Isoperimetric inequalities and spectral consequences in warped product manifolds

TL;DR

This paper explores isoperimetric inequalities in warped product manifolds, establishing geometric conditions, improving Cheeger inequalities, and providing bounds for eigenvalues, thereby linking geometry and spectral theory.

Contribution

It introduces new geometric conditions for isoperimetric inequalities, improves classical Cheeger bounds, and derives eigenvalue estimates in warped product manifolds.

Findings

Geodesic balls minimize perimeter among sets of fixed volume.

Derived necessary and sufficient geometric conditions for inequalities.

Established lower bounds for the first Dirichlet eigenvalue.

Abstract

In this article, we investigate the centered isoperimetric inequality on Cartan-Hadamard manifolds endowed with a warped product structure, namely, among all bounded measurable sets of finite perimeter and prescribed volume, the geodesic ball centered at the pole minimizes the perimeter. Exploiting the interplay between this inequality and the underlying warped product structure, we derive several necessary geometric conditions, some of which are closely related to and comparable with phenomena identified in the work of Simon Brendle [Publ. Math. Inst. Hautes \'Etudes Sci. 117 (2013)]. We also establish a sufficient condition ensuring the validity of the centered isoperimetric inequality in this setting. Furthermore, by introducing a suitable isoperimetric-type quotient, we obtain an improvement of the classical Cheeger inequality for a broad class of manifolds. Finally, we derive a quantitative lower bound for the first nonzero Dirichlet eigenvalue of geodesic balls centered at the pole, valid for a certain class of Riemannian manifolds.