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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2310.00778 |
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| _version_ | 1866914692480368640 |
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| author | Sagman, Nathaniel |
| author_facet | Sagman, Nathaniel |
| contents | We prove that for any two Riemannian metrics $σ_1, σ_2$ on the unit disk, a homeomorphism $\partial\mathbb{D}\to\partial\mathbb{D}$ extends to at most one quasiconformal minimal diffeomorphism $(\mathbb{D},σ_1)\to (\mathbb{D},σ_2)$ with $L^1$ Hopf differential. For minimal Lagrangian diffeomorphisms between hyperbolic disks, the result is known, but this is the first proof that does not use anti-de Sitter geometry. We show that the result fails without the $L^1$ assumption in variable curvature. The key input for our proof is the uniqueness of solutions for a certain Plateau problem in a product of trees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_00778 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Minimal diffeomorphisms with $L^1$ Hopf differentials Sagman, Nathaniel Differential Geometry We prove that for any two Riemannian metrics $σ_1, σ_2$ on the unit disk, a homeomorphism $\partial\mathbb{D}\to\partial\mathbb{D}$ extends to at most one quasiconformal minimal diffeomorphism $(\mathbb{D},σ_1)\to (\mathbb{D},σ_2)$ with $L^1$ Hopf differential. For minimal Lagrangian diffeomorphisms between hyperbolic disks, the result is known, but this is the first proof that does not use anti-de Sitter geometry. We show that the result fails without the $L^1$ assumption in variable curvature. The key input for our proof is the uniqueness of solutions for a certain Plateau problem in a product of trees. |
| title | Minimal diffeomorphisms with $L^1$ Hopf differentials |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2310.00778 |