Lower bounds of Lipschitz constants on foliations (Q2272955)

The normalized scalar curvature \(\tilde{k}_g\) of an \(n\)-dimensional Riemannian manifold \((M, g)\) is defined by \(\tilde{k}_g=\frac{k_g}{n(n-1)}\), where \(k_g\) is the usual scalar curvature. In [Math. Ann. 310, No. 1, 55--71 (1998; Zbl 0895.53037)], \textit{M. Llarull} proved the following theorem which confirms an older conjecture of M. Gromov: If \((M, g)\) is a compact spin manifold of dimension \(n\), and if \(f:(M, g)\rightarrow (S^n, g_0)\) is \(1\)-contracting (that is, \(\|\mathrm{d}f(v)\|\le \|v\|\) for all \(v\in \mathrm{T}M\)) and of non-zero degree, then either there exists \(x\in M\) with \(\tilde{k}_g(x)<1\), or \(M\equiv S^n\) and \(f\) is an isometry. (Here, \((S^n, g_0)\) denotes the \(n\)-dimensional Euclidean unit sphere with its standard metric.) In this paper, the author generalizes this theorem to the case in which \(M\) has a foliation \(F\) and \(f\) is \(1\)-contracting on \(F\). Then, there exists \(x\in M\) such that the normalized leafwise scalar curvature at \(x\) (see [\textit{W. Zhang}, Ann. Math. (2) 185, No. 3, 1035--1068 (2017; Zbl 1404.53038)]) is less or equal to \(1\).