Harmonic maps to Hadamard spaces and a universal higher Teichm\"{u}ller space
J. Maxwell Riestenberg, Peter Smillie

TL;DR
This paper establishes a criterion for harmonic maps from manifolds with Ricci curvature bounds to Hadamard spaces, generalizes a conjecture on quasi-isometric embeddings, and introduces a universal higher Teichmüller space.
Contribution
It introduces the stability criterion for harmonic maps, generalizes the Schoen-Li-Wang conjecture, and defines a universal Hitchin component extending Teichmüller space.
Findings
Harmonic maps are close to Lipschitz maps under stability.
Generalization of the Schoen-Li-Wang conjecture to higher rank spaces.
Definition of a universal higher Teichmüller space for PGL_d(R).
Abstract
We give a sufficient criterion, which we call stability, for a coarse Lipschitz map from a complete manifold with Ricci curvature bounded below to a proper Hadamard space to be within bounded distance of a harmonic map. We prove uniqueness of the harmonic map under additional assumptions on and . Using this criterion, we prove a significant generalization of the Schoen-Li-Wang conjecture on quasi-isometric embeddings between rank 1 symmetric spaces. In particular, under a natural generalization of the quasi-isometric condition, we remove the assumption that the target has rank 1. This allows us to define a universal Hitchin component for each , generalizing universal Teichm\"uller space, and show that it can be described both as a space of quasi-symmetric positive maps from to the flag variety, and as a space of harmonic…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAnalytic and geometric function theory · Geometry and complex manifolds · Geometric and Algebraic Topology
