Nonnegative Scalar Curvature and Area Decreasing Maps

Let $(M,g^{TM})$ be a noncompact complete spin Riemannian manifold of even dimension $n$, with $k^{TM}$ denote the associated scalar curvature. Let $f\text{\hspace{0.17em}⁣}:M\to S^n(1)$ 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 \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.