S. M. Natanzon
The Riemann Theorem states, that for any nontrivial connected and simply connected domain on the Riemann sphere there exists some its conformal bijection to the exterior of the unit disk. In this paper we find an explicit form of this map for a broad class of domains with analytic boundaries.
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Consider a closed analytic curve $γ$ in the complex plane and denote by > $D_+$ and $D_-$ the interior and exterior domains with respect to the curve. The point $z=0$ is assumed to be in $D_+$. Then according to Riemann theorem there exists a function $w(z)=\frac 1r z+\sum_{j=0}^\infty p_j z^{-j}$, mapping $D_-$ to the exterior of the unit disk $\{w\in C|| w | >1\}$. It is follow from [arXiv : hep-th /0005259] that this function is described by formula $\log w=\log z-\partial_{t_0} (\frac 12\partial_{t_0}+\sum\limits_{k\geqslant 1}\frac{z^{-k}}{k} \partial_{t_k})v$, where $v=v(t_0, t_1, \bar t…
This article presents a clear proof of the Riemann Mapping Theorem via Riemann's method, uncompromised by any appeals to topological intuition.
In this paper we give a new proof of Riemann's well known mapping theorem. The suggested method permits to prove an analog of that theorem for the three dimensional case.
In this paper we give results about projective embeddings of Riemann surfaces, smooth or nodal, which we apply to the inverse Dirichlet-to-Neumann problem and to the inversion of a Riemann-Klein theorem. To produce useful embeddings, we adapt a technique of Bishop in the open bordered case and use Runge type harmonic approximation theorem in the compact case.
We prove that the space of dominant/non-constant holomorphic mappings from a product of hyperbolic Riemann surfaces of finite type into certain hyperbolic manifolds with universal cover a bounded domain is a finite set.
Using the notion of commutative operator vessels, this work investigates de Branges-Rovnyak spaces whose elements are sections of a line bundle of multiplicative half-order differentials on a compact real Riemann surface. As a special case, we obtain a Beurling-Lax type theorem in the setting of the corresponding Hardy space on a finite bordered Riemann surface.
We prove a Poincaré-Bendixson theorem describing the asymptotic behavior of geodesics for a meromorphic connection on a compact Riemann surface. We shall also briefly discuss the case of non-compact Riemann surfaces, and study in detail the geodesics for a holomorphic connection on a complex torus.
The Riemann-Roch theorem is of utmost importance in the algebraic geometric theory of compact Riemann surfaces. It tells us how many linearly independent meromorphic functions there are having certain restrictions on their poles. The aim of this article is to present a simple direct proof of this theorem and explore some of its numerous consequences. We also give an analytic proof of the Riemann-Hurwitz formula. As an application, we compute the genus of some interesting algebraic curves.
In this paper continuing our work started in our earlier papers we prove the corona theorem for the algebra of bounded holomorphic functions defined on an unbranched covering of a Caratheodory hyperbolic Riemann surface of finite type.
We discuss applications of an improvement on the Riemann mapping theorem which replaces the unit disc by another "double quadrature domain," i.e., a domain that is a quadrature domain with respect to both area and boundary arc length measure. Unlike the classic Riemann Mapping Theorem, the improved theorem allows the original domain to be finitely connected, and if the original domain has nice boundary, the biholomorphic map can be taken to be close to the identity, and consequently, the double quadrature domain close to the original domain. We explore some of the parallels between this new th…
Let $M$ be a non-compact connected Riemann surface of finite type, and $R\subset\subset M$ be a relatively compact domain such that $H_{1}(M,\Z)=H_{1}(R,\Z)$. Let $\tilde R\longrightarrow R$ be a covering. We study the algebra $H^{\infty}(U)$ of bounded holomorphic functions defined in some domains $U\subset\tilde R$. Our main result is a Forelli type theorem on projections in $H^{\infty}(\Di)$.
We generalize the H. Cartan's theory of holomorphic curves for a general open Riemann surface. Besides, a vanishing theorem for jet differentials and a Bloch's theorem for Riemann surfaces are obtained.
We prove that if $X = X_1 \times \dots \times X_n$ is a product of hyperbolic Riemann surfaces of finite type and $Y = Ω/Γ$ is a complex manifold, where $Ω$ is a bounded simply-connected domain in $\mathbb{C}^m$, then the space of dominant holomorphic mappings from $X$ to $Y$ is a finite set. As corollaries, we obtain the finiteness of the space of dominant holomorphic mappings into products and symmetric products of hyperbolic Riemann surfaces.
We introduce the heat equation relative to a positive dd-bar-closed current and apply it to the invariant currents associated with Riemann surface laminations possibly with singularities. The main examples are holomorphic foliations by Riemann surfaces in projective spaces. We prove two kinds of ergodic theorems for such currents: one associated to the heat diffusion and one close to Birkhoff's averaging on orbits of a dynamical system. The heat diffusion theorem with respect to a harmonic measure is also developed for real laminations.
Newman's Tauberian theorem, the Riemann-Lebesgue Lemma, and abstract analytic number theory
本文改进了新曼泰贝定理，并应用其研究抽象解析半群，证明了素数定理及梅特恩函数的估计。
We prove a compactness theorem for embedded measured hyperbolic Riemann surface laminations in a compact almost complex manifold $(X, J)$. To prove compactness result, we show that there is a suitable topology on the space of measured Riemann surface laminations induced by Levy-Prokhorov metric. As an application of the compactness theorem, we show that given a biholomorphism of $ϕ$ of a closed complex manifold $X$, some power $ϕ^k $ ($k>0$) fixes a measured Riemann surface lamination in $X$.
We give a general formula for the defect appearing in the Verdier-type Riemann-Roch formula for Chern-Schwartz-MacPherson classes in the case of a regular embedding. Our proof of this formula uses the constructible function version of Verdier's specialization functor, together with a specialization property of Chern-Schwartz-MacPherson classes and the corresponding Riemann-Roch theorem for smooth morphisms. As a very special case we get a formula for the Milnor-class of a local complete intersection in a smooth manifold, which in the case of a hypersurface gives back a result of Parusinski-Pra…
We study cohomology groups of the Lie algebra of vector fields on the complex line, $W_1$, with values in the tensor fields in several variables. From a generalization by Scheja of the second Riemann (Hartogs) continuation theorem, we deduce a cohomology exact sequence of the subalgebra of $W_1$ consisting of vectors having a zero at the origin. As applications, we compute the cohomology algebra of $W_1$ with values in the functions on $\Bbb C^n$ explicitly, and establish a certain vanishing theorem for the cohomology of $W_1$ with values in the quadratic differentials in several variables, wh…
We prove a generalization of Kawai theorem for the case of orbifold Riemann surface. The computation is based on a formula for the differential of a holomorphic map from the cotangent bundle of the Teichmüller space to the $\mathrm{PSL}(2,\mathbb{C})$-character variety, which allows to evaluate explicitly the pullback of Goldman symplectic form in the spirit of Riemann bilinear relations. As a corollary, we obtain a generalization of Goldman's theorem that the pullback of Goldman symplectic from on $\mathrm{PSL}(2,\mathbb{R})$-character variety is a symplectic form of the Weil-Petersson metric…
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