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Fine properties of nonlinear potentials and a unified perspective on monotonicity formulas

Luca Benatti, Alessandra Pluda, Marco Pozzetta

TL;DR

The work provides a unified framework showing that monotone quantities along weak IMCF can be obtained as limits of monotone quantities along level sets of p-capacitary potentials as p→1+. Through sharp convergence of p-harmonic potentials to the IMCF and a detailed analysis of curvature varifolds, the authors derive a broad family of monotonicity formulas, Gauss–Bonnet-type results, and Willmore/Minkowski inequalities, with applications to Hawking mass and Geroch monotonicity in 3D. The approach hinges on strong W^{1,q} convergence, varifold convergence of level sets, and robust regularization techniques that avoid reliance on ε-regularization, yielding both existence and rigidity statements. The results illuminate deep connections between nonlinear potential theory, curvature flows, and geometric inequalities on manifolds with nonnegative Ricci curvature or scalar curvature, offering tools for Penrose-type estimates and rigidity phenomena. Overall, the paper provides a comprehensive, technically rigorous bridge between nonlinear potentials and IMCF-driven geometric inequalities, with broad implications for geometric analysis and mathematical relativity.

Abstract

We rigorously show that a large family of monotone quantities along the weak inverse mean curvature flow is the limit case of the corresponding ones along the level sets of $p$-capacitary potentials. Such monotone quantities include Willmore and Minkowski-type functionals on Riemannian manifolds with nonnegative Ricci curvature. In $3$-dimensional manifolds with nonnegative scalar curvature, we also recover the monotonicity of the Hawking mass and its nonlinear potential theoretic counterparts. This unified view is built on a refined analysis of $p$-capacitary potentials. We prove that they strongly converge in $W^{1,q}_{\mathrm{loc}}$ as $p\to 1^+$ to the inverse mean curvature flow and their level sets are curvature varifolds. Finally, we also deduce a Gauss-Bonnet-type theorem for level sets of $p$-capacitary potentials.

Fine properties of nonlinear potentials and a unified perspective on monotonicity formulas

TL;DR

The work provides a unified framework showing that monotone quantities along weak IMCF can be obtained as limits of monotone quantities along level sets of p-capacitary potentials as p→1+. Through sharp convergence of p-harmonic potentials to the IMCF and a detailed analysis of curvature varifolds, the authors derive a broad family of monotonicity formulas, Gauss–Bonnet-type results, and Willmore/Minkowski inequalities, with applications to Hawking mass and Geroch monotonicity in 3D. The approach hinges on strong W^{1,q} convergence, varifold convergence of level sets, and robust regularization techniques that avoid reliance on ε-regularization, yielding both existence and rigidity statements. The results illuminate deep connections between nonlinear potential theory, curvature flows, and geometric inequalities on manifolds with nonnegative Ricci curvature or scalar curvature, offering tools for Penrose-type estimates and rigidity phenomena. Overall, the paper provides a comprehensive, technically rigorous bridge between nonlinear potentials and IMCF-driven geometric inequalities, with broad implications for geometric analysis and mathematical relativity.

Abstract

We rigorously show that a large family of monotone quantities along the weak inverse mean curvature flow is the limit case of the corresponding ones along the level sets of -capacitary potentials. Such monotone quantities include Willmore and Minkowski-type functionals on Riemannian manifolds with nonnegative Ricci curvature. In -dimensional manifolds with nonnegative scalar curvature, we also recover the monotonicity of the Hawking mass and its nonlinear potential theoretic counterparts. This unified view is built on a refined analysis of -capacitary potentials. We prove that they strongly converge in as to the inverse mean curvature flow and their level sets are curvature varifolds. Finally, we also deduce a Gauss-Bonnet-type theorem for level sets of -capacitary potentials.

Paper Structure

This paper contains 21 sections, 44 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Acknowledgments
  4. Preliminaries and notation
  5. Nonlinear potential theory
  6. Inverse mean curvature flow
  7. Local approximation of IMCF
  8. Monotonicity formulas in nonlinear potential theory
  9. Main computations
  10. Regularity of level sets of wp
  11. Well-posedness of the monotone quantity
  12. Proof of Theorem 3.1
  13. Monotonicity formula for the IMCF
  14. Improved convergence to the weak IMCF
  15. Lower semicontinuity of Fp'
  16. ...and 6 more sections

Key Result

Theorem 1

Let $(M,g)$ be a complete noncompact Riemannian manifold of dimension $n\ge 3$. Let $\Omega \subset M$ be a closed bounded set with $\mathscr{C}^{1,1}$-boundary. Suppose that $\Omega$ admits a proper solution $w_p$ to p-Laplaciano for $p\in[1,2]$ and let $\Omega^{\scaleobj{.9}{(p)}}_t \coloneqq \lbr

Theorems & Definitions (55)

  • Theorem 1: Monotonicity formulas
  • Theorem 2: Improved convergence
  • Theorem 3: Gauss--Bonnet-type theorem
  • Proposition 4
  • Proposition 5
  • Lemma 6
  • Proposition 7
  • Proposition 8
  • Lemma 9
  • Proposition 10: Uniform gradient bound
  • ...and 45 more