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Geometric inequalities and the Alexandrov-Bakelman-Pucci technique

S. Brendle

Abstract

In this expository paper, we discuss a unified framework for proving various geometric inequalities, based on the so-called Alexandrov-Bakelman-Pucci technique. Examples include Cabré's proof of the classical isoperimetric inequality in Euclidean space; the Fenchel-Willmore-Chen inequality for the mean curvature of a submanifold; the sharp version of the Michael-Simon Sobolev inequality for submanifolds; the sharp version of Ecker's logarithmic Sobolev inequality for submanifolds; and the Sobolev inequality for complete manifolds with nonnegative Ricci curvature and Euclidean volume growth. Finally, we discuss a connection to the work of Heintze and Karcher on the volume of a tubular neighborhood of a hypersurface in a manifold with nonnegative Ricci curvature.

Geometric inequalities and the Alexandrov-Bakelman-Pucci technique

Abstract

In this expository paper, we discuss a unified framework for proving various geometric inequalities, based on the so-called Alexandrov-Bakelman-Pucci technique. Examples include Cabré's proof of the classical isoperimetric inequality in Euclidean space; the Fenchel-Willmore-Chen inequality for the mean curvature of a submanifold; the sharp version of the Michael-Simon Sobolev inequality for submanifolds; the sharp version of Ecker's logarithmic Sobolev inequality for submanifolds; and the Sobolev inequality for complete manifolds with nonnegative Ricci curvature and Euclidean volume growth. Finally, we discuss a connection to the work of Heintze and Karcher on the volume of a tubular neighborhood of a hypersurface in a manifold with nonnegative Ricci curvature.
Paper Structure (6 sections, 32 theorems, 136 equations)

This paper contains 6 sections, 32 theorems, 136 equations.

Table of Contents

  1. The classical isoperimetric inequality and the Sobolev inequality
  2. The Fenchel-Willmore-Chen inequality for submanifolds of Euclidean space
  3. The Sobolev inequality for submanifolds of Euclidean space
  4. The logarithmic Sobolev inequality for submanifolds of Euclidean space
  5. The Sobolev inequality for manifolds with nonnegative Ricci curvature
  6. The Fenchel-Willmore-Chen inequality for hypersurfaces in manifolds with nonnegative Ricci curvature

Key Result

Theorem 1.1

Let $D$ be a compact domain in $\mathbb{R}^n$ with smooth boundary. Then $|\partial D| \geq n \, |B^n|^{\frac{1}{n}} \, |D|^{\frac{n-1}{n}}$, where $B^n$ denotes the open unit ball in $\mathbb{R}^n$.

Theorems & Definitions (32)

  • Theorem 1.1: Isoperimetric inequality
  • Theorem 1.2: Sobolev inequality
  • Lemma 1.3
  • Lemma 1.4
  • Theorem 2.1: Fenchel-Willmore-Chen inequality
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Theorem 3.1: cf. Brendle1
  • ...and 22 more