On the generalized Geroch conjecture for complete spin manifolds

TL;DR

This paper proves that certain noncompact spin manifolds, when connected summed with enlargeable manifolds, cannot support complete metrics of positive scalar curvature, confirming a generalized Geroch conjecture in the spin case.

Contribution

It establishes a new obstruction to positive scalar curvature on noncompact spin manifolds involving enlargeable manifolds, extending the Geroch conjecture to the spin setting.

Findings

Connected sum with enlargeable manifolds obstructs positive scalar curvature.

Confirms the generalized Geroch conjecture for spin manifolds when W=T^n.

Provides new insights into scalar curvature obstructions in noncompact manifolds.

Abstract

Let $W$ be a closed area enlargeable manifold in the sense of Gromov-Lawson and $M$ be 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.