Nonnegative Scalar Curvature and Area Decreasing Maps

Abstract

Let (M, gᵀᴹ) be a noncompact complete spin Riemannian manifold of even dimension n, with kᵀᴹ denoting the associated scalar curvature. Let 𝑓: M → Sⁿ(1) be a smooth area decreasing map, which is locally constant near infinity and of nonzero degree. We show that if kᵀᴹ ≥ n(n−1) on the support of d𝑓, then inf(kᵀᴹ) < 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.