The Yamabe problem on stratified spaces

Abstract

We introduce new invariants of a Riemannian singular space, the local Yamabe and Sobolev constants, and then go on to prove a general version of the Yamabe theorem under that the global Yamabe invariant of the space is strictly less than one or the other of these local invariants. This rests on a small number of structural assumptions about the space and of the behavior of the scalar curvature function on its smooth locus. The second half of this paper shows how this result applies in the category of smoothly stratified pseudomanifolds, and we also prove sharp regularity for the solutions on these spaces. This sharpens and generalizes the results of Akutagawa and Botvinnik [3] on the Yamabe problem on spaces with isolated conic singularities.

An Obata singular theorem for stratified spaces

“…A standard proof of this can be found in [18]. In [1], the usual Sobolev embeddings which hold on compact Riemannian manifolds are proven in the setting of stratified spaces as well. In particular we have the following Sobolev inequality: there exist positive constants A and B such that for any u in W 1,2 (X)…”

“…We refer to [15] for a description of the Yamabe problem on compact smooth manifolds, and to [1] for the same in the setting of stratified spaces. The Yamabe problem has a variational formulation depending on a conformal invariant, called the Yamabe constant: this latter is defined as the infimum of the integral of the scalar curvature among conformal metrics of volume one.…”

“…For this purpose, we precise that for a truncated cone C(Z) over a compact metric space Z we mean the product Z × [0, 1] with the equivalence relation (z 1 , 0) ∼ (z 2 , 0) for all z 1 , z 2 in Z: we identify all the points in Z × {0} to a unique point, called the vertex of the cone. We say that a truncated cone is of size δ if we consider the interval [0, δ] instead of [0,1]. If Z is a compact manifold endowed with the Riemannian metric k, then a conic metric on C(Z) has the form dr 2 + r 2 k. Definition 1.1.…”

“…One can define an admissible metric on a stratified space: for a precise discussion we refer to section 3 of [3] and section 2.1 of [1]. For the purposes of this paper, the reader only needs to know that an admissible metric g is a smooth Riemannian metric on the regular set Ω and near to the stratum Σ j it is a perturbation of the model metric g 0 = ξ j + dr 2 + r 2 k j , where ξ j is the Euclidean metric on R j and k j is an iterated edge metric on the link Z j .…”

“…We refer to α has the cone angle of the stratum Σ n−2 . On a stratified space we can consider an iterate edge metric, as defined in [3] or [1], which is a smooth Riemannian metric on the regular set Ω, and define the usual tools of geometric analysis.…”