一类具有纤维化的一般型极小曲面的斜率

设 S 是一般型极小的代数曲面,f 是纤维化,则有自然的正合列 π_1(F)→π_1(S)→π→1.在本文中我们证明,如果 f 为非局部平凡,且 S 具有一个2阶挠元,那么λ(f)≥4-1/(g-1),这里 g 是 f 的纤维 F 的亏格.     

R~3中自共轭极小曲面的特征

本文研究R~3中与其共轭曲面合同的极小曲面,并给出这种极小曲面的度量特征。     

关于仿射极小平移超曲面

本文属于仿射微分几何。在3-维欧氏空间 E~3中,F.Scherk 定理告诉我们,极小平移曲面必需是平面或 Scherk 曲面az=1n(cos ax/cos ay),a=constant。在一般(n+1)维仿射空间 A~(n+1)中,仿射极小平移超曲面是什么曲面?本文得到了这种曲面共有两类的结果(见定理1)。当 n=2时,这就是引文[3]中的结果(见定理2)。     

一类极小曲面的不存在性

本文考虑三维 Euclid 空间 R~3中拓扑型为球面的有限全曲率完备嵌入极小曲面.通过对伪嵌入极小曲面的研究,证明了一类嵌入极小曲面的不存在性,并明确了伪嵌入与嵌入极小曲面的差异.     

任意曲面求交问题综述

曲面求交是ＣＡＤ／ＣＡＭ领域中最重要也最为复杂的问题之一，它是曲面造型中各种几何处理的关键。本文就已有的任意曲面求交算法进行了系统地分析与评价，指出了目前所存在的问题。     

具有平行非退化截面的极小曲面

利用「１」的方法及Ｓｉｍｏｎｓ型公式对常曲率空间中具有平行非退化截面的极小曲面进行了探讨,得到了一个较简洁的结果。     

计算机辅助几何设计中极小曲面造型的研究进展

极小曲面在工程领域有着广泛应用,因此将其引入计算机辅助几何设计领域具有重要意义.详细概述了近年来计算机辅助几何设计领域中极小曲面造型的研究工作,按照造型方法的不同,可将现有工作分为精确造型方法和逼近造型方法两类.精确造型方法主要包括两个部分:某些特殊极小曲面的控制网格表示与构造;等温参数多项式极小曲面的挖掘与性质.逼近造型方法主要包括3个部分t基于数值计算的逼近方法;基于线性偏微分方程的逼近方法;基于能量函数最优化的逼近方法.最后对这些方法进行了分析比较,并讨论了极小曲面造型中有待进一步解决的问题.     

超曲面上极小与极小凸点的分布

本文继续文［1］中的讨论,给出超曲面上点的极小与极小凸性更一般的判别方式,并且对超曲面上极小与极小凸点的分布有了更深刻的认识.作为应用,还证明了超曲面上一个极小点的传递性定理.     

超曲面上极小与极小凸点的分布

本文继续文［１］中的讨论,给出超曲面上点的极小与极小凸性更一般的判别方式,并且对超曲面上极小与极小凸点的分布有了更深刻的认识．作为应用,还证明了超曲面上一个极小点的传递性定理．     

S~6中具有常数Kahler角的极小曲面

球面Ｓ６具有典型的近复结构，因而对于其中的曲面可引入Ｋａｈｌｅｒ角的概念．本文研究Ｓ６内具有常数Ｋａｈｌｅｒ角的极小曲面，得到了刚性定理．结合已有结果，易知Ｓｉｍｏｎ猜想对于这类曲面成立．     

On the Galois covers of degenerations of surfaces of minimal degree

We investigate the topological structures of Galois covers of surfaces of minimal degree (i.e., degree n) in ${\mathbb{CP}}^{n+1}$. We prove that for $n\ge 5$, the Galois covers of any surfaces of minimal degree are simply-connected surfaces of general type.     

The classification of bi-quintic parametric polynomial minimal surfaces

Parametric polynomial surface is a fundamental element in CAD systems. Since the most of the classic minimal surfaces are represented by non-parametric polynomial, it is interesting to study the minimal surfaces represented in parametric polynomial form. Recently,Ganchev presented the canonical principal parameters for minimal surfaces. The normal curvature of a minimal surface expressed in these parameters determines completely the surface up to a position in the space. Based on this result, in this paper, we study the bi-quintic isothermal minimal surfaces. According to the condition that any minimal isothermal surface is harmonic,we can acquire the relationship of some control points must satisfy. Follow up, we obtain two holomorphic functions f(z) and g(z) which give the Weierstrass representation of the minimal surface. Under the constrains that the minimal surface is bi-quintic, f(z) and g(z) can be divided into two cases. One case is that f(z) is a constant and g(z) is a quadratic polynomial, and another case is that the degree of f(z) and g(z) are 2 and 1 respectively. For these two cases,we transfer the isothermal parameter to canonical principal parameter, and then compute their normal curvatures and analyze the properties of the corresponding minimal surfaces. Moreover,we study some geometric properties of the bi-quintic harmonic surfaces based on the B′ezier representation. Finally, some numerical examples are demonstrated to verify our results.     

Designing approximation minimal parametric surfaces with geodesics

Geodesic is an important curve in practical application, especially in shoe design and garment design. In practical applications, we not only hope the shoe and garment surfaces possess characteristic curves, but also we hope minimal cost of material to build surfaces. In this paper, we combine the geodesic and minimal surface. We study the approximation minimal surface with geodesics by using Dirichlet function. The extremal of such a function can be easily computed as the solutions of linear systems, which avoid the high nonlinearity of the area function. They are not extremal of the area function but they are a fine approximation in some cases.     

Existence of algebraic minimal surfaces for an arbitrary puncture set

We will show that any punctured Riemann surface can be conformally immersed into a Euclidean -space as a branched complete minimal surface of finite total curvature called an algebraic minimal surface.

     

There exist no 2-type surfaces in E 3 which are images under stereographic projection of minimal surfaces in S 3

In this paper we deal with the following particular case of a weaker conjecture by B. Y. Chen: Are there 2-type Willmore surfaces in E ³? In particular we prove that the above question has a negative answer when the surface is the image under stereographic projection of a minimal surface in S ³.     

Some remarks on the symmetry of self-conjugate minimal surfaces inR 3

Symmetry properties of self-conjugate minimal surfaces, i.e. minimal surfaces which are congruent with their conjugate ones inR ³ are studied.Supported by ICTP and NNSF of China.Supported by ICTP and SAREC.     

The classification of singly periodic minimal surfaces with genus zero and Scherk-type ends

Given an integer , let be the space of complete embedded singly periodic minimal surfaces in , which in the quotient have genus zero and Scherk-type ends. Surfaces in can be proven to be proper, a condition under which the asymptotic geometry of the surfaces is well known. It is also known that consists of the -parameter family of singly periodic Scherk minimal surfaces. We prove that for each , there exists a natural one-to-one correspondence between and the space of convex unitary nonspecial polygons through the map which assigns to each the polygon whose edges are the flux vectors at the ends of (a special polygon is a parallelogram with two sides of length and two sides of length ). As consequence, reduces to the saddle towers constructed by Karcher.