Nonnegative scalar curvature and area decreasing maps on complete foliated manifolds

Abstract

Let $(M,g^{TM})$ be a noncompact complete Riemannian manifold of dimension $n$, and let $F\subseteq TM$ be an integrable subbundle of $TM$. Let $g^F=g^{TM}{|}_F$ be the restricted metric on $F$ and let $k^F$ be the associated leafwise scalar curvature. Let $f:M\to S^n(1)$ be a smooth area decreasing map along $F$, which is locally constant near infinity and of non-zero degree. We show that if $k^F>\mathrm{rk}(F)(\mathrm{rk}(F)-1)$ on the support of $\mathrm{d}f$, and either $TM$ or $F$ is spin, then $\inf (k^F)<0$. As a consequence, we prove Gromov's sharp foliated ${\otimes }_{\epsilon }$-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.

29 pages, 3 figures, the published version

Cited by in corpus (4)

• Rigidity of 3D spherical caps via $\mu$-bubbles
• The odd-dimensional long neck problem via spectral flow
• Product Inequalities for ${\mathbb{T}}^{⋊}$-Stabilized Scalar Curvature
• $\mathrm{K}$-cowaist on complete foliated manifolds