三维欧氏空间的仿射乘积曲面

定义了三维欧氏空间中的仿射乘积曲面,给出了极小仿射乘积曲面以及高斯曲率为零的仿射乘积曲面的分类,同时还给出了平均曲率为非零常数的仿射乘积曲面的分类.     

完备α度量的相对抛物型仿射球(英文)

摘要:设y:M→Rⁿ⁺¹是一个光滑连通流形到实仿射空间Rⁿ⁺¹的局部强凸浸入超曲面,而且是一个定义在区域Ω（?）Rⁿ上的严格凸函数x_n+1=f（x₁,x₂,…,x_n）的图.在α相对法化下,相对抛物型仿射球满足一个四阶非线性偏微分方程组.本文证明了这类抛物型仿射球的一个新的Bernstein性质.     

关于A^3中仿射球面的两个定理

本文讨论仿射凸曲面为仿射球面或其一部分的问题。利用椭圆型偏微分方程组解的唯一性定理(或称“拟解析函数法”),文中证明了两个较为广泛的定理(见定理1与2),它们进一步推广了诸如 H-定理,K-定理以及许多关于特殊的 Weingarten 曲面为仿射球面的定理。     

R4中关于Klingenberg确定的横截丛的仿射全脐曲面

如果对R4中属于仿射法平面的每个向量相应的形状算子是恒等算子的倍数 ,就称该曲面是全脐的 .本文关于Klingenberg确定的横截丛将仿射全脐曲面进行分类     

类空和类时曲线的仿射性质研究

本文研究类空和类时曲线的中心仿射曲率,中心仿射挠率,曲线的曲率和挠率满足的关系以及两类曲线的正交标架和仿射标架之间的关系的问题.利用仿射空间和Minkowski空间中曲线的基本理论,讨论当类空和类时曲线的弧长与仿射弧长相同时,类空和类时曲线的仿射性质.根据得到的结论,通过变量代换讨论当类空和类时曲线的曲率κ(s)和挠率τ(s)满足τ(s)=aκ^λ(s)(a≠0,λ∈R)时,曲线的曲率所满足的特殊微分方程.     

差异张量平行和半平行的仿射超曲面

本文用活动标架法证明了：若 Ｍｎ（ｎ≥２）是 ｎ＋１维仿射空间 Ａｎ＋１中非退化的仿射超曲面,（１）若■Ｋ＝０（即差异张量平行）,则Ｍ是仿射球,且Ｊ＝０和Ｇ是一个Ｅｉｎｓｔｅｉｎ度量,这里Ｊ是Ｍ的 Ｐｉｃｋ不变量,Ｇ是Ｂｌａｓｃｈｋｅ度量;（２）Ｒ·Ｋ＝０（即差异张量半平行）当且仅当Ｓ＝０（Ｍ为虚仿射球）,或者Ｋ＝０（Ｍ为非退化的二次超曲面）,这里 Ｒ为诱导仿射联络 ■的黎曼曲率算子．     

R^4中关于K1ingenberg确定的横截丛的仿射全脐曲面

如果对R^4中属于仿射法平面的每个向量相应的形状算子是恒等算子的倍数，就称该曲面是全脐的，本文关于K1ingenberg确定的横截丛将仿射全脐曲面进行分类。     

R^(5)中具有平行Fubini-Pick形式的Calabi超曲面的分类北大核心CSCD

Fubini-Pick形式关于Blaschke-Berwald度量(或中心仿射度量)平行的局部强凸等仿射(或中心仿射)超曲面分类问题,在过去十年内已被完全解决.文献[20]对Fubini-Pick形式关于Calabi度量平行的2,3维Calabi超曲面进行了分类,该文将其推广到4维的情形.     

平面上的星形仿射弹性曲线

次仿射弹性曲线是全平方次仿射曲率泛函的临界点.该文对平面上的星形仿射曲线进行了研究，用椭圆函数的方法解出了次仿射弹性曲线的次仿射曲率，并运用Killing 场和sl(2, R)的共轭类分类用积分给出了次仿射弹性曲线的完全解.     

齐次函数的一个仿射球定理

本文研究了相关齐次函数的仿射球定理.利用Hopf极大值原理,对任意给定的带凹性条件的初等对称曲率问题,获得了此类仿射球定理.特别地,这也给出了Deicke齐次函数定理的一个新证明.     

Evolving a convex closed curve to another one via a length-preserving linear flow

Motivated by a recent curvature flow introduced by Professor S.-T. Yau [S.-T. Yau, Private communication on his “Curvature Difference Flow”, 2007], we use a simple curvature flow to evolve a convex closed curve to another one (under the assumption that both curves have the same length). We show that, under the evolution, the length is preserved and if the curvature is bounded above during the evolution, then an initial convex closed curve can be evolved to another given one.     

Affine Invariant Detection: Edge Maps, Anisotropic Diffusion, and Active Contours

In this paper we undertake a systematic investigation of affine invariant object detection and image denoising. Edge detection is first presented from the point of view of the affine invariant scale-space obtained by curvature based motion of the image level-sets. In this case, affine invariant maps are derived as a weighted difference of images at different scales. We then introduce the affine gradient as an affine invariant differential function of lowest possible order with qualitative behavior similar to the Euclidean gradient magnitude. These edge detectors are the basis for the extension of the affine invariant scale-space to a complete affine flow for image denoising and simplification, and to define affine invariant active contours for object detection and edge integration. The active contours are obtained as a gradient flow in a conformally Euclidean space defined by the image on which the object is to be detected. That is, we show that objects can be segmented in an affine invariant manner by computing a path of minimal weighted affine distance, the weight being given by functions of the affine edge detectors. The gradient path is computed via an algorithm which allows to simultaneously detect any number of objects independently of the initial curve topology. Based on the same theory of affine invariant gradient flows we show that the affine geometric heat flow is minimizing, in an affine invariant form, the area enclosed by the curve.     

Evolution of hypersurfaces by the mean curvature minus an external force field

In this paper, we study the evolution of hypersurface moving by the mean curvature minus an external force field. It is shown that the flow will blow up in a finite time if the mean curvature of the initial surface is larger than some constant depending on the boundness of derivatives of the external force field. For a linear force, we prove that the convexity of the hypersurface is preserved during the evolution and the flow has a unique smooth solution in any finite time and expands to infinity as the time tends to infinity if the initial curvature is smaller than the slope of the force.     

The hyperbolic mean curvature flow

We introduce a geometric evolution equation of hyperbolic type, which governs the evolution of a hypersurface moving in the direction of its mean curvature vector. The flow stems from a geometrically natural action containing kinetic and internal energy terms. As the mean curvature of the hypersurface is the main driving factor, we refer to this model as the hyperbolic mean curvature flow (HMCF). The case that the initial velocity field is normal to the hypersurface is of particular interest: this property is preserved during the evolution and gives rise to a comparatively simpler evolution equation. We also consider the case where the manifold can be viewed as a graph over a fixed manifold. Our main results are as follows. First, we derive several balance laws satisfied by the hypersurface during the evolution. Second, we establish that the initial-value problem is locally well-posed in Sobolev spaces; this is achieved by exhibiting a convexity property satisfied by the energy density which is naturally associated with the flow. Third, we provide some criteria ensuring that the flow will blow-up in finite time. Fourth, in the case of graphs, we introduce a concept of weak solutions suitably restricted by an entropy inequality, and we prove that a classical solution is unique in the larger class of entropy solutions. In the special case of one-dimensional graphs, a global-in-time existence result is established.     

The hyperbolic mean curvature flow

We introduce a geometric evolution equation of hyperbolic type, which governs the evolution of a hypersurface moving in the direction of its mean curvature vector. The flow stems from a geometrically natural action containing kinetic and internal energy terms. As the mean curvature of the hypersurface is the main driving factor, we refer to this model as the hyperbolic mean curvature flow (HMCF). The case that the initial velocity field is normal to the hypersurface is of particular interest: this property is preserved during the evolution and gives rise to a comparatively simpler evolution equation. We also consider the case where the manifold can be viewed as a graph over a fixed manifold. Our main results are as follows. First, we derive several balance laws satisfied by the hypersurface during the evolution. Second, we establish that the initial-value problem is locally well-posed in Sobolev spaces; this is achieved by exhibiting a convexity property satisfied by the energy density which is naturally associated with the flow. Third, we provide some criteria ensuring that the flow will blow-up in finite time. Fourth, in the case of graphs, we introduce a concept of weak solutions suitably restricted by an entropy inequality, and we prove that a classical solution is unique in the larger class of entropy solutions. In the special case of one-dimensional graphs, a global-in-time existence result is established.     

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.     

Gauss maps of the Ricci-mean curvature flow

In this paper, we investigate the Gauss maps of a Ricci-mean curvature flow. A Ricci-mean curvature flow is a coupled equation of a mean curvature flow and a Ricci flow on the ambient manifold. Ruh and Vilms (Trans Am Math Soc 149: 569–573, 1970) proved that the Gauss map of a minimal submanifold in a Euclidean space is a harmonic map, and Wang (Math Res Lett 10(2–3):287–299, 2003) extended this result to a mean curvature flow in a Euclidean space by proving its Gauss maps satisfy the harmonic map heat flow equation. In this paper, we deduce the evolution equation for the Gauss maps of a Ricci-mean curvature flow, and as a direct corollary we prove that the Gauss maps of a Ricci-mean curvature flow satisfy the vertically harmonic map heat flow equation when the codimension of submanifolds is 1.     

Bakry–Emery Curvature Operator and Ricci Flow

In this paper, we introduce some techniques of Bakry–Emery curvature operator to Ricci flow and prove the evolution equation for the Bakry–Emery scalar curvature. As its application, we can easily derive the Perelman’s entropy functional monotonicity formula. We also discuss some gradient estimates of Ricci curvature and L ²– estimates of scalar curvature.Project partially supported by Yumiao Fund of Putian University.     

曲率流的参数化有限元逼近

许多物理现象可以在数学上描述为受曲率驱动的自由界面运动,例如薄膜和泡沫的演变、晶体生长,等等.这些薄膜和界面的运动常依赖于其表面曲率,从而可以用相应的曲率流来描述,其相关自由界面问题的数值计算和误差分析一直是计算数学领域中的难点.参数化有限元法是曲率流的一类有效计算方法,已经能够成功模拟一些曲面在几类基本的曲率流下的演化过程.本文重点讨论曲率流的参数化有限元逼近,它的产生、发展和当前的一些挑战.     

关于Pan—Yang不等式的新证明

曲率积分不等式是研究平面曲线的演化问题的重要组成部分,Pan—Yang在研究一类缩短流时得到一个关于曲率积分的不等式．本文主要利用傅里叶分析的方法给出了此不等式的一种简单的证明方法．