Cartan uniqueness theorem on nonopen sets
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866915159305355264 |
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| author | Lebl, Jiri Noell, Alan Ravisankar, Sivaguru |
| author_facet | Lebl, Jiri Noell, Alan Ravisankar, Sivaguru |
| contents | Cartan's uniqueness theorem does not hold in general for CR mappings, but it does hold under certain conditions guaranteeing extendibility of CR functions to a fixed neighborhood. These conditions can be defined naturally for a wide class of sets such as local real-analytic subvarieties or subanalytic sets, not just submanifolds. Suppose that $V$ is a locally connected and locally closed subset of ${\mathbb{C}}^n$ such that the hull constructed by contracting analytic discs close to arbitrarily small neighborhoods of a point always contains the point in the interior. Then restrictions of holomorphic functions uniquely extend to a fixed neighborhood of the point. Using this extension, we obtain a version of Cartan's uniqueness theorem for such sets. When $V$ is a real-analytic subvariety, we can generalize the concept of infinitesimal CR automorphism and also prove an analogue of the theorem. As an application of these two results we show that, for circular subvarieties satisfying the condition, the only automorphisms, CR or infinitesimal, are linear. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_07585 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Cartan uniqueness theorem on nonopen sets Lebl, Jiri Noell, Alan Ravisankar, Sivaguru Complex Variables 32H02 (Primary), 32V40 32B20 (Secondary) Cartan's uniqueness theorem does not hold in general for CR mappings, but it does hold under certain conditions guaranteeing extendibility of CR functions to a fixed neighborhood. These conditions can be defined naturally for a wide class of sets such as local real-analytic subvarieties or subanalytic sets, not just submanifolds. Suppose that $V$ is a locally connected and locally closed subset of ${\mathbb{C}}^n$ such that the hull constructed by contracting analytic discs close to arbitrarily small neighborhoods of a point always contains the point in the interior. Then restrictions of holomorphic functions uniquely extend to a fixed neighborhood of the point. Using this extension, we obtain a version of Cartan's uniqueness theorem for such sets. When $V$ is a real-analytic subvariety, we can generalize the concept of infinitesimal CR automorphism and also prove an analogue of the theorem. As an application of these two results we show that, for circular subvarieties satisfying the condition, the only automorphisms, CR or infinitesimal, are linear. |
| title | Cartan uniqueness theorem on nonopen sets |
| topic | Complex Variables 32H02 (Primary), 32V40 32B20 (Secondary) |
| url | https://arxiv.org/abs/2112.07585 |