gradient flow, and it has been a long open problem to construct suitable weak solutions in the singular setting. In this talk, I shall describe an elliptic approach (which goes back to De Giorgi and also T. Ilmanen in the 1990s) to this problem both in the smooth and the singular settings, i.e. when the target is CAT(0) space. During the first talk, I will explain the strategy to build such a regularization and derive the gradient flow estimates. I will apply this approach to the standard case of smooth targets and provide yet another proof of Eells-Sampson theorem.
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Title：Heat flow of harmonic maps and singular spaces II
Time&Venue：2026年8月11日（星期二）14:00-15:00 & 南楼N204
Abstract:During the second talk, I will concentrate on singular targets and explain how to modify the strategy to deal with CAT(0) spaces. I will explain in particular how to get Lipschitz bounds in the space variables (hence a suitable solution of the flow) via Almgren-Poon monotonicity. I will also draw several open problems and possible directions of research. This is a joint work with FH Lin, A. Segatti and C. Wang.
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