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I am currently reading the book by Professor Jost, "Riemannian geometry and geometric analysis", the last chapter on harmonic maps. It talks mainly about existence and regularity of the theory of harmonic mappings between Riemannian manifolds (with certain constraint on the target manifolds, e.g., non-positive curvature). I wonder whether there is a book with a comprehensive treatment on the theory of harmonic mappings between Riemannian manifolds and with applications in other problems.

Another related question that I would like to know is how many different methods that we have to prove the existence of harmonic mappings from a Riemannian manifolds to a Riemannian manifolds with non-positive curvature. How about the regularity part?

The motivation behind the question is that I would like to study the theory of harmonic mappings between singular metric spaces (e.g. the survey book of Professor Lin Fanghua). I know the earlier works of Korevaar-Schoen and Jost around the 1990s, considering such problems in general metric space setting. Any comments and suggestions are warmly welcome!

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  • $\begingroup$ I think that (reference-request) tag would be a reasonable tag for this question (but for some reason the software does not allow me to make edit suggestion adding the tag). $\endgroup$ – Martin Sleziak Nov 6 '16 at 7:45
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I can imagine that you have different books in mind, but harmonic maps are widely used in the differential geometry of surfaces, e.g. the Gauss map of constant mean curvature surfaces is harmonic (into the positive curved 2-sphere), and the conformal gauss map of willmore surfaces is harmonic into the semi-riemannian space of round 2-spheres in the 3-space, and there are many books about this topic, e.g."Harmonic maps, conservation laws and moving frames" by Helein.

Another non-recent article which might be of interest for you is "Harmonic maps" by Helein and Wood.

For the question concerning different methods for the proof of existence, I would like to mention Hitchin's self-duality equations and generalizations thereof. In this work, Hitchin proved, by using Uhlenbeck compactness and a moment map interpretation of the self-duality equations, the existence of solutions out of algebraic geometric initial data (stable Higgs pairs). The solutions (in the classical case studied by Hitchin) can be interpreted as (equivariant) harmonic maps into hyperbolic 3-space.

This story continues to more complicated situations. Also, there is some related work about harmonic maps to buildings, which might be of interest for you, see for example recent work by Katzarkov, Noll,Pandit and Simpson (https://arxiv.org/pdf/1311.7101v1.pdf).

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Maybe the following book is useful for you:

The analysis of harmonic maps and their heat flows (Fanghua Lin & Changyou Wang)

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