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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: The stability operator of a compact oriented minimal hypersurface $M^{n-1}\subset S^n$ is given by $J = -\Delta -\Vert A\Vert^2-(n-1)$, where $\Vert A\Vert$ is the norm of the second fundamental form. Let $\lambda_1$ be the first eigenvalue of $J$ and define $\beta = -\lambda_1-2(n-1)$. In 1968 Simons proved that $\beta\ge0$ for any non-equatorial minimal hypersurface $M\subset S^n$. In this paper we will show that $\beta=0$ only for Clifford hypersurfaces. For minimal surfaces in $S^3$, let $\vert M\vert$ denote the area of $M$ and let $g$ denote the genus of $M$. We will prove that $\beta\vert M\vert\ge 8\pi(g-1)$. Moreover, if $M$is embedded, then we will prove that $\beta \ge \frac{g-1}{g+1}$. If in addition to the embeddeness condition we have that $\beta<1$, then we will prove that $\vert M\vert\le \frac{16 \pi}{1-\beta} $.

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  • [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

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