Nonnegative scalar curvature and area decreasing maps on complete foliated manifolds

Abstract

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