preprint
Inserted: 21 jan 2021
Last Updated: 21 jan 2021
Year: 2021
Abstract:
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 \[ \left(\frac{\vert{\partial \Omega^*}\vert}{\vert{\mathbb{S}^{n-1}}\vert}\right)^{\!\frac{n-2}{n-1}}\mathrm{AVR}(g)^{\frac{1}{n-1}}\leq \frac{1}{\vert{\mathbb{S}^{n-1}}\vert}\int\limits_{\partial \Omega} \left\vert{ \frac{\mathrm{H}}{n-1}} \right\vert\,\mathrm{d} \sigma , \] 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 $\Omega$.