On complete noncompact submanifolds with constant mean curvature and finite total curvature in Euclidean spaces

In this paper, we consider a complete noncompact n-submanifold M with parallel mean curvature vector h in an Euclidean space. If M has finite total curvature, we prove that M must be minimal, so that M is an affine n-plane if it is strongly stable. This is a generalization of the result on strongly stable complete hypersurfaces with constant mean curvature in Received: 30 June 2005     

The mean curvature of cylindrically bounded submanifolds

We give an estimate of the mean curvature of a complete submanifold lying inside a closed cylinder in a product Riemannian manifold . It follows that a complete hypersurface of given constant mean curvature lying inside a closed circular cylinder in Euclidean space cannot be proper if the circular base is of sufficiently small radius. In particular, any possible counterexample to a conjecture of Calabi on complete minimal hypersurfaces cannot be proper. As another application of our method, we derive a result about the stochastic incompleteness of submanifolds with sufficiently small mean curvature. Dedicated to Professor Manfredo P. do Carmo on the occasion of his 80th birthday.     

Area-stationary surfaces inside the sub-Riemannian three-sphere

We consider the sub-Riemannian metric g _h on given by the restriction of the Riemannian metric of curvature 1 to the plane distribution orthogonal to the Hopf vector field. We compute the geodesics associated to the Carnot–Carathéodory distance and we show that, depending on their curvature, they are closed or dense subsets of a Clifford torus. We study area-stationary surfaces with or without a volume constraint in (). By following the ideas and techniques by Ritoré and Rosales (Area-stationary surfaces in the Heisenberg group , arXiv:math.DG/0512547) we introduce a variational notion of mean curvature, characterize stationary surfaces, and prove classification results for complete volume-preserving area-stationary surfaces with non-empty singular set. We also use the behaviour of the Carnot–Carathéodory geodesics and the ruling property of constant mean curvature surfaces to show that the only C ² compact, connected, embedded surfaces in () with empty singular set and constant mean curvature H such that is an irrational number, are Clifford tori. Finally we describe which are the complete rotationally invariant surfaces with constant mean curvature in (). A. Hurtado has been partially supported by MCyT-Feder research project MTM2004-06015-C02-01. C. Rosales has been supported by MCyT-Feder research project MTM2004-01387.     

A Ricci inequality for hypersurfaces in the sphere

Let Mⁿ be a complete Riemannian manifold immersed isometrically in the unity Euclidean sphere In [9], B. Smyth proved that if Mⁿ, n ≧ 3, has sectional curvature K and Ricci curvature Ric, with inf K > −∞, then sup Ric ≧ (n − 2) unless the universal covering of Mⁿ is homeomorphic to Rn or homeomorphic to an odd-dimensional sphere. In this paper, we improve the result of Smyth. Moreover, we obtain the classification of complete hypersurfaces of with nonnegative sectional curvature.Received: 11 November 2003     

Stable complete noncompact hypersurfaces with constant mean curvature

This article concerns the structure of complete noncompact stable hypersurfaces M ⁿ with constant mean curvature H > 0 in a complete noncompact oriented Riemannian manifold N ⁿ⁺¹. In particular, we show that a complete noncompact stable constant mean curvature hypersurface M ⁿ , n = 5, 6, in the Euclidean space must have only one end. Any such hypersurface in the hyperbolic space with , respectively, has only one end.     

Complete surfaces of constant curvature in H<Superscript>2</Superscript> × R and S<Superscript>2</Superscript> × R

We study isometric immersions of surfaces of constant curvature into the homogeneous spaces and . In particular, we prove that there exists a unique isometric immersion from the standard 2-sphere of constant curvature c > 0 into and a unique one into when c > 1, up to isometries of the ambient space. Moreover, we show that the hyperbolic plane of constant curvature c < −1 cannot be isometrically immersed into or . J.A. Aledo was partially supported by Ministerio de Education y Ciencia Grant No. MTM2004-02746 and Junta de Comunidades de Castilla-La Mancha, grant no. PAI-05-034. J.M. Espinar and J.A. Gálvez were partially supported by Ministerio de Education y Ciencia grant no. MTM2004-02746 and Junta de Andalucía Grant No. FQM325.     

Skew-symmetric derivations of a complex Lie algebra

Let be a complex Lie algebra, its underlying real Lie algebra, a real form of and ·, · the euclidean product induced by the real part of an hermitian inner product on . Let aut be the Lie algebra of skew-symmetric derivations of . We give necessary and sufficient conditions to ensure that aut is composed of skew-hermitian derivations. As an application, we study holomorphy in large subgroups of isometries of Lie groups.     

Mean curvature of extrinsic spheres in submanifolds of real space forms

Given a submanifold P^m with the Hilbert-Schmidt norm of its second fundamental form bounded from above, in a real space form of constant curvature we have obtained a lower bound for the norm of the mean curvature normal vector field of extrinsic spheres with sufficiently small radius in P^m in terms of the mean curvature of the geodesic spheres in with same radius, and the mean curvature of P^m.Received: 4 April 2003     

On 3-dimensional asymptotically harmonic manifolds

Let (M,g) be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that M is a hyperbolic manifold of constant sectional curvature , provided M is asymptotically harmonic of constant h > 0. Received: 4 October 2007     

On partially null and pseudo null curves in the semi-euclidean space R^{4}_{2}

In this paper, we obtain the Frenet equations of a pseudo null and a partially null curves, lying fully in the semi–Euclidean space , and classify all such curves with constant curvatures.     

Caccioppoli’s inequalities on constant mean curvature hypersurfaces in Riemannian manifolds

We prove some Caccioppoli’s inequalities for the traceless part of the second fundamental form of a complete, noncompact, finite index, constant mean curvature hypersurface of a Riemannian manifold, satisfying some curvature conditions. This allows us to unify and clarify many results scattered in the literature and to obtain some new results. For example, we prove that there is no stable, complete, noncompact hypersurface in ${{\mathbb R}^{n+1}, n \leq 5}$ , with constant mean curvature ${H \not=0}$ , provided that, for suitable p, the L ^p norm of the traceless part of second fundamental form satisfies some growth condition.     

The Dirichlet problem for constant mean curvature surfaces in Heisenberg space

We study constant mean curvature graphs in the Riemannian three- dimensional Heisenberg spaces . Each such is the total space of a Riemannian submersion onto the Euclidean plane with geodesic fibers the orbits of a Killing field. We prove the existence and uniqueness of CMC graphs in with respect to the Riemannian submersion over certain domains taking on prescribed boundary values. L. J. Alías was partially supported by MEC/FEDER project MTM2004-04934-C04-02 and Fundación Séneca project 00625/PI/04, Spain.     

Mean curvature flow of monotone Lagrangian submanifolds

We use holomorphic disks to describe the formation of singularities in the mean curvature flow of monotone Lagrangian submanifolds in . Supported by DFG, priority program SPP 1154, SM 78/1-1, SCHW 892/1-1.     

Volume and Projective Equivalence Between Riemannian Manifolds

Let (M, g) and (M, ) be two Riemannian metrics which are pointwise projectively equivalent, i.e. they have the same geodesics as point sets. We prove that the pointwise projective equivalence is trivial, if (M, g) is a noncompact complete manifold which has at most quadratic volume growth and nonnegative total scalar curvature, and (M, ) has nonpositive Ricci curvature. Mathematics Subject Classifications (2000): 53C22, 58J05     

On periodic groups of automorphisms of extremal groups

It is proved that if a periodic group has an extremal normal divisor , determining a complete abelian factor group , then the center of the group contains a complete abelian subgroup , satisfying the relation and intersecting on a finite subgroup. It is also established with the aid of this proposition that every periodic group of automorphisms of an extremal group is a finite extension of a contained in it subgroup of inner automorphisms of the group .Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp. 91–96, July, 1968.     

Non-existence of regular algebraic minimal surfaces of spheres of degree 3

In this paper we prove that if is a closed minimal surface, then, , for any homogeneous polynomial f of degree 3 with 0 a regular value of the function .     

A characterization of the Clifford torus

We provide a characterization of the Clifford torus via a Ricci type condition among minimal surfaces in S⁴. More precisely, we prove that a compact minimal surface in S⁴, with induced metric ds² and Gaussian curvature K, for which the metric is flat away from points where K = 1, is the Clifford torus, provided that m is an integer with m > 2.Received: 8 September 2004     

Codimension two submanifolds with 2-nonnegative curvature operator

In this paper we obtain a classification of compact n-submanifolds of the Euclidean space with 2-nonnegative curvature operator. Received: 4 April 2007     

A remark on left invariant metrics on compact Lie groups

We show that a left invariant metric on a compact Lie group G with Lie algebra has some negative sectional curvature if it is obtained by enlarging a biinvariant metric on a subalgebra , unless the semi-simple part of is an ideal of This answers a question raised in [8]. Received: 7 May 2007     

Complete Hypersurfaces with Constant Mean Curvature in a Unit Sphere

By investigating hypersurfaces M ⁿ in the unit sphere S ⁿ⁺¹(1) with constant mean curvature and with two distinct principal curvatures, we give a characterization of the torus S ¹(a) × , where . We extend recent results of Hasanis et al. [5] and Otsuki [10].