Minimal Surfaces in a Cone

We prove the convex hull property for properly immersed minimal hypersurfaces in a cone of ⁿ . We deal with the existence of new barriers for the maximum principle application in noncompact truncated tetrahedral domains of ³, describing the space of such domainsadmitting barriers of this kind. Nonexistence results for nonflatminimal surfaces whose boundary lies in opposite faces of a tetrahedraldomain are obtained. Finally, new simple closed subsets of ³ whichhave the property of intersecting any properly immersed minimal surfaceare shown.     

Minimal Surfaces Obtained by Ribaucour Transformations

We consider Ribaucour transformations between minimal surfaces and we relate such transformations to generating planar embedded ends. Applying Ribaucour transformations to Enneper's surface and to the catenoid, we obtain new families of complete, minimal surfaces, of genus zero, immersed in R ³, with infinitely many embedded planar ends or with any finite number of such ends. Moreover, each surface has one or two nonplanar ends. A particular family is obtained from the catenoid, for each pair (n,m), nm, such that n m0 is an irreducible rational number. For any such pair, we get a 1-parameter family of finite total curvature, complete minimal surfaces with n+2 ends, n embedded planar ends and two nonplanar ends of geometric index m, whose total curvature is –4(n+m). The analytic interpretation of a Ribaucour transformation as a Bäcklund type transformation and a superposition formula for the nonlinear differential equation = e^-2 is included.     

Timelike Minimal Surfaces via Loop Groups

This work consists of two parts. In Part I, we shall give a systematic study of Lorentz conformal structure from structural viewpoints. We study manifolds with split-complex structure. We apply general results on split-complex structure for the study of Lorentz surfaces.In Part II, we study the conformal realization of Lorentz surfaces in the Minkowski 3-space via conformal minimal immersions. We apply loop group theoretic Weierstrass-type representation of timelike constant mean curvature for timelike minimal surfaces. Classical integral representation formula for timelike minimal surfaces will be recovered from loop group theoretic viewpoint.     

三维Minkowski空间中的极小曲面

给出了三维Minkowski空间中类空曲面活动标价的四元素表示,并利用四元素既适合于旋转结构的侵入又适合于2×2矩阵处理极小曲面的分析特性,经研究得到了R12中的极小曲面Weierstrass表示和曲面的可积条件.     

Minimal Surfaces in the Heisenberg Group

We investigate the minimal surface problem in the three dimensional Heisenberg group, H, equipped with its standard Carnot–Carathéodory metric. Using a particular surface measure, we characterize minimal surfaces in terms of a sub-elliptic partial differential equation and prove an existence result for the Plateau problem in this setting. Further, we provide a link between our minimal surfaces and Riemannian constant mean curvature surfaces in H equipped with different Riemannian metrics approximating the Carnot–Carathéodory metric. We generate a large library of examples of minimal surfaces and use these to show that the solution to the Dirichlet problem need not be unique. Moreover, we show that the minimal surfaces we construct are in fact X-minimal surfaces in the sense of Garofalo and Nhieu.     

Density Theorems for Complete Minimal Surfaces in {\mathbb{R}}^{3}

In this paper we have proved several approximation theorems for the family of minimal surfaces in that imply, among other things, that complete minimal surfaces are dense in the space of all minimal surfaces endowed with the topology of C ^k convergence on compact sets, for any . As a consequence of the above density result, we have been able to produce the first example of a complete proper minimal surface in with uncountably many ends. This research is partially supported by MEC-FEDER Grant no. MTM2004 - 00160.     

Compact Embedded Minimal Surfaces of Positive Genus Without Area Bounds

Let M ³ be a three-manifold (possibly with boundary). We will show that, for any positive integer , there exists an open nonempty set of metrics on M (in the C ²-topology on the space of metrics on M) for each of which there are compact embedded stable minimal surfaces of genus with arbitrarily large area. This extends a result of Colding and Minicozzi, who proved the case =1.     

一种改进边界元法在弹性扭转问题中的应用

本文用一种改进边界元法分析与计算了椭圆截面等直杆的扭转问题.并与边界元法的解进行比较,其结果极为符合.然而,改进边界元法较边界元法所需要的数据量少得多,计算时间也将大大减少了.因此,本文方法对求解Poisson方程问题是一种经济而行之有效的数值计算方法.     

极小曲面中的若干问题

本文综述了Ｅ＾ｎ和Ｓ＾ｎ中极小曲面的若干经典结果和最新发展，指出了一些尚未解决的问题，在第４节中，对Ｅ＾３中极小曲面的Ｆｕｊｉｍｏｔｏ定理给出了一个更直接的证明。     

矩形弹性夹杂与裂纹相互干扰的边界元分析

使用边界元法研究了无限弹性体中矩形弹性夹杂对曲折裂纹的影响,导出了新的复边界积分方程．通过引入与界面位移密度和面力有关的未知复函数H(t),并使用分部积分技巧,使得夹杂和基体界面处的面力连续性条件自动满足,而边界积分方程减少为2个,且只具有1/r阶奇异性．为了检验该边界元法的正确性和有效性,对典型问题进行了数值计算．所得结果表明:裂纹的应力强度因子随着夹杂弹性模量的增大而减小,软夹杂有利于裂纹的扩展,而刚性较大的夹杂对裂纹有抑制作用．     

Uniqueness of PL Minimal Surfaces

Using a standard fact in hyperbolic geometry, we give a simple proof of the uniqueness of PL minimal surfaces, thus filling in a gap in the original proof of Jaco and Rubinstein. Moreover, in order to clarify some ambiguity, we sharpen the definition of PL minimal surfaces, and prove a technical lemma on the Plateau problem in the hyperbolic space.     

S6中具有常数K(a)hler角和常数曲率的极小曲面

本文给出了近Ｋａｈｌｅｒ球面Ｓ６中具有常数Ｋａｈｌｅｒ角和常数曲率的极小曲面的例子,同时证明了两个唯一性定理．     

最小Q-过程中断时Martin流出边界与Ray-Knight紧化的关系

在最小Q-过程中断条件下证明了Martin流出边界与Ray-Knight紧化之间存在一定的映射关系.并举例说明如果最小Q-过程中断且无限流出,则Ray-Knight紧化可能比Martin流出边界要多出一些点.因此在Markov链的构造论中不能只考虑Martin边界.     

Closed Geodesics on Incomplete Surfaces

We consider the problem of finding embedded closed geodesics on the two-sphere with an incomplete metric defined outside a point. Various techniques including curve shortening methods are used The authors acknowledge the support of the Australian Research Council     

具有特殊类型端口的极小曲面的构造

We proved that there exists a family of complete oriented minimal surfaces in R3 with finite total curvature-4nπ,each of which has 0 genus and two ends,and both of the ends have winding order n,where n ∈ N,and discussed the symmetric property for special parameters.     

表面承受常静水载荷下弹性半无限体的表面稳定性分析

本文应用小变形叠加到大变形上的分析方法,考察了表面受常静水载荷作用下可压缩和不可压缩弹性半无限体的表面平面应变失稳问题,得到了各自的临界屈曲条件的一般形式.在此基础上.分别对由Blatz-Ko材料和谐和材料所组成的可压缩弹性半无限体以及由Mooney材料所组成的不可压缩弹性半无限体,详细讨论了表面所受的静水压力对临界屈曲条件的影响.     

Stabilization of an Elastic Plate with Viscoelastic Boundary Conditions

The boundary control problem of an elastic thin plate with boundary viscoelasticity is formulated in the standard form of a linear infinite-dimensional systems in the energy Hilbert space. The feedback control is designed so that the input and output are collocated. The frequency-domain approach is adopted in investigating the exponential stability of the closed-loop system. Finally, by considering the boundary viscoelasticity as damping, we establish a strong stability result based on the LaSalle invariance principle and the Hömander uniqueness theorem.     

单位球中的极小曲面

Let M be a closed surface with positive Gauss curvature minimally immersed in a standard Euclidean unit sphere S~n.In this paper,we choose a local orthonormal frame field on M,under which the shape operators have very convenient form.We also give some applications of this kind of frame field.     

空间形式中具平行非退化极小截面的曲面

利用文「１」的方法及Ｓｉｍｏｎｓ型公式对空间形式中具有平行非退化极小截面曲面进行了研究,从而将文「１」中一系列定理推广到２＋Ｐ维空间形式中。     

Weierstrass Representation of Some Simply-Periodic Minimal Surfaces

We prove a desingularization result for minimal surfaces inEuclidean space using Weierstrass representation. We solve the periodproblem using the implicit function theorem at a degenerate point.