Isoperimetric profile comparisons and Yamabe constants

TL;DR

This paper estimates the isoperimetric profile of the product space $S^2 imes e^2$ and uses it to derive lower bounds for the Yamabe constants of such products, impacting the understanding of Yamabe invariants for various surfaces.

Contribution

It provides new lower bounds for the Yamabe constants of product manifolds involving $S^2$ and any closed Riemann surface, using isoperimetric profile estimates.

Findings

Lower bound for the Yamabe constant of $S^2 imes M^2$ is greater than (2/3) of that of $S^4$.

Explicit estimates of the isoperimetric profile of $S^2 imes e^2$.

Implications for Yamabe invariants of product manifolds.

Abstract

We estimate from below the isoperimetric profile of $S^2 \times \re^2$ and use this information to obtain lower bounds for the Yamabe constant of $S^2 \times \re^2$. This provides a lower bound for the Yamabe invariants of products $S^2 \times M^2$ for any closed Riemann surface $M$. Explicitly we show that $Y(S^2 \times M^2) > (2/3) Y(S^4)$.