On an inequality of mean curvatures of higher degree (original) (raw)
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American Journal of Mathematics, 1972
an (n -fp) -dimensional Eiemannian manifold Nn+p. Let h be the second fundamental form of this immersion; it is a certain symmetric bilinear mapping T^XTx-* Tw-L for x ? M, where T^ is the tangent space of M at x and IVthe normal space to M at x. We let 8 denote the length of h, H the mean curvature vector of this immersion, and a the length of H. If a = 0 identically on M, then M is called a minimal submanifold of Nn+p. In the first paper of this series [1], the author proved that the integral of an satisfies
A new characterization of submanifolds with parallel mean curvature vector in Sn+pS^{n+p}Sn+p
Kodai Mathematical Journal, 2004
In this work we will consider compact submanifold M n immersed in the Euclidean sphere S nþp with parallel mean curvature vector and we introduce a Schrö dinger operator L ¼ ÀD þ V , where D stands for the Laplacian whereas V is some potential on M n which depends on n; p and h that are respectively, the dimension, codimension and mean curvature vector of M n . We will present a gap estimate for the first eigenvalue m 1 of L, by showing that either m 1 ¼ 0 or m 1 a Ànð1 þ H 2 Þ. As a consequence we obtain new characterizations of spheres, Cli¤ord tori and Veronese surfaces that extend a work due to Wu [W] for minimal submanifolds. Key words and phrases: Mean curvature vector, first eigenvalue, Cli¤ord torus. We would like to thank FINEP and FAPERJ for financial support.
Embedded Constant Mean Curvature Hypersurfaces on Spheres
Asian Journal of Mathematics, 2010
Let m ≥ 2 and n ≥ 2 be any pair of integers. In this paper we prove that if H lies between cot(π m) and bm,n = (m 2 −2) √ n−1 n √ m 2 −1 , there exists a non isoparametric, compact embedded hypersurface in S n+1 with constant mean curvature H that admits O(n) × Zm in its group of isometries. These hypersurfaces therefore have exactly 2 principal curvatures. When m = 2 and H is close to the boundary value 0 = cot(π 2), such a hypersurface looks like two very close n-dimensional spheres with two catenoid necks attached, similar to constructions made by Kapouleas. When m > 2 and H is close to cot(π m), it looks like a necklace made out of m spheres with m + 1 catenoid necks attached, similar to constructions made by Butscher and Pacard. In general, when H is close to bm,n the hypersurface is close to an isoparametric hypersurface with the same mean curvature. For hyperbolic spaces we prove that every H ≥ 0 can be realized as the mean curvature of an embedded CMC hypersurface in H n+1. Moreover we prove that when H > 1 this hypersurface admits O(n)× Z in its group of isometries. As a corollary of the properties we prove for these hypersurfaces, we construct, for any n ≥ 6, non-isoparametric compact minimal hypersurfaces in S n+1 whose cones in R n+2 are stable. Also, we prove that the stability index of every non-isoparametric minimal hypersurface with two principal curvatures in S n+1 exceeds n + 3.
Minimal immersions into space forms with two principal curvatures
Mathematische Zeitschrift, 1984
In [2] Chern, do Carmo and Kobayashi consider minimal immersions of an ndimensional manifold M" into the unit sphere S "+N of the (n+N+l)dimensional euclidean space IR n+N+l with the property that the second fundamental form has constant length n/(2-1/N). (If M is compact, this is the smallest possible value for a non-totally geodesic minimal immersion as above.) They prove that locally M is either a piece of a Veronese surface (n = 2, N =2) or the product of two spaces of dimensions m and n-m and of constant curvatures n/m and n/(n-m). They also prove the corresponding global result under a compactness assumption for M. The main feature of the situation is that the second fundamental form in any normal direction has two distinct eigenvalues and, in general, the respective eigenspaces will determine involutive distributions whose integral manifolds will give the local product structure. This leads naturally to the study of minimal immersions into spaces of constant curvature such that the second fundamental form has, in any normal direction, at most two distinct eigenvalues. This problem was studied by Otsuki in [9] for the case of hypersurfaces. He proves the following result (see also Remark (3.7)).