The Yamabe problem on stratified spaces

Kazuo Akutagawa (1) , Gilles Carron (2) , Rafe Mazzeo (3)

1 TITECH - Tokyo Institute of Technology [Tokyo]
2 LMJL - Laboratoire de Mathématiques Jean Leray
3 Stanford University

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 \cite{AB} on the Yamabe problem on spaces with isolated conic singularities.

Mots clés: Yamabe problem; Stratified spaces; Sobolev inequality

Domaines: Géométrie différentielle [math.DG]; Equations aux dérivées partielles [math.AP]

Kazuo Akutagawa, Gilles Carron, Rafe Mazzeo. The Yamabe problem on stratified spaces. Geometry and Functional Analysis, 2014, 24 (4), pp.1039-1079. ⟨10.1007/s00039-014-0298-z⟩. ⟨hal-00747322⟩