Firstly, we consider the unitary geometry of two exceptional Cartan domains $\Re_{V}(16)$ and $\Re_{VI}(27)$. We obtain the explicit formulas of Bergman kernal funtion, Cauchy-Szegö kernel, Poinsson kernel and Bergman metric for $\Re_{V}(16)$ and $\Re_{VI}(27)$. Secondly, we give a class of invariant differential operators for Cartan domain $\Re$ of dimension n: If the Bergman metric of $\Re$ is $$ds^{2}=\sum\limits_{i,j=1}^{n}g_{ij}dz_{i}d\bar{z}_{j}, T(z,\bar{z})=(g_{ij})$$ and $$L(u)=T^{-1}(z,\bar{z}) [\frac{\partial^2u}{\partial z_i\partial\bar{z}_j}],$$then $$L_j(u)=\{\mbox {The sum of all prinipal minors of degree} j {for} L(u)\}$$ is invariant under the biholomorphic mapping of $\Re$. Let $D$ be the irreducible bounded homogeneous domain in $C^n$, $P=P(z,*)$ the Poisson kernel of $D$, then for any fixed $J(1\leq j \leq n)$ one has $L_j(P^{1/j})=0$ iff $D$ is a symmetric domain.
In this paper we reduce the Lu Qikeng problem for Cartan-Hartogs domains to an algebraic problem on Hua polynomials and we give a complete solution when the Cartan-Hartogs domain is built on a bounded symmetric domain of dimension at most 4.
In this paper we study the complete invariant metrics on Cartan-Hartogs domains which are the special types of Hua domains. Firstly, we introduce a class of new complete invariant metrics on these domains, and prove that these metrics are equivalent to the Bergman metric. Secondly, the Ricci curvatures under these new metrics are bounded from above and below by the negative constants. Thirdly, we estimate the holomorphic sectional curvatures of the new metrics, we prove that the holomorphic sectional curvatures are bounded from above and below by the negative constants. Finally, by using these…
This paper consists of two results dealing with balanced metrics (in S. Donaldson terminology) on nonconpact complex manifolds. In the first one we describe all balanced metrics on Cartan domains. In the second one we show that the only Cartan-Hartogs domain which admits a balanced metric is the complex hyperbolic space. By combining these results with those obtained in [13] (Kaehler-Einstein submanifolds of the infinite dimensional projective space, to appear in Mathematische Annalen) we also provide the first example of complete, Kaehler-Einstein and projectively induced metric g such that $…
Let $ϕ$ be a globally defined real Kähler potential on a domain $Ω\subset \mathbb{C}^d$, and $g_{F}$ be a Kähler metric on the Hartogs domain $ M=\{(z,w)\in Ω\times\mathbb{C}^{d_0}: \|w\|^2<e^{-ϕ(z)}\}$ associated with the Kähler potential $Φ_{F}(z,w)=ϕ(z)+F(ϕ(z)+\ln\|w\|^2)$. Firstly, we obtain explicit formulas of the coefficients $\mathbf{a}_j\;(j=1,2)$ of the Bergman function expansion for the Hartogs domain $( M,g_F)$ in a momentum profile $φ$. Secondly, using explicit expressions of $\mathbf{a}_j\;(j=1,2)$, we obtain necessary and sufficient conditions for the coefficients $\mathbf{a}_j\…
The Cartan-Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. For a Cartan-Hartogs domain $Ω^{B}(μ)$ endowed with the natural Kähler metric $g(μ),$ Zedda conjectured that the coefficient $a_2$ of the Rawnsley's $\varepsilon$-function expansion for the Cartan-Hartogs domain $(Ω^{B}(μ), g(μ))$ is constant on $Ω^{B}(μ)$ if and only if $(Ω^{B}(μ), g(μ))$ is biholomorphically isometric to the complex hyperbolic space. In this paper, following Zedda's argument, we give a geometric proof of the Zedda's conjecture by computing the curvature tenso…
This paper treats a holomorphic self-mapping f: Omega --> Omega of a bounded domain Omega in a separable Hilbert space H with a fixed point p. In case the domain is convex, we prove an infinite-dimensional version of the Cartan-Caratheodory-Kaup-Wu Theorem. This is basically a rigidity result in the vein of the uniqueness part of the classical Schwarz lemma. The main technique, inspired by an old idea of H. Cartan, is iteration of the mapping f and its derivative. A normality result for holomorphic mappings in the compact-weak-open topology, due to Kim and Krantz, is used.
In this paper we address two problems concerning a family of domains $M_Ω(μ) \subset \C^n$, called Cartan-Hartogs domains, endowed with a natural Kaehler metric $g(μ)$. The first one is determining when the metric $g(μ)$ is extremal (in the sense of Calabi), while the second one studies when the coefficient $a_2$ in the Engliš expansion of Rawnsley $ε$-function associated to $g(μ)$ is constant.
Inspired by the work of Z. Lu and G. Tian [21] in the compact setting, in this paper we address the problem of studying the Szegö kernel of the disk bundle over a noncompact Kähler manifold. In particular we compute the Szegö kernel of the disk bundle over a Cartan-Hartogs domain based on a bounded symmetric domain. The main ingredients in our analysis are the fact that every Cartan-Hartogs domain can be viewed as an "iterated" disk bundle over its base and the the ideas given in [4] for the computation of the Szegö kernel of the disk bundle over an Herimitian symmetric space of compact type.
For a domain $Ω$ in the complex plane, we consider the domain $S_n(Ω)$ consisting of those $n\times n$ complex matrices whose spectrum is contained in $Ω$. Given a holomorphic self-map $Ψ$ of $S_n(Ω)$ such that $Ψ(A)=A$ and the derivative of $Ψ$ at $A$ is identity for some $A\in S_n(Ω)$, we investigate when the map $Ψ$ would be spectrum-preserving. We prove that if the matrix $A$ is either diagonalizable or non-derogatory then for most domains $Ω$, $Ψ$ is spectrum-preserving on $S_n(Ω)$. Further, when $A$ is arbitrary, we prove that $Ψ$ is spectrum-preserving on a certain analytic subset of $S…
A theorem due to Cartan asserts that every origin-preserving automorphism of bounded circular domains with respect to the origin is linear. In the present paper, by employing the theory of Bergman's representative domain, we prove that under certain circumstances Cartan's assertion remains true for quasi-circular domains in $\mathbb C^n$. Our main result is applied to obtain some simple criterions for the case $n=3$ and to prove that Braun-Kaup-Upmeier's theorem remains true for our class of quasi-circular domains.
The Cartan-Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. The purpose of this paper is twofold. Firstly, for a Cartan-Hartogs domain $Ω^{B^{d_0}}(μ)$ endowed with the canonical metric $g(μ)$, we obtain an explicit formula for the Bergman kernel of the weighted Hilbert space $\mathcal{H}_α$ of square integrable holomorphic functions on $(Ω^{B^{d_0}}(μ), g(μ))$ with the weight $\exp\{-αφ\}$ (where $φ$ is a globally defined Kähler potential for $g(μ)$) for $α>0$, and, furthermore, we give an explicit expression of the Rawnsley's $\vareps…
In this paper, we prove some fundamental theorems for holomorphic curves on angular domain intersecting a hypersurface, finite set of fixed hyperplanes in general position and finite set of fixed hypersurfaces in general position on complex projective variety with the level of truncation. As applications of the second main theorems for an angle, we will discuss the uniqueness problem of holomorphic curves in an angle instead of the whole complex plane. Detail, we establish a result for uniqueness problem of holomorphic curve by inverse image of a hypersurface. In my knowledge, this is the firs…
We prove that if $D\subset C^n$ is a bounded domain with real analytic boundary and D is pseudoconvex then the compact open topology in the group of holomorphic automorphisms of D is the topology of uniform convergence on D.
We study the equivalence problem for CR-manifolds belonging to general class III_2, i.e. the 5-dimensional CR-manifolds of CR-dimension 1 and codimension 3 whose CR-bundle satisfies a certain degeneracy condition. For such a CR-manifold M, we construct a canonical Cartan connection on a 6-dimensional principal bundle P on M. This provides a complete set of biholomorphic invariants for M.
Cartan-Thullen theorem is a basic one in the theory of analytic functions of several complex variables. It states that for any open set $U$ of ${\mathbb C}^k$, the following conditions are equivalent: (a) $U$ is a domain of existence, (b) $U$ is a domain of holomorphy and (c) $U$ is holomorphically convex. On the other hand, when $f \, (\, =(f_1,f_2,\cdots,f_n)\, )$ is a $\mathbb C^n$-valued function on an open set $U$ of $\mathbb C^{k_1}\times\mathbb C^{k_2}\times\cdots\times\mathbb C^{k_n}$, $f$ is said to be $\mathbb C^n$-analytic, if $f$ is complex analytic and for any $i$ and $j$, $i\not=…
This volume contains a mildly expanded version of lectures and talks at seminars and conferences, as well as review papers on subjects listed in the title of the volume. A great deal of these texts have already been published or sent to press. However, the only way to provide a good exposition of the field we are interested in is to collect all of those papers together. Bounded symmetric domains form a favorite subject of research in function theory, non-commutative harmonic analysis, and representation theory of real reductive Lie groups. The authors introduce the notions of q-analogues of bo…
We study the equivalence problem for $4$-dimensional CR-manifolds of CR-dimension $1$ and codimension $2$ which are referred to as Engel CR-manifolds. We construct a canonical Cartan connection on such CR-manifolds through Cartan equivalence's method. In particular, we give the explicit expression of $4$ biholomorphic invariants, the annulation of which is a necessary and sufficient condition for an Engel manifold to be locally biholomorphic to Beloshapka's cubic in $\mathbb{C}^3$.
We reduce to various absolute parallelisms, namely to certain {e}-structures on manifolds of dimensions 7, 6, 5, the biholomorphic equivalence problem or the intrinsic CR equivalence problem for generic submanifolds M^5 in C^4 of CR dimension 1 and of codimension 3 that are maximally minimal and are geometry-preserving deformations of one natural cubic model of Beloshapka, somewhere else called the General Class III-1 of 5-dimensional CR manifolds. Some inspiration links exist with the treatment of the General Class II previously done in 2007 by Beloshapka, Ezhov, Schmalz, and also with the cl…
