Harmonic Maps into Euclidean Buildings and Non-Archimedean Superrigidity
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915971212509184 |
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| author | Breiner, Christine Dees, Ben K. Mese, Chikako |
| author_facet | Breiner, Christine Dees, Ben K. Mese, Chikako |
| contents | We prove that harmonic maps into Euclidean buildings, which are not necessarily locally finite, have singular sets of Hausdorff codimension 2, extending the locally finite regularity result of Gromov and Schoen. As an application, we prove superrigidity for algebraic groups over fields with non-Archimedean valuation, thereby generalizing the rank 1 $p$-adic superrigidity results of Gromov and Schoen and casting the Bader-Furman generalization of Margulis' higher rank superrigidity result in a geometric setting. We also prove an existence theorem for a pluriharmonic map from a Kähler manifold to a Euclidean building. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_02783 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Harmonic Maps into Euclidean Buildings and Non-Archimedean Superrigidity Breiner, Christine Dees, Ben K. Mese, Chikako Differential Geometry 53C43, 58E20 (Primary) 22E40, 22E50 (Secondary) We prove that harmonic maps into Euclidean buildings, which are not necessarily locally finite, have singular sets of Hausdorff codimension 2, extending the locally finite regularity result of Gromov and Schoen. As an application, we prove superrigidity for algebraic groups over fields with non-Archimedean valuation, thereby generalizing the rank 1 $p$-adic superrigidity results of Gromov and Schoen and casting the Bader-Furman generalization of Margulis' higher rank superrigidity result in a geometric setting. We also prove an existence theorem for a pluriharmonic map from a Kähler manifold to a Euclidean building. |
| title | Harmonic Maps into Euclidean Buildings and Non-Archimedean Superrigidity |
| topic | Differential Geometry 53C43, 58E20 (Primary) 22E40, 22E50 (Secondary) |
| url | https://arxiv.org/abs/2408.02783 |