Continuous analytic capacity and holomorphic motions
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866916592686727168 |
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| author | Younsi, Malik |
| author_facet | Younsi, Malik |
| contents | We construct a compact set whose continuous analytic capacity does not vary continuously under a certain holomorphic motion, thereby answering a question of Paul Gauthier. Our example is inspired by holomorphic dynamics and relies on the works of Bishop--Carleson--Garnett--Jones and Browder--Wermer relating tangent points of Jordan curves, harmonic measure and Dirichlet algebras. Our approach also provides a new proof of a result of Ransford, Younsi and Ai on the variation of analytic capacity under holomorphic motions. In addition, we show that extremal functions for continuous analytic capacity may not exist. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_05198 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Continuous analytic capacity and holomorphic motions Younsi, Malik Complex Variables We construct a compact set whose continuous analytic capacity does not vary continuously under a certain holomorphic motion, thereby answering a question of Paul Gauthier. Our example is inspired by holomorphic dynamics and relies on the works of Bishop--Carleson--Garnett--Jones and Browder--Wermer relating tangent points of Jordan curves, harmonic measure and Dirichlet algebras. Our approach also provides a new proof of a result of Ransford, Younsi and Ai on the variation of analytic capacity under holomorphic motions. In addition, we show that extremal functions for continuous analytic capacity may not exist. |
| title | Continuous analytic capacity and holomorphic motions |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2207.05198 |