Isoperimetric regions in spherical cones and Yamabe constants of $M\times S^1$

TL;DR

This paper investigates isoperimetric regions in spherical cones over positively curved manifolds and uses these results to estimate Yamabe constants and invariants for related product manifolds.

Contribution

It provides new insights into isoperimetric problems on spherical cones and derives bounds for Yamabe constants of product manifolds involving $S^1$.

Findings

Characterization of isoperimetric regions in spherical cones

Explicit computation of Yamabe constants for certain product manifolds

Lower bounds for Yamabe invariants of $M imes S^1$

Abstract

Given closed Riemannian manifold $(M^n, g)$ of positive Ricci curvature $Ricci(g) \geq (n-1)g$ we study isoperimetric regions on the spherical cone over $M$. When $g$ is Einstein we use this to compute the Yamabe constant of $(M \times {\bf R}, g + dt^2)$ and so to obtain lower bounds for the Yamabe invariant of $M\times S^1$.