Rigidity Theorems of Clifford Torus

Let M be an n-dimensional closed minimally immersed hypersurface in the unit sphere Sn+1. Assume in addition that M has constant scalar curvature or constant Gauss-Kronecker curvature. In this note we announce that if M has(n−1) principal curvatures with the same sign everywhere, thenM is isometric to a Clifford TorusS1 (√ 1 n ) × Sn−1(√n−1 n ) .

It follows from Cartan's Lemma that w n+1,i = j h ij w j , h ij = h ji . (1.1) The second fundamental form h and the mean curvature H of M are defined by h ij w i w j e n+1 and H = i h ii .
(1.2) E-mail: amancio@impa.br An. Acad.Bras. Cienc., (2001) 73 (3) We recall that M is by definition a minimal hypersurface if the mean curvature of M is identically zero.The connection form w ij is characterized by the structure equations where ij (resp.R ij kl ) denotes the curvature form (resp. the components of the curvature tensor) of M. The Gauss equation is given by (1.4) The covariant derivative h of the second fundamental form h of M with components h ij k is given by Then the exterior derivative of (1.1) together with the structure equations yield the following Codazzi equation (1.5) Similarly, we have the covariant derivative 2 h of h with components h ij kl as follows and it is easy to get the following Ricci formula (1.6) From now on, we assume that M is minimal.Denote by S = i,j h 2 ij the square of length of h.The components of the Ricci curvature and the scalar curvature are given respectively by (1.9) The following formulas can be obtained by a direct computation (Peng and Terng 1983). (1.10) (1.11) The Gauss-Kronecker curvature K of M is defined by (1.12) In this note we give a sketch of the proof of the following results.
Theorem 1.1.Let M be a closed minimal hypersurface with constant scalar curvature in S n+1 .If M has (n − 1) principal curvatures with the same signal everywhere, then Corollary 1.2.Let M be a closed minimal hypersurface with constant scalar curvature in S 5 .Assume in addition that M has Gauss-Kronecker curvature K negative everywhere.Then M is isometric to a Clifford Torus S 1 1 2 × S 3 Corollary 1.3.Let M be a closed minimal hypersurface with constant scalar curvature in S 5 .Assume S > 4. Then there exists a point p on M such that K(p) ≥ 0.
Theorem 1.4.Let M be a closed minimal hypersurface with constant Gauss-Kronecker curvature in S n+1 .If M has (n−1) principal curvatures with the same signal everywhere, then M is isometric to a Clifford Torus S 1 1 n × S n−1 n−1 n .

SKETCH OF THE PROOF OF THEOREM 1.1
By changing the orientation for M and renumbering e 1 , ..., e n if necessary, we may assume (2.1) Notice that K = 0, hence the following function is well defined We compute the Laplacian of F obtaining (2.2) Since S is constant, from (1.10), (1.11) and using the symmetry of h ij k for indices, we get We need the following algebraic Lemma.

SKETCH OF THE PROOF OF THEOREM 1.4
The proof of Theorem 1.4 follows essentially the pattern of the proof of Theorem 1.1 taking into account the presence of one term that contains S. We only stress those points which may lead to some differences.By using (1.10), (1.11), (2.2) and the fact that K is constant, we obtain 0 Since M is compact, we can find a point p on M such that S(p) = max S. In particular, ∇S(p) = 0 and S(p) ≤ 0 at p. Since ∇S(p) = 0, we have Remark.Assume that S is bounded from above or that K is bounded from below.In this situation, by applying the Generalized Maximum Principle due to Yau and Omori (Yau 1975), and a result due to (Cheng 1993), we can generalize all the results in this note to complete hypersurfaces in S n+1 .