Integral Theorems Based on a New Gradient Operator Derived from Biomembranes （Part Ⅱ）： Applications

Based on the second gradient operator and corresponding integral theorems such as the second divergence theorem, the second gradient theorem, the second curl theorem, and the second circulation theorem on curved surfaces, a few new scalar differential operators are defined and a series of integral transformations are derived. Interesting transformations between the average curvature and the Gauss curvature are presented. Various conserved integrals related to the Gauss curvature and the second fundamental tensor are disclosed. The important applications of the results in disciplines such as the geometry, physics, mechanics, and biology are briefly discussed.     

Integral Theorems Based on a New Gradient Operator Derived from Biomembranes （Part I）： Fundamentals

A new gradient operator was derived in recent studies of topological structures and shape transitions in biomembranes. Because this operator has widespread potential uses in mechanics, physics, and biology, the operator‘s general mathematical characteristics should be investigated. This paper explores the integral characteristics of the operator. The second divergence and the differential properties of the operator are used to demonstrate new integral transformations for vector and scalar fields on curved surfaces, such as the second divergence theorem, the second gradient theorem, the second curl theorem, and the second circulation theorem. These new theorems provide a mathematical basis for the use of this operator in many disciplines.     

Symmetrical Fundamental Tensors, Differential Operators, and Integral Theorems in Differential Geometry

To make the geometrical basis for soft matters with curved surfaces such as biomembranes as simple as possible, a symmetrical analytical system was developed in conventional differential geometry. The conventional second fundamental tensor is replaced by the so-called conjugate fundamental tensor. Because the differential properties of the conjugate fundamental tensor and the first fundamental tensor are symmetrical, the symmetrical analytical system including the symmetrical differential operators, symmetrical differential characteristics, and symmetrical integral theorems for tensor fields defined on curved surfaces can be constructed. From the symmetrical analytical system, the symmetrical integral theorems for mean curvature and Gauss curvature, with which the symmetrical Minkowski integral formulas are easily deduced just as special cases, can be derived. The applications of this symmetrical analytical system to biology not only display its simplicity and beauty, but also show its powers in depicting the symmetrical patterns of networks of biomembrane nanotubes. All these symmetrical patterns in soft matters should be just the reasonable and natural results of the symmetrical analytical system.     

常曲率黎曼流形中具平行平均曲率向量的二维浸入曲面

构造了常曲率黎曼流形中具平行平均曲率向量的二维浸入曲面上的全纯微分形式,在同胚球时,给出了其上的Frenet-Boruvka公式及维数定理;估计了浸入的高斯曲率及像的面积;并研究了该浸入的Pinching问题.     

四维欧氏空间中的曲面

研究四维欧氏空间中的一类曲面,给出了具有常平均曲率、常高斯曲率及相关平均曲率和高斯曲率的这类曲面的分类·具有常平均曲率的曲面有4种;具有常高斯曲率的曲面有3种;具有相关非常数平均曲率和非常数高斯曲率的曲面有3种     

R2,1中曲面的伪球线汇

线汇通过线产生于曲面间变换的经典方法中.如果保留原始一些曲面的几何性质,这些转变是特别有趣的.线汇的两个参数族作为线空间的的曲面来研究.利用活动标架来研究线汇,给出了3维闵氏空间R2,1中常Gauss曲率曲面间统一的Backlund变换和Bianchi's置换定理的证明.最后,利用定理的结果构造了一些伪球曲面.     

基于高斯曲率改进的PM模型

在Perona和Malik的各项异性扩散方程的消噪模型中,一些小梯度的细节信息和噪声会被扩散掉,而在这些细节信息点往往具有零高斯曲率的特征,根据这一特点,对PM模型加以改进。新的模型不但可以保持零高斯曲率的图像特征,如:直线边缘、曲线边缘、角点、斜坡和小尺度特征,还可以增强尖锐的边缘。实验表明,改进的模型比PM保留了更多的图像信息,因此该模型可以大量地应用于图像处理和计算机视觉。     

Invariants for Parallel Mapping

This paper analyzes the geometric quantities that remain unchanged during parallel mapping (i.e., mapping from a reference curved surface to a parallel surface with identical normal direction). The second gradient operator, the second class of integral theorems, the Gauss-curvature-based integral theorems, and the core property of parallel mapping are used to derive a series of parallel mapping invariants or geometrically conserved quantities. These include not only local mapping invariants but also global mapping invariants found to exist both in a curved surface and along curves on the curved surface. The parallel mapping invariants are used to identify important transformations between the reference surface and parallel surfaces. These mapping invariants and transformations have potential applications in geometry, physics, biomechanics, and mechanics in which various dynamic processes occur along or between parallel surfaces.     

完整非保守力学系统的Noether逆定理与Lie对称性逆问题

研究完整非保守系统的Ｌｉｅ对称性逆问题：由已知积分寻求相应的Ｌｉｅ对称变换。主要方法是,按Ｎｏｅｔｈｅｒ逆定理由已知积分找到相应的Ｎｏｅｔｈｅｒ广义准对称变换,再将所得变换代入Ｌｉｅ对称性的确定方程来判断变换是否Ｌｉｅ的,举例说明方法的应用。     

三维Minkowski空间中具有逐点1型高斯映射的时间轴旋转曲面

讨论R31 中具有逐点1型高斯映射的第一类和第二类时间轴旋转曲面。证明了时间轴旋转曲面具有第一类逐点1型高斯映射,等价于该曲面的平均曲率为常数;非类光洛伦兹圆锥面是惟一具有第二类逐点1型高斯映射的有理类时间轴旋转曲面。     

H3( -c2)中的CMC-c曲面

讨论了对称空间SL(n,C)/SU(n)中的曲面.首先,讨论了H3(-c2)中的CMC-c曲面(常中曲率为c的曲面)与R3中的极小曲面的关系,利用初等方法证明了H3(-c2)中的一个CMC-c曲面族,当c趋于零时,收敛到R3中的一个极小曲面的结论;其次,把经典的Ricci定理推广到对称空间SL(n,C)/Su(n)上.证明了单连通黎曼曲面(M2,ds2)可以共形等距地浸入到SL(n,C)/SU(n)上,且有全纯右Gauss映射的充分必要条件是ds2的截面曲率K<0及Ricci条件——-K·ds2的截面曲率为 1.     

de Sitter空间中常曲率的类空超曲面

研究de Sitter空间中具有常数量曲率的类空超曲面, 将Cheng-Yau的自共轭算子□作用在对称张量T上, 得到了这类超曲面关于第二基本形式模长平方的一个拼挤定理, 加强了已有的相应结果。     

关于黎曼流形的1/4对称度量联络

本文在黎曼流形为紧致可定向的假设下,给出了关于黎曼联络和1/4对称度量联络的数量曲率之间关系的一个积分公式及其某些应用。同时研究了1/4对称度量联络的曲率张量、利齐张量和数量曲率的性质,给出 C.C.Hwang和 C.Y.Ma的一个定理的推广。     

非线性中立双曲型偏泛函微分方程组的振动性

研究一类具非线性扩散系数的中立型双曲泛函偏微分方程组的振动性,利用Gauss散度定理、积分不等式和泛函微分方程的某些结果,获得了该类方程组在第一类边值条件下所有解振动的若干充分判据.结论充分表明振动是由时滞量引起的,同时也揭示该类方程组与普通双曲型偏微分方程组质的差异.     

子流形的高斯映照(Ⅰ)

设Ｎ是欧氏空间Ｅ＾ｎ＋１中的超曲面，Ｍ是Ｎ的子流形，本文研究Ｍ上的高斯映照，计算高映照的微分，由此建立起Ｍ的Ｒｉｃｃｉ形式与第二、第三基本形式之间的关系。     

黎曼流形的抛物性与度量的共形形变

借助于黎曼流形的抛物性概念研究黎曼度量的共形形变问题, 证明了Gauss曲率小于某负常数的非紧完备2维黎曼流形其度量不可能共形形变到具有非负Gauss曲率的完备度量.     

完备非紧梯度扩张Ricci孤立子的刚性

利用已有梯度Ricci孤立子的刚性定理, 讨论完备非紧梯度扩张Ricci孤立子, 在Ricci曲率非负、 径向曲率为0及Weyl张量的四阶散度非负的条件下, 得到了其刚性的结果.     

离散求交算法中的点元采样

从要进行求交的曲面方程中提取出曲面的高斯曲率、平均曲率、边界点等几何信息，根据这些信息将曲面离散成点元的形式，然后进行求交运算。实验表明，通过对曲面特征分析，缩小了点元动态重采样范围，能够有效地避免一些特征点丢失导致交点遗漏的情况，使离散求交算法更加稳定可靠。     

The comparison theorem for Bergman and Kobayashi metrics on Cartan-Hartogs domain of the second type

In this paper, the holomorphic sectional curvature under invariant metric on a Cartan-Hartogs domain of the second type YII(N,p,K) is presented and an invariant K?]lher metric which is complete and not less than the Bergman metric is constructed, such that its holomorphic sectional curvature is bounded above by a negative constant. Hence a comparison theorem for the Bergman and Kobayashi metrics on YII(N,p,K) is obtained.     

A kind of integral representation on Riemannian manifods

We generalized the Bochner-Martinelli integral representation to that on Riemannian manifolds. Things become quite different in such case. First we define a kind of Newtonian potential and take the interior product of its gradient to be the integral kernel. Then we prove that this kernel is harmonic in some sense. At last an integral representative theorem is proved. Hu Jicheng, born in July, 1965, Ph.D. Current research interest is in function theory, including wavelet analysis and singular integral equations Supported by the National Natural Science Foundation of China