Llarull's theorem on odd dimensional manifolds: the noncompact case

TL;DR

This paper extends Llarull's theorem to odd-dimensional, noncompact spin manifolds, showing that under certain curvature and map conditions, the scalar curvature must be negative somewhere, answering a question posed by Gromov.

Contribution

It generalizes Llarull's theorem to noncompact, odd-dimensional manifolds with specific curvature and map conditions, providing a new insight into scalar curvature constraints.

Findings

If the scalar curvature is bounded below by a positive constant on the support of the differential of the map, then the scalar curvature must be negative somewhere.

The result applies to noncompact manifolds with a smooth area decreasing map of nonzero degree.

Answers Gromov's question regarding scalar curvature on noncompact odd-dimensional manifolds.

Abstract

Let $(M,g^{TM})$ be an odd dimensional ($\dim M\geq 3$) connected oriented noncompact complete spin Riemannian manifold. Let $k^{TM}$ be the associated scalar curvature. Let $f:M\to S^{\dim M}(1)$ be a smooth area decreasing map which is locally constant near infinity and of nonzero degree. Suppose $k^{TM}\geq ({\dim M})({\dim M}-1)$ on the support of ${\rm d}f$, we show that $\inf(k^{TM})<0$. This answers a question of Gromov.