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Minkowski Inequality on complete Riemannian manifolds with nonnegative Ricci curvature

Luca Benatti, Mattia Fogagnolo, Lorenzo Mazzieri

Abstract

In this paper we consider Riemannian manifolds of dimension at least $3$, with nonnegative Ricci curvature and Euclidean Volume Growth. For every open bounded subset with smooth boundary we establish the validity of an optimal Minkowski Inequality. We also characterise the equality case, provided the domain is strictly outward minimising and strictly mean convex. Along with the proof, we establish in full generality sharp monotonicity formulas, holding along the level sets of $p$-capacitary potentials in $p$-nonparabolic manifolds with nonnegative Ricci curvature.

Minkowski Inequality on complete Riemannian manifolds with nonnegative Ricci curvature

Abstract

In this paper we consider Riemannian manifolds of dimension at least , with nonnegative Ricci curvature and Euclidean Volume Growth. For every open bounded subset with smooth boundary we establish the validity of an optimal Minkowski Inequality. We also characterise the equality case, provided the domain is strictly outward minimising and strictly mean convex. Along with the proof, we establish in full generality sharp monotonicity formulas, holding along the level sets of -capacitary potentials in -nonparabolic manifolds with nonnegative Ricci curvature.

Paper Structure

This paper contains 8 sections, 7 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Statements of the main results
  3. Outline of the proof.
  4. Further Monotonicity-Rigidity results
  5. Summary
  6. The p-capacitary potential in Riemannian manifolds
  7. p-harmonic functions and regularity
  8. p-nonparabolic manifolds and the p-capacitary potential

Key Result

Theorem 1.1

Let $(M,g)$ be complete Riemannian manifold with $\mathop{\mathrm{Ric}}\nolimits \geq 0$ and Euclidean Volume Growth. Let $\Omega \subseteq M$ be a open bounded set with smooth boundary. Then where $\mathop{\mathrm{AVR}}\nolimits(g)$ is the asymptotic volume ratio of $(M,g)$, ${\rm H}$ is the mean curvature of $\partial \Omega$ with respect to the outward normal unit vector and $\Omega^*$ is the

Theorems & Definitions (11)

  • Theorem 1.1: Extended Minkowski Inequality
  • Theorem 1.2: Rigidity for the Minkowski Inequality
  • Theorem 1.3: Volumetric Minkowski inequality
  • Proposition 2.1: Coarea formula
  • Remark 2.2
  • Lemma 2.3
  • Definition 2.4: $p$-Green's function
  • Definition 2.5: $p$-nonparabolicity
  • Theorem 2.6: Existence of the $p$-capacitary potential
  • Definition 2.7: $p$-capacity and normalised $p$-capacity
  • ...and 1 more