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

TL;DR

This paper characterizes the structure of weighted Riemannian manifolds with negative effective dimension where the isoperimetric inequality attains equality, showing they are isometric to a specific warped product with hyperbolic features.

Contribution

It proves a rigidity theorem for isoperimetric equality cases on manifolds with negative effective dimension, identifying their precise warped product structure.

Findings

Manifolds with Ric_N ≥ K > 0 and N < -1 are isometric to a hyperbolic warped product.

Equality in the isoperimetric inequality implies the manifold's structure is uniquely determined.

Isoperimetric minimizers are isometric to half-spaces in the warped product setting.

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.