In this paper we prove that a compact oriented hypersurface of a Euclidean sphere with nonnegative Ricci curvature and infinite fundamental group is isometric to an H (r)-torus with constant mean curvature. Furthermore, we generalize, whithout any hypothesis about the mean curvature, a characterization of Clifford torus due to Hasanis and Vlachos.
Introduction.Let M n be an n-dimensional hypersurface of the (n + 1)-dimensional unit Euclidean sphere S n+1 . If M n is compact, minimal and 0 S n, then Simons [16] proved that S = 0 or S = n, where S is the square of the length of the second fundamental form of M n . Chern, Do Carmo and Kobayashi [3] and Lawson [11] proved, independently, that the Clifford Tori are the only minimal hypersurfaces with S = n. Peng and Terng [15] studied the case where S is constant and n = 3, and proved that if S > 3, then S 6. Jorge and Mercuri [10] proved that if M n is minimal with two distinct principal curvatures of multiplicities m and (n − m) and 2 m (n − 2), then M n is locally S m (c 1 ) × S n−m (c 2 ). Otsuki in [14] gives necessary conditions for a minimal hypersurface of the sphere to be a product of spheres, namely that the second fundamental form have just two eigenvalues, each one of constant multiplicity. Recently, Hasanis and Vlachos [9] proved that if M n is minimal and compact with two principal curvatures, one of them has multiplicity 1 and S n, then S = n and M n is a Clifford Torus. Alencar and do Carmo [1] proved that if M n is compact with constant mean curvature H and S −nH 2 B H , where B H is a constant that depends only on H and n, then S − nH 2 = 0 or S − nH 2 = B H . They also proved that the H (r)-tori S n−1 (r) × S 1 ( √ 1 − r 2 ) with r 2
The aim of this paper is to give an estimate for the squared norm S of the second fundamental form A of a compact minimal hypersurface M n ⊂ S n+1 in terms of the gap n − λ 1 , where λ 1 stands for the first eigenvalue of the Laplacian of M. More precisely we will show that there exists a constant k n n−1 such that S k n−1 n (n − λ 1 ).
Abstract. The aim of this paper is to prove that the Ricci curvature Ric M of a complete hypersurface M n , n ≥ 3, of the Euclidean sphere ޓ n+1 , with two distinct principal curvatures of multiplicity 1 and n − 1, satisfies sup Ric M ≥ inf f (H), for a function f depending only on n and the mean curvature H. Supposing in addition that M n is compact, we will show that the equality occurs if and only if H is constant and M n is isometric to a Clifford torus2000 Mathematics Subject Classification. Primary 53C42. Secondary 53B30, 53C50.
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