3-D涡块边界曲面高阶正则性的传播

本文讨论了涡块问题边界曲面高阶正则性的传播，同时解决了［８］中所提出的关于边界曲面Σ０的高阶正则性的传播问题．     

de Sitter空间中的类空超曲面在平均曲率下的形变

本文考虑了deSitter空间中紧致的类空扭曲面在平均曲率下的形变,证明了如果初始曲面满足某一Pinching条件,超曲面将收敛到一个球面．这一结论类似于球面上超曲面的情形,而与欧氏空间中的超曲面完全不同．     

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．     

裁剪B样条曲面间过渡曲面的构造

1 引言在CAD曲面造型中,对于复杂的模型,通常无法用一张曲面将其表示出来．当采用多张曲面表示的时候,就涉及到曲面的裁剪、过渡、拼接等问题(见[1],[2],[3])．在本文中, 我们对于基曲面是裁剪B样条曲面,裁剪线是B样条曲线的情况,提出两种构造光滑过渡曲面的方法．其中第一种方法能够较快的生成近似G1连续的过渡曲面,而第二种方法是对Filip在[2]中的工作的发展,所生成的过渡曲面不但是G1连续的,而且可以灵活地调整形状．     

有理曲线和曲面的分片多项式逼近

本文利用摄动的思想,以摄动有理曲线（曲面）的系数的无穷模作为优化目标,给出了用多项式曲线（曲面）逼近有理曲线（曲面）的一种新方法．同以前的各种方法相比,该方法不仅收敛而且具有更快的收敛速度,并且可以与细分技术相结合,得到有理曲线与曲面的整体光滑、分片多项式的逼近．     

三维Minkowski空间中曲面的广义Weierstrass公式

本文给出了三维Ｍｉｎｋｏｗｓｋｉ空间中一般类空曲面与类时曲面的广义Ｗｅｉｅｒ－ｓｔｒａｓｓ表示公式．     

双曲空间中的共形平坦极小超曲面

本文证明了双曲空间中的共形平坦极小超曲面必为旋转超曲面或由一些旋转超曲面片用全测地超曲面片粘合而成．将这结果与王新民及许志才的一个已发表定理相组合，推广了Ｂｌａｉｒ关于推广悬链面的一个定理．     

基于NURBS的极小曲面造型

1 引言极小曲面问题是微分几何领域中一个古老而活跃的问题．在微分几何学中,极小曲面的研究已十分成熟．如何把极小曲面引入CAGD领域,是一个极有价值的课题．文献 [1]提出一种几何构造法,得到了一类三次多项式形式的负高斯曲率极小曲面,将其表示为三次B-B曲面,并将其用到房顶曲面设计当中．文献[2]讨论参数多项式极小曲面,证明了只存在一类三次等温参数极小曲面,并研究了这类曲面的一些基本性质．虽然这些多     

关于3维欧氏空间中一族嵌入极小曲面的一个注记

本文证明了在 Ｄ． Ｈｏｆｆｍａｎ和 Ｗ．Ｈ． Ｍｅｅｋｓ， Ⅲ［４］给出的 ３维欧氏空间的一族嵌入极小曲面中，每一个曲面与其自身的和曲面是平凡的极小核心．     

简单曲面的顶点对应算法及其关键帧动画实现

本文利用曲面参数化的思想设计了３维关键帧动画中的顶点对应简化算法．在这个基础上，本文把Ｓｅｄｅｒｂｅｒｇ关于２维闭曲线帧插值的方法推广到３维闭曲面帧插值，并进行了大量的动画编程实践．试验表明，本文的方法对许多简单曲面都有较好的插值效果．     

Volume-preserving flow by powers of the mth mean curvature

We consider the evolution of a closed convex hypersurface under a volume preserving curvature flow. The speed is given by a power of the mth mean curvature plus a volume preserving term, including the case of powers of the mean curvature or of the Gauss curvature. We prove that if the initial hypersurface satisfies a suitable pinching condition, the solution exists for all times and converges to a round sphere.     

Some remarks on the level set flow by anisotropic curvature

We show that almost all level sets of the unique viscosity solution for general anisotropic mean curvature flow satisfy a weak form of the flow equation. This generalizes the case of isotropic mean curvature flow studied by Evans and Spruck, in which a relation between the viscosity solution and Brakke's varifold mean curvature flow is established. Received June 4, 1998 / Accepted February 26, 1999     

带有线性外力场的双曲平均曲率流Cauchy问题经典解的生命跨度

本文提出并研究带有线性外力场的双曲平均曲率流,通过凸曲线的支撑函数,导出一个双曲型Monge-Ampère 方程并将其转化成Riemann 不变量满足的拟线性双曲方程组。利用拟线性双曲方程组Cauchy 问题的局部解理论,讨论带有线性外力场的双曲平均曲率流Cauchy 问题经典解的生命跨度（即局部解存在的最大时间区间）。     

A Gradient Estimate for the Ricci–Kähler Flow

We show that for a complete solution to theRicci–Kähler flow where the curvature, the potential andscalar curvature functions and their gradients are bounded depending ontime, the absolute value of both the scalar curvature and the gradientsquared of a modified potential function are bounded byC/t.     

Evolution by Non-Convex Functionals

We establish a semi-group solution concept for flows that are generated by generalized minimizers of non-convex energy functionals. We use relaxation and convexification to define these generalized minimizers. The main part of this work consists in exemplary validation of the solution concept for a non-convex energy functional. For rotationally invariant initial data it is compared with the solution of the mean curvature flow equation. The basic example relates the mean curvature flow equation with a sequence of iterative minimizers of a family of non-convex energy functionals. Together with the numerical evidence this corroborates the claim that the non-convex semi-group solution concept defines, in general, a solution of the mean curvature equation.     

Expanding convex immersed closed plane curves

We study the evolution driven by curvature of a given convex immersed closed plane curve. We show that it will converge to a self-similar solution eventually. This self-similar solution may or may not contain singularities. In case it does, we also have estimate on the curvature blow-up rate.     

Stability of mean convex cones under mean curvature flow

We consider graphical solutions to mean curvature flow and obtain a stability result for homothetically expanding solutions coming out of cones of positive mean curvature. If another solution is initially close to the cone at infinity, then the difference to the homothetically expanding solution becomes small for large times. The proof involves the construction of appropriate barriers.     

Space-time formulation of Harnack inequalities for curvature flows of hypersurfaces

We consider the motion of hypersurfaces in Riemannian manifolds by their curvature vectors. We show that the Harnack quadratic is an affine second fundamental form of the space-time track of the hypersurface. Given a solution to the Ricci flow, we show that with respect to an appropriate metric on space-time, the space-slices evolve by mean curvature flow. This enables us to identify the Harnack quadratic for the mean curvature flow with the trace Harnack quadratic for the Ricci flow.     

Mean curvature flow with obstacles

We consider the evolution of fronts by mean curvature in the presence of obstacles. We construct a weak solution to the flow by means of a variational method, corresponding to an implicit time-discretization scheme. Assuming the regularity of the obstacles, in the two-dimensional case we show existence and uniqueness of a regular solution before the onset of singularities. Finally, we discuss an application of this result to the positive mean curvature flow.     

On the Topology of Vacuum Spacetimes

We prove that there are no restrictions on the spatial topology of asymptotically flat solutions of the vacuum Einstein equations in (n + 1)-dimensions. We do this by gluing a solution of the vacuum constraint equations on an arbitrary compact manifold Sⁿ \Sigma^n to an asymptotically Euclidean solution of the constraints on \mathbbRⁿ \mathbb{R}^n . For any Sⁿ \Sigma^n which does not admit a metric of positive scalar curvature, this provides for the existence of asymptotically flat vacuum spacetimes with no maximal slices. Our main theorem is a special case of a more general gluing construction for nondegenerate solutions of the vacuum constraint equations which have some restrictions on the mean curvature, but for which the mean curvature is not necessarily constant. This generalizes the construction [16], which is restricted to constant mean curvature data.