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On Yamabe constants of Riemannian products

Kazuo Akutagawa·Luis A. Florit·Jimmy Petean
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Abstract

For a closed Riemannian manifold $(M^m,g)$ of constant positive scalar curvature and any other closed Riemannian manifold $(N^n,h)$, we show that the limit of the Yamabe constants of the Riemannian products $(M\times N,g+rh)$ as $r$ goes to infinity is equal to the Yamabe constant of $(M^m \times R^n, [g+g_E])$ and is strictly less than the Yamabe invariant of $S^{m+n}$ provided $n\geq 2$. We then consider the minimum of the Yamabe functional restricted to functions of the second variable and we compute the limit in terms of the best constants of the Gagliardo-Nirenberg inequalities.

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Kazuo Akutagawa, Luis A. Florit, Jimmy Petean. 2006-03-20. On Yamabe constants of Riemannian products. https://arxiv.org/abs/math/0603486

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