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First stability eigenvalue characterization of Clifford hypersurfaces

Authors:	Oscar Perdomo

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

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 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 , let $\vert M\vert$ denote the area of and let denote the genus of . We will prove that $\beta\vert M\vert\ge 8\pi(g-1)$ . Moreover, if 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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