Foliations and Chern-Heinz inequalities
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 $Ω\subset \mathbb{R}^{2}$ in terms of its inradius) for embedded tubular neighborhoods of simple curves of $\mathbb{R}^{n}$.