Harmonic maps to Hadamard spaces and a universal higher Teichmüller space
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917096834727936 |
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| author | Riestenberg, J. Maxwell Smillie, Peter |
| author_facet | Riestenberg, J. Maxwell Smillie, Peter |
| contents | We give a sufficient criterion, which we call stability, for a coarse Lipschitz map $f$ from a complete manifold $X$ with Ricci curvature bounded below to a proper Hadamard space $Y$ to be within bounded distance of a harmonic map. We prove uniqueness of the harmonic map under additional assumptions on $X$ and $Y$.
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 $\mathrm{PGL}_d(\mathbb{R})$, generalizing universal Teichmüller space, and show that it can be described both as a space of quasi-symmetric positive maps from $\mathbb{RP}^1$ to the flag variety, and as a space of harmonic maps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_11469 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Harmonic maps to Hadamard spaces and a universal higher Teichmüller space Riestenberg, J. Maxwell Smillie, Peter Differential Geometry Geometric Topology Metric Geometry 53C43, 53C35, 58E20 We give a sufficient criterion, which we call stability, for a coarse Lipschitz map $f$ from a complete manifold $X$ with Ricci curvature bounded below to a proper Hadamard space $Y$ to be within bounded distance of a harmonic map. We prove uniqueness of the harmonic map under additional assumptions on $X$ and $Y$. 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 $\mathrm{PGL}_d(\mathbb{R})$, generalizing universal Teichmüller space, and show that it can be described both as a space of quasi-symmetric positive maps from $\mathbb{RP}^1$ to the flag variety, and as a space of harmonic maps. |
| title | Harmonic maps to Hadamard spaces and a universal higher Teichmüller space |
| topic | Differential Geometry Geometric Topology Metric Geometry 53C43, 53C35, 58E20 |
| url | https://arxiv.org/abs/2511.11469 |