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

By Virginia Agostiniani, Mattia Fogagnolo, L. Mazzieri. Inventiones Mathematicae, 2018

120 citations recorded by the source corpus.

In this paper we consider complete noncompact Riemannian manifolds (M, g) with nonnegative Ricci curvature and Euclidean volume growth, of dimension n≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 3$$\end{document}. For every bounded open subset Ω⊂M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset M$$\end{document} with smooth boundary, we prove that ∫∂ΩHn-1n-1dσ≥AVR(g)|Sn-1|,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int \limits _{\partial \Omega } \left| \frac{\mathrm{H}}{n-1}\right| ^{n-1} \!\!\!\!\!{\mathrm{d}}\sigma \,\,\ge \,\,{\mathrm{AVR}}(g)\,\big |\mathbb {S}^{n-1}\big |, \end{aligned}$$\end{document}where H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{H}}$$\end{document} is the mean curvature of ∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega $$\end{document} and AVR(g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{AVR}}(g)$$\end{document} is the asymptotic volume ratio of (M, g). Moreover, the equality holds true if and only if (M\Ω,g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(M{{\setminus }}\Omega , g)$$\end{document} is isometric to a truncated cone over ∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega $$\end{document}. 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.

Subfields: Applied Mathematics, Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/74df597dbca3c67b36def2f951cd8a3216ef8241

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Citations in the corpus

On the 1/H-flow by p-Laplace approximation: New estimates via fake distances under Ricci lower bounds (2019, methodology): “In the linear case, similar identities were obtained in [24] and in [3, 13].”
Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature (2020, methodology): “4 The smoothness of the boundary of the isoperimetric set is an essential requirement not only in our argument, but also inAgostiniani, Fogagnolo andMazzieri [1, 25] and Brendle [7].”
Minkowski Inequalities via Nonlinear Potential Theory (2019, background): “As shown in [1,3,4] and [26], it is very convenient to work in the conformally related Riemannian manifold (Rn \ , g), where g is given by g = u(2) ( p−1 n−p ) gRn .”