First stability eigenvalue characterization of Clifford hypersurfaces

Author:
Oscar Perdomo

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

MSC (2000):
Primary 53A10

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

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 .

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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