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Monotone Quantities for $p$-Harmonic functions and the Sharp $p$-Penrose inequality

Liam Mazurowski, Xuan Yao

Abstract

Consider a complete asymptotically flat 3-manifold $M$ with non-negative scalar curvature and non-empty minimal boundary $Σ$. Fix a number $1 < p < 3$. We derive monotone quantities for $p$-harmonic functions on $M$ which become constant on Schwarzschild. These monotonicity formulas imply a sharp mass-capacity estimate relating the ADM mass of $M$ with the $p$-capacity of $Σ$ in $M$, which was first proved by Xiao using weak inverse mean curvature flow.

Monotone Quantities for $p$-Harmonic functions and the Sharp $p$-Penrose inequality

Abstract

Consider a complete asymptotically flat 3-manifold with non-negative scalar curvature and non-empty minimal boundary . Fix a number . We derive monotone quantities for -harmonic functions on which become constant on Schwarzschild. These monotonicity formulas imply a sharp mass-capacity estimate relating the ADM mass of with the -capacity of in , which was first proved by Xiao using weak inverse mean curvature flow.
Paper Structure (13 sections, 16 theorems, 153 equations)

This paper contains 13 sections, 16 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Outline of proof
  3. Further Discussion
  4. A Differential Inequality for W
  5. ODE Analysis on Schwarzschild
  6. Asymptotic Expansions on Schwarzschild
  7. The Decaying Solution
  8. The Growing Solution
  9. Monotonicity in the Presence of Critical Points
  10. No Critical points
  11. Monotonicity with negligible critical values
  12. An approximate monotonicity formula
  13. Asymptotic behavior of $Q_*$ and $Q^{*}$

Key Result

Theorem 3

Fix a number $1<p<3$. There exist functions $f_*,g_*,h_*\colon [0,\infty)\to \mathbb{R}$ with the following properties. Let $(M^3,g)$ be a complete asymptotically flat manifold with non-negative scalar curvature and non-empty minimal boundary $\Sigma$. Assume that $H_2(M,\Sigma) = 0$ and let $u$, $w Then $Q_*(s) \le Q_*(t)$ for any regular values $s \le t$ of $w$. Moreover, $Q_*$ is constant if an

Theorems & Definitions (43)

  • Definition 1
  • Remark 2
  • Theorem 3
  • Theorem 4
  • Definition 5
  • Corollary 6: Sharp $p$-Penrose inequality
  • Conjecture 7
  • Remark 8
  • Remark 9
  • Proposition 10
  • ...and 33 more