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Globally subanalytic CMC surfaces in ℝ3 with singularities

Part of: Surfaces and higher-dimensional varieties Classical differential geometry

Published online by Cambridge University Press:  30 March 2020

José Edson Sampaio
José Edson Sampaio*
Affiliation:
Departamento de Matemática, Universidade Federal do Ceará, Rua Campus do Pici, s/n, Bloco 914, Pici, 60440-900, Fortaleza, CE, Brazil BCAM – Basque Center for Applied Mathematics, Mazarredo, 14 E48009Bilbao, Basque Country, Spain ([email protected])
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Abstract

In this paper we present a classification of a class of globally subanalytic CMC surfaces in ℝ3 that generalizes the recent classification made by Barbosa and do Carmo in 2016. We show that a globally subanalytic CMC surface in ℝ3 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 ℝ3 that is a topological manifold does not have isolated singularities. It is also proved that a connected closed globally subanalytic CMC surface in ℝ3 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 also presented some results on regularity of semialgebraic sets and, in particular, it proved a real version of Mumford's Theorem on regularity of normal complex analytic surfaces and a result about C1 regularity of minimal varieties.

Keywords

Classification of surfacesConstant mean curvature surfacesSemialgebraic Sets

MSC classification

Primary: 14J17: Singularities
Secondary: 14P10: Semialgebraic sets and related spaces 53A10: Minimal surfaces, surfaces with prescribed mean curvature
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 1 , February 2021 , pp. 407 - 424
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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