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

Abstract: We study, on a weighted Riemannian manifold of Ric${N} \geq K > 0$ 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 Poincar\'e inequality. Moreover, every isoperimetric minimizer set is isometric to a half-space in an appropriate sense.