Ricci curvature and Yamabe constants

Details Ricci curvature and Yamabe constants Ricci curvature and Yamabe constants

2005-10-14

arxiv.org/abs/math/0510308v2

Mathematics

Differential Geometry

The author was informed that the inequality in the main theorem has already been proved by S. Ilias in Constantes explicites p

Scientific paper

We prove that if a closed unit volume Riemannian manifold, $(M^n, g)$, has Ricci curvature bounded from below by r>0 then the Yamabe constant of the conformal class of $g$ is at least $n.r$. This inequality has already been proved by S. Ilias (Constantes explicites pour les inegalites de Sobolev sur les varietes riemannienes compactes, Ann. Inst. Fourier 33, 151-165). The equality is achieved if the metric is Einstein (with Ricci curvature r). This implies for instance that if $h$ is the Fubini-Study metric on $CP^2$ and $g$ is any other metric on $CP^2$ with $Ricci(g) \geq Ricci(h)$ then $Vol(CP^2, g) \leq Vol(CP^2, h)$.

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