Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Available in electronic format
Available in print format
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

Eigenvalue estimates for minimal surfaces in hyperbolic space

Author(s): Alberto Candel
Journal: Trans. Amer. Math. Soc. 359 (2007), 3567-3575.
MSC (2000): Primary 53A10, 53C21
Posted: March 7, 2007
MathSciNet review: 2302506
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: This paper gives an upper bound for the first eigenvalue of the universal cover of a complete, stable minimal surface in hyperbolic space, and a sharper one for least area disks.


References:

1.
Michael T. Anderson, Complete minimal varieties in hyperbolic space, Invent. Math. 69 (1982), no. 3, 477-494.MR 0679768 (84c:53005)

2.
-, Complete minimal hypersurfaces in hyperbolic $ n$-manifolds, Comment. Math. Helv. 58 (1983), no. 2, 264-290.MR 0705537 (85e:53076)

3.
João Lucas Barbosa and Manfredo do Carmo, Stability of minimal surfaces and eigenvalues of the Laplacian, Math. Z. 173 (1980), no. 1, 13-28.MR 0584346 (81h:53007)

4.
Robert Brooks, A relation between growth and the spectrum of the Laplacian, Math. Z. 178 (1981), no. 4, 501-508.MR 0638814 (83a:58089)

5.
Manfredo do Carmo and M. Dajczer, Rotation hypersurfaces in spaces of constant curvature, Trans. Amer. Math. Soc. 277 (1983), no. 2, 685-709.MR 0694383 (85b:53055)

6.
Manfredo do Carmo and C. K. Peng, Stable complete minimal surfaces in $ {\bf R}\sp{3}$ are planes, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 6, 903-906.MR 0546314 (80j:53012)

7.
Isaac Chavel, Isoperimetric inequalities, Cambridge Tracts in Mathematics, vol. 145, Cambridge University Press, Cambridge, 2001, Differential geometric and analytic perspectives. MR 1849187 (2002h:58040)

8.
Shiu Yuen Cheng, Peter Li, and Shing-Tung Yau, Heat equations on minimal submanifolds and their applications, Amer. J. Math. 106 (1984), no. 5, 1033-1065. MR 0761578 (85m:58171)

9.
Jaigyoung Choe and Robert Gulliver, Isoperimetric inequalities on minimal submanifolds of space forms, Manuscripta Math. 77 (1992), no. 2-3, 169-189. MR 1188579 (93k:53059)

10.
Doris Fischer-Colbrie and Richard Schoen, The structure of complete stable minimal surfaces in $ 3$-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), no. 2, 199-211. MR 0562550 (81i:53044)

11.
Robert Hermann, Focal points of closed submanifolds of Riemannian spaces, Indag. Math. 25 (1963), 613-628.MR 0158333 (28:1558)

12.
Shigeo Kawai, Operator $ \Delta-aK$ on surfaces, Hokkaido Math. J. 17 (1988), no. 2, 147-150. MR 0945852 (89j:58149)

13.
Masatoshi Kokubu, Weierstrass representation for minimal surfaces in hyperbolic space, Tohoku Math. J. (2) 49 (1997), no. 3, 367-377.MR 1464184 (98f:53008)

14.
Hiroshi Mori, Remarks on the paper of Barbosa and do Carmo, Arch. Math. (Basel) 37 (1981), no. 2, 173-178. MR 0640804 (83e:53061)

15.
Geraldo de Oliveira Filho and Marc Soret, Complete minimal surfaces in hyperbolic space, Math. Ann. 311 (1998), no. 3, 397-419. MR 1637915 (2000a:53013)

16.
A. V. Pogorelov, On the stability of minimal surfaces, Dokl. Akad. Nauk SSSR 260 (1981), no. 2, 293-295. MR 0630142 (83b:49043)

17.
Konrad Polthier, Geometric a priori estimates for hyperbolic minimal surfaces, Bonner Mathematische Schriften [Bonn Mathematical Publications], 263, Universität Bonn Mathematisches Institut, Bonn, 1994, Dissertation, Universität Bonn, Bonn, 1993. MR 1293964 (95h:53011)

18.
Richard Schoen, Estimates for stable minimal surfaces in three-dimensional manifolds, Seminar on minimal submanifolds, Ann. of Math. Stud., vol. 103, Princeton Univ. Press, Princeton, NJ, 1983, pp. 111-126.MR 0795231 (86j:53094)

19.
Karen K. Uhlenbeck, Closed minimal surfaces in hyperbolic $ 3$-manifolds, Seminar on minimal submanifolds, Ann. of Math. Stud., vol. 103, Princeton Univ. Press, Princeton, NJ, 1983, pp. 147-168.MR 0795233 (87b:53093)


Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53A10, 53C21

Retrieve articles in all Journals with MSC (2000): 53A10, 53C21


Additional Information:

Alberto Candel
Affiliation: Department of Mathematics, California State University, Northridge, Northridge, California 91330
Email: alberto.candel@csun.edu

DOI: 10.1090/S0002-9947-07-04104-9
PII: S 0002-9947(07)04104-9
Received by editor(s): February 14, 2005
Posted: March 7, 2007
Additional Notes: This research was supported by N.S.F. Grant 0205825
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia