Ricci curvature and Yamabe constants

2005/10/14 by Jimmy Petean, Petean, Jimmy · 1 citation
Mathematics · Physics and Astronomy · #53C21 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.math/0510308

openalex publication_date 2005/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that if a closed unit volume Riemannian manifold, (Mⁿ, 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 CP² and g is any other metric on CP² with Ricci(g) ≥ Ricci(h) then Vol(CP², g) ≤ Vol(CP², h).

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