Harmonic Maps and Rigidity

Explore a comprehensive lecture on harmonic maps and rigidity presented by Chikako Mese from Johns Hopkins University at the Fields Institute. Delve into topics such as harmonic maps between Riemannian manifolds, existence theory, Mostow and Margulis Rigidity, and Siu's Holomorphic Rigidity. Examine the differences between uniform and non-uniform lattices, and investigate non-compact domains. Learn about the Bochner method, infinite energy harmonic maps, and the proof of existence for such maps in various dimensions. Gain insights into Teichmüller theory, integral rigidity, and the fundamental group of quasi-projective varieties. This 51-minute talk, part of the Workshop on Geometry of Spaces with Upper and Lower Curvature Bounds, offers a deep dive into the mathematical concepts surrounding harmonic maps and their applications in rigidity theory.

Syllabus

Intro
Harmonic maps between Riemannian manifolds
Harmonic maps into CAT(0) spaces
Existence theory of harmonic maps
Mostow and Margulis Rigidity
Mostow's strong rigidity - geometric version
Harmonic maps approach to Rigidity
Siu's Holomorphic Rigidity
Mok-Siu-Yeung's Geometric Rigidity
Uniform lattice vs. Non-uniform lattice
Non-compact domains
Idea of the proof
The Bochner method
Existence of harmonic map for the compact case
Infinite energy harmonic maps in dime=1
Proof of existence of infinite energy maps
Proof in dimensions 2
Sketch of the existence of harmonic map
Sketch of the proof of pluriharmonicity
Proof in higher dimensions
Teichmüller theory
Integral rigidity
Fundamental group of a quasi-projective variety