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.
We study a particular class of open manifolds. In the category of Riemannian manifolds these are complete manifolds with cylindrical ends. We give a natural setting for the conformal geometry on such manifolds including an appropriate notion of the cylindrical Yamabe constant/invariant. This leads to a corresponding version of the Yamabe problem on cylindrical manifolds. We affirmatively solve this Yamabe problem: we prove the existence of minimizing metrics and analyze their singularities near infinity. These singularities turn out to be of very particular type: either almost conical or almost cusp singularities. We describe the supremum case, i.e. when the cylindrical Yamabe constant is equal to the Yamabe invariant of the sphere. We prove that in this case such a cylindrical manifold coincides conformally with the standard sphere punctured at a finite number of points. In the course of studying the supremum case, we establish a Positive Mass Theorem for specific asymptotically flat manifolds with two almost conical singularities. As a by-product, we revisit known results on surgery and the Yamabe invariant.
Abstract. For a closed Riemannian manifold (M m , g) of constant positive scalar curvature and any other closed Riemannian manifold (N n , h), we show that the limit of the Yamabe constants of the Riemannian products (M × N, g + rh) as r goes to infinity is equal to the Yamabe constant of (M m × R n , [g + g E ]) and is strictly less than the Yamabe invariant of S m+n provided n ≥ 2. We then consider the minimum of the Yamabe functional restricted to functions of the second variable and we compute the limit in terms of the best constants of the Gagliardo-Nirenberg inequalities.
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