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Let(M,gT⁢M){(M,g^{TM})}be a noncompact complete Riemannian manifold of dimension  n , and letF⊆T⁢M{F\subseteq TM}be an integrable subbundle of  TM . LetgF=gT⁢M|F{g^{F}=g^{TM}|_{F}}be the restricted metric on  F  and letkF{k^{F}}be the associated leafwise scalar curvature. Letf:M→Sn⁢(1){f:M\to 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 ifkF&gt;rk⁢(F)⁢(rk⁢(F)-1){k^{F}&gt;{\rm rk}(F)({\rm rk}(F)-1)}on the support ofd⁢f{{\rm d}f}, and either  TM  or  F  is spin, theninf⁡(kF)&lt;0{\inf(k^{F})&lt;0}. As a consequence, we prove Gromov’s sharp foliated⊗ε{\otimes_{\varepsilon}}-twisting conjecture. Using the same method, we also extend two famous non-existence results due to Gromov and Lawson aboutΛ2{\Lambda^{2}}-enlargeable metrics (and/or manifolds) to the foliated case.

Nonnegative scalar curvature and area decreasing maps on complete foliated manifolds