# Removability of isolated singularities on surfaces with bounded mean curvature

In this talk, I will prove the following generalized version of Bers’ Theorem on isolated singularities of minimal surfaces. Let S be a topological surface in a 3-dimensional Riemannian manifold M. Assume that the set where S is not a smooth surface, denoted by Sing(S), is a set of isolated points. If the mean curvature of Reg(S)=S\Sing(S) is bounded around each point of Sing(S), then S is a C^{1,b} smooth surface, for all b in (0,1). Moreover, if the mean curvature of Reg(S) extends to S as a locally Lipschitz function then S is C^{2,b} smooth surface, for all b in (0,1). With this result, we obtain several consequences, for instance, we obtain the following: (1) A classification of a class of algebraic CMC surfaces in the 3-dimensional Euclidean space: A connected algebraic CMC surface in the 3-dimensional Euclidean space 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; (2) A CMC surface in the 3-dimensional Euclidean space that is a topological manifold does not have isolated singularities; (3) A connected algebraic CMC surface in the 3-dimensional Euclidean space with isolated singularities which is locally Lipschitz normally embedded (or quasi-convex) must be a plane or a round sphere or a right circular cylinder.