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

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

2012-02-06

arxiv.org/abs/1202.1022v1

Mathematics

Differential Geometry

Scientific paper

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 $Y(S^2 \times \re^3 ,[g_0^2 +dx^2]) > 0.63 Y(S^5)$. The techniques are similar to those used by the same authors in [15] and the results are a complement to the work developed by B. Ammann, M. Dahl and E. Humbert to obtain an explicit gap theorem for the Yamabe invariants of simply connected 5-manifolds. We also obtain explicit lower bounds in higher dimensions.

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