The Yamabe problem on stratified spaces.

Abstract : We study a class of singular metric spaces, stratified spaces, with an approach whose goal is to extend to these latter some tools and results of Riemannian geometry and analysis on smooth manifolds. In a first part, we show the existence of a lower bound for the bottom of the spectrum of the Laplacian, under the assumption that the Ricci curvature is bounded by below. This allows us to prove also the existence of a Sobolev inequality whose constants only depend on the volume and of the dimension of the space, and of an upper bound for the diameter. Furthermore, we prove that the bound for the diameter is attained if and only if the one for the bottom of the spectrum is attained as well. The second part is devoted to the direct consequences of the previous results on the Yamabe problem on a stratified space: this problem consists in looking for a conformal metric with constant scalar curvature, and the existence of a solution depends on a conformal invariant, the local Yamabe constant, whose value is generally unknown. We show that this latter can be computed in a large number of cases, when a geometric hypothesis on the singular set is verified. We use techniques which are related to the Sobolev and the isoperimetric inequalities. Finally, we give a class of examples for which we can prove the existence of a conformal metric with constant scalar curvature.
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Submitted on : Thursday, September 24, 2015 - 1:40:07 PM
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Ilaria Mondello. The Yamabe problem on stratified spaces.. Differential Geometry [math.DG]. Université de Nantes, 2015. English. 〈tel-01204671〉

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