Smooth Yamabe invariant and surgery

Abstract

We prove a surgery formula for the smooth Yamabe invariant σ(M ) of a compact manifold M . Assume that N is obtained from M by surgery of codimension at least 3. We prove the existence of a positive constant Λn, depending only on the dimension n of M , such that σ(N ) ≥ min{σ(M ), Λn}.

The Yamabe problem on stratified spaces

“…We identify the local Yamabe invariants at all point p ∈ M as higher versions of the cylindrical/conic Yamabe invariants discussed above; these are simply the global Yamabe invariants for the model spaces R k × C(Z), or (conformally) equivalently, H k+1 × Z, where Z is a compact iterated edge space with lower singular 'depth' than the original space M . The special cases of these invariants when Z = S n−k−1 play an interesting role in the work of Ammann, Dahl and Humbert [5], [6], where quantitative estimates of the change of the σ-Yamabe invariant (which is the supremum of the Yamabe constants over all conformal classes) under surgeries are obtained. Finally, using the more specialized analytic tools available for the study of PDE on smoothly stratified spaces, we prove sharp regularity results about the behaviour of the minimizer u (or indeed any solution of the Yamabe equation) at the singular strata of M .…”

The Yamabe equation on manifolds of bounded geometry

“…In Theorem 13, we will show that for almost homogeneous manifolds (for the Definition see 13) with uniformly positive scalar curvature one can drop the assumption on Q. This was shown to the author by Akutagawa who proved this by exhaustion of the manifold at infinity, similarly as in [ Then, as an application we will apply this result in Example 15 to products of spheres with hyperbolic spaces that are the non-compact model spaces that appear in the surgery results for the Yamabe invariant in [3].…”

“…Those spaces appeared in [3] and have the symmetries required in the last remark. Their scalar curvature is constant and given by scal gc…”

Aubin’s Lemma for the Yamabe constants of infinite coverings and a positive mass theorem

“…[1][2][3][4]6,7,39]). In this article, we will present some extensions to Aubin's Lemma which will include, as a particular case, the first equality in the above working hypothesis (4).…”