Notes on Harmonic Maps

TL;DR

This paper provides an overview of harmonic maps between Riemannian manifolds, discussing existence, regularity, pluriharmonic maps, and key theorems relevant to non-Abelian Hodge theory.

Contribution

It compiles and explains fundamental concepts and recent results in harmonic maps, serving as educational notes for a summer school on non-Abelian Hodge theory.

Findings

Discussion of existence and regularity results

Introduction to pluriharmonic maps and Siu-Sampson formula

Explanation of Donaldson-Corlette theorem

Abstract

This is set of notes prepared for the Summer School on non-Abelian Hodge theory in Abbaye de Saint-Jacut de la Mer June, 6-19, 2022. We cover the following topics: Lecture 1. Harmonic Maps Between Riemannian Manifolds Lecture 2. Existence and Regularity Lecture 3. Pluriharmonic Maps and the Siu-Sampson Formula Lecture 4. Donaldson Corlette Theorem