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.

中文翻译：

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具有有限 Willmore 能量的曲面拓扑

在本文中，我们研究了阿拉德正则定理的临界情况。结合 Reifenberg 的拓扑盘定理，我们得到了一个临界的 Allard-Reifenberg 型正则性定理。作为主要结果，我们得到了具有有限 Willmore 能量的 $\mathbb{R}^n$ 中一类正确浸入表面的拓扑有限性。特别是，我们证明了具有有限 Willmore 能量的表面的重数奇异性的可移除性和在没有先验拓扑有限性假设的情况下悬链线的唯一性定理。