Topology of surfaces with finite Willmore energy

TL;DR

This paper establishes topological finiteness and singularity removability for surfaces with finite Willmore energy, combining regularity theorems and topological results to advance understanding of such surfaces.

Contribution

It introduces a new regularity theorem linking Allard and Reifenberg results and proves topological finiteness and uniqueness results for surfaces with finite Willmore energy.

Findings

Proves topological finiteness for properly immersed surfaces with finite Willmore energy.

Establishes removability of singularities for multiplicity one surfaces with finite Willmore energy.

Demonstrates uniqueness of the catenoid without topological assumptions.

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 a removability of singularity of multiplicity one surface with finite Willmore energy and a uniqueness theorem of the catenoid under no a priori topological finiteness assumption.