The local Yamabe constant of Einstein stratified spaces

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• 关键词：Geometric analysis ; Stratified spaces ; Yamabe constant ; Sobolev inequality ; Eigenvalues of the Laplacian ; Isoperimetric profiles
• 刊名：Annales de l'Institut Henri Poincare (C) Non Linear Analysis
• 出版年：2017
• 出版时间：January-February 2017
• 年：2017
• 卷：34
• 期：1
• 页码：249-275
• 全文大小：450 K
• 卷排序：34

文摘

On a compact stratified space (X,g)(X,g), a metric of constant scalar curvature exists in the conformal class of g   if the scalar curvature SgSg satisfies an integrability condition and if the Yamabe constant of X   is strictly smaller than the local Yamabe constant Yℓsub>(X)Yℓ(X). This latter is a conformal invariant introduced in the recent work of K. Akutagawa, G. Carron and R. Mazzeo. It depends on the local structure of X  , in particular on its links, but its explicit value is unknown. We show that if the links satisfy a Ricci positive lower bound, then we can compute Yℓsub>(X)Yℓ(X). In order to achieve this, we prove a lower bound for the spectrum of the Laplacian, by extending a well-known theorem by A. Lichnerowicz, and a Sobolev inequality, inspired by a result due to D. Bakry. A particular stratified space, with one stratum of codimension 2 and cone angle bigger than 2π, must be handled separately – in this case we prove the existence of an Euclidean isoperimetric inequality.