Enlargeable metrics on nonspin manifolds

TL;DR

This paper extends Gromov and Lawson's result to show that enlargeable metrics on nonspin manifolds cannot have positive scalar curvature, and demonstrates that all noncompact manifolds admit nonenlargeable metrics, using minimal hypersurface techniques.

Contribution

It generalizes the nonexistence of positive scalar curvature metrics to enlargeable nonspin manifolds and introduces methods involving minimal hypersurfaces and degree maps.

Findings

Enlargeable metrics on nonspin manifolds cannot have positive scalar curvature.

Every noncompact manifold admits a nonenlargeable metric.

Inequalities relating scalar curvature bounds and contracting maps to tori and spheres.

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 from a manifold X to the product of the k-sphere with the n-k dimensional torus, 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.