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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: [email protected], [email protected], [email protected]
G. P. BESSA
Affiliation:
Department of Mathematics, Universidade Federal do Ceará, 60455-760 Fortaleza-CE, Brazil. e-mail: [email protected], [email protected], [email protected]
J. F. MONTENEGRO
Affiliation:
Department of Mathematics, Universidade Federal do Ceará, 60455-760 Fortaleza-CE, Brazil. e-mail: [email protected], [email protected], [email protected]

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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References

REFERENCES

[1]Bañuelos, R. and Carroll, T.. Brownnian motion and the fundamental frequence of a drum. Duke Math. J. 75 (1994), 575602.CrossRefGoogle 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.CrossRefGoogle Scholar
[3]Barbosa, J. L. M., Kenmotsu, K. and Oshikiri, O.. Foliations by hypersurfaces with constant mean curvature. Math. Z. 207 (1991), 97108.CrossRefGoogle Scholar
[4]Bernstein, S.Sur la généralisation du problème de Dirichlet. Math. Ann. 69 (1910), 82136.CrossRefGoogle Scholar
[5]Bernstein, S.Sur les surfaces définies au moyen de leur courbure moyenne ou totale. Ann. École Norm. Sup. 27 (1909), 233256.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar
[10]Chern, S. S.On the curvature of a piece of hypersurface in Euclidean space. Abh. Math. Sem Hamburg 29 (1965), 7791.CrossRefGoogle Scholar
[11]Elbert, M. F.Constant positive 2-mean curvature hypersurfaces. Illinois J. Math. 46 (2002), 247267.CrossRefGoogle 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.CrossRefGoogle Scholar
[13]Flanders, H.Remark on mean curvature. J. London Math. Soc. (2) 41 (1966), 364366.CrossRefGoogle 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.CrossRefGoogle Scholar
[15]Haymann, W.Some bounds for principal frequency. Applicable Anal. 7 (1978), 247254.CrossRefGoogle Scholar
[16]Heinz, E.Über Flächen mit eindeutiger projektion auf eine ebene, deren krümmungen durc ungleichungen eingschränkt sind. Math. Ann. 129 (1955), 451454.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar
[21]Ros, A.Compactness of spaces of properly embedded minimal surfaces with finite total curvature. Indiana Univ. Math. J. 44 (1995), 139152.CrossRefGoogle 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