Topological invariants and Holomorphic Mappings

A holomorphic mapping $H$ between two real-analytic CR manifolds $M$ and $M'$ is said to be locally rigid if any other holomorphic map $F\colon M \to M'$ which is close enough to $H$ is obtained by composing $H$ with suitable automorphisms of $M$ and $M'$. With the aim of reducing the local rigidity problem to a linear one, we provide sufficient infinitesimal conditions. Furthermore we study some topological properties of the action of the automorphism group on the space of nondegenerate mappings from $M$ to $M'$.

Peschl defined invariant higher-order derivatives of a holomorphic or meromorphic function on the unit disk. Here, the invariance is concerned with the hyperbolic metric of the source domain and the canonical metric of the target domain. Minda and Schippers extended Peschl’s invariant derivatives to the case of general conformal metrics. We introduce similar invariant derivatives for smooth functions on a Riemann surface and show a complete analogue of Faa di Bruno’s formula for the composition of a smooth function with a holomorphic map with respect to the derivatives. An interpretation of these derivatives in terms of intrinsic geometry and some applications will be also given.

In the study of holomorphic maps, the term "rigidity" refers to certain types of results that give us very specific information about a general class of holomorphic maps owing to the geometry of their domains or target spaces. Under this theme, we begin by studying when, given two compact connected complex manifolds X and Y, a degree-one holomorphic map f : Y → X is a biholomorphism. Given that the real manifolds underlying X and Y are diffeomorphic, we provide a condition under which f is a biholomorphism. Using this result, we deduce a rigidity result for holomorphic self-maps of the total space of a holomorphic fiber space. Lastly, we consider products X = X1 × X2 and Y = Y1 × Y2 of compact connected complex manifolds. When X1 is a Riemann surface of genus ≥ 2, we show that any non-constant holomorphic map F : Y → X is of a special form.

The richly developed theory of complex manifolds plays important roles in our understanding of holomorphic functions in several complex variables. It is natural to consider manifolds that will play similar roles in the theory of holomorphic functions in several non-commuting variables. In this paper we introduce the class of \emph{nc-manifolds}, the mathematical objects that at each point possess a neighborhood that has the structure of an \emph{nc-domain} in the \emph{$d$-dimensional nc-universe $\m^d$}. We illustrate the use of such manifolds in free analysis through the construction of the non-commutative Riemann surface for the matricial square root function. A second illustration is the construction of a non-commutative analog of the elementary symmetric functions in two variables. For any symmetric domain in $\m^2$ we construct a 2-dimensional non-commutative manifold such that the symmetric holomorphic functions on the domain are in bijective correspondence with the holomorphic functions on the manifold. We also derive a version of the classical Newton-Girard formulae for power sums of two non-commuting variables.

In this chapter, we consider some elementary properties of Riemann surfaces, as well as a fundamental technique called the L 2 \(\bar{\partial}\)-method, Rado’s theorem on second countability of Riemann surfaces, and analogues of the Mittag-Leffler theorem and the Runge approximation theorem for open Riemann surfaces. Viewing holomorphic functions as solutions of the homogeneous Cauchy–Riemann equation \(\partial f/\partial\bar{z}=0\) in ℂ allows one to very efficiently obtain their basic properties (see Chap. 1). The intrinsic form of the homogeneous Cauchy–Riemann equation on a Riemann surface is given by \(\bar{\partial}f=0\) (see Sect. 2.5). In order to obtain holomorphic functions (and holomorphic 1-forms) on a Riemann surface (even on an open subset of ℂ), it is useful to consider the inhomogeneous Cauchy–Riemann equation \(\bar{\partial}\alpha=\beta\). One well-known approach to solving this differential equation (as well as differential equations in many other contexts) is to consider weak solutions in L 2. This is the approach taken in this book. In order to do so, we must develop suitable versions of an L 2 space of differential forms (see Sect. 2.6) and an (intrinsic) distributional \(\bar{\partial}\) operator (see Sect. 2.7). The relatively simple approaches to the above appearing in this book are, in part, special to Riemann surfaces; but they do contain important elements of the higher-dimensional versions (see, for example, L. Hormander, An Introduction to Complex Analysis in Several Variables, North-Holland, 1990 or J.-P. Demailly, Complex Analytic and Differential Geometry, online book, for the higher-dimensional versions).

Suppose M is a real-analytic submanifold of complex Euclidean n = space and consider the following question: Given a real-analytic function/ defined on M, is/ the restriction to M of an ambient holomorphic function?If M is a C.R. submanifold the question has been answered completely.Namely,/is the trace of a holomorphic function if and only if/ is a C.R. function.The more general situation in which M need not be a C.R. submanifold is discussed in this paper.A complete answer is obtained in case the dimension of M is larger than or equal to n and M is generic in some neighborhood of each point off its C.R. singularities.The solution is of infinite order and follows from a consideration of the following problem: Given a holomorphic function/and a holomorphic mapping $, when does there exist a holomorphic mapping F such that/= F$?

International Journal of MathematicsVol. 10, No. 06, pp. 697-706 (1999) No AccessON THE TORELLI THEOREM FOR MINIMAL IMMERSIONSCLAUDIO AREZZOCLAUDIO AREZZOMathematics Institute, University of Warwick, Coventry CV4 7AL, UKThe author was supported by European Community Grant TRM N. ERBFMBICT 950079.https://doi.org/10.1142/S0129167X99000288Cited by:0 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail FiguresReferencesRelatedDetails Recommended Vol. 10, No. 06 Metrics History Received 5 August 1998 PDF download

The purpose of the present note is to show that holomorphic fiber bundles with hyperbolic fiber are almost trivial : They are trivial if the base is simply connected, and have a simple structure otherwise. Kobayashi [2],[3] has introduced the notion of a hyperbolic complex manifold.We need not be concerned with the definition here, but only with the following consequence [3], [4]: If M is a hyperbolic complex manifold, p a point of M, and U a neighborhood of /, then there is a <5>0 such that any holomorphic map <f> of the unit disk into M with (f,(0)=p maps the disk {z:|z|<} into U. From this, it follows that, if iv1, , wn are coordinates at/, and / an integer, then there are constants Al depending only on / and the choice of coordinates such that, if-p is any map of the unit disk into M with cf>(0)=p, then \dlwidzl\<Al at z=0.The crux of our argument is the following proposition, which generalizes a result of Kobayashi's that a complex Lie group (of dimension greater than 0) cannot act holomorphically on a hyperbolic manifold. Proposition.Let M and N be complex manifolds with M hyperbolic.Let F=F(w, z) be a holomorphic map of MxN into M. Suppose that, for some z0 e N, the restriction of F to Mx{z0} is a biholomorphic map onto M. Then F is independent of z; i.e., F(w, z)=F(w, z0). Proof.It suffices to prove the proposition for the case that N is a ball, and hence for the case that Nis a disk A, which we take to be the unit disk with z0=0.Without loss of generality, we may assume F(w, z0)=w.Near a point (w0, z0), the map F has a power series expansion of the form F\w, z) = w' + a\w)zl + 0(zl+1).-

Holomorphic mappings between real analytic submanifolds in complex space

The present paper is devoted to the further development of the discrete theory of Riemann surfaces, which was started in the papers by M. Baker and S. Norine and their followers at the beginning of the century. This theory considers finite graphs as analogs of Riemann surfaces and branched coverings of graphs as holomorphic mappings. The genus of a graph is defined as the rank of its fundamental group. The main object of investigation in the paper is automorphism groups of a graph acting freely on the set of half-edges of the graph. These groups are discrete analogs of groups of conformal automorphisms of a Riemann surface. The celebrated Hurwitz theorem (1893) states that the order of the group of conformal automorphisms of a compact Riemann surface of genus g > 1 does not exceed 84(g — 1). Later, K. Oikawa and T. Arakawa refined this bound in the case of groups that fix several finite sets of prescribed cardinalities. This paper provides proofs of discrete versions of the mentioned theorems. In addition, a discrete analog of the E. Bujalance and G. Gromadzki theorem improving one of Arakawa’s results is obtained.

In this paper, we have extended S.S. Chern′s second basic theorem about holomorphic mapping between two Riemann surfaces to more general case, and also obtained two similar results.

Let be a smooth Riemann surface foliation on , where M is a complex manifold and is a closed set. Fix a hermitian metric g on and assume that all leaves of are hyperbolic. For each leaf , the ratio of , the restriction of g to L, and the Poincaré metric on L defines a positive function η that is known to be continuous on under suitable conditions on M, E. For a domain , we consider , the restriction of to U and the corresponding positive function by considering the ratio of g and the Poincaré metric on the leaves of . First, we study the variation of as U varies in the Hausdorff sense motivated by the work of Lins Neto–Martins. Secondly, Minda had shown the existence of a domain Bloch constant for a hyperbolic Riemann surface S, which in other words shows that every holomorphic map from the unit disc into S, whose distortion at the origin is bounded below, must be locally injective in some hyperbolic ball of uniform radius. We show how to deduce a version of this Bloch constant for .

Let X be a Riemann surface, and let a ∈ X. We consider pairs (U, f), where U is an open neighbourhood of a and f is a holomorphic function on U. Two such pairs (U, f) and (V, g) are said to be equivalent, and define the same germ of holomorphic function at a, if there exists an open neighbourhood W of a, W ⊂ U ∩ V, such that f|W = g|W. An equivalence class is called a germ of holomorphic function at a; the class of a pair (U, f) is called the germ of f at a and denoted by f a. The value at a of f a is defined by f a(a) = f(a) for any pair (U, f) defining f a.

We perform a complete classification of AdS2 solutions of Type II supergravity realising N = 6 supersymmetry and OSp(6|2) superconformal symmetry on backgrounds that are foliations of AdS2 × ℂℙ3 over a Riemann surface Σ2. Such solutions only exist in type IIB supergravity and are in 1 to 1 correspondence with a fourth order PDE that can be locally solved in terms of two holomorphic functions. Particular solutions in the class are the T-duals of the AdS3 × ℂℙ3 and AdS2 × S7 solutions to massive Type IIA supergravity found in the literature. We discuss the field theory interpretation of the two sub-classes of solutions related to AdS3 × ℂℙ3 by Abelian and non-Abelian T-duality, which provide explicit examples for the Riemann surface an annulus or a strip. In the second case we interpret the solutions as holographic duals to baryon vertex configurations realised in D2-brane box models.

The complete Type IIB supergravity solutions with 16 supersymmetries are obtained on the manifold AdS4 ? S2 ? S2 ? ? with SO(2, 3) ? SO(3) ? SO(3) symmetry in terms of two holomorphic functions on a Riemann surface ?, which generally has a boundary. This is achieved by reducing the BPS equations using the above symmetry requirements, proving that all solutions of the BPS equations solve the full Type IIB supergravity field equations, mapping the BPS equations onto a new integrable system akin to the Liouville and Sine-Gordon theories, and mapping this integrable system to a linear equation which can be solved exactly. Amongst the infinite class of solutions, a non-singular Janus solution is identified which provides the AdS/CFT dual of the maximally supersymmetric Yang-Mills interface theory discovered recently. The construction of general classes of globally non-singular solutions, including fully back-reacted AdS5 ? S5 and supersymmetric Janus doped with D5 and/or NS5 branes, is deferred to a companion paper [1].