Euclidean Domains with Nearly Maximal Yamabe Quotient

TL;DR

This paper investigates domains in three-dimensional space with nearly maximal Yamabe quotient, showing they are close to a ball in shape and metric, and explores related conformal properties.

Contribution

It establishes geometric and conformal closeness of nearly extremal domains to the ball, extending previous maximal Yamabe quotient results.

Findings

Domains are diffeomorphic to a ball.

Domains are geometrically close to a ball in the Gromov-Hausdorff sense.

Near-maximal Yamabe quotient implies the domain's shape and metric are close to a ball.

Abstract

Let $\Omega$ be a smooth, bounded domain in $\mathbb R^3$ with connected boundary. It follows from work of Escobar that the Yamabe quotient of $\Omega$ is at most the Yamabe quotient of a ball, and equality holds if and only if $\Omega$ is a ball. We show that if equality almost holds then the following things are true: (i)$\Omega$ is diffeomorphic to a ball; (ii) There is a small number $\epsilon > 0$ such that $B(x,r) \subset \Omega \subset B(x,r(1+\epsilon))$; (iii) After suitable scaling, $\Omega$ is Gromov-Hausdorff close to the unit ball when considered as a metric space with its induced length metric. We also give a qualitative comparison between $Q$ and the coefficient of quasi-conformality studied in the theory of quasi-conformal maps.