Euclidean Domains with Nearly Maximal Yamabe Quotient
Liam Mazurowski, Xuan Yao
TL;DR
The work establishes quantitative stability for the Yamabe quotient of smooth bounded domains in $\mathbb{R}^3$ with connected boundary: if $Q(B)-Q(\Omega)$ is small, then $\Omega$ is uniformly ball-like after translation and scaling, with inclusions $B(x,r)\subset\Omega\subset B(x,r(1+\varepsilon))$ and a small Gromov–Hausdorff distance to the unit ball. The authors develop a two-step strategy, first proving a stability result for exterior domains via the capacitary potential and a carefully chosen test function, then transferring to interior domains using conformal inversion and medial-axis analysis to show the boundary is close to a sphere; they also prove analogous results for the Sobolev quotient and compare $Q$ with the quasi-conformality coefficient $K$. A qualitative correspondence is drawn between obstructions to $K$ being near 1 (spikes, ridges, hair) and obstructions to $Q$ approaching $Q(B)$, highlighting geometric features that enforce deviation from the ball. The results yield explicit, though nontrivial, quantitative rigidity: $\delta=O(\varepsilon^9)$ in the main stability estimates, with extensions to related invariants.
Abstract
Let $Ω$ be a smooth, bounded domain in $\mathbb R^3$ with connected boundary. It follows from work of Escobar that the Yamabe quotient of $Ω$ is at most the Yamabe quotient of a ball, and equality holds if and only if $Ω$ is a ball. We show that if equality almost holds then the following things are true: (i)$Ω$ is diffeomorphic to a ball; (ii) There is a small number $ε> 0$ such that $B(x,r) \subset Ω\subset B(x,r(1+ε))$; (iii) After suitable scaling, $Ω$ 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.