The local Yamabe constant of Einstein stratified spaces

Abstract

On a compact stratified space $(X,g)$, a metric of constant scalar curvature exists in the conformal class of $g$ if the scalar curvature $S_g$ satisfies an integrability condition and if the Yamabe constant of $X$ is strictly smaller than the local Yamabe constant $Y_{\ell }(X)$. 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 $Y_{\ell }(X)$. 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 $2\pi$, must be handled separately – in this case we prove the existence of an Euclidean isoperimetric inequality.