In this presentation, we talk about a 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 with finite Willmore energy.  Especially, we prove the removability of isolated singularity of multiplicity one surfaces with finite Willmore energy and a uniqueness theorem of the catenoid  under no a priori topological finiteness assumption.

 

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