On the Principal Curvatures of Complete Minimal Hypersurfaces in Space Forms
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On the Principal Curvatures of Complete Minimal Hypersurfaces in Space Forms Rosa M. B. Chaves, L. A. M. Sousa Jr. , and B. C. Val´erio Abstract. In recent decades, there has been an increase in the number of publications related to the hypersurfaces of real space forms with two principal curvatures. The works focus mainly on the case when one of the two principal curvatures is simple. The purpose of this paper is to study a slightly more general class of complete minimal hypersurfaces in real space forms of constant curvature c, namely those with n − 1 principal curvatures having the same sign everywhere. From assumptions on the scalar curvature R and the Gauss–Kronecker curvature K we characterize Clifford tori if c > 0 and prove that K is identically zero if c ≤ 0. Mathematics Subject Classification. Primary 53C42, 53A10. Keywords. Complete minimal hypersurfaces, Gauss–Kronecker curvature, scalar curvature.
1. Introduction Throughout this paper, Qn+1 (c) will denote an n + 1-dimensional space form, that is, a complete and simply connected n + 1-dimensional Riemannian manifold with constant sectional curvature c. Although our results hold for any value of c, in the statements and proofs we will have that c = 0 or c = ±1, that is, Qn+1 (c) is the unit sphere Sn+1 if c > 0, Qn+1 (c) is the n + 1-dimensional Euclidean space Rn+1 , if c = 0, and Qn+1 (c) is the n + 1-dimensional hyperbolic space Hn+1 of constant sectional curvature −1, if c < 0. We say that M n is a minimal hypersurface of Qn+1 (c) if its mean curvature vanishes identically. Let M n be a minimal submanifold of Sn+p and let S be the square of the length of the second fundamental form of M n . In his pioneering work, 0123456789().: V,-vol
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Simons [24] proved the following inequality for the Laplacian of S 1 1 ΔS ≥ S n − 2 − S . 2 p As an application, he deduced that if M n is a closed (i.e. compact and without boundary) minimal submanifold of Sn+p , then either M n is totally geodesic, n n , or sup S > 2−1/p . or S = 2−1/p Later, Chern et al. [11], in their famous paper, determined all the minimal n submanifolds of Sn+p satisfying S = 2−1/p . Theorem. Let M n be a closed minimal submanifold of Sn+p . Assume that S ≤ n , then: 2 − 1/p n (i) Either S = 0 (and M n is totally geodesic) or S = 2−1/p . n if and only if: (ii) S = 2 − 1/p k n−k n−k (a) p = 1 and M n is locally a Clifford torus Sk ×S . n n (b) p = n = 2 and M 2 is locally a Veronese surface in S4 . We would like to point out that the result in (ii) is local. Let M n be a hypersurface of Qn+1 (c) and let λ1 , . . . , λn be the principal curvatures of M n . The k-th symmetric function of the principal curvatures σk is defined as follows: λi1 . . . λik , 1 ≤ k ≤ n. σk = 1≤i1 0. Then F = log |det(hij )| is well defined and bounded from below. Using the same reasoning already done in Theorem 1.1 we obtain 0 ≤ ΔF ≤ n(cn − S). If c = 0 we conclude that −nS ≥ 0 which implies S = 0 and contradicts our assump
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