Foliations and Chern-Heinz inequalities

Abstract

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 ${\mathbb{R}}^n$.

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

References in corpus (1)

• Eigenvalue estimates for submanifolds with locally bounded mean curvature

Cited by in corpus (3)

• Complete Foliations of Space Forms by Hypersurfaces
• Heinz type estimates for graphs in Euclidean space
• Bernstein-Heinz-Chern results in calibrated manifolds