Enlargeable metrics on nonspin manifolds

Abstract:

We show that an enlargeable Riemannian metric on a (possibly nonspin) manifold cannot have uniformly positive scalar curvature. This extends a well-known result of Gromov and Lawson to the nonspin setting. We also prove that every noncompact manifold admits a nonenlargeable metric. In proving the first result, we use the main result of the recent paper by Schoen and Yau on minimal hypersurfaces to obstruct positive scalar curvature in arbitrary dimensions. More concretely, we use this to study nonzero degree maps $f\colon X^n\rightarrow S^k\times T^{n-k}$, with $k=1,2,3$. When $X$ is a closed oriented manifold endowed with a metric $g$ of positive scalar curvature and the map $f$ is (possibly area) contracting, we prove inequalities relating the lower bound of the scalar curvature of $g$ and the contracting factor of the map $f$.