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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.06175 |
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| _version_ | 1866916830544658432 |
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| author | Emam, Christian El Sagman, Nathaniel |
| author_facet | Emam, Christian El Sagman, Nathaniel |
| contents | We prove that, given a path of Beltrami differentials on $\mathbb C$ that live in and vary holomorphically in the Sobolev space $W^{l,\infty}_{loc}(Ω)$ of an open subset $Ω\subset \mathbb C$, the canonical solutions to the Beltrami equation vary holomorphically in $W^{l+1,p}_{loc}(Ω)$ for admissible $p > 2$. This extends a foundational result of Ahlfors and Bers (the case $l = 0$). As an application, we deduce that Bers metrics on surfaces depend holomorphically on their input data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_06175 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Holomorphic dependence for the Beltrami equation in Sobolev spaces Emam, Christian El Sagman, Nathaniel Complex Variables Analysis of PDEs Differential Geometry 30C62 35J46 53C15 We prove that, given a path of Beltrami differentials on $\mathbb C$ that live in and vary holomorphically in the Sobolev space $W^{l,\infty}_{loc}(Ω)$ of an open subset $Ω\subset \mathbb C$, the canonical solutions to the Beltrami equation vary holomorphically in $W^{l+1,p}_{loc}(Ω)$ for admissible $p > 2$. This extends a foundational result of Ahlfors and Bers (the case $l = 0$). As an application, we deduce that Bers metrics on surfaces depend holomorphically on their input data. |
| title | Holomorphic dependence for the Beltrami equation in Sobolev spaces |
| topic | Complex Variables Analysis of PDEs Differential Geometry 30C62 35J46 53C15 |
| url | https://arxiv.org/abs/2410.06175 |