We study fixed point sets for holomorphic automorphisms (and endomorphisms) on complex manifolds. The main object of our interest is to determine the number and configuration of fixed points that forces an automorphism (endomorphism) to be the identity. These questions have been examined in a number of papers for a bounded domain in ${\Bbb C}^n$. Here we resolve the case for a general finite dimensional hyperbolic manifold. We also show that the results for non-hyperbolic manifolds are notably different.
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.
We show, using the Kobayashi and Caratheodory metrics on special holomorphic disks in the universal Teichmuller space, that a wide class of holomorphic functionals on the space of univalent functions in the disk is maximized by the Koebe function or by its root transforms; their extremality is forced by hyperbolic features. As consequences, this implies the proofs of the famous Zalcman and Bieberbach conjectures.
This paper studies the determining sets for analytic functions from the symmetrized bidisk into the open unit disk in $\mathbb C$. It relates the idea to the uniqueness of the solutions of a Nevanlinna-Pick interpolation problem. It also investigates when certain thin sets can be determining.
We prove a result on the structure of finite proper holomorphic mappings between complex manifolds that are products of hyperbolic Riemann surfaces. While an important special case of our result follows from the ideas developed by Remmert and Stein, the proof of the full result relies on the interplay of the latter ideas and a finiteness theorem for Riemann surfaces.
The Fock-Bargmann-Hartogs domain $D_{n,m}(μ)$ ($μ>0$) in $\mathbb{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-μ\|z\|^2},$ where $(z,w)\in \mathbb{C}^n\times \mathbb{C}^m$, which is an unbounded non-hyperbolic domain in $\mathbb{C}^{n+m}$. Recently, Tu-Wang obtained the rigidity result that proper holomorphic self-mappings of $D_{n,m}(μ)$ are automorphisms for $m\geq 2$, and found a counter-example to show that the rigidity result isn't true for $D_{n,1}(μ)$. In this article, we obtain a classification of proper holomorphic mappings between $D_{n,1}(μ)$ and $D_{N,1}(μ)$ with $N<2n$.
We determine all connected homogeneous Kobayashi-hyperbolic manifolds of dimension $n\ge 2$ whose holomorphic automorphism group has dimension $n^2-2$. This result complements an existing classification for automorphism group dimension $n^2-1$ and greater obtained without the homogeneity assumption.
Let $f:\mathbb{C}\sp n\to\mathbb{C}\sp n$, $n\geq2$, be a biholomorphism and let $Λ\subseteq \mathbb{C}\sp n$ be a compact $f$-invariant set such that $f|Λ$ is partially hyperbolic. We give equivalent conditions to hyperbolicity on $Λ$. In the particular case of generalized Hénon map with dominated splitting in the Julia set $J$, we characterize the hyperbolicity of $J$.
We construct canonical intertwining semi-models with Kobayashi hyperbolic base space for holomorphic self-maps of complex manifolds which are univalent on some absorbing cocompact hyperbolic domain. In particular, in the unit ball we solve the Valiron equation for hyperbolic univalent self-maps and for hyperbolic semigroups.
In this paper, we study a notion of hyperbolicity for hyperbolicity foliations with 1-dimensional parabolic leaves, namely the non-existence of holomorphic cylinders along the foliation - holomorphic maps from $\D^{n-1} \times \C$ to the manifold sending each $\{*\} \times \C$ to a leaf. We construct a tensor, that is a leaf-wise holomorphic section of a bundle above the foliated manifold, which vanishes on an open saturated set if and only if there exists such a cylinder through each point of this open set. Thanks to this study, we prove that open sets saturated by compact leaves are union of…
Let $M$ be a Kobayashi hyperbolic homogenous manifold. Let $\mathcal F$ be a holomorphic foliation on $M$ invariant under a transitive group $G$ of biholomorphisms. We prove that the leaves of $\mathcal F$ are the fibers of a holomorphic $G$-equivariant submersion $π\colon M \to N$ onto a $G$-homogeneous complex manifold $N$. We also show that if $\mathcal Q$ is an automorphism family of a hyperbolic convex (possibly unbounded) domain $D$ in $\mathbb C^n$, then the fixed point set of $\mathcal Q$ is either empty or a connected complex submanifold of $D$.
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…
Based on a well known Sh.-T. Yau theorem we obtain that the real part of a holomorphic function on a Kähler manifold with the Ricci curvature bounded from below by $-1$ is contractive with respect to the distance on the manifold and the hyperbolic distance on $(-1,1)$ inhered from the domain $(-1,1)\times\mathbb{R}$. Moreover, in the case of bounded holomorphic functions we prove that the modulus is contractive with respect to the distance on the manifold and the hyperbolic distance on the unit disk.
We study the laminarity of the Green current of endomorphisms of $P^2C$ near hyperbolic measures of saddle type. When these measures are supported by attracting sets, we prove that the Green current is laminar in the basin of attraction and we obtain new ergodic properties. This generalizes some results of Bedford and Jonsson on regular polynomial mappings in $C^2$.
We initiate a parametric study of holomorphic families of polynomial skew products, i.e., polynomial endomorphisms of $\mathbb{C}^2$ of the form $F(z,w)= (p(z), q(z,w))$ that extend to holomorphic endomorphisms of $\mathbb{P}^2(\mathbb{C})$. We prove that dynamical stability in the sense of arXiv:1403.7603 preserves hyperbolicity within such families, and give a complete classification of the hyperbolic components that are the analogue, in this setting, of the complement of the Mandelbrot set for the family $z^2 +c$. We also precisely describe the geometry of the bifurcation locus and current …
The construction of sections of bundles with prescribed jet values plays a fundamental role in problems of algebraic and complex geometry. When the jet values are prescribed on a positive dimensional subvariety, it is handled by theorems of Ohsawa-Takegoshi type which give extension of line bundle valued square-integrable top-degree holomorphic forms from the fiber at the origin of a family of complex manifolds over the open unit 1-disk when the curvature of the metric of line bundle is semipositive. We prove here an extension result when the curvature of the line bundle is only semipositive o…
In this notes, we characterize discrete subgroups of PU(2,1), holomorphic isometric group of complex hyperbolic space, which have an invariant totally geodesic submanifold.
Let \Fc be a holomorphic foliation by curves defined in a neighborhood of 0 in \C^2 having 0 as a hyperbolic singularity. Let T be a harmonic current directed by \Fc which does not give mass to any of the two separatrices. Then we show that the Lelong number of T at 0 vanishes. Next, we apply this local result to investigate the global mass-distribution for directed harmonic currents on singular holomorphic foliations living on compact complex surfaces. Finally, we apply this global result to study the recurrence phenomenon of a generic leaf.
The Fock-Bargmann-Hartogs domain $D_{n,m}(μ)$ ($μ>0$) in $\mathbf{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-μ\|z\|^2},$ where $(z,w)\in \mathbf{C}^n\times \mathbf{C}^m$, which is an unbounded non-hyperbolic domain in $\mathbf{C}^{n+m}$. Recently, Yamamori gave an explicit formula for the Bergman kernel of the Fock-Bargmann-Hartogs domains in terms of the polylogarithm functions and Kim-Ninh-Yamamori determined the automorphism group of the domain $D_{n,m}(μ)$. In this article, we obtain rigidity results on proper holomorphic mappings between two equidimensional Fock-Bargmann-Hartogs …
This article is concerned with the geometry of germs of real analytic surfaces in $(\mathbb{C}^2,0)$ having an isolated Cauchy-Riemann (CR) singularity at the origin. These are perturbations of {\it Bishop quadrics}. There are two kinds of CR singularities stable under perturbation~: {\it elliptic} and {\it hyperbolic}. Elliptic case was studied by Moser-Webster \cite{moser-webster} who showed that such a surface is locally, near the CR singularity, holomorphically equivalent to {\it normal form} from which lots of geometric features can be read off. In this article we focus on perturbations o…
Chern number formulas for holomorphic jet bundles are computed for projective curves and for projective surfaces. These formulas are used to show that certain minimal surfaces of general type (generic hypersurfaces of degree at least 5 in projective 3-spaces in particular) are hyperbolic in the sense of complex hyperbolic geometry.
The goal of this paper is to prove the conjecture of Krzyz posed in 1968 that for nonvanishing holomorphic functions $f(z) = c_0 + c_1 z + ...$ in the unit disk with $|f(z)| \le 1$, we have the sharp bound $|c_n| \le 2/e$ for all $n \ge 1$, with equality only for the function $f(z) = \exp [(z^n - 1)/(z^n + 1)]$ and its rotations. The problem was considered by many researchers, but only partial results have been established. The desired estimate has been proved only for $n \le 5$. Our approach is completely different and relies on complex geometry and pluripotential features of convex domains i…
We develop Nevanlinna's theory for a class of holomorphic maps when the source is a disc. Such maps appear in the theory of foliations by Riemann Surfaces.
We study the possible dimensions that the groups of holomorphic automorphisms of hyperbolic Reinhardt domains can have. We are particularly interested in the problem of characterizing Reinhardt domains with automorphism group of prescribed dimension.
The McMullen-Sullivan holomorphic motion for topologically conjugate, complex polynomials with connected Julia set follows level sets of the Bottcher coordinate. The Buzzard-Verma holomorphic motion for hyperbolic, unstably connected, polynomial diffeomorphisms of C^2 follows level sets of the Bedford-Smillie solenoid map. It follows that this solenoid map is a conjugacy for those Henon maps that are perturbations of (one-dimensional) hyperbolic maps with connected Julia set.
The main result is that for a connected hyperbolic complete Kähler manifold with bounded geometry of order two and exactly one end, either the first compactly supported cohomology with values in the structure sheaf vanishes or the manifold admits a proper holomorphic mapping onto a Riemann surface.
We introduce a notion of hyperbolicity and parabolicity for a holomorphic self-map $f: Δ^N \to Δ^N$ of the polydisc which does not admit fixed points in $Δ^N$. We generalize to the polydisc two classical one-variable results: we solve the Valiron equation for a hyperbolic $f$ and the Abel equation for a parabolic nonzero-step $f$. This is done by studying the canonical Kobayashi hyperbolic semi-model of $f$ and by obtaining a normal form for the automorphisms of the polydisc. In the case of the Valiron equation we also describe the space of all solutions.
We study monodromy of holomorphic motions and show the equivalence of triviality of monodromy of holomorphic motions and extensions of holomorphic motions to continuous motions of the Riemann sphere. We also study liftings of holomorphic maps into certain Teichmüller spaces. We use this "lifting property" to prove that, under the condition of trivial monodromy, any holomorphic motion of a closed set in the Riemann sphere, over a hyperbolic Riemann surface, can be extended to a holomorphic motion of the sphere, over the same parameter space. We conclude that this extension can be done in a conf…
The aim of this article is twofold: First we study holomorphic germs of parabolic diffeomorphisms of $(\mathbb{C}^2,0)$ that are reversed by a holomorphic reflection and posses an analytic first integral with non-degenerate critical point at the origin. We find a canonical formal normal form and provide a complete analytic classification (in formal generic cases) in terms of a collection of functional invariants. Their restriction to an irreductible component of the zero locus of the first integral reduces to the Birkhoff--Écalle--Voronin modulus of the 1-dimensional restricted parabolic germ.…
In this paper, we generalize the Kobayashi pseudo-distance to complex manifolds which admit holomorphic bracket generating distributions. The generalization is based on Chow's theorem in sub-Riemannian geometry. Let G be a linear semisimple Lie group. For a complex $G$-homogeneous manifold M with a G-invariant holomorphic bracket generating distribution D, we prove that (M,D) is Kobayashi hyperbolic if and only if the universal covering of M is a canonical flag domain and the induced distribution is the superhorizontal distribution.
In our previous work, for a given Riemann surface $Y_0$ with marked handle, we investigated geometric properties of the set of marked once-holed tori $X$ allowing holomorphic mappings of $X$ into $Y_0$. It turned out that it is a closed domain with Lipschitz boundary. In the present paper we show that the boundary is never smooth. Also, we evaluate the critical extremal length for the existence of holomorphic mappings in terms of hyperbolic lengths.
We introduce a model for Hermitian holormorphic Deligne cohomology on a projective algebraic manifold which allows to incorporate singular hermitian structures along a normal crossing divisor. In the case of a projective curve, the cup-product in cohomology is shown to correspond to a generalization of the Deligne pairing to line bundles with "good" hermitian metrics in the sense of Mumford and others. A particular case is that of the tangent bundle of the curve twisted by the negative of the singularity divisor of a hyperbolic metric: its cup square (corrected by the total area) is shown to b…
We show that there does not exist a Kobayashi hyperbolic complex manifold of dimension $n\ne 3$, whose group of holomorphic automorphisms has dimension $n^2+1$ and that, if a 3-dimensional connected hyperbolic complex manifold has automorphism group of dimension 10, then it is holomorphically equivalent to the Siegel space. These results complement earlier theorems of the authors on the possible dimensions of automorphism groups of domains in comlex space. The paper also contains a proof of our earlier result on characterizing $n$-dimensional hyperbolic complex manifolds with automorphism grou…
Let the function $φ$ be holomorphic in the unit disk $\mathbb{D}$ of the complex plane $\mathbb{C}$ and let $φ(\mathbb{D})\subset \mathbb{D}$. We study the level sets and the critical points of the hyperbolic derivative of $φ$, $$|D_φ(z)|:=\frac{(1-|z|^2)|φ'(z)|}{1-|φ(z)|^2}.$$ In particular, we show how the Schwarzian derivative of $φ$ reveals the nature of the critical points.
Let $(ϕ_t)_{t \geq 0}$ be a semigroup of holomorphic functions in the unit disk $\mathbb{D}$ and $K$ a compact subset of $\mathbb{D}$. We investigate the conditions under which the backward orbit of $K$ under the semigroup exists. Subsequently, the geometric characteristics, as well as, potential theoretic quantities for the backward orbit of $K$ are examined. More specifically, results are obtained concerning the asymptotic behavior of its hyperbolic area and diameter, the harmonic measure and the capacity of the condenser that $K$ forms with the unit disk.
We develop an isotopy principle for holomorphic motions. Our main result concerns the extendability of a holomorphic motion of a finite subset $E$ of a Riemann surface $Y$ parameterized by a point $t$ in a pointed hyperbolic surface $(X, t_{0})$. If a holomorphic motion from $E$ to $E_{t}$ in $Y$ has a guiding quasiconformal isotopy, then there is a holomorphic extension to any new point $p$ in $Y-E$ that follows the guiding isotopy. The proof gives a canonical way to replace a quasiconformal motion of the $(n+1)^{th}$ point by a holomorphic motion while leaving unchanged the given holomorphic…
Let X be a Kobayashi hyperbolic complex manifold, and assume that X does not contain compact complex submanifolds of positive dimension (e.g., X Stein). We shall prove the following generalization of Ritt's theorem: every holomorphic self-map f of X such that f(X) is relatively compact in X has a unique fixed point p(f) in X, which is attracting. Furthermore, we shall prove that p(f) depends holomorphically on f in a suitable sense, generalizing results by Heins, Joseph-Kwack and the second author.
We obtain a complete classification of complex Kobayashi-hyperbolic manifolds of dimension $n\ge 2$, for which the dimension of the group of holomorphic automorphisms is equal to $n^2$.
Let $f: Y \rightarrow X$ be a continuous map between a compact real analytic Kähler manifold $(Y,g)$ and a compact complex {hyperbolic manifold} $(X,g_0)$. In this paper we give a lower bound of the diastatic entropy of $(Y,g)$ in terms of the diastatic entropy of $(X,g_0)$ and the degree of $f$. When the lower bound is attained we get geometric rigidity theorems for the diastatic entropy analogous to the ones obtained by G. Besson, G. Courtois and S. Gallot [2] for the volume entropy. As a corollary, when $X=Y$, we show that the minimal diastatic entropy is achieved if and only if $g$ is holo…
Let $Δ\subsetneq \mathbb C$ be a simply connected domain, let $f:\mathbb D \to Δ$ be a Riemann map and let $\{z_k\}\subset Δ$ be a compactly divergent sequence. Using Gromov's hyperbolicity theory, we show that $\{f^{-1}(z_k)\}$ converges non-tangentially to a point of $\partial \mathbb D$ if and only if there exists a simply connected domain $U\subsetneq \mathbb C$ such that $Δ\subset U$ and $Δ$ contains a tubular hyperbolic neighborhood of a geodesic of $U$ and $\{z_k\}$ is eventually contained in a smaller tubular hyperbolic neighborhood of the same geodesic. As a consequence we show that i…
We prove that the stable manifold of every point in a compact hyperbolic invariant set of a holomorphic automorphism of a complex manifold is biholomorphic to a complex vector space, provided that a bunching condition, which is weaker than the classical bunching condition for linearizability, holds.
We prove that evolution families on complex complete hyperbolic manifolds are in one to one correspondence with certain semicomplete non-autonomous holomorphic vector fields, providing the solution to a very general Loewner type differential equation on manifolds.
We prove that a finite family of commuting holomorphic self-maps of the unit ball $\mathbb{B}^q\subset \mathbb{C}^q$ admits a simultaneous holomorphic conjugacy to a family of commuting automorphisms of a possibly lower dimensional ball, and that such conjugacy satisfies a universal property. As an application we describe when a hyperbolic and a parabolic holomorphic self-map of $\mathbb{B}^q$ can commute.
We determine all connected homogeneous Kobayashi-hyperbolic manifolds of dimension $n\ge 2$ whose holomorphic automorphism group has dimension $n^2-3$. This result complements existing classifications for automorphism group dimension $n^2-2$ (which is in some sense critical) and greater.
Let $(f_λ)_{λ\in Λ}$ be a holomorphic family of polynomial automorphisms of $\mathbb{C}^2$. Following previous work of Dujardin and Lyubich, we say that such a family is weakly stable if saddle periodic orbits do not bifurcate. It is an open question whether this property is equivalent to structural stability on the Julia set $J^*$ (that is, the closure of the set of saddle periodic points). In this paper we introduce a notion of regular point for a polynomial automorphism, inspired by Pesin theory, and prove that in a weakly stable family, the set of regular points moves holomorphically. It f…
