On the Yamabe constants of $S^2 \times \re^3$ and $S^3 \times \re^2$

TL;DR

This paper compares isoperimetric profiles of specific product manifolds with a sphere to derive explicit lower bounds for their Yamabe constants, extending previous methods and contributing to the understanding of Yamabe invariants.

Contribution

The authors establish new explicit lower bounds for Yamabe constants of certain product manifolds using isoperimetric profile comparisons, extending prior techniques.

Findings

Yamabe constant of S^3 x R^2 exceeds (3/4) of Y(S^5)

Yamabe constant of S^2 x R^3 exceeds 0.63 of Y(S^5)

Derived bounds apply to higher dimensions and other product manifolds

Abstract

We compare the isoperimetric profiles of $S^2 \times \re^3$ and of $S^3 \times \re^2$ with that of a round 5-sphere (of appropriate radius). Then we use this comparison to obtain lower bounds for the Yamabe constants of $S^2 \times \re^3$ and $S^3 \times \re^2$. Explicitly we show that $Y(S^3 \times \re^2, [g_0^3 +dx^2]) > (3 /4) Y(S^5)$ and $Y(S^2 \times \re^3, [g_0^2 +dx^2]) > 0.63 Y(S^5)$. We also obtain explicit lower bounds in higher dimensions and for products of Euclidean space with a closed manifold of positive Ricci curvature. The techniques are a more general version of those used by the same authors in previous work and the results are a complement to the work developed by B. Ammann, M. Dahl and E. Humbert to obtain explicit gap theorems for the Yamabe invariants in low dimensions.