Nonnegative Scalar Curvature and Area Decreasing Maps

TL;DR

This paper proves that on certain noncompact spin manifolds with scalar curvature bounds, area decreasing maps of nonzero degree force the scalar curvature to be negative somewhere, answering a question posed by Gromov.

Contribution

It introduces a deformation of the Dirac operator to establish scalar curvature constraints related to area decreasing maps, extending Gromov's question.

Findings

If scalar curvature is bounded below by n(n-1) on the support of the differential of an area decreasing map, then scalar curvature must be negative somewhere.

The method involves a simple deformation of the Dirac operator to derive the result.

The paper also presents an odd-dimensional analogue of the main theorem.

Abstract

Let $\big(M,g^{TM}\big)$ be a noncompact complete spin Riemannian manifold of even dimension $n$, with $k^{TM}$ denote the associated scalar curvature. Let $f\colon M\rightarrow S^{n}(1)$ be a smooth area decreasing map, which is locally constant near infinity and of nonzero degree. We show that if $k^{TM}\geq n(n-1)$ on the support of ${\rm d}f$, then $ \inf \big(k^{TM}\big)< 0$. This answers a question of Gromov. We use a simple deformation of the Dirac operator to prove the result. The odd dimensional analogue is also presented.