A domain in R^3 whose Yamabe quotient is close to the maximal ball value is close to a ball: diffeomorphic, nearly round, and Gromov-Hausdorff close after scaling.
Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative ricci curvature
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Euclidean Domains with Nearly Maximal Yamabe Quotient
A domain in R^3 whose Yamabe quotient is close to the maximal ball value is close to a ball: diffeomorphic, nearly round, and Gromov-Hausdorff close after scaling.