Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 091, 29 pages      arXiv:2305.01453      https://doi.org/10.3842/SIGMA.2023.091
Contribution to the Special Issue on Differential Geometry Inspired by Mathematical Physics in honor of Jean-Pierre Bourguignon for his 75th birthday

Nonlinear Isocapacitary Concepts of Mass in 3-Manifolds with Nonnegative Scalar Curvature

Luca Benatti a, Mattia Fogagnolo b and Lorenzo Mazzieri c
a) Università degli Studi di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
b) Università di Padova, via Trieste 63, 35121 Padova, Italy
c) Università degli Studi di Trento, via Sommarive 14, 38123 Povo (TN), Italy

Received May 03, 2023, in final form October 23, 2023; Published online November 10, 2023

Abstract
We deal with suitable nonlinear versions of Jauregui's isocapacitary mass in $3$-manifolds with nonnegative scalar curvature and compact outermost minimal boundary. These masses, which depend on a parameter $1$ < $p\leq 2$, interpolate between Jauregui's mass $p=2$ and Huisken's isoperimetric mass, as $p \to 1^+$. We derive positive mass theorems for these masses under mild conditions at infinity, and we show that these masses do coincide with the ADM mass when the latter is defined. We finally work out a nonlinear potential theoretic proof of the Penrose inequality in the optimal asymptotic regime.

Key words: Penrose inequality; positive mass theorem; isoperimetric mass; nonlinear potential theory; nonlinear potential theory.

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