The local Yamabe constant of Einstein stratified spaces

On a compact stratified space (X,g), a metric of constant scalar curvature exists in the conformal class of g if the scalar curvature Sg satisfies an integrability condition and if the Yamabe constant of X is strictly smaller than the local Yamabe constant 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 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.