A note on the generalized Geroch conjecture for complete spin manifolds

Abstract

Let $W$ be a closed area enlargeable manifold in the sense of Gromov-Lawson and $M$ a noncompact spin manifold, we show that the connected sum $M\# W$ admits no complete metric of positive scalar curvature. When $W=T^n$, this provides a positive answer to the generalized Geroch conjecture in the spin setting.Comment: 4 pages, comments are welcome