Translation surfaces with constant mean curvature in 3-dimensional spaces

We give the classification of the translation surfaces with constant mean curvature or constant Gauss curvature in 3-dimensional Euclidean space E³ and 3-dimensional Minkowski space E ₁ ³ .The author is supported by the EDU. COMM. of CHINA, the NSF of Liaoning and the Northeastern University.Dedicated to Professor Udo Simon on the occation of his sixtieth birthday     

A sextic holomorphic form of affine surfaces with constant affine mean curvature

We extend an original idea of Calabi for affine maximal surfaces and define a sextic holomorphic differential form for affine surfaces with constant affine mean curvature. We get some rigidity results for affine complete surfaces by using this sextic holomorphic form. Received: 17 May 2003     

On two-dimensional immersions that are stable for parametric functionals of constant mean curvature type

We consider two-dimensional immersions in Euclidean 3-space that are stable for parametric functionals of constant mean curvature type. We develop analytical and geometric concepts to give a perturbation result to estimate the principle curvatures of such mappings via uniformization.     

Hypersurfaces with constant mean curvature and prescribed area

Summary We consider—in the setting of geometric measure theory—hypersurfacesT (of codimension one) with prescribed boundaryB in Euclideann+1 space which maximize volume (i.e.T together with a fixed hypersurfaceT ₀ encloses oriented volume) subject to a mass constraint. We prove existence and optimal regularity of solutionsT of such variational problems and we show that, on the regular part of its support,T is a classical hypersurface of constant mean curvature. We also prove that the solutionsT become more and more spherical as the valuem of the mass constraint approaches ∞. This work was done at the Centre for Mathematics and its Applications at the Australian National University, Canberra while the author was a visiting member This article was processed by the author using the LAT_EX style filecljour1 from Springer-Verlag.     

Holomorphic representation of constant mean curvature surfaces in Minkowski space: Consequences of non-compactness in loop group methods

We give an infinite dimensional generalized Weierstrass representation for spacelike constant mean curvature (CMC) surfaces in Minkowski 3-space R2,1. The formulation is analogous to that given by Dorfmeister, Pedit and Wu for CMC surfaces in Euclidean space, replacing the group SU₂ with SU1,1. The non-compactness of the latter group, however, means that the Iwasawa decomposition of the loop group, used to construct the surfaces, is not global. We prove that it is defined on an open dense subset, after doubling the size of the real form SU1,1, and prove several results concerning the behavior of the surface as the boundary of this open set is encountered. We then use the generalized Weierstrass representation to create and classify new examples of spacelike CMC surfaces in R2,1. In particular, we classify surfaces of revolution and surfaces with screw motion symmetry, as well as studying another class of surfaces for which the metric is rotationally invariant.     

Existence of constant mean curvature graphs in hyperbolic space

We give an existence result for constant mean curvature graphs in hyperbolic space . Let be a compact domain of a horosphere in whose boundary is mean convex, that is, its mean curvature (as a submanifold of the horosphere) is positive with respect to the inner orientation. If H is a number such that , then there exists a graph over with constant mean curvature H and boundary . Umbilical examples, when is a sphere, show that our hypothesis on H is the best possible. Received July 18, 1997 / Accepted April 24, 1998     

Maximal surfaces in a certain 3-dimensional homogeneous spacetime

A generalized integral representation formula for spacelike maximal surfaces in a certain 3-dimensional homogeneous spacetime is obtained. This spacetime has a solvable Lie group structure with left invariant metric. The normal Gauß map of maximal surfaces in the homogeneous spacetime is discussed and the harmonicity of the normal Gauß map is studied.     

Killing graphs with prescribed mean curvature and Riemannian submersions

It is proved the existence and uniqueness of graphs with prescribed mean curvature in Riemannian submersions fibered by flow lines of a vertical Killing vector field.     

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.     

Hypersurfaces with constant mean curvature in hyperbolic space form

In this article, we prove the following theorem: A complete hypersurface of the hyperbolic space form, which has constant mean curvature and non-negative Ricci curvature Q, has non-negative sectional curvature. Moreover, if it is compact, it is a geodesic distance sphere; if its soul is not reduced to a point, it is a geodesic hypercylinder; if its soul is reduced to a point p, its curvature satisfies Q<, and the geodesic spheres centered at p are convex, then it is a horosphere.A part of this work has been done when the second author visited Université Claude Bernard Lyon 1, and was supported by a grant of the People's Republic of China.     

On the area of constant mean curvature discs and annuli with circular boundaries

It is still an open question whether a constant mean curvature (CMC) disc which is bounded by a circle is necessarily a spherical cap or a flat disc. The authors together with López [1] recently showed that the only stable CMC discs which are bounded by a circle are spherical caps. In this paper we derive lower bounds for the area of constant mean curvature discs and annuli with circular boundaries in 3-dimensional space forms. Received November 8, 1999; in final form January 18, 2000 / Published online March 12, 2001     

Embedded constant mean curvature surfaces with special symmetry

We give sharp, necessary conditions on complete embedded CMC surfaces with three ends and an extra reflection symmetry. The respective submoduli space is a two-dimensional variety in the moduli space of general CMC surfaces. Fundamental domains of our CMC surfaces are characterized by associated great circle polygons in the three-sphere. Received: 23 January 1998 / Revised version: 23 October 1998     

Absolute gradient bounds for nonparametric hypersurfaces of constant mean curvature

We consider the problem of determining the existence of absolute apriori gradient bounds of nonparametric hypersurfaces of constant mean curvature in ann-dimensional sphereB _R, 1>R>R ₀ ⁽ⁿ⁾ , (R ₀ ⁽ⁿ⁾ being a constant depending only onn), without imposing boundary conditions or bounds of any sort.

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Sunto Consideriamo il problema di determinare stime a priori di gradienti di ipersuperfici non parametriche di curvatura media costante in una sferan-dimensionaleB _R, 1>R>R ₀ ⁽ⁿ⁾, (R ₀ ⁽ⁿ⁾ essendo una costante che dipende solo dan), senza imporre condizioni al contorno o limiti di altro tipo.

     

Complete hypersurfaces with constant mean curvature and finite L p norm curvature in Euclidean spaces

In this paper, we consider complete hypersurfaces in R ⁿ⁺¹ with constant mean curvature H and prove that M ⁿ is a hyperplane if the L ² norm curvature of M ⁿ satisfies some growth condition and M ⁿ is stable. It is an improvement of a theorem proved by H. Alencar and M. do Carmo in 1994. In addition, we obtain that M ⁿ is a hyperplane (or a round sphere) under the condition that M ⁿ is strongly stable (or weakly stable) and has some finite L ^p norm curvature. Received: 14 July 2007     

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     

Rigidity theorems for complete spacelike hypersurfaces with constant scalar curvature in De Sitter space

In this paper we give a partially affirmative answer to the following question posed by Haizhong Li: is a complete spacelike hypersurface in De Sitter space , n?3, with constant normalized scalar curvature R satisfying totally umbilical?     

On constant mean curvature hypersurfaces with finite index

We discuss the non-existence of complete noncompact constant mean curvature hypersurfaces with finite index in a 4- or 5-dimensional manifold. As a consequence, we obtain that a complete noncompact constant mean curvature hypersurface in with finite index must be minimal. Received: 30 May 2005