Title: The Sharp $p$-Penrose Inequality
\nAuthors: Liam Mazurowski, Xuan Yao
\nCategories: math.DG math-ph math.CA math.MP
\nComments: 19 pages, comments are welcome!
\n\\\\
\n Consider a complete asymptotically flat 3-manifold $M$ with non-negative
\nscalar curvature and non-empty minimal boundary $\\Sigma$. Fix a number $1 < p <\n2$. We prove a sharp mass-capacity inequality relating the ADM mass of $M$ with\nthe $p$-capacity of $\\Sigma$ in $M$. Equality holds if and only if $M$ is\nisometric to a spatial Schwarzschild manifold with horizon boundary. This\ninequality interpolates between the Riemannian Penrose inequality when $p\\to 1$\nand Bray's mass-capacity inequality for harmonic functions when $p\\to 2$. To\nprove the mass-capacity inequality, we derive monotone quantities for\n$p$-harmonic functions on asymptotically flat manifolds which become constant\non Schwarzschild.\n\\\\ ( https:\/\/arxiv.org\/abs\/2305.19784 , 20kb)\n<\/p>\n","protected":false},"excerpt":{"rendered":"