Nonnegative scalar curvature and area decreasing maps on complete foliated manifolds

Autor: Su, Guangxiang, Wang, Xiangsheng, Zhang, Weiping
Rok vydání: 2021
Předmět:
Zdroj: J. reine angew. Math. 790 (2022), 85-113
Druh dokumentu: Working Paper
DOI: 10.1515/crelle-2022-0038
Popis: 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 $k^F> {\rm rk}(F)({\rm rk}(F)-1)$ on the support of ${\rm d}f$, and either $TM$ or $F$ is spin, then $\inf (k^F)<0$. As a consequence, we prove Gromov's sharp foliated $\otimes_\varepsilon$-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.
Comment: 29 pages, 3 figures, the published version
Databáze: arXiv