In this paper we develop the compactness theorem for $\lambda $-surface in $\mathbb R^3$ with uniform $\lambda $, genus, and area growth. This theorem can be viewed as a generalization of Colding–Minicozzi’s compactness theorem for self-shrinkers in $\mathbb R^3$. As an application of this compactness theorem, we prove a rigidity theorem for convex $\lambda $-surfaces.

中文翻译：

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λ表面的紧实度和刚度

在本文中，我们开发了在$ mathbb R ^ 3 $中的$ \ lambda $-曲面的紧致性定理，具有统一的$ \ lambda $，属和面积增长。这个定理可以看作是Colding–Minicozzi在$ \ mathbb R ^ 3 $中对自收缩器的紧性定理的推广。作为该紧性定理的一个应用，我们证明了凸$ \ lambda $-曲面的刚性定理。