First stability eigenvalue characterization of Clifford hypersurfaces

Author:
Oscar Perdomo

Journal:
Proc. Amer. Math. Soc. **130** (2002), 3379-3384

MSC (2000):
Primary 53A10

Published electronically:
April 11, 2002

MathSciNet review:
1913017

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Abstract | References | Similar Articles | Additional Information

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 non-equatorial 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. J. Almgren Jr.,*Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem*, Ann. of Math. (2)**84**(1966), 277–292. MR**0200816****[C]**Isaac Chavel,*Eigenvalues in Riemannian geometry*, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR**768584****[C-D-K]**S. S. Chern, M. do Carmo, and S. Kobayashi,*Minimal submanifolds of a sphere with second fundamental form of constant length*, Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968) Springer, New York, 1970, pp. 59–75. MR**0273546****[C-W]**Hyeong In Choi and Ai Nung Wang,*A first eigenvalue estimate for minimal hypersurfaces*, J. Differential Geom.**18**(1983), no. 3, 559–562. MR**723817****[L]**H. Blaine Lawson Jr.,*Complete minimal surfaces in 𝑆³*, Ann. of Math. (2)**92**(1970), 335–374. MR**0270280****[L1]**H. Blaine Lawson Jr.,*Local rigidity theorems for minimal hypersurfaces*, Ann. of Math. (2)**89**(1969), 187–197. MR**0238229****[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 62-105.**[SL]**Leon Simon,*Lectures on geometric measure theory*, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. MR**756417****[SS]**Leon Simon and Bruce Solomon,*Minimal hypersurfaces asymptotic to quadratic cones in 𝑅ⁿ⁺¹*, Invent. Math.**86**(1986), no. 3, 535–551. MR**860681**, 10.1007/BF01389267**[Y-Y]**Paul C. Yang and Shing Tung Yau,*Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**7**(1980), no. 1, 55–63. MR**577325**

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Additional Information

**Oscar Perdomo**

Affiliation:
Departamento de Matematicas, Universidad del Valle, Cali, Colombia

Email:
osperdom@mafalda.univalle.edu.co

DOI:
https://doi.org/10.1090/S0002-9939-02-06451-1

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