Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On A. Hurwitz' method in isoperimetric inequalities


Author: Isaac Chavel
Journal: Proc. Amer. Math. Soc. 71 (1978), 275-279
MSC: Primary 53C40
DOI: https://doi.org/10.1090/S0002-9939-1978-0493885-2
MathSciNet review: 0493885
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that if M is complete simply connected with nonpositive sectional curvatures, $ \Omega $ a minimal submanifold of M with connected suitably oriented boundary $ \Gamma $ then $ {\lambda ^{1/2}}V/A \leqslant {(n - 1)^{1/2}}/n$ where V is the volume of $ \Omega $, A the volume of $ \Gamma ,\lambda $ the first nonzero eigenvalue of the Laplacian of $ \Gamma $, and $ n( \geqslant 2)$ is the dimension of $ \Omega $.


References [Enhancements On Off] (What's this?)

  • [1] R. L. Bishop and R. J. Crittenden, Geometry of Manifolds, Academic Press, New York, 1964. MR 0169148 (29:6401)
  • [2] T. Carleman, Zur theorie der Minimalflächen, Math. Z. 9 (1921), 154-160. MR 1544458
  • [3] I. Chavel and E. A. Feldman, Isoperimetric inequalities on curved surfaces, Advances in Math. (to appear). MR 591721 (82i:53056)
  • [4] R. E. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II, Interscience, New York, 1962.
  • [5] D. Hoffman and J. Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math. 29 (1974), 715-727. MR 0365424 (51:1676)
  • [6] C. C. Hsiung, Isoperimetric inequalities for two dimensional manifolds with boundary, Ann. of Math. 73 (1961), 213-220. MR 0130637 (24:A497)
  • [7] A. Hurwitz, Sur le problème des isopérimètres, C. R. Acad. Sci. Paris 132 (1901), 401-403.
  • [8] J. H. Jellet, Sur la surface dont la courbure moyenne est constante, J. Math. Pure Appl. 18 (1853), 163-167.
  • [9] W. T. Reid, The isoperimetric inequality and associated boundary problems, J. Math. Mech. 8 (1959), 571-581. MR 0130623 (24:A483)
  • [10] R. C. Reilly, On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space, Comment. Math. Helv. 52 (1977). MR 0482597 (58:2657)
  • [11] J. Simons, Minimal varieties in Riemannian manifolds, Ann. Math. 88 (1968), 62-105. MR 0233295 (38:1617)
  • [12] T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380-385. MR 0198393 (33:6551)
  • [13] H. F. Weinberger, An isoperimetric inequality for the n-dimensional free membrane problem, J. Rational Mech. Anal. 5 (1956), 633-636. MR 0079286 (18:63c)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 53C40

Retrieve articles in all journals with MSC: 53C40


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1978-0493885-2
Keywords: Sectional curvatures, minimal submanifold, volume, area, eigenvalues of Laplacian
Article copyright: © Copyright 1978 American Mathematical Society

American Mathematical Society