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)

 
 

 

Volume growth of submanifolds and the Cheeger isoperimetric constant


Authors: Vicent Gimeno and Vicente Palmer
Journal: Proc. Amer. Math. Soc. 141 (2013), 3639-3650
MSC (2010): Primary 53C20, 53C42
DOI: https://doi.org/10.1090/S0002-9939-2013-11664-3
Published electronically: June 14, 2013
MathSciNet review: 3080186
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We obtain an estimate of the Cheeger isoperimetric constant in terms of the volume growth for a properly immersed submanifold in a Riemannian manifold which possesses at least one pole and sectional curvature bounded from above.


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

  • [1] M. T. Anderson, The compactification of a minimal submanifold in Euclidean space by the Gauss map, I.H.E.S. preprint, 1984.
  • [2] I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press (1984). MR 768584 (86g:58140)
  • [3] I. Chavel, Isoperimetric inequalities. Differential geometric and analytic perspectives, Cambridge Tracts in Mathematics, 145. Cambridge University Press (2001). MR 1849187 (2002h:58040)
  • [4] I. Chavel, Riemannian geometry: A modern introduction, Cambridge Tracts in Mathematics, 108. Cambridge University Press (1993). MR 1271141 (95j:53001)
  • [5] J. Cheeger, A Lower Bound for the Smallest Eigenvalue of the Laplacian, Problems in Analysis (Papers dedicated to Salomon Bochner, 1969), pp 195-199, Princeton Univ. Press, NJ (1970). MR 0402831 (53:6645)
  • [6] Q. Chen, On the volume growth and the topology of complete minimal submanifolds of a Euclidean space, J. Math. Sci. Univ. Tokyo 2 (1995), 657-669. MR 1382525 (97g:53074)
  • [7] Q. Chen, On the area growth of minimal surfaces in $ \mathbb{H}^n$, Geometriae Dedicata 75 (1999), 263-273. MR 1689272 (2000h:53081)
  • [8] Q. Chen Qing and Y. Cheng, Chern-Osserman inequality for minimal surfaces in $ \mathbb{H}^n$, Proc. Amer. Math Soc. 128 (1999), 2445-2450. MR 1664325 (2000k:53006)
  • [9] V. Gimeno and V. Palmer, Extrinsic isoperimetry and compactification of minimal surfaces in Euclidean and hyperbolic spaces, Israel Jour. Math. 194 (2013), 539-553. MR 3047082
  • [10] R. Greene and S. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Math., 699 (1979), Springer Verlag, Berlin. MR 521983 (81a:53002)
  • [11] A. Grigor'yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. 36 (1999), 135-249. MR 1659871 (99k:58195)
  • [12] D. Hoffman and J. Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. of Pure and App. Mathematics 27 (1974), 715-727. MR 0365424 (51:1676)
  • [13] A. Hurtado, S. Markvorsen and V. Palmer, Torsional rigidity of submanifolds with controlled geometry, Math. Ann. 344 (2009), 511-542. MR 2501301 (2010g:53108)
  • [14] L.P. Jorge and D. Koutroufiotis, An estimate for the curvature of bounded submanifolds, Amer. Journal of Math. 103, n. 4 (1981), 711-725. MR 623135 (83d:53041b)
  • [15] S. Markvorsen and V. Palmer, The relative volume growth of minimal submanifolds, Archiv der Mathematik (Basel) 79 (2002), 507-514. MR 1967269 (2004a:53078)
  • [16] S. Markvorsen and V. Palmer, Torsional rigidity of minimal submanifolds, Proc. London Math. Soc. 93 (2006), 253-272. MR 2235949 (2008a:53060)
  • [17] S. Markvorsen and V. Palmer, Extrinsic isoperimetric analysis on submanifolds with curvatures bounded from below, J. Geom. Anal. 20 (2010), 388-421. MR 2579515 (2011c:53131)
  • [18] B. O'Neill, Semi-Riemannian Geometry; With Applications to Relativity, Academic Press (1983). MR 719023 (85f:53002)
  • [19] V. Palmer, Isoperimetric inequalities for extrinsic balls in minimal submanifolds and their applications, J. London Math. Soc. (2) 60 (1999), 607-616. MR 1724821 (2000j:53050)
  • [20] V. Palmer, On deciding whether a submanifold is parabolic or hyperbolic using its mean curvature, Simon Stevin Transactions on Geometry, vol. 1, 131-159, Simon Stevin Institute for Geometry, Tilburg, The Netherlands, 2010.
  • [21] S.T. Yau, Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold, Annales Scientifiques de L'E.N.S., 8, n. 4 (1975), 487-507. MR 0397619 (53:1478)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53C20, 53C42

Retrieve articles in all journals with MSC (2010): 53C20, 53C42


Additional Information

Vicent Gimeno
Affiliation: Departament de Matemàtiques-INIT, Universitat Jaume I, Castelló, Spain
Email: gimenov@guest.uji.es

Vicente Palmer
Affiliation: Departament de Matemàtiques-INIT, Universitat Jaume I, Castelló, Spain
Email: palmer@mat.uji.es

DOI: https://doi.org/10.1090/S0002-9939-2013-11664-3
Keywords: Cheeger isoperimetric constant, volume growth, submanifold, Chern-Osserman inequality.
Received by editor(s): April 29, 2011
Received by editor(s) in revised form: December 12, 2011
Published electronically: June 14, 2013
Additional Notes: This work was supported by Fundació Caixa Castelló-Bancaixa Grants P1.1B2006-34 and P1.1B2009-14 and by MICINN grant No. MTM2010-21206-C02-02.
Communicated by: Michael Wolf
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society