In this paper, we study the minimizers of curvature functionals (Willmore functional and extrinsic energy functional) subject to an area constraint in asymptotically flat manifolds. Under some certain conditions, we prove that such minimizers exist. Besides the surface theory related to the Willmore functional, the proofs also rely on the inverse mean curvature flow developed by Huisken and Ilmanen, on the positive mass theorem due to Schoen–Yau, and on the positive mass theorem for asymptotically flat manifolds with corners due to Miao and Shi–Tam. Our results may be of some interest in the General Relativity.

中文翻译：

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渐近平坦流形中的曲率泛函的极小化

在本文中，我们研究在渐近平坦流形中受面积限制的曲率泛函的最小化函数（Willmore泛函和外在能量泛函）。在某些条件下，我们证明存在这种最小化器。除了与Willmore泛函有关的表面理论外，证明还依赖于Huisken和Ilmanen提出的平均逆曲率流，Schoen-Yau所产生的正质量定理，以及渐近平流形的正质量定理。到苗族和石潭。我们的结果可能与广义相对论有关。