Hostname: page-component-7c8c6479df-fqc5m Total loading time: 0 Render date: 2024-03-19T09:17:16.468Z Has data issue: false hasContentIssue false

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

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bañuelos, R. and Carroll, T.. Brownnian motion and the fundamental frequence of a drum. Duke Math. J. 75 (1994), 575602.Google Scholar
[2]Barbosa, J. L. M., Gomes, J. M. and Silveira, A. M.. Foliations of 3-dimensional space forms by surfaces with constant mean curvature Bol. Soc. Bras. Mat. 18 (1987), 112.Google Scholar
[3]Barbosa, J. L. M., Kenmotsu, K. and Oshikiri, O.. Foliations by hypersurfaces with constant mean curvature. Math. Z. 207 (1991), 97108.Google Scholar
[4]Bernstein, S.Sur la généralisation du problème de Dirichlet. Math. Ann. 69 (1910), 82136.Google Scholar
[5]Bernstein, S.Sur les surfaces définies au moyen de leur courbure moyenne ou totale. Ann. École Norm. Sup. 27 (1909), 233256.Google Scholar
[6]Bessa, G. P., Jorge, L. P. and Oliveira–Filho, G.Half-space theorems for minimal surfaces. J. Diff. Geom. 57 (2001), 493508.Google Scholar
[7]Bessa, G. P. and Montenegro, J. F.. Eigenvalue estimates for submanifolds with locally bounded mean curvature. Ann. Global Anal. Geom. 24 (2003), 279290.Google Scholar
[8]Cheeger, J. A lower bound for the smallest eigenvalue of the Laplacian. Problems in Analysis (Princenton Univ. Press, 1970), 195199.Google Scholar
[9]Cheng, S. Y.Eigenvalue comparison theorems and its geometric applications. Math. Z. 143 (1975), 289297.Google Scholar
[10]Chern, S. S.On the curvature of a piece of hypersurface in Euclidean space. Abh. Math. Sem Hamburg 29 (1965), 7791.Google Scholar
[11]Elbert, M. F.Constant positive 2-mean curvature hypersurfaces. Illinois J. Math. 46 (2002), 247267.Google Scholar
[12]Fischer–Colbrie, D. and Schoen, R.The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature. Comm. Pure. Appl. Math. 33 (1980), 199211.Google Scholar
[13]Flanders, H.Remark on mean curvature. J. London Math. Soc. (2) 41 (1966), 364366.Google Scholar
[14]Fontenele, F. and Silva, S. L.Sharp estimates for size of balls in the complement of a hypersurface. Geom. Dedicata 115 (2005), 163179.Google Scholar
[15]Haymann, W.Some bounds for principal frequency. Applicable Anal. 7 (1978), 247254.Google Scholar
[16]Heinz, E.Über Flächen mit eindeutiger projektion auf eine ebene, deren krümmungen durc ungleichungen eingschränkt sind. Math. Ann. 129 (1955), 451454.Google Scholar
[17]Gårding, L.. An inequality for hyperbolic polinomials. J. Math. Mech. 8 (1959), 957965.Google Scholar
[18]Makai, E.A lower estimation of the principal frequencies of simply connected membranes. Acta Math. Acad. Sci. Hungar. 16 (1965), 319366.Google Scholar
[19]Montiel, S. and Ros, A. Compact hypersurfaces: the Alexandrov theorem for higher order mean curvatures. Differential Geometry, a symposium in honor of Manfredo do Carmo. Lawson, B. and Teneblat, K., (Eds.). Pitman Monographs, 52 (Longman Scientific and Technical, 1991), 179–296.Google Scholar
[20]Osserman, R.A note on Hymann's theorem on the bass note of a drum. Comment. Math. Helv. 52 (1977), 545555.Google Scholar
[21]Ros, A.Compactness of spaces of properly embedded minimal surfaces with finite total curvature. Indiana Univ. Math. J. 44 (1995), 139152.Google Scholar
[22]Isabel, M. C. Salavessa.Graphs with parallel mean curvature. Proc. Amer. Math. Soc. 107 (1989), 449458.Google Scholar
[23]Schoen, R.Estimates for stable minimal surfaces in three dimensional manifolds. Ann. of Math. Stud. 103 (1983), 111126.Google Scholar