Spectral Geroch conjecture and noncompact area enlargeable summands

Abstract

We prove that the connected sum of a possibly noncompact area enlargeable manifold $M_1$ with an arbitrary spin manifold $M_2$ of the same dimension admits no complete Riemannian metric of uniformly positive scalar curvature. This extends a theorem of Wang--Zhang, where $M_1$ is assumed closed, to noncompact enlargeable summands; in this generality the uniform positivity hypothesis enters the argument in an essential way. We also prove a spectral analogue of the generalized Geroch conjecture in terms of the $γ$-spectral constant: for $γ>(\dim M_1-1)/(4\dim M_1)$, such a connected sum carries no complete metric with positive $γ$-spectral constant. The proofs are based on a covering connected sum construction together with the scalar-cowaist and spectral-cowaist inequalities.