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．     

三维Minkowski空间中的新型平移曲面

In this paper we study translation surfaces of some new types in 3-Minkowski space E13 and give some classifications of such surfaces whose mean curvature and Gauss curvature satisfy certain conditions.     

三维Minkowski空间中的曲面及其可积性

利用可积系统的思想,借助三维Minkowski空间L3的矩阵模型,研究了L3中具有调和逆平均曲率的类空曲面和洛伦兹调和逆平均曲率类时曲面的可积性及其形变.     

三维Minkowski空间中的曲面及其可积性

利用可积系统的思想,借助三维Minkowski空间L3的矩阵模型,研究了L3中具有调和逆平均曲率的类空曲面和洛伦兹调和逆平均曲率类时曲面的可积性及其形变.     

三维Minkowski空间中类空轴旋转曲面的分类

研究了三维Minkowski空间中满足ΔG=φ(G+C)条件的类空轴旋转曲面,并给出了该类曲面的分类.主要结论为上述条件中当C为零向量时该曲面拥有常值平均曲率;当C为非零向量时,该曲面或是一类圆锥面、或是二类圆锥面.     

四维Minkowski空间中的类空曲面的一类高斯映射的奇异性

Cuong在文献中引入了四维Minkowski空间中类空曲面的LSr高斯映射的概念并研究了该空间中全脐类空曲面的微分几何性质.发现该类高斯映射存在奇异性并利用Lagrangian奇点理论和切触理论具体刻画了LSr值高斯映射的奇点.     

三维Minkowski空间中的一类直线汇

研究了由三维Minkowski空间$E^3_1$中一个类空曲面$S_1$上一个单参数测地曲线族的切线所构成的直线汇$T$,它以$S_1$为一个焦曲面.证明了$T$的两个可展曲面族沿着第二个焦曲面$S_2$的正交曲线网相交的充要条件是$S_1$是可展曲面.对于$T$的两个焦曲面$S_1$和$S_2$之间沿着同一光线的对应,还证明了其保持渐近曲线网的充要条件.最后,研究了$T$的正交曲面$S$,并且证明了如果$S$是$E^3_1$中的一个极大曲面,那么,$T$的焦曲面$S_1$和$S_2$之间沿着同一光线的对     

三维Minkowski空间R^2.1中混合型时空曲面的等温参数

对于三维Minkowski空间中的混合型时空曲面，证明了其在类空部分和类时部分上分别存在光滑的，并且连续到交界线上的等温参数．进一步，给出了混合型时空曲面存在在其类空部分和类时部分上是光滑的，而在交界线上是连续的等温参数的—个必要条件．     

非平坦的Randers空间中的极小旋转超曲面

本文的研究分为两部分.第一部分是在特殊的Randers空间中得到了等距浸入的HT-极小旋转超曲面,这些特殊的Rander空间是非Minkowski的,但是它们的旗曲率为零.第二部分刻画了Funk空间中的各向异性极小旋转超曲面.     

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.     

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.     

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.     

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.     

A Note on Minimal Surfaces in Euclidean 3-Space

In this note, a construction of minimal surfaces in Euclidean 3-space is given. By using the product of Weierstrass data of two known minimal surfaces, one gets a new Weierstrass data and a corresponding minimal surface from the Weierstrass representation.     

Minimal Surfaces with an Elastic Boundary

LetD:= { C ³ (

Totally real minimal 2-spheres in quaternionic projective space

Let HPn be the quaternionic projective space with constant quaternionic sectional curvature 4. Then locally there exists a tripe {I, J, K} of complex structures on HPn satisfying U = -JI = K,JK = -KJ = /, KI = -IK = J. A surface M(？) HPn is called totally real, if at each point p ∈M the tangent plane TPM is perpendicular to I(TPM), J(TPM) and K(TPM). It is known that any surface M(？)RPn(？) HPn is totally real, where RPn (?) HPn is the standard embedding of real projective space in HPn induced by the inclusion R in H, and that there are totally real surfaces in HPn which don't come from this way. In this paper we show that any totally real minimal 2-sphere in HPn is isometric to a full minimal 2-sphere in Rp2m (？) RPn(？) HPn with 2m≤n. As a consequence we show that the Veronese sequences in KP2m (m≥1) are the only totally real minimal 2-spheres with constant curvature in the quaternionic projective space.     

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.     

Time-like Willmore surfaces in Lorentzian 3-space

Let R13 be the Lorentzian 3-space with inner product (, ). Let Q3 be the conformal compactification of R13, obtained by attaching a light-cone C∞ to R13 in infinity. Then Q3 has a standard conformal Lorentzian structure with the conformal transformation group O(3,2)/{±1}. In this paper, we study local conformal invariants of time-like surfaces in Q3 and dual theorem for Willmore surfaces in Q3. Let M (?) R13 be a time-like surface. Let n be the unit normal and H the mean curvature of the surface M. For any p ∈ M we define S12(p) = {X ∈ R13 (X - c(P),X - c(p)) = 1/H(p)2} with c(p) = P 1/H(p)n(P) ∈ R13. Then S12 (p) is a one-sheet-hyperboloid in R3, which has the same tangent plane and mean curvature as M at the point p. We show that the family {S12(p),p ∈ M} of hyperboloid in R13 defines in general two different enveloping surfaces, one is M itself, another is denoted by M (may be degenerate), and called the associated surface of M. We show that (i) if M is a time-like Willmore surface in Q3 with non-degenerate associated surface M, then M is also a time-like Willmore surface in Q3 satisfying M = M; (ii) if M is a single point, then M is conformally equivalent to a minimal surface in R13.