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arXiv · 1202.1022

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

Jimmy Petean·Juan Miguel Ruiz
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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.

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Jimmy Petean, Juan Miguel Ruiz. 2012-02-06. On the Yamabe constants of $S^2 \times \re^3$ and $S^3 \times \re^2$. https://arxiv.org/abs/1202.1022

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