Abstract
In this paper we consider compact oriented hypersurfaces M with constant mean curvature and two principal curvatures immersed in the Euclidean sphere. In the minimal case, Perdomo (Perdomo 2004) andWang (Wang 2003) obtained an integral inequality involving the square of the norm of the second fundamental form of M, where equality holds only if M is the Clifford torus. In this paper, using the traceless second fundamental form of M, we extend the above integral formula to hypersurfaces with constant mean curvature and give a new characterization of the H(r)-torus.
Highlights
Let M be a compact minimal hypersurface of the (n + 1)-dimensional unit Euclidean sphere Sn+1
In (Otsuki 1970) Otsuki proved that minimal hypersurfaces of Sn+1 having distinct principal curvatures of multiplicities k and m = n − k ≥ 2 are locally product of spheres of the type Sm(c1) × Sn−m(c2), and he constructed examples of compact minimal hypersurfaces in Sn+1 with two distinct principal
Let M be a compact oriented hypersurface immersed in the sphere Sn+1, with two distinct principal curvatures λ and μ with multiplicities 1 and n − 1, respectively
Summary
Let M be a compact minimal hypersurface of the (n + 1)-dimensional unit Euclidean sphere Sn+1. 2000) proved that if M is a compact minimal hypersurface with two principal curvatures, one of them with multiplicity 1 and S ≥ n, S = n and M is a Clifford torus. Let M be a compact oriented hypersurface immersed in the sphere Sn+1, with two distinct principal curvatures λ and μ with multiplicities 1 and n − 1, respectively. Perdomo (Perdomo 2004) and Wang (Wang 2003) independently obtained the following integral formula for compact minimal hypersurfaces with two principal curvatures immersed in Sn+1. In this paper we will extend the integral formula (1) for compact hypersurfaces √with constant mean curvature and obtain a new characterization of the H (r)-torus Sn−1(r) × S1( 1 − r2 ). √ with equality only if M is an H (r)-torus Sn−1(r) × S1( 1 − r2)
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