The Local Yamabe Constant of Einstein Stratified Spaces

TL;DR

This paper investigates the local Yamabe constant on Einstein stratified spaces, providing explicit computations under Ricci positive bounds and establishing spectral and Sobolev inequalities to understand conformal geometry in singular spaces.

Contribution

It introduces methods to compute the local Yamabe constant on stratified spaces with Ricci positive links and extends spectral and Sobolev inequalities to these singular settings.

Findings

Computed the local Yamabe constant under Ricci positive bounds.

Extended Lichnerowicz's theorem to stratified spaces.

Established Euclidean isoperimetric inequality for specific stratified spaces.

Abstract

On a compact stratified space (X, g) there exists a metric of constant scalar curvature in the conformal class of g, if the scalar curvature satisfies an integrability condition and if the Yamabe constant of X is strictly smaller than the local Yamabe constant , another conformal invariant introduced in the recent work of K. Akutagawa, G. Carron and R. Mazzeo. Such invariant depends on the local structure of X, in particular on the links, but its explicit value is not known. We are going to show that if the links satisfy a Ricci positive lower bound, then we can compute the local Yamabe constant. In order to achieve this, we prove a lower bound for the spectrum of the Laplacian, by extending a well-known theorem by Lichenrowicz, and a Sobolev inequality, inspired by a result due to D. Bakry. Furthermore, we prove the existence of an Euclidean isoperimetric inequality on particular stratified space, with one stratum of codimension 2 and cone angle bigger than 2$\pi$.