Euclidean Domains with Nearly Maximal Yamabe Quotient

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.