In this paper we consider complete noncompact Riemannian manifolds $(M, g)$ with nonnegative Ricci curvature and Euclidean volume growth, of dimension $n \geq 3$. We prove a sharp Willmore-type inequality for closed hypersurfaces $\partial \Omega$ in $M$, with equality holding true if and only if $(M{\setminus}\Omega, g)$ is isometric to a truncated cone over $\partial\Omega$. An optimal version of Huisken's Isoperimetric Inequality for $3$-manifolds is obtained using this result. Finally, exploiting a natural extension of our techniques to the case of parabolic manifolds, we also deduce an enhanced version of Kasue's non existence result for closed minimal hypersurfaces in manifolds with nonnegative Ricci curvature.

中文翻译：

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具有非负 Ricci 曲率的流形中闭合超曲面的尖锐几何不等式

在本文中，我们考虑具有非负 Ricci 曲率和欧几里德体积增长的完全非紧黎曼流形 $(M, g)$，维度 $n \geq 3$。我们证明了 $M$ 中闭合超曲面 $\partial \Omega$ 的尖锐 Willmore 型不等式，当且仅当 $(M{\setminus}\Omega, g)$ 与截锥$\部分\欧米茄$。使用该结果获得了 3 美元流形的 Huisken 等周不等式的最佳版本。最后，利用我们的技术对抛物线流形情况的自然扩展，我们还推导出了 Kasue 不存在结果的增强版本，用于具有非负 Ricci 曲率的流形中的封闭最小超曲面。