Topology of Surfaces with Finite Willmore Energy

Abstract

Abstract In this paper, we study the critical case of the Allard regularity theorem. Combining with Reifenberg’s topological disk theorem, we get a critical Allard–Reifenberg-type regularity theorem. As a main result, we get the topological finiteness for a class of properly immersed surfaces in $\mathbb{R}^n$ with finite Willmore energy. Especially, we prove the removability of the isolated singularity of multiplicity one surfaces with finite Willmore energy and a uniqueness theorem of the catenoid under no a priori topological finiteness assumption.