First stability eigenvalue characterization of Clifford hypersurfaces
Author:
Oscar Perdomo
Journal:
Proc. Amer. Math. Soc. 130 (2002), 33793384
MSC (2000):
Primary 53A10
Published electronically:
April 11, 2002
MathSciNet review:
1913017
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Abstract: The stability operator of a compact oriented minimal hypersurface is given by , where is the norm of the second fundamental form. Let be the first eigenvalue of and define . In 1968 Simons proved that for any nonequatorial minimal hypersurface . In this paper we will show that only for Clifford hypersurfaces. For minimal surfaces in , let denote the area of and let denote the genus of . We will prove that . Moreover, if is embedded, then we will prove that . If in addition to the embeddeness condition we have that , then we will prove that .
 [A]
F. Almgren, Some interior regularity theorems for minimal surfaces and an extension of Bernstein theorem, Ann. of Math. 85 (1966) pp 277292. MR 34:702
 [C]
Isaac Chavel, Eigenvalues in Riemannian Geometry, Pure and Applied Mathematics. 115, Academic Press, 1984. MR 86g:58140
 [CDK]
S.S. Chern, M. DoCarmo, and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, Functional Analysis and Related Fields, Proc. Conf. M. Stone, Springer, 1970, pp 5975. MR 42:8424
 [CW]
H. Choi and A. Wang, A first eigenvalue estimate for minimal hypersurfaces, J. Differential Geometry 18 (1983) pp 559562. MR 85d:53028
 [L]
H. B. Lawson, Complete minimal surfaces in , Ann. of Math. (2) 92 (1970) pp 335374. MR 42:5170
 [L1]
H. B. Lawson, Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (2) 89 (1969) pp 187197. MR 38:6505
 [P]
O. Perdomo, First eigenvalue and index: Two characterizations of minimal Clifford hypersurfaces of spheres, Ph.D. Thesis, Indiana University, 2000.
 [S]
J. Simons, Minimal Varieties in Riemannian manifolds, Ann. of Math. 88 (1968), pp 62105.
 [SL]
L. Simon, Lectures on Geometry Measure Theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, 3, 1983. MR 87a:49001
 [SS]
L. Simon, B. Solomon, Minimal hypersurfaces asymptotic to quadratic cones in , Invent. Math. 86 (1986), pp 535551. MR 87k:49047
 [YY]
P. Yang and S.T. Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa 7 (1980) pp 5563. MR 81m:58084
 [A]
 F. Almgren, Some interior regularity theorems for minimal surfaces and an extension of Bernstein theorem, Ann. of Math. 85 (1966) pp 277292. MR 34:702
 [C]
 Isaac Chavel, Eigenvalues in Riemannian Geometry, Pure and Applied Mathematics. 115, Academic Press, 1984. MR 86g:58140
 [CDK]
 S.S. Chern, M. DoCarmo, and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, Functional Analysis and Related Fields, Proc. Conf. M. Stone, Springer, 1970, pp 5975. MR 42:8424
 [CW]
 H. Choi and A. Wang, A first eigenvalue estimate for minimal hypersurfaces, J. Differential Geometry 18 (1983) pp 559562. MR 85d:53028
 [L]
 H. B. Lawson, Complete minimal surfaces in , Ann. of Math. (2) 92 (1970) pp 335374. MR 42:5170
 [L1]
 H. B. Lawson, Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (2) 89 (1969) pp 187197. MR 38:6505
 [P]
 O. Perdomo, First eigenvalue and index: Two characterizations of minimal Clifford hypersurfaces of spheres, Ph.D. Thesis, Indiana University, 2000.
 [S]
 J. Simons, Minimal Varieties in Riemannian manifolds, Ann. of Math. 88 (1968), pp 62105.
 [SL]
 L. Simon, Lectures on Geometry Measure Theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, 3, 1983. MR 87a:49001
 [SS]
 L. Simon, B. Solomon, Minimal hypersurfaces asymptotic to quadratic cones in , Invent. Math. 86 (1986), pp 535551. MR 87k:49047
 [YY]
 P. Yang and S.T. Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa 7 (1980) pp 5563. MR 81m:58084
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Additional Information
Oscar Perdomo
Affiliation:
Departamento de Matematicas, Universidad del Valle, Cali, Colombia
Email:
osperdom@mafalda.univalle.edu.co
DOI:
http://dx.doi.org/10.1090/S0002993902064511
PII:
S 00029939(02)064511
Received by editor(s):
September 8, 2000
Received by editor(s) in revised form:
June 6, 2001
Published electronically:
April 11, 2002
Communicated by:
Bennett Chow
Article copyright:
© Copyright 2002 American Mathematical Society
