Abstract
In this paper, we provide various Sobolev-type inequalities for smooth nonnegative functions with compact support on a submanifold with variable mean curvature in a Riemannian manifold whose sectional curvature is bounded above by a constant. We further obtain the corresponding linear isoperimetric inequalities involving mean curvature. We also provide various first Dirichlet eigenvalue estimates for submanifolds with bounded mean curvature.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Allard W.: On the first variation of a varifold. Ann. Math. 95(2), 417–491 (1972)
Almgren F.: Optimal isoperimetric inequalities. Indiana Univ. Math. J. 35(3), 451–547 (1986)
Bessa G., Montenegro J.: Eigenvalue estimates for submanifolds with locally bounded mean curvature. Ann. Global Anal. Geom. 24(3), 279–290 (2003)
Cheung L.F., Leung P.F.: The mean curvature and volume growth of complete noncompact submanifolds. Differ. Geom. Appl. 8(3), 251–256 (1998)
Cheung L.F., Leung P.F.: Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space. Math. Z. 236, 525–530 (2001)
Choe J.: The isoperimetric inequality for a minimal surface with radially connected boundary. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17(4), 583–593 (1990)
Choe J.: The isoperimetric inequality for minimal surfaces in a Riemannian manifold. J. reine angew. Math. 506, 205–214 (1999)
Choe J., Gulliver R.: Isoperimetric inequalities on minimal submanifolds of space forms. Manuscripta. Math. 77, 169–189 (1992)
Feinberg J.: The isoperimetric inequality for doubly-connected minimal surfaces in R n. J. Analyse Math. 32, 249–278 (1977)
Hoffman D., Spruckc J.: Sobolev and isoperimetric inequalities for Riemannian submanifolds. Comm. Pure Appl. Math. 27, 715–727 (1974)
Karp L.: Differential inequalities on complete Riemannian manifolds and applications. Math. Ann. 272(4), 449–459 (1985)
Li P., Schoen R., Yau S.T.: On the isoperimetric inequality for minimal surfaces. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 11(2), 237–244 (1984)
McKean H.P.: An upper bound to the spectrum of Δ on a manifold of negative curvature. J. Diff. Geom. 4, 359–366 (1970)
Michael J., Simon L.M.: Sobolev and mean-value inequalities on generalized submanifolds of \({\mathbb{R}^n}\) . Comm. Pure. Appl. Math. 26, 361–379 (1973)
Osserman R.: The isoperimetric inequality. Bull. Amer. Math. Soc. 84, 1182–1238 (1978)
Osserman R., Schiffer M.: Doubly-connected minimal surfaces. Arch. Rational Mech. Anal. 58, 285–307 (1975)
Schoen, R., Yau, S.-T.: Lectures on differential geometry. Conference Proceedings and Lecture Notes in Geometry and Topology. I. International Press (1994)
Stone A.: On the isoperimetric inequality on a minimal surface. Calc. Var. Partial Differ. Equ. 17, 369–391 (2003)
Yau S.-T.: Isoperimetric constants and the first eigenvalue of a compact manifold. Ann. Sci. Ecole Norm. Sup. 8, 487–507 (1975)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Andreas Cap.
This research was supported by the Sookmyung Women’s University Research Grants 2011.
Rights and permissions
About this article
Cite this article
Seo, K. Isoperimetric inequalities for submanifolds with bounded mean curvature. Monatsh Math 166, 525–542 (2012). https://doi.org/10.1007/s00605-011-0332-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-011-0332-2