Nonnegative Scalar Curvature and Area Decreasing Maps

Abstract

Let $(M,g^{TM})$ be a noncompact complete spin Riemannian manifold of even dimension $n$, with $k^{TM}$ denote the associated scalar curvature. Let $f:M\to S^n(1)$ be a smooth area decreasing map, which is locally constant near infinity and of nonzero degree. We show that if $k^{TM}\ge n(n-1)$ on the support of $\mathrm{d}f$, then $\inf \left(k^{TM}\right)<0$. 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.

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