Globally subanalytic CMC surfaces in $\mathbb{R}^3$ with singularities
In this paper we present a classification of a class of globally subanalytic CMC surfaces in $\mathbb{R}^3$ that generalizes the recent classification made by Barbosa and do Carmo in 2016. We show that a globally subanalytic CMC surface in $\mathbb{R}^3$ with isolated singularities and a suitable co...
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| Tipo de documento: | artigo |
| Estado: | Versión enviada para evaluación y publicación |
| Data de publicação: | 2020 |
| País: | España |
| Recursos: | Basque Center for Applied Mathematics (BCAM) |
| Repositório: | BIRD. BCAM's Institutional Repository Data |
| OAI Identifier: | oai:bird.bcamath.org:20.500.11824/1093 |
| Acesso em linha: | http://hdl.handle.net/20.500.11824/1093 |
| Access Level: | Acceso aberto |
| Palavra-chave: | Classification of surfaces Constant mean curvature surfaces Semialgebraic Sets |
| Resumo: | In this paper we present a classification of a class of globally subanalytic CMC surfaces in $\mathbb{R}^3$ that generalizes the recent classification made by Barbosa and do Carmo in 2016. We show that a globally subanalytic CMC surface in $\mathbb{R}^3$ with isolated singularities and a suitable condition of local connectedness is a plane or a finite union of round spheres and right circular cylinders touching at the singularities. As a consequence, we obtain that a globally subanalytic CMC surface in $\mathbb{R}^3$ that is a topological manifold does not have isolated singularities. It is also proved that a connected closed globally subanalytic CMC surface in $\mathbb{R}^3$ with isolated singularities which is locally Lipschitz normally embedded needs to be a plane or a round sphere or a right circular cylinder. A result in the case of non-isolated singularities is also presented. It is also presented some results on regularity of semialgebraic sets and, in particular, it is proved a real version of Mumford's Theorem on regularity of normal complex analytic surfaces and a result about $C^1$ regularity of minimal varieties. |
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