Semialgebraic CMC surfaces in $\mathbb{R}^3$ with singularities

Abstract

In this paper we present a classification of a class of semialgebraic CMC surfaces in $\mathbb{R}^3$ that generalizes the recent classification made by Barbosa and do Carmo in 2016 (complete reference is in the paper), we show that a semialgebraic CMC surface in $\mathbb{R}^3$ with isolated singularities and suitable conditions on the singularities and of local connectedness is a plane or a finite union of round spheres and cylinders touching at the singularities. As a consequence, we obtain that a semialgebraic good CMC surface in $\mathbb{R}^3$ that is a topological manifold does not have isolated singularities and, moreover, it is a plane or a round sphere or a cylinder. A result in the case non-isolated singularities also is presented.