Affine translation surfaces with constant sectional curvature

In this paper we characterize affine translation surfaces with constant Gaussian curvature. We show that such surfaces must be flat and that one of the defining curves must be planar.     

Helicoidal surfaces in Minkowski space with constant mean curvature and constant Gauss curvature

We classify all helicoidal non-degenerate surfaces in Minkowski space with constant mean curvature whose generating curve is a the graph of a polynomial or a Lorentzian circle. In the first case, we prove that the degree of the polynomial is 0 or 1 and that the surface is ruled. If the generating curve is a Lorentzian circle, we prove that the only possibility is that the axis is spacelike and the center of the circle lies on the axis.     

Invariant surfaces in the homogeneous space Sol with constant curvature

A surface in homogeneous space is said to be an invariant surface if it is invariant under some of the two 1‐parameter groups of isometries of the ambient space whose fix point sets are totally geodesic surfaces. In this work we study invariant surfaces that satisfy a certain condition on their curvatures. We classify invariant surfaces with constant mean curvature and constant Gaussian curvature. Also, we characterize invariant surfaces that satisfy a linear Weingarten relation.     

Complete surfaces in the hyperbolic space with a constant principal curvature

In this paper we study complete orientable surfaces with a constant principal curvature R in the 3‐dimensional hyperbolic space H ³. We prove that if R² > 1, such a surface is totally umbilical or umbilically free and it can be described in terms of a complete regular curve in H ³. When R² ≤ 1, we show that this result is not true any more by means of several examples. This contradicts a previous statement by Zhisheng [6]. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Translation surfaces of constant curvatures in a simply isotropic space

In this paper we describe, up to a congruence, translation surfaces in a simply isotropic space having constant isotropic Gaussian or mean curvature. It turns out that, contrary to the Euclidean case, there exist translation surfaces with constant Gaussian curvature \(K\ne 0\) and translation surfaces with constant mean curvature \(H\ne 0\) that are not cylindrical. Furthermore, we investigate a class of Weingarten translation surfaces.     

On stable surfaces of constant gauss curvature in space forms

We prove that a domain on a surface of constant curvature is stable provided the integral of the mean curvature is small enough.     

The moduli space of complete embedded constant mean curvature surfaces

We examine the space of finite topology surfaces in ³ which are complete, properly embedded and have nonzero constant mean curvature. These surfaces are noncompact provided we exclude the case of the round sphere. We prove that the spaceM _k of all such surfaces withk ends (where surfaces are identified if they differ by an isometry of ³) is locally a real analytic variety. When the linearization of the quasilinear elliptic equation specifying mean curvature equal to one has noL ²-nullspace, we prove thatM _k is locally the quotient of a real analytic manifold of dimension 3k–6 by a finite group (i.e. a real analytic orbifold), fork 3. This finite group is the isotropy subgroup of the surface in the group of Euclidean motions. It is of interest to note that the dimension ofM _k is independent of the genus of the underlying punctured Riemann surface to which is conformally equivalent. These results also apply to hypersurfaces of H ⁿ⁺¹ with nonzero constant mean curvature greater than that of a horosphere and whose ends are cylindrically bounded.Research of the first author supported in part by NSF grant # DMS9404278 and an NSF Postdoctoral Fellowship, of the second auther by NSF Young Investigator Award, a Sloan Foundation Postdoctoral Fellowship and NSF grant # DMS9303236, and of the third author by NSF grant # DMS9022140 and an NSF Postdoctoral Fellowship.     

Biminimal Lagrangian surfaces of constant mean curvature in complex space forms

We determine all biminimal Lagrangian surfaces of non-zero constant mean curvature in 2-dimensional complex space forms.     

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.     

Stable complete surfaces with constant mean curvature

Letx:M ²N ³ be a stable immersion with constant mean curvatureH of a complete orientable surfaceM ² into a complete oriented three dimensional Riemannian manifoldN ³. In this paper we prove that, ifM ² is compact andH ²> –1/2 inf _M Ricc _N , thenM ² has genusg3, here Ricc _N is the Ricci curvature ofN ³. We also prove that, ifM ² is complete non compact andN ² has bounded geometry, the area ofM ² is infinite in the metric induced byx. In this case, ifH ²–1/2 inf _M Ricc _N thenx is umbilic and the equality holds.     

Estimates in surfaces with positive constant Gauss curvature

We give optimal bounds of the height, curvature, area and volume of -surfaces in bounding a planar curve. The spherical caps are characterized as the unique -surfaces achieving these bounds.

     

关于SOL流形具有常平均曲率旋转曲面的注记

研究非欧流形SOL 空间上共形平均曲率方程的可解性,通过研究轮廓曲线对具有平均曲率的旋转曲面进行分类。当这些旋转曲面的平均曲率为给定函数时,计算出相应轮廓曲线的微分方程。通过求解这些微分方程,给出旋转函数是其上共形平均曲率的充分条件。