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.

中文翻译：

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ℝ3 中具有奇点的全局子解析 CMC 曲面

在本文中，我们在ℝ中提出了一类全局子解析 CMC 曲面的分类3概括了 Barbosa 和 do Carmo 在 2016 年所做的最新分类。我们展示了 ℝ 中的全局子分析 CMC 曲面3具有孤立奇点和局部连通性的合适条件是在奇点处接触的平面或圆球和直圆柱的有限联合。因此，我们获得了 ℝ 中的全局子解析 CMC 曲面3那是一个拓扑流形没有孤立的奇点。还证明了 ℝ 中的连通封闭全局子解析 CMC 曲面3具有局部 Lipschitz 通常嵌入的孤立奇点需要是平面或圆球或直圆柱。还给出了非孤立奇点情况下的结果。它还给出了关于半代数集正则性的一些结果，特别是，它证明了关于正则复解析曲面正则性的 Mumford 定理的一个真实版本和一个关于C1最小品种的规律性。