Rigidity for the isoperimetric inequality of negative effective dimension on weighted Riemannian manifolds

Abstract

We study, on a weighted Riemannian manifold of Ric${}_N\ge K>0$ for $N<-1$, when equality holds in the isoperimetric inequality. Our main theorem asserts that such a manifold is necessarily isometric to the warped product $\mathbb{R}{\times }_{\cosh (\sqrt{K/(1-N)}t)}{\Sigma }^{n-1}$ of hyperbolic nature, where ${\Sigma }^{n-1}$ is an $(n-1)$-dimensional manifold with lower weighted Ricci curvature bound and $\mathbb{R}$ is equipped with a hyperbolic cosine measure. This is a similar phenomenon to the equality condition of Poincaré inequality. Moreover, every isoperimetric minimizer set is isometric to a half-space in an appropriate sense.

arXiv commentResubmit adding the rigidity for the isoperimetric inequality of negative effective dimension on weighted Riemannian manifolds

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• Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds
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• Quantitative estimates for the Bakry-Ledoux isoperimetric inequality. II