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

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