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 Yℓ(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 Yℓ(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.
