Minkowski Inequality on Asymptotically Conical manifolds

Luca Benatti, Mattia Fogagnolo, Lorenzo Mazzieri

Published 2021-01-15Version 1

In this paper we consider Asymptotically Conical Riemannian manifolds $(M,g)$ of dimension $n\geq 3$ with nonnegative Ricci curvature. For every open bounded subset $\Omega\subseteq M$ with smooth boundary we prove that \begin{equation} \left(\frac{\vert{\partial \Omega^*}\vert}{\vert{\mathbb{S}^{n-1}}\vert}\right)^{\!\frac{n-2}{n-1}}\mathrm{AVR}(g)^{\frac{1}{n-1}}\leq \frac{1}{\vert{\mathbb{S}^{n-1}}\vert}\int\limits_{\partial \Omega} \left\vert{ \frac{\mathrm{H}}{n-1}} \right\vert\,\mathrm{d} \sigma , \end{equation} where $\mathrm{H}$ is the mean curvature of $\partial \Omega$, $\mathrm{AVR}(g)$ is the asymptotic volume ratio of $(M,g)$ and $\Omega^*$ is the strictly outward minimising hull of $\Omega$.