A conformally invariant generalization of the Willmore energy for compact immersed submanifolds of even dimension in a Riemannian manifold is derived and studied. The energy arises as the coefficient of the log term in the renormalized area expansion of a minimal submanifold in a Poincaré–Einstein space with prescribed boundary at infinity. Its first variation is identified as the obstruction to smoothness of the minimal submanifold. The energy is explicitly identified for the case of submanifolds of dimension four. Variational properties of this four-dimensional energy are studied in detail when the background is a Euclidean space or a sphere, including identifications of critical embeddings, questions of boundedness above and below for various topologies, and second variation.

中文翻译：

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通过最小的子流形渐近性提高高维Willmore能量

导出并研究了黎曼流形中偶数维的紧凑沉浸子流形的Willmore能量的保形不变概化。能量以对数项的系数出现在庞加莱-爱因斯坦空间中具有规定边界为无穷大的最小子流形的重新规格化区域扩展中。它的第一个变化被认为是对最小子流形平滑性的阻碍。对于四维子流形的情况，能量被明确标识。当背景为欧几里德空间或球体时，将详细研究此四维能量的变分性质，包括确定关键嵌入，各种拓扑上下界的问题以及第二种变化。