Kobayashi-hyperbolic Manifolds Research Articles

Continuing the Classification of Homogeneous Kobayashi-Hyperbolic Manifolds with High-Dimensional Automorphism Group

We classify all homogeneous Kobayashi-hyperbolic manifolds of dimension n ge 2 whose group of holomorphic automorphisms has dimension either n^2 - 7 or n^2 - 8. This paper continues the work of A. Isaev, who classified all such manifolds with automorphism group dimension n^2 - 6 and greater.

A characterization of the bidisc by a subgroup of its automorphism group

We make a connection between the structure of the bidisc and a distinguished subgroup of its automorphism group. The automorphism group of the bidisc, as we know, is of dimension six and acts transitively. We observe that it contains a subgroup that is isomorphic to the automorphism group of the open unit disc and this subgroup partitions the bidisc into a complex curve and a family of strongly pseudo-convex hypersurfaces that are non-spherical as CR-manifolds. Our work reverses this process and shows that any 2-dimensional Kobayashi-hyperbolic manifold whose automorphism group (which is known, from the general theory, to be a Lie group) has a 3-dimensional subgroup that is non-solvable (as a Lie group) and that acts on the manifold to produce a collection of orbits possessing essentially the characteristics of the concretely known collection of orbits mentioned above, is biholomorphic to the bidisc.The distinguished subgroup is interesting in its own right. It turns out that if we consider any subdomain of the bidisc that is a union of a proper sub-collection of the collection of orbits mentioned above, then the automorphism group of this subdomain can be expressed very simply in terms of this distinguished subgroup.

Horosphere topology

We introduce a prime end-type theory on complete Kobayashi hyperbolic manifolds using horosphere sequences. This allows to introduce a new notion of boundary-new even in the unit disc in the complex space-the horosphere boundary, and a topology on the manifold together with its horosphere boundary, the horosphere topology. We prove that a bounded strongly pseudoconvex domain endowed with the horosphere topology is homeomorphic to its Euclidean closure, while for the polydisc such a horosphere topology is not even Hausdorff and is different from the Gromov topology. We use this theory to study boundary behavior of univalent maps from bounded strongly pseudoconvex domains. Among other things, we prove that every univalent map of the unit ball whose image is bounded and convex, extends as a homeomorphism up to the closure. Such a result, relying in an essential way on our theory and on the Gromov hyperbolicity theory, is completely new, dealing with non smooth domains.

Further Steps Towards Classifying Homogeneous Kobayashi-Hyperbolic Manifolds with High-Dimensional Automorphism Group

We determine all connected homogeneous Kobayashi-hyperbolic manifolds of dimension $$n\ge 4$$ whose group of holomorphic automorphisms has dimension either $$n^2-4$$, or $$n^2-5$$, or $$n^2-6$$. This paper continues a series of articles that achieve classifications for automorphism group dimension $$n^2-3$$ and greater.

Algebraically hyperbolic manifolds have finite automorphism groups

A projective manifold [Formula: see text] is algebraically hyperbolic if there exists a positive constant [Formula: see text] such that the degree of any curve of genus [Formula: see text] on [Formula: see text] is bounded from above by [Formula: see text]. A classical result is that Kobayashi hyperbolicity implies algebraic hyperbolicity. It is known that Kobayashi hyperbolic manifolds have finite automorphism groups. Here, we prove that, more generally, algebraically hyperbolic projective manifolds have finite automorphism groups.

A comparison of two biholomorphic invariants

The Fridman invariant, which is a biholomorphic invariant on Kobayashi hyperbolic manifolds, can be seen as the dual of the much studied squeezing function. We compare this pair of invariants by showing that they are both equally capable of determining the boundary geometry of a bounded domain if their boundary behavior is a priori known.

Homogeneous Kobayashi-hyperbolic manifolds with high-dimensional group of holomorphic automorphisms

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.

Homogeneous Kobayashi-hyperbolic manifolds with automorphism group of subcritical dimension

We determine all connected homogeneous Kobayashi-hyperbolic manifolds of dimension whose holomorphic automorphism group has dimension . This result complements existing classifications for automorphism group dimension (which is in some sense critical) and greater.

Canonical Models for the Forward and Backward Iteration of Holomorphic Maps

We prove the existence and the essential uniqueness of canonical models for the forward (resp., backward) iteration of a holomorphic self-map f of a cocompact Kobayashi hyperbolic complex manifold, such as the ball \(\mathbb {B}^q\) or the polydisc \(\Delta ^q\). This is done by performing a time-dependent conjugacy of the autonomous dynamical system defined by f, obtaining in this way a non-autonomous dynamical system admitting a relatively compact forward (resp., backward) orbit, and then proving the existence of a natural complex structure on a suitable quotient of the direct limit (resp., subset of the inverse limit). As a corollary we prove the existence of a holomorphic solution with values in the upper half-plane of the Valiron equation for a hyperbolic holomorphic self-map of the unit ball.

Invariant holomorphic foliations on Kobayashi hyperbolic homogeneous manifolds

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 $\pi \colon M \to N$ 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 $D$.

Proper actions of high-dimensional groups on complex manifolds

Proper group actions are ubiquitous in mathematics and have many of the attractive features of actions of compact groups. In this survey, we discuss proper actions of Lie groups on smooth manifolds. If the group dimension is sufficiently high, all proper effective actions can be explicitly determined, and our principal goal is to provide a comprehensive exposition of known classification results in the complex setting. They include a complete description of Kobayashi-hyperbolic manifolds with high-dimensional automorphism group, which is a case of special interest.

Oka properties of some hypersurface complements

Oka manifolds can be viewed as the opposite of Kobayashi hyperbolic manifolds. Kobayashi asked whether the complement in projective space of a generic hypersurface of sufficiently high degree is hyperbolic. Therefore it is natural to investigate Oka properties of complements of low degree hypersurfaces. We determine which complements of hyperplane arrangements in projective space are Oka. A related question is which hypersurfaces in affine space have Oka complements. We give some results for graphs of meromorphic functions.

Semi-contractions des espaces localement compacts et cas des variétés complexes

Inspired by papers of Beardon, we give results for fixed points and orbits of contractions and semi-contractions of locally compact connected spaces. More precise results are obtained for the case of complex Kobayashi hyperbolic manifolds.

Inégalité d’Ahlfors en dimension supérieure

We prove an Ahlfors’ like inequality for the holomorphic curves with boundary of a complex compact Kobayashi-hyperbolic manifold.

Stein spaces in connection with El Naschie’s exceptional Lie groups hierarchies in high energy physics

In this work we introduce Kobayashi hyperbolic manifolds with their automorphism groups which are Lie groups. Bounded Stein manifolds are Kobayashi hyperbolic with Lie groups as automorphism groups. One and two-Stein spaces are in close connection with dimensions of El Naschie’s exceptional Lie groups hierarchies. The sum of seventeen one and two-Stein spaces can be expressed in terms of the dimensions of certain Lie groups. The number of high energy elementary particles in an extended standard model and their fuzzy values are found.

On proper actions of Lie groups of dimension [formula omitted] on n-dimensional complex manifolds

We explicitly classify all pairs ( M , G ) , where M is a connected complex manifold of dimension n ⩾ 2 and G is a connected Lie group acting properly and effectively on M by holomorphic transformations and having dimension d G satisfying n 2 + 2 ⩽ d G < n 2 + 2 n . We also consider the case d G = n 2 + 1 . In this case all actions split into three types according to the form of the linear isotropy subgroup. We give a complete explicit description of all pairs ( M , G ) for two of these types, as well as a large number of examples of actions of the third type. These results complement a theorem due to W. Kaup for the maximal group dimension n 2 + 2 n and generalize some of the author's earlier work on Kobayashi-hyperbolic manifolds with high-dimensional holomorphic automorphism group.

Proper actions of high-dimensional groups on complex manifolds

We explicitly classify all pairs (M, G), where M is a connected complex manifold of dimension n≥2 and G is a connected Lie group acting properly and effectively on M by holomorphic transformations and having dimension dG satisfying n2+2≤dG<n2+2n. These results extend—in the complex case—the classical description of manifolds admitting proper actions of groups of sufficiently high dimensions. They also generalize some of the author’s earlier work on Kobayashi-hyperbolic manifolds with high-dimensional holomorphic automorphism group.

On a Kobayashi hyperbolic manifold N modulo a closed subset ΔN and its applications

We show that the degeneration locus of the Kobayashi pseudodistance on a complex manifold is always a pseudoconcave set of order 1. We give some results cocerning the degeneration locus of the Kobayashi pseudodistance. Next we prove a generalization of the little Picard theorem relevantly. Finally, we consider the case N = ΔN.

Hyperbolic n-dimensional manifolds with automorphism group of dimension n2

We obtain a complete classification of complex Kobayashihyperbolic manifolds of dimension n ≥ 2, for which the dimension of the group of holomorphic automorphisms is equal to n2.

Hyperbolic manifolds with high-dimensional automorphism group

We survey our recent classification results for Kobayashi-hyperbolic n-dimensional manifolds with holomorphic automorphism group of dimension at least n 2 − 1 for n ≥ 2.