Foliations and Chern-Heinz inequalities

Details Foliations and Chern-Heinz inequalities Foliations and Chern-Heinz inequalities

2006-01-09

arxiv.org/abs/math/0601198v2

Math. Proc. Camb. Phil. Soc. 144, (2008), 457-464

Mathematics

Differential Geometry

This paper is an improvment of an earlier paper titled On Chern-Heinz Inequalities. 8 Pages, Latex

Scientific paper

10.1017/S0305004107000643

We extend the Chern-Heinz inequalities about mean curvature and scalar curvature of graphs of $C^{2}$-functions to leaves of transversally oriented codimension one $C^{2}$-foliations of Riemannian manifolds. That extends partially Salavessa's work on mean curvature of graphs and generalize results of Barbosa-Kenmotsu-Oshikiri \cite{barbosa-kenmotsu-Oshikiri} and Barbosa-Gomes-Silveira \cite{barbosa-gomes-silveira} about foliations of 3-dimensional Riemannian manifolds by constant mean curvature surfaces. These Chern-Heinz inequalities for foliations can be applied to prove Haymann-Makai-Osserman inequality (lower bounds of the fundamental tones of bounded open subsets $\Omega \subset \mathbb{R}^{2}$ in terms of its inradius) for embedded tubular neighborhoods of simple curves of $\mathbb{R}^{n}$.

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