Geometric inequalities and the Alexandrov-Bakelman-Pucci technique

TL;DR

This paper presents a unified approach using the Alexandrov-Bakelman-Pucci technique to prove various geometric inequalities, connecting classical and modern results in differential geometry and analysis.

Contribution

It introduces a comprehensive framework that simplifies and unifies proofs of multiple geometric inequalities involving submanifolds and manifolds with nonnegative Ricci curvature.

Findings

Unified proof technique for geometric inequalities

Extension of inequalities to manifolds with nonnegative Ricci curvature

Connections between volume estimates and geometric inequalities

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 Cabr\'e'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.