We consider a smooth CR mapping $f$ from a real-analytic generic submanifold $M$ in $\bC^N$ into $\bC^N$. For $M$ of finite type and essentially finite at a point $p\in M$, and $f$ formally finite at $p$, we give a necessary and sufficient condition for $f$ to extend as a holomorphic mapping in some neighborhood of $p$. In a similar vein, we consider a formal holomorphic mapping $H$ and give a necessary and sufficient condition for $H$ to be convergent.
We study the problem of holomorphic extension of a smooth CR mapping from a real analytic hypersurface to a real algebraic set in complex spaces of different dimensions.
We construct a (non Kähler) compact complex 3-dimensional manifold $X$ having two following properties: 1) for any domain $D$ in $C^2$ every meromorphic map $f$ from this domain into $X$ extends to a meromorphic map from the envelope of meromorphy $\hat D$ of $D$ into $X$; 2) but there exist a meromorphic map $F$ from a punctured ball $B_*$ in $C^3$ into $X$ which doens't extend meromorphically to the origin. In other words, one can allways remove the singularities of complex codimesion two for the meromorphic maps into this $X$, but only up to some subset of complex codimension three. A descr…
Iterated Segre mappings of real analytic generic submanifolds in complex space have been an essential tool in the study of holomorphic, formal, and CR mappings between such manifolds. In this paper we present a theory of iterated Segre mappings for smooth or formal submanifolds. In particular, we characterize finite type in terms of the rank of the iterated Segre mappings. The approach given here is new, simplifies proofs, and extends results previously known for the real analytic case.
We give some results concerning the smoothness of the image of a real-analytic submanifold in complex space under the action of a finite holomorphic mapping. For instance, if the submanifold is not contained in a proper complex subvariety, we give a necessary and sufficient condition guaranteeing that its image is smooth and the mapping is transversal to the image.
In this paper I survey some recent results on finite determination, convergence, and approximation of formal mappings between real submanifolds in complex spaces. A number of conjectures are also given.
We show that a holomorphic mapping sending one generic submanifold into another of the same dimension is CR transversal to the target submanifold provided that the source manifold is of finite type and the map is of generic full rank. This result and its corollaries completely resolve two questions posed by Linda P. Rothschild and the first author in a paper from 2006.
It is shown that a germ of a holomorphic mapping sending a real-analytic generic submanifold of finite type into another is determined by its projection on the Segre variety of the target manifold. A necessary and sufficient condition is given for a germ of a mapping into the Segre variety of the target manifold to be the projection of a holomorphic mapping sending the source manifold into the target. An application to the biholomorphic equivalence problem is also given.
In this paper, we study formal mappings between smooth generic submanifolds in multidimensional complex space and establish results on finite determination, convergence and local biholomorphic and algebraic equivalence. Our finite determination result gives sufficient conditions to guarantee that a formal map as above is uniquely determined by its jet at a point of a preassigned order. For real-analytic generic submanifolds, we prove convergence of formal mappings under appropriate assumptions and also give natural geometric conditions to assure that if two germs of such submanifolds are forma…
Let M be a connected real-analytic hypersurface in two dimensional complex space, $\mathbb C^2$, containing a connected complex hypersurface E, and let f be a smooth CR mapping sending M into another real-analytic hypersurface M' in $\mathbb C^2$. In this paper, we prove that if f does not collapse E to a point and does not collapse M into the image of E, and if the Levi form of M vanishes to first order along E, then f is real-analytic in a neighborhood of E. In general, the corresponding statement is false if the Levi form of M vanishes to second order or higher, in view of an example due to…
This paper is devoted to the study of semigroups of composition operators and semigroups of holomorphic mappings. We establish conditions under which these semigroups can be extended in their parameter to sector given a priori. We show that the size of this sector can be controlled by the image properties of the infinitesimal generator, or, equivalently, by the geometry of the so-called associated planar domain. We also give a complete characterization of all composition operators acting on the Hardy space $H^p$ on the right half-plane.
We establish results on holomorphic extension of CR-mappings of class $C^\infty$ between a real-analytic CR-submanifold of $\C^N$ and a real-algebraic CR-submanifold of $\C^{N'}$.
We consider CR submersive mappings between generic submanifolds in complex space. We show that, under suitable conditions on the manifolds, there is an integer k such that any jet of the CR mapping at a given point is a rational function of its k-jet at that point. As a consequence, it is shown that the stability group of a (suitably nondegenerate) real-analytic generic submanifold is a real algebraic Lie group. Moreover, it is shown that a formal equivalence between two such real-analytic submanifolds is necessarily convergent.
It is shown that a formal mapping between two real-analytic hypersurfaces in complex space is convergent provided that neither hypersurface contains a nontrivial holomorphic variety. For higher codimensional generic submanifolds, convergence is proved e.g. under the assumption that the source is of finite type, the target does not contain a nontrivial holomorphic variety, and the mapping is finite. Finite determination (by jets of a predetermined order) of formal mappings between smooth generic submanifolds is also established.
Let H:(M,p)->(M',p') be a formal mapping between two germs of real-analytic generic submanifolds in C^N with nonvanishing Jacobian. Assuming M to be minimal at p and M' holomorphically nondegenerate at p', we prove the convergence of the mapping H. As a consequence, we obtain a new convergence result for arbitrary formal maps between real-analytic hypersurfaces when the target does not contain any holomorphic curve. In the case when both M and M' are hypersurfaces, we also prove the convergence of the associated reflection function when M is assumed to be only minimal. This allows us to derive…
We prove the following regularity result: If M and M' are smooth generic submanifolds of C^N and C^N' respectively, where N and N' are not necessarily equal, and if M is minimal, then every C^k-CR-map from M into M^\prime which is k-nondegenerate is smooth. As an application, every CR diffeomorphism of k-nondegenerate minimal submanifolds in C^N of class C^k is smooth.
The purpose of this article is threefold. The first is to construct a Nevanlina theory for meromorphic mappings from a polydisc to a compact complex manifold. In particular, we give a simple proof of Lemma on logarithmic derivative for nonzero meromorphic functions on $\mathbb{C}^l.$ The second is to improve the definition of the non-integrated defect relation of H. Fujimoto \cite{F2} and to show two theorems on the new non-integrated defect relation of meromorphic maps from a closed submanifold of $\mathbb{C}^l$ to a compact complex manifold. The third is to give a unicity theorem for meromor…
We construct a family of integral kernels for solving the \bar\partial equation with C^k and Holder estimates in thin tubes around totally real submanifolds in complex Eulidean spaces (theorems 1.1 and 3.1). Combining this with the proof of a theorem of Serre we solve the d-equation with estimates for holomorphic forms in such tubes (theorem 5.1). We apply these techniques and a method of Moser to approximate C^k-diffeomorphisms between totally real submanifolds in C^n in the C^k-topology by biholomorphic mappings in tubes, by unimodular and symplectic biholomorphic mappings, and by holomorphi…
Let f : (M,p)\to (M',p') be a formal (holomorphic) nondegenerate map, i.e. with formal holomorphic Jacobian J_f not identically vanishing, between two germs of real analytic generic submanifolds in \C^n, p'=f(p). Assuming the target manifold to be real algebraic, and the source manifold to be minimal at p\in M in the sense of Tumanov, we prove the convergence of the so-called reflection mapping associated to f. From this, we deduce the convergence of such mappings from minimal real analytic generic submanifolds onto real algebraic holomorphically nondegenerate ones, as well as related results …
In this paper we continue our study of local rigidity for maps of CR submanifolds of the complex space. We provide a linear sufficient condition for local rigidity of finitely nondegenerate maps between minimal CR manifolds. Furthermore, we show higher order infinitesimal conditions can be used to give a characterization of local rigidity.
We prove the following finite jet determination result for CR mappings: Given a smooth generic submanifold M of C^N, N >= 2, which is essentially finite and of finite type at each of its points, for every point p on M there exists an integer l(p), depending upper-semicontinuously on p, such that for every smooth generic submanifold M' of C^N of the same dimension as M, if h_1 and h_2: (M,p)->M' are two germs of smooth finite CR mappings with the same l(p) jet at p, then necessarily their k-jets agree for all positive integers k. In the hypersurface case, this result provides several new unique…
We prove the following Artin type approximation theorem for smooth CR mappings: given M a connected real-analytic CR submanifold in C^N that is minimal at some point, M' a real-analytic subset of C^N', and H:M->M' a smooth CR mapping, there exists a dense open subset O in M such that for any q in O and any positive integer k there exists a germ at q of a real-analytic CR mapping H^k:(M,q)->M' whose k-jet at q agrees with that of H up to order k.
We study the real analyticity of a CR mapping on the hypersurface $\h$ defined by $\h=\{z\in\C^n,\Re(z_n)+|z'|^2=0\}$ in model almost complex manifolds. We make use of a method of prolongation for the tangential Cauchy-Riemann equations and a result about complete systems. We also prove that a CR mapping is real analytic when the almost complex structure of the codomain is a deformation of a model structure and the function defining the hypersurface in the codomain has a particular form.
Invariants for Riemann surfaces covered by the disc and for hyperbolic manifolds in general involving minimizing the measure of the image over the homotopy and homology classes of closed curves and maps of the $k$-sphere into the manifold are investigated. The invariants are monotonic under holomorphic mappings and strictly monotonic under certain circumstances. Applications to holomorphic maps of annular regions in $\mathbb{C}$ and tubular neighborhoods of compact totally real submanifolds in general in $\mathbb{C}^n$, $n \geq 2$, are given. The contractibility of a hyperbolic domain with con…
In this paper we consider basic hypergeometric functions introduced by Heine. We study mapping properties of certain ratios of basic hypergeometric functions having shifted parameters and show that they map the domains of analyticity onto domains convex in the direction of the imaginary axis. In order to investigate these mapping properties, few useful identities are obtained in terms of basic hypergeometric functions. In addition, we find conditions under which the basic hypergeometric functions are in $q$-close-to-convex family.
We prove here new results about transversality and related geometric properties of a holomorphic, formal, or CR mapping, sending one generic submanifold of $\bC^N$ into another. One of our main results is that a finite mapping is transversal to the target manifold provided this manifold is of finite type. For the case of hypersurfaces, transversality in this context was proved by Baouendi and the second author in 1990. The general case of generic manifolds of higher codimension, which we treat in this paper, had remained an open problem since then. Applications of this result include a suffici…
Let S be a generic C-infinity smooth CR manifold in C^n, n > 1, and let M be a generic C-infinity CR submanifold of S X C^m. We prescribe conditions on M so that it is the disjoint union of graphs of CR maps f:S-->C^m. We also consider the special case where S is the boundary of a bounded, smoothly bounded open set D in C^n, n > 1. In this case we obtain conditions which guarantee that such an f above extends to be analytic on D. This provides a solution to a particular nonlinear Riemann-Hilbert boundary value problem for analytic functions.
We give conditions under which a germ of a holomorphic mapping in $\Bbb C^N$, mapping an irreducible real algebraic set into another of the same dimension, is actually algebraic. Let $A\subset \bC^N$ be an irreducible real algebraic set. Assume that there exists $\po \in A$ such that $A$ is a minimal, generic, holomorphically nondegenerate submanifold at $\po$. We show here that if $H$ is a germ at $p_1 \in A$ of a holomorphic mapping from $\bC^N$ into itself, with Jacobian $H$ not identically $0$, and $H(A)$ contained in a real algebraic set of the same dimension as $A$, then $H$ must extend …
We construct closed complex submanifolds of dimension three in C^5 which are differential complete intersections but not holomorphic complete intersections. We also prove a homotopy principle concerning the removal of intersections of holomorphic mappings from Stein manifolds to complex Euclidean spaces C^d with certain closed complex subvarities of C^d.
Let $M$ be a generic CR submanifold in $\C^{m+n}$, $m= CRdim M \geq 1$,$n=codim M \geq 1$, $d=dim M = 2m+n$. A CR meromorphic mapping (in the sense of Harvey-Lawson) is a triple $(f,{\cal D}_f, [Γ_f])$, where: 1. $f: {\cal D}_f \to Y$ is a ${\cal C}^1$-smooth mapping defined over a dense open subset ${\cal D}_f$ of $M$ with values in a projective manifold $Y$; 2. The closure $Γ_f$ of its graph in $\C^{m+n} \times Y$ defines a oriented scarred ${\cal C}^1$-smooth CR manifold of CR dimension $m$ (i.e. CR outside a closed thin set) and 3. Such that $d[Γ_f]=0$ in the sense of currents. We prove in…
In this paper we obtain, by the method of Loewner chains, some sufficient conditions for the analyticity and the univalence of the functions defined by an integral operator. These conditions in- volves Ruscheweyh and Salagean derivative operator in the open unit disk. In particular cases, we find the well-known conditions for univalency established by Becker [3], Ahlfors [2], Kanas and Srivastava [8] and others for analytic mappings f : U ! C: Also, we obtain the corresponding new, useful and simpler conditions for this integral operator.
Let $Ω$ be a smooth real analytic submanifold of a complex manifold $X$. We establish and study the link between the following 3 subjects: 1) topological properties of smooth families of attached analytic discs, the manifold $Ω$ admits, 2) lower bounds for dimensions of complex tangent spaces of $Ω$, 3) a generalization of the argument principle for smooth families of holomorphic mappings from the standard complex disc to $X$. In particular, we obtain characterization of complex manifolds and their boundaries in terms of attached analytic discs. The special case when $Ω$ is the graph, leads to…
