Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature

• 成果类型: Article
• 署名作者: Agostiniani, Virginia; Fogagnolo, Mattia; Mazzieri, Lorenzo
• 署名单位: University of Verona; University of Trento
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-020-00985-4
• 发表日期: 2020
• 页码: 1033-1101
• 关键词: mean-curvature; harmonic-functions; RIEMANNIAN-MANIFOLDS; greens-functions; FLOW; MONOTONICITY; CONSTRUCTION; RIGIDITY; THEOREM; kernel

摘要

In this paper we consider complete noncompact Riemannian manifolds (M, g) with nonnegative Ricci curvature and Euclidean volume growth, of dimension n >= 3. For every bounded open subset Omega subset of M with smooth boundary, we prove that integral(partial derivative Omega)vertical bar H/n-1 vertical bar(n-1) d sigma >= AVR(g)vertical bar Sn-1 vertical bar, where H is the mean curvature of partial derivative Omega and AVR(g) is the asymptotic volume ratio of (M, g). Moreover, the equality holds true if and only if (M\Omega,g) is isometric to a truncated cone over partial derivative 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.