A characterization of the Clifford Torus

Abstract

In this paper, we prove that an n-dimensional closed minimal hypersurface M with Ricci curvature Ric(M) ≥ n 2 of a unit sphere S n+1 (1) is isometric to a Clifford torus if n ≤ S ≤ n + 14(n+4) 9n+30 , where S is the squared norm of the second fundamental form of M .

A new bound on the Morse index of constant mean curvature tori of revolution in $${{\mathbb{S}}^{3}}$$

“…Later on, F. Urbano [24] proved that any compact minimal surface (not totally geodesic) in S 3 has index greater than or equal to five, with equality uniquely for the Clifford torus. The analogous result in the n-dimensional case has been partially demonstrated in [9], [11], [16]. A nice review of these results can be found in [1].…”

“…Moreover, since the mean curvature H is constant, the following relations hold ([12, eq. ( 4)]): (11) x…”

Bifurcation of Constant Mean Curvature Tori in Euclidean Spheres

“…In [44], Urbano showed that the conjecture is true when m = 2. Later on, Guadalupe, Brasil Jr. and Delgado [22] showed that the conjecture is true for every dimension m, under the additional hypothesis of constant scalar curvature of M . More recently, Perdomo [33] proved that the conjecture is also true for every dimension m with an additional assumption about the symmetries of M , and, in particular, the conjecture is true for minimal hypersurfaces with antipodal symmetry.…”

Spherical Ruled Surfaces in S3 Characterized by the Spherical Gauss Map

“…In particular, the Clifford torus S 1 (1/ √ 2) × S 1 (1/ √ 2) is minimal and flat in S 3 (1) and its closed geodesics are mapped onto closed curves of finite-type in S 3 (1). There are many papers devoted to characterize the Clifford torus with different view points by dealing with minimal surfaces of 3-sphere [2][3][4].…”

“…Then, M is part of the ruled surface parameterized byM : x(s, t) = cos tα(s) + sin tβ(s) = (cos t + u sin t)α(s) + sin tN (51)satisfying the function ξ(s) = α , γ is a solution of the non-linear differentiable equationξ 4 ln |bξ| = (1 + u 2 )ξ 2 + ξ 4 + 2(ξ ) 2 − ξξ .Proof. Let M be a ruled surface in S 3 parameterized by(4). We suppose that M has generalized 1-type spherical Gauss map.…”