R3内具给定平均曲率的闭曲面

设H^*是R³内某个开区域内的实值函数,在H^*上加两个条件,在_R³内能找到同胚于给定闭曲面(任意亏格)的曲面,其平均曲率由H^*给出。     

某类振荡积分算子在Lebesgue空间及Wiener共合空间上的映射性质*

假设 β₁ > 3α₁ > 0, β₂ > 3α₂ > 0,给定函数f(x) ∈ S(R³), 定义算子T_α,β如下:T_α,βf(x,y,z) = p.v.ZT_Q2f(x- t, y-s, z-γ(t)h(s)) e^{-2πit-β₁ s-β₂}/t^1+α₁ s^1+α₂dtds.本文主要考虑如上定义的算子T_α,β在Lebesgue空间L^p(R³)及Wiener共合空间W(FL^p, L^q)(R³)上的有界性. 这里 Q² = [0, 1] × [0, 1], γ(t), h(s)满足适当的条件.作为应用, 本文还考虑了带粗糙核的奇异积分算子在乘积空间上的有界性.     

R_1~3中混合型曲面保主曲率等距变形

本文研究了Minkowski空间R_(1)~3曲面的等距变形问题.建立了R_(1)~3中曲面的共形、等距等概念.推广了O.Bonnet和S.S.Chern关于欧氏空间的结论.对R_(1)~3出现的新情况——曲面的中曲率梯度类光作了一定探讨,得出的主要结果为:非平坦的、允许保主曲率等距变形的曲面一定不是W-曲面.     

局部对偶的黎曼空间和引力瞬子解

四维正定黎曼空间R₄能局部地生成两个SU₂规范场和,如果,至少有一个具有自对偶性或反自对偶性,那末空间称为具局部对偶性的.我们证明它们是Einstein空间、数量曲率为0的共形平坦空间以及只R⁺⁺=0(或R^--=0)的空间.文中得出了R⁺⁺=0(R^--≠0)的一类黎曼线素.对曲率张量平方可积的情形,作出了规范场作用量,Euler示性数,Pontrjagin示性数之间的一个不等式,证明它的等号在而且只在R₄具局部对偶性时达到,这结果改进了文献[7]中关于引力瞬子解的研究.并以Hitchin关于4维紧致Einstein形流的一个不等式作为特殊情况.     

de Sitter空间S1m+1中具有平行Blaschke张量的正则类空超曲面

本文引入两个以de Sitter空间为模型的非齐性坐标来覆盖共形空间Q₁^m+1.利用球面S^m+1中超曲面的Möbius几何的方法,本文研究了Q₁^m+1中正则类空超曲面的共形几何.作为其结果,本文对所有具有平行Blaschke张量的正则类空超曲面进行了完全分类.     

S4中等参超曲面的谱刻画

本文证明了如果Ｓ⁴中的具常平均曲率ｈ的超曲面Ｍ与其具平均曲率ｈ的等参超曲面Ｍ_０（强）等谱，则Ｍ＝Ｍ_０．     

S4(1)中常数量曲率的完备超曲面

本文把[1]的结论推广到超曲面是完备的情形,即我们证明了:设M³是单位球面S⁴(1)中常平均曲率及常数量曲率的完备超曲面。若S≤H²+6,则S只能等于1/3H²,3/4H²—1/4(H⁴+8H²)^1/2+3,(3/4)H²+1/4(H⁴+8H²)^{1/2相似文献}   

R31中混合型曲面保主曲率等距变形

本文研究了Minkowski空间R31曲面的等距变形问题.建立了R31中曲面的共形、等距等概念.推广了O.Bonnet和S.S.Chern关于欧氏空间的结论.对R31出现的新情况--曲面的中曲率梯度类光作了一定探讨,得出的主要结果为:非平坦的、允许保主曲率等距变形的曲面一定不是W-曲面.     

范畴RMnl上的一个函子

Let _RM_n^l′ be a category which is equivalent to the category of left R-modules.In this paper,we define afunction F:_RM_n^l→_RM_n^l′ and prove that the functor Fpreserves products,direct limits,injections,surjectios and total esactness.Finally,we show that the functor F is a left-adjoint of the inclusion functor I:_RM_n^l′→_RM_n^l. Hence I:_RM_n^l′ is a renective subcategory of _RM_n^l.     

积域上某些奇异积分算子的L2-有界性

本文给出积域上一类 T(b)定理.粗略地说,设T是 R^d₁)×R^d₂上的一个奇异积分算子,T^t是T关于变量(x,u)或(y,v)或两者的转置,T^t(j)是T^t在R^d_j上的限制,j=1,2,则T的L²-有界性由下述条件推出:T有WBP, 以及T^t(j)(b_j)=0,其中b_j是R^d_j上的任意强仿增长函数,j=1,2.     

R4(C)中平均曲率方向平行的Bonnet曲面

本文利用[5]的方法,对R⁴(C)中平均曲率方向平行的Bonnet曲面引入了半测地等温 参数,并给出了一个分类结果.     

Complete surfaces in the hyperbolic space with a constant principal curvature

In this paper we study complete orientable surfaces with a constant principal curvature R in the 3‐dimensional hyperbolic space H ³. We prove that if R² > 1, such a surface is totally umbilical or umbilically free and it can be described in terms of a complete regular curve in H ³. When R² ≤ 1, we show that this result is not true any more by means of several examples. This contradicts a previous statement by Zhisheng [6]. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A halfspace theorem for proper,negatively curved immersions

Generalizing the classical halfspace theorem for minimal surfaces (Hoffman and Meeks in Invent Math 101:373–377, 1990), we prove such a result for two-dimensional surfaces in \mathbbR³{\mathbb{R}^3} of negative Gaussian curvature. Instead of requiring an elliptic differential equation, we merely assume some inequality involving the principal curvatures of the surface to be satisfied, see assumption (1). Surfaces of this type arise naturally as critical points of weighted area functionals defined in (2).     

The Minding formula and its applications

We apply the Minding Formula for geodesic curvature and the Gauss-Bonnet Formula to calculate the total Gaussian curvature of certain 2-dimensional open complete branched Riemannian manifolds, the M\cal M surfaces. We prove that for an M\cal M surface, the total curvature depends only on its Euler characteristic and the local behaviour of its metric at ends and branch points. Then we check that many important surfaces, such as complete minimal surfaces in \Bbb Rⁿ{\Bbb R}^n with finite total curvature, complete constant mean curvature surfaces in hyperbolic 3-space H³ (–1) with finite total curvature, are actually branch point free M\cal M surfaces. Therefore as corollaries we give simple proofs of some classical theorems such as the Chern-Osserman theorem for complete minimal surfaces in \Bbb Rⁿ{\Bbb R}^n with finite total curvature. For the reader's convenience, we also derive the Minding Formula.     

Four Classes of Surfaces with Constant Mean Curvature in S 3 × R and H 3 × R

Deforming rotation surfaces with constant mean curvature in S ³ and H ³ to S ³ × R and H ³ × R respectvely, we give four classes of surfaces with mean curvature vector of constant length in S ³ × R and H ³ × R. We have complete minimal surfaces in S ³ × R and H ³ × R. Also we obtain minimal 2-tori in S ³ × S ¹, some of which are embedded.     

Compact Surfaces with Constant Gaussian Curvature in Product Spaces

We prove that the only compact surfaces of positive constant Gaussian curvature in \mathbbH²×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} (resp. positive constant Gaussian curvature greater than 1 in \mathbbS²×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}}) whose boundary Γ is contained in a slice of the ambient space and such that the surface intersects this slice at a constant angle along Γ, are the pieces of a rotational complete surface. We also obtain some area estimates for surfaces of positive constant Gaussian curvature in \mathbbH²×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} and positive constant Gaussian curvature greater than 1 in \mathbbS²×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}} whose boundary is contained in a slice of the ambient space. These estimates are optimal in the sense that if the bounds are attained, the surface is again a piece of a rotational complete surface.     

Geodesics in minimal surfaces

Abstract—In this paper, we consider connected minimal surfaces in R³ with isothermal coordinates and with a family of geodesic coordinates curves, these surfaces will be called GICM-surfaces. We give a classification of the GICM-surfaces. This class of minimal surfaces includes the catenoid, the helicoid and Enneper’s surface. Also, we show that one family of this class of minimal surfaces has at least one closed geodesic and one 1-periodic family of this class has finite total curvature. As application we show other characterization of catenoid and helicoid. Finally, we show that the class of GICM-surfaces coincides with the class of minimal surfaces whose the geodesic curvature k _g ¹ and k _g ² of the coordinates curves satisfy αk _g ¹ + βk _g ² = 0, α, β ∈ R.     

Spectral transformation of constant mean curvature surfaces in H3 and Weierstrass representation

By using the method of integrable system, we study the deformation of constant mean curvature surfaces in three-dimensional hyperbolic space form H₃. We also obtain a Weierstrass representation formula of the constant mean curvature surfaces with mean curvature greater than 1     

K0 of Purely Infinite Simple Regular Rings

We extend the notion of a purely infinite simple C ^*-algebra to the context of unital rings, and we study its basic properties, specially those related to K-theory. For instance, if R is a purely infinite simple ring, then K ₀(R)⁺ = K ₀(R), the monoid of isomorphism classes of finitely generated projective R-modules is isomorphic to the monoid obtained from K ₀(R) by adjoining a new zero element, and K ₁(R) is the Abelianization of the group of units of R. We develop techniques of construction, obtaining new examples in this class in the case of von Neumann regular rings, and we compute the Grothendieck groups of these examples. In particular, we prove that every countable Abelian group is isomorphic to K ₀ of some purely infinite simple regular ring. Finally, some known examples are analyzed within this framework.