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On Bernstein–Heinz–Chern–Flanders inequalities

Published online by Cambridge University Press:  01 March 2008


J. L. M. BARBOSA
Affiliation:
Department of Mathematics, Universidade Federal do Ceará, 60455-760 Fortaleza-CE, Brazil. e-mail: lucas@mat.ufc.br, bessa@mat.ufc.br, fabio@mat.ufc.br
G. P. BESSA
Affiliation:
Department of Mathematics, Universidade Federal do Ceará, 60455-760 Fortaleza-CE, Brazil. e-mail: lucas@mat.ufc.br, bessa@mat.ufc.br, fabio@mat.ufc.br
J. F. MONTENEGRO
Affiliation:
Department of Mathematics, Universidade Federal do Ceará, 60455-760 Fortaleza-CE, Brazil. e-mail: lucas@mat.ufc.br, bessa@mat.ufc.br, fabio@mat.ufc.br
Corresponding

Abstract

We give an interpretation of the Chern–Heinz inequalities for graphs in order to extend them to transversally oriented codimension one C2-foliations of Riemannian manifolds. It contains Salavessa's work on mean curvature of graphs and fully generalizes results of Barbosa–Kenmotsu–Oshikiri [3] and Barbosa–Gomes–Silveira [2] about foliations of 3-dimensional Riemannian manifolds by constant mean curvature surfaces. This point of view of the Chern–Heinz inequalities can be applied to prove a Haymann–Makai–Osserman inequality (lower bounds of the fundamental tones of bounded open subsets Ω ⊂ ℝ2 in terms of its inradius) for embedded tubular neighbourhoods of simple curves of ℝn.


Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2008

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