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Lower bounds of Lipschitz constants on foliations. (English) Zbl 1428.53035
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)], 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 [W. Zhang, Ann. Math. (2) 185, No. 3, 1035–1068 (2017; Zbl 1404.53038)]) is less or equal to $$1$$.
##### MSC:
 53C12 Foliations (differential geometric aspects) 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 53C20 Global Riemannian geometry, including pinching 57N65 Algebraic topology of manifolds
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##### References:
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