An extension of Barta’s Theorem and geometric applications

Annals of Global Analysis and Geometry - Tập 31 Số 4 - Trang 345-362 - 2007
G. Pacelli Bessa1, J. Fábio Montenegro1
1Departamento de Matematica, Universidade Federal do Ceará-UFC, Campus do Pici, Bloco. 914, 60455-760, Fortaleza-Ceará, Brazil

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Barta J. (1937) Sur la vibration fundamentale d’une membrane. C. R. Acad. Sci. 204, 472–473

Berger M., Gauduchon P., Mazet E. (1974) Le Spectre d’une Variété Riemannienes. Lect. Notes Math. 194, Springer-Verlag

Bessa G.P., Montenegro J.F. (2003) Eigenvalue estimates for submanifolds with locally bounded mean curvature. Ann. Global Anal. and Geom. 24, 279–290

Bishop R., Crittenden R. (1964) Geometry of Manifolds. Academic Press, New York

Chavel I. (1984) Eigenvalues in Riemannian Geometry. Pure and Applied Mathematics, Academic Press, INC

Cheng S.Y. (1975) Eigenfunctions and eigenvalues of the Laplacian. Am. Math. Soc Proc. Symp. Pure Math. 27, part II, 185–193

Cheung L.-F., Leung P.-F. (2001) Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space. Math. Z. 236, 525–530

Cheng S.Y., Li P., Yau S.T. (1984) Heat equations on minimal submanifolds and their applications. Amer. J. Math. 106, 1033–1065

Fischer-Colbrie D., Schoen R. (1980) The structure of complete stable minimal surfaces in 3-manifolds of non negative scalar curvature. Comm. Pure and Appl. Math. 33, 199–211

Grigoryan A. (1999) Analytic and geometric background of recurrence and non-explosion of the brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. 36(2):135–249

Kasue A. (1984) On a lower bound for the first eigenvalue of the Laplace operator on a Riemannian manifold. Ann. Sci. éc. Norm. Sup. 17, 31–44

Kazdan J., Kramer R. (1978) Invariant Criteria for existence of solutions to second-order quasilinear elliptic equations. Comm. Pure. Appl. Math. 31(5):619–645

Kazdan J., Warner F.W. (1975) Remarks on some quasilinear elliptic equations. Comm. Pure Appl. Math. 28, 567–597

Jorge L., Koutrofiotis D. (1980) An estimate for the curvature of bounded submanifolds. Amer. J. Math. 103(4):711–725

McKean H.P. (1970) An upper bound for the spectrum of $${\triangle}$$ on a manifold of negative curvature. J. Differ. Geom. 4, 359–366

Matilla P. Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press (1995).

Nadirashvili N. (1996) Hadamard’s and Calabi-Yau’s conjectures on negatively curved and minimal surfaces. Invent. Math. 126, 457–465

Schoen R.: Stable minimal surfaces in three manifolds. Seminar on Minimal Submanifolds. Annals of Math Studies. Princeton University Press.

Schoen, R., Yau, S.T.: Lectures on Differential Geometry. Conference Proceedings and Lecture Notes in Geometry and Topology, vol. 1, 1994.

Yau S.T. (2000) Review of Geometry and Analysis Asian, J Math 4(1): 235–278

Whitney H. (1957) Geometric Integration Theory. Princeton Mathematical Series, Princeton University Press