Nonnegative scalar curvature and area decreasing maps on complete foliated manifolds

Let.M; g(TM)/be a noncompact complete Riemannian manifold of dimension n, and let F subset of TM be an integrable subbundle of TM. Let g(F) D g(TM)vertical bar F be the restricted metric on F and let k(F) be the associated leafwise scalar curvature. Let f : M -> S-n(1) be a smooth area decreasing map along F, which is locally constant near infinity and of non-zero degree. We show that if k(F) > rk(F)(rk(F) - 1) on the support of df, and either TM or F is spin, then inf (k(F)) < 0. As a consequence, we prove Gromov's sharp foliated circle times(epsilon)-twisting conjecture. Using the same method, we also extend two famous non-existence results due to Gromov and Lawson about Lambda(2)-enlargeable metrics (and/or manifolds) to the foliated case.