Analytic dependence of riemann mappings for bounded domains and minimal surfaces

In this paper, we prove that the functional which takes a closed Lavrentiev curve to the corresponding Riemann mapping is locally Lipl on the set Ω of all closed Lavrentiev curves. This set is a subset of BMO(T). It is, however, not open in BMO(T). We also prove that the previous functional is analytic for certain classes of closed Lavrentiev curves, including the class of curves which have some symmetry with respect to the unit circle. These classes of curves are submanifolds of BMO(T). Finally, we consider the functional which takes a Lavrentiev curve (closed or not) in n-dimensional Euclidean space to the corresponding minimal surface, and we study the differentiability and analyticity of this functional on certain function spaces. © 1993 John Wiley & Sons, Inc.     

关于解析函数的一个唯一性定理及其应用

<正> §1.在近代解析函数论中边界值的唯一性定理有许多的研究,其中有我们所习知著名的(?)氏唯一性定理,即:若 D 是某一可求长约当曲线Γ所范围的内域,而 f(z)是 D 内的半纯函数,如在Γ上存在某一测度大于零的集 E_z,对 E_z 任一点 z_0 上,f(z)的角形边界值为零.则必致f(z)≡0于 D 内.同时,卢洵(?)与普里瓦洛夫还指出:存在有单位圆域内非常数的解析函数,而在一个正测度的集(?)上具有等于零的射形边界值.除此之外还有一个很有用的 Koebe 氏定理.即:     

Zur approximation des transfiniten durchmessers bei bis auf ecken analytischen geschlossenen jordankurven

LetC be a closed Jordan curve in the complex plane and letf(z)=dz+a ₀+a ₁ z ^?1+… be the analytic function mapping |z|>1 schlicht onto the exterior ofC (d>0 is the transfinite diameter ofC). Similar to the Fekete points a point system will be defined calledextremal points. One can use the Fekete points or the extremal points to approximated. The author has proved [3] that in the case of an analytic closed Jordan curve the approximation ofd by means of extremal points is much better than the approximation ofd by the use of Fekete points. Here we show how to approximated by means of extremal points in the case of a piecewise analytic, closed Jordan curve possessing corners of openingαπ (0<α<2).     

On the Convergence of Rational Functions of Best Lp-Approximation

In the present paper, for sequences of rational functions, analytic in some domain, a theorem of Montel’s type is proved. As an application, sequences of rational functions of best Lp — approximation of a function ?(z) ∈ L_p(Γ), with Γ being a closed rectifiable analytic curve, are considered. The case ?(z) ∈ Hp is discussed, too.     

Focal approximation on the complex plane

The problem of analytic approximation of a smooth closed curve specified by a set of its points on the complex plane is proposed. An algorithmic method for constructing an approximating lemniscate is proposed and investigated. This method is based on a mapping of the curve to be approximated onto the phase circle; the convergence of the method is proved. The location of the lemniscate foci inside the curve provides the degrees of freedom for the focal approximation.     

Funktionentheoretische Bestimmung des Aussenfeldes zu einer zweidimensionalen magnetohydrostatischen Konfiguration

Summary Two-dimensional analogues of the torus-like Meyer-Schmidt plasma configurations without interior field are treated by functiontheoretical methods. If the boundary of the doubly connected plasma domain consists of two closed analytic curves, the structure of the exterior magnetic field and its singularities (which determine the currents) can be found by a combination of analytic continuation and conformal mapping. An example is discussed in detail. The problem is essentially equivalent to a problem in hydrodynamics: to find the structure of an incompressible irrotational flow in two dimensions which has a given closed analytic curve as a free streamline.

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Diese Arbeit wurde im Rahmen des Vertrags zwischen dem Institut für Plasmaphysik GmbH und der Europäischen Atomgemeinschaft über die Zusammenarbeit auf dem Gebiet der Plasmaphysik durchgeführt. Sie ist eine kondensierte Darstellung des Berichts [6]. Auf der Tagung der Gesellschaft für Angewandte Mathematik und Mechanik in Giessen (Deutschland) vom 8. bis 10. April 1964 wurde über ihren Inhalt ein Vortrag gehalten (man vergleiche [7]).     

An analytic version of the De Rham theorem

LetV be a real analytic paracompact variety; in §1 of this paper we prove that: $$H^q (V,R) \approx \frac{{closed analytic differentiable q - forms on V}}{{exact analytic differentiable q - forms on V}}$$ Then we prove that the closed (exact) analytic differentiableq-forms onV are dense, in the Whitney topology, in the set of closed (exact) differentiableq-forms onV. We also consider the problem of extending closed (exact) analytic differentiableq-forms, defined on a subvarietyX ofV, to closed (exact) analytic forms defined onV.     

Real structures of Teichmüller spaces, Dehn twists, and moduli spaces of real curves

An orientation reversing involution of a topological compact genus surface induces an antiholomorphic involution of the Teichmüller space of genus g Riemann surfaces. Two such involutions and are conjugate in the mapping class group if and only if the corresponding orientation reversing involutions and of are conjugate in the automorphism group of . This is equivalent to saying that the quotient surfaces and are homeomorphic. Hence the Teichmüller space has distinct antiholomorphic involutions, which are also called real structures of ([7]). This result is a simple fact that follows from Royden's theorem ([4]) stating that the the mapping class group is the full group of holomorphic automorphisms of the Teichmüller space (). Let and be two real structures that are not conjugate in the mapping class group. In this paper we construct a real analytic diffeomorphism such that This mapping d is a product of full and half Dehn–twists around certain simple closed curves on the surface . This has applications to the moduli spaces of real algebraic curves. A compact Riemann surface admitting an antiholomorphic involution is a real algebraic curve of the topological type . All fixed–points of the real structure of the Teichmüller space , are real curves of the above topological type and every real curve of that topological type is represented by an element of the fixed–point set of . The fixed–point set is the Teichmüller space of real algebraic curves of the corresponding topological type. Given two different real structures and , let d the the real analytic mapping satisfying (1). It follows that d maps onto and is an explicit real analytic diffeomorphism between these Teichmüller spaces. Received 8 December 1997; accepted 12 August 1998     

A Tridimensional Inverse Shaping Problem

We study the following tridimensional magnetostatic inverse shaping problem: can one find a distribution of currents around a levitating liquid metal bubble so that it takes a given shape? It leads to the resolution of an Eilonal equation on the surface of the bubble which has self-contained interest. We answer the question for closed smooth surface which are homeomorphic to a sphere. We give a necessary and sufficient condition on the data for existence and uniqueness of a C¹ solution. When the desired shape is axisymmetric and analytic, the solution is also analytic and the problem can be completely solved. A counterexample proves that not all analytic perturbations of such surface are shapable.     

A two-parameter spectral theorem

Summary We study the characteristic set of a couple (A, B) of selfadjoint compact operators on a real Hilbert spaceH. We prove thatC is the union of a sequence of characteristic curvesC _n in the (, ) plane. Each curve is the analytic image of an open interval and it is either closed or it goes to infinity at both ends of the interval. Moreover, it may intersect either itself or other characteristic curves in an at most countable set of points, which may accumulate only at infinity. Finally, to each characteristic curve one can associate an analytic function E_n, which gives the eigenprojection onto the eigenspace attached to each point of the characteristic curve, except at the intersection points, where the eigenspace is the direct sum of the projection relevant to each branch passing through the point. The dimension of the eigenprojection is constant along each curve and it is called the multiplicity of the characteristic curve.     

Curves as measured foliation on noncompact surfaces

In the present work, that regards the Thurston's theory, we prove that, if we choose a closed curve, how we wish, on a noncompact surface, it is always possible to construct a particular masured foliation that has the choosed curve like a leaf; we also prove this foliation has a remarkable property that makes very easy to mesure all homotopy classes of closed curves of our surface. To prove this statement we need some Propositions and some Lemma that we also demonstre.     

On the problems of Hartshorne and Serre for some -analytic surfaces

Let M be a compact -analytic surface, let Γ ⊂ M be a compact analytic subvariety, and let X := Mx00393;. We are interested in the following two problems: Assume that X does not contain any compact curve and that Γ is an irreducible compact curve such that Γ² ≥ 0 (resp. assume that the analytic cohomology groups H¹ (X, Ω^p) = 0, for all 0 ≤ p ≤ 2). Is X always Stein? It is our main purpose here to provide an affirmative answer to those two problems, provided M is either a (minimal) ruled surface or a non-algebraic compact surface. Also, the affine structure of such Stein surfaces will be discussed.     

On Superheight Conditions for the Affineness of Open Subsets

In this paper we consider the open complement U of a hypersurface Y = V(a) in an affine scheme X. We study the relations between the affineness of U, the intersection of Y with closed subschemes, the property that every closed surface in U is affine, the property that every analytic closed surface is Stein, and the superheight of a defining ideal a.     

關閉撓曲線的全曲率

<正> §1.引言.大家知道,微分幾何學所討論的一般是關於圖形的局部性質.但是這些局部性質與圖形的整個性質間常存在有某些關係.討論圖形的整個性質的微分幾何學叫做整體性的.關於整體性微分幾何學有這樣的一個著名定理:設一關閉撓曲線C     

Weak solutions of one inverse problem in geometric optics

We consider the problem of recovering a closed convex reflecting surface such that for a given point source of light (inside the convex body bounded by the surface) the reflected directions cover a unit sphere with prespecified in advance density. In analytic formulation, the problem leads to an equation of Monge-Ampère type on a unit sphere. We formulate the problem in terms of certain associated measures and establish the existence of weak solutions. Bibliography: 11 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 37, 2008, pp. 37–46.     

Free interpolation in Hölder classes in quasiconformal domains

The free interpolation problem for Hölder classes on a simply connected bounded domain whose boundary is a quasiconformal curve is studied. The necessary and sufficient conditions on a closed set $E \subset \overline G $ under which the whole Hölder space on E is generated by the restrictions of analytic functions of the Hölder class to the domain G are discussed. Bibliography: 7 titles.     

The ring of differential Fourier expansions

For a fixed prime we prove structure theorems for the kernel and the image of the map that attaches to any differential modular function its differential Fourier expansion. The image of this map, which is the ring of differential Fourier expansions, plays the role of ring of functions on a “differential Igusa curve”. Our constructions are then used to perform an analytic continuation between isogeny covariant differential modular forms on the differential Igusa curves belonging to different primes.     

两个各向同性半平面焊接的界面共线裂纹问题

复合材料焊接的界面裂纹问题是工程中碰到的实际问题,这方面已有不少有意义的研究工作（如献[3]-[6]等等）。本讨论两个各同性半平面焊接的界面共线裂纹问题,利用复变方法和解析函数边值问题题的基本理论,给出了弹性体应力分布封闭形式的解。     

插值逼近具边界数据的调和函数

该文考虑光滑闭Jondan曲线Γ围成的单连区域D,证明了在Γ上具有已知导数数据的D内调和函数u(x,y)的存在性.继而构造了一个调和插值多项式序列在(?)=D∪Γ上一致收敛于u(x,y),且具理想的收敛速度.此外,以往同类研究工作中的边界Γ是解析曲线,而在该文中已减少边界限制为Γ∈J_0.     

近似可展曲面的构造及应用

1引言经典微分几何中Gauss曲率为零的曲面称为可展曲面,它是一种特殊的直纹面．可展曲面有且只有三种,即锥面、柱面和切线面,它对于自由曲面造型具有重要的意义．例如,如果物体外壳是可展曲面,那么它可以没有形变地展开到平面上,从而可以用平板材料无形变地设计出来．这一性质对于造船业、航空业中的外形设计具有重要的意义．关于可展曲面的微分几何性质,可以在任何一本微分几何教材中找到,例如[3]．可展曲面可以说是微分几何中比较简单的一类曲面,但是在计算机辅助几何设计(CAGD)和计算机图形学中至今还不存在简单有效的设计方法．在[1,2,5]以及它们的参考文献中