Towards Conformal Capacities in Euclidean Spaces

TL;DR

This paper compares three definitions of conformal capacities in higher-dimensional Euclidean spaces and explores their geometric inequalities and connections to various classical problems in geometry and analysis.

Contribution

It provides a comprehensive comparison of existing conformal capacity definitions and investigates their associated inequalities and links to fundamental geometric and physical problems.

Findings

Comparison of three conformal capacity definitions.

Establishment of iso-capacitary inequalities related to geometric measures.

Connections to classical problems like Minkowski and Yau problems.

Abstract

This paper addresses the so-called conformal capacities in $\mathbb R^n$, $n\ge 3$, through comparing three existing definitions (due to Betsakos, Colesanti-Cuoghi, Anderson-Vamananmurthy-Fuglede respectively) and studying their associated iso-capacitary inequalities with connection to half-diameter, mean-width, mean-curvature and ADM-mass, Hadamard type variational formula, Minkowski type problem, and Yau type problem.