Euclidean Complete Affine Surfaces with Constant Affine Mean Curvature

The purpose of this paper is to prove that alocally strongly convex, Euclidean complete surface with constantaffine mean curvature is also affine complete. Consequently weobtain a classification of locally strongly convex, Euclideancomplete surfaces with constant affine mean curvature.     

Constant Gaussian Curvature Surfaces with Parallel Mean Curvature Vector in Two-Dimensional Complex Space Forms

We classify constant Gaussian curvature surfaces with nonzero parallel mean curvature vector in two-dimensional complex space forms. As a result, we find new examples of such surfaces.     

Hypersurfaces in E4 with Harmonic Mean Curvature Vector Field

The minimal hypersurfaces in E⁴ are the only hypersurfaces possessing the following property: Its mean curvature vector field is harmonic.     

On Discrete Constant Mean Curvature Surfaces

Recently, a curvature theory for polyhedral surfaces has been established that associates with each face a mean curvature value computed from areas and mixed areas of that face and its corresponding Gaussian image face. Therefore, a study of constant mean curvature (cmc) surfaces requires studying pairs of polygons with some constant nonvanishing value of the discrete mean curvature for all faces. We focus on meshes where all faces are planar quadrilaterals or planar hexagons. We show an incidence geometric characterization of a pair of parallel quadrilaterals having a discrete mean curvature value of ?1. This characterization yields an integrability condition for a mesh being a Gaussian image mesh of a discrete cmc surface. Thus, we can use these geometric results for the construction of discrete cmc surfaces. In the special case where all faces have a circumcircle, we establish a discrete Weierstrass-type representation for discrete cmc surfaces.     

Caylay射影平面的全实曲面

It has been shown, under certain conditions on the Gauss curvature, every totally real surface of the Cayley projective plane with parallel mean curvature vector is either flat or totally geodesic.     

调和映照与常平均曲率的曲面

本文利用三维欧氏空间Ｒ^３或三维Ｍｉｎｋｏｗｓｋｉ空间Ｒ^2.1中常平均曲率的曲面与Ｓｉｎｈ－Ｌａｐｌａｃｅ方程和Ｓｉｎｈ－Ｇｏｒｄｏｎ方程之间的关系，研究了常平均曲率的曲面与Ｒ²到Ｓ²（或Ｈ²）及Ｒ^1.1到Ｓ^1.1（＋１）的调和映照之间的内在联系，并且提供了一种构造到球面Ｓ²，Ｈ²和Ｓ^1.1     

Timelike Constant Mean Curvature Surfaces with Singularities

We use integrable systems techniques to study the singularities of timelike non-minimal constant mean curvature (CMC) surfaces in the Lorentz–Minkowski 3-space. The singularities arise at the boundary of the Birkhoff big cell of the loop group involved. We examine the behavior of the surfaces at the big cell boundary, generalize the definition of CMC surfaces to include those with finite, generic singularities, and show how to construct surfaces with prescribed singularities by solving a singular geometric Cauchy problem. The solution shows that the generic singularities of the generalized surfaces are cuspidal edges, swallowtails, and cuspidal cross caps.     

Constant Mean Curvature Surfaces of any Positive Genus

The paper shows the existence of several new families of noncompactconstant mean curvature surfaces: (i) singly punctured surfacesof arbitrary genus g1, (ii) doubly punctured tori, and (iii)doubly periodic surfaces with Delaunay ends.     

Constant Mean Curvature Surfaces with Boundaryin Hyperbolic Space

 We study constant mean curvature compact surfaces immersed in hyperbolic space with non-empty boundary (=H-surfaces). We prove that the only H-surfaces with boundary circular and 0≤∣H∣≤1, are the umbilical examples. When the surface is embedded, conditions to be umbilical are given. Finally, we characterize umbilical surfaces bounded by a circle among all H-discs with small area. Received 27 March 1997; in final form 11 June 1998     

Complete Embedded Self-Translating Surfaces Under Mean Curvature Flow

We describe a construction of complete embedded self-translating surfaces under mean curvature flow by desingularizing the intersection of a finite family of grim reapers in general position.     

Constant Mean Curvature Surfaces with Boundaryin Hyperbolic Space

 We study constant mean curvature compact surfaces immersed in hyperbolic space with non-empty boundary (=H-surfaces). We prove that the only H-surfaces with boundary circular and 0≤∣H∣≤1, are the umbilical examples. When the surface is embedded, conditions to be umbilical are given. Finally, we characterize umbilical surfaces bounded by a circle among all H-discs with small area.     

Curvature Estimates for Strongly Stable Domains on Surfaces with Constant Mean Curvature in Space Forms

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Generalized Symplectic Mean Curvature Flows in Almost Einstein Surfaces

The authors mainly study the generalized symplectic mean curvature flow in an almost Einstein surface, and prove that this flow has no type-I singularity. In the graph case, the global existence and convergence of the flow at infinity to a minimal surface with metric of the ambient space conformal to the original one are also proved.     

Simulation of Mean Curvature Flows on Surfaces of Revolution

Computational Mathematics and Mathematical Physics - This paper is devoted to the simulation of the mean curvature flow on a surface of revolution. The surface is discretized using the icosahedral...