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