We extend the result of Z. Feng and Z. Tu in \cite{fengtu by showing that if one of the coefficients $a_j$, $2\leq j\leq n$, of Rawnlsey's epsilon function associated to a $n$-dimensional Cartan--Hartogs domain is constant, then the domain is biholomorphically equivalent to the complex hyperbolic space.
1409.8660.tex
\documentclass[11pt,reqno]{amsart}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{amsthm}
\usepackage{enumerate}
\usepackage{lscape}
\usepackage{dsfont}
\usepackage{color}

\newcommand{\sse}{\Longleftrightarrow}             
\newcommand{\R}{\mathds{R}}                   
\newcommand{\z}{\mathds{Z}}
\newcommand{\q}{\mathds{Q}}
\newcommand{\CP}{\mathds{C}\mathrm{P}}
\newcommand{\N}{\mathds{N}}
\newcommand{\Chi}{\mathfrak{X}}
\newcommand{\CH}{\mathds{C}\mathrm{H}}
\newcommand{\isom}{\mathrm{Isom}}
\newcommand{\aut}{\mathrm{Aut}}
\newcommand{\ol}{\mathrm{Hol}}
\newcommand{\po}{\forall}                       
\newcommand{\f}{\rightarrow}                  
\newcommand{\C}{\mathds{C}}            
\newcommand{\de}{\partial}          
\newcommand{\dd}{\mathrm{D}}


\newtheorem{theor}{Theorem}
\newtheorem{prop}[theor]{Proposition}
\newtheorem{defin}[theor]{Definition}
\newtheorem{lem}[theor]{Lemma}
\newtheorem{cor}[theor]{Corollary}
\newtheorem{note}{Note}
\newtheorem{ex}{Example}
\newtheorem{remark}[theor]{Remark}

\linespread{1.2}
\begin{document}
\title[On the epsilon function of Cartan--Hartogs domains]{A note on the coefficients of Rawnsley's epsilon function of Cartan--Hartogs domains}
\author[M. Zedda]{Michela Zedda}
\address{Dipartimento di Matematica ``G. Peano", Universit\`{a} di Torino}
\email{michela.zedda@gmail.com}

\thanks{
The author was supported by the project FIRB ``Geometria Differenziale e teoria geometrica delle funzioni'' and by INdAM-GNSAGA - Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni.
}
\date{}
\subjclass[2010]{32Q15; 32A25; 32M15} %32Q15  	K�hler manifolds. 53C55 Hermitian and K�hlerian manifolds. 32A25: Integral representations; canonical kernels (Szeg?, Bergman, etc.) 32A07  	Special domains (Reinhardt, Hartogs, circular, tube) 32M15  	Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras
\keywords{Cartan-Hartogs domains, Engli\v{s} expansion, Rawnsley's epsilon function}

\begin{abstract}
We  extend the result of Z. Feng and Z. Tu in \cite{fengtu} by showing that if one of the coefficients $a_j$, $2\leq j\leq n$, of  Rawnlsey's epsilon function associated to a $n$-dimensional Cartan--Hartogs domain is constant, then the domain is biholomorphically equivalent to the complex hyperbolic space. 
\end{abstract}

\maketitle


\section{Introduction and statement of main result}
Consider an $n$-dimensional complex manifold $(M,g)$ endowed with a K\"ahler metric $g$ and assume that there exists a globally defined  K\"ahler potential $\varphi\!:M\rightarrow \mathds{R}$ for $g$, i.e. if $\omega$ is the K\"ahler form associated to $g$, we have $\omega=\frac i2\partial\bar\partial \varphi$.
Let $\mathcal{H}_\alpha$ be the weighted Bergman space of square integrable holomorphic functions on $M$ with respect to the measure $e^{-\alpha\varphi}\frac{\omega^n}{n!}$, i.e.:
$$
\mathcal{H}_\alpha=\left\{f\in {\rm Hol(M)}|\,\int_Me^{-\alpha\varphi}|f|^2\frac{\omega^n}{n!}<\infty \right\}.
$$
If $\mathcal{H}_\alpha\neq \{0\}$, choose an orthonormal basis $\{f_j\}$ with respect to the product:
$$
(f,h)_\alpha=\int_Me^{-\alpha\varphi}f\bar h \frac{\omega^n}{n!},
$$
and denote by $K_\alpha(x,y)$ the reproducing kernel of $\mathcal{H}_\alpha$, namely:
$$
K_\alpha(x,y)=\sum_j f_j(x)\bar f_j(y),\qquad x,y\in M.
$$
Define the $\epsilon$-function associated to $g$ to be the function:
$$\epsilon_{\alpha g}(x)=e^{-\alpha\varphi(x)}K_{\alpha}(x,x), \quad x\in M.$$

In the literature the function $\epsilon_{\alpha g}$ was first introduced by J. Rawnsley under the name of $\eta$-{\em function} in \cite{rawnsley} and later as $\theta$-{\em function} in \cite{CGR}.

We say that  $\epsilon_{\alpha g}$ admits the {\em Engli\v{s} expansion}: 
\begin{equation}\label{englisexp}
\epsilon_{\alpha g}(x)\sim\sum_{j=0}^\infty a_j(x)\alpha^{n-j},\quad x\in M,
\end{equation}
for $\alpha\rightarrow+\infty$, if for every integers $l$, $r$ and every compact $H \subseteq M$,
\begin{equation}\label{TalphaMEANS}
\| \epsilon_{\alpha}(x) -
\sum_{j=0}^l  a_j(x){\alpha}^{n-j} \|_{C^r} \leq \frac{C(l, r, H)}{\alpha^{l+1}},
\end{equation}
for some constant $C(l, r, H) >0$. Such expansion is the counterpart for noncompact manifolds of the the celebrated TYZ (Tian-Yau-Zelditch) expansion
of Kempf's distortion function for polarized compact K\"ahler manifolds (see \cite{ze}
and  also \cite{arlquant}).
In \cite{englisasymp} M. Engli\v{s}
proved  that each of the coefficients $a_j(x)$ in \eqref{englisexp} 
 is a polynomial of the curvature of the metric $g$ and its covariant
derivatives at $x$, which can be
found by finitely many steps of algebraic operations, and gives an explicit expression of the coefficients $a_j$ for $j\leq 3$. \\





In this paper we consider the case of Cartan-Hartogs domains, which are defined as follows.
Consider a Cartan domain $\Omega\subset \C^d$, i.e. an irreducible bounded symmetric domain, of rank $r$ and numerical invariants $a$, $b$. Recall that the triple $\{r,a,b\}$ uniquely determines $\Omega$ and in particular it defines the dimension $d=\frac{r(r-1)}2a+rb+r$ and the genus $\gamma=(r-1)a+b+2$ of $\Omega$. Let $K(z, z)$ be the Bergman kernel of $\Omega$ and $N_\Omega(z, z)$ its {\em generic norm}, i.e.
\begin{equation}\label{genericnorm}
N_\Omega(z, z)=(V(\Omega)K(z, z))^{-\frac{1}{\gamma}},
\end{equation}
where $V(\Omega)$ is the total volume of $\Omega$ with respect to the Euclidean measure of the ambient complex Euclidean space.

For all positive real numbers $\mu$, a Cartan--Hartogs domains is given by $\left(M^{d_0}_{\Omega}(\mu),g(\mu)\right)$ where:
\begin{equation}\label{defm}
M^{d_0}_{\Omega}(\mu)=\left\{(z,w)\in \Omega\times\C^{d_0},\ ||w||^2<N_\Omega(z, z)^\mu\right\},
\end{equation}
and $g(\mu)$ is the K\"ahler metric whose associated K\"ahler form $\omega(\mu)$ can be described by the (globally defined) K\"ahler potential centered at the origin:
\begin{equation}\label{diastM}
\Phi(z,w)=-\log(N_{\Omega}(z, z)^\mu-||w||^2).
\end{equation}
The domain $\Omega$ is called  the {\em  base} of the Cartan--Hartogs domain 
$M^{d_0}_{\Omega}(\mu)$ (one also  says that 
$M^{d_0}_{\Omega}(\mu)$  is based on $\Omega$).
These domains have been considered by several authors (see e.g. \cite{roos} and references therein). In \cite{roos} it is shown that 
for $\mu_0=\gamma/(d+1)$, $(M^1_{\Omega}(\mu_0),g(\mu_0))$ is a complete K\"ahler-Einstein manifold which is homogeneous if and only if $\Omega$ is the complex hyperbolic space.
In \cite{articwall} the authors of the present paper proved that for $\Omega\neq\CH^d$, the metric $\alpha g(\mu)$ on $M^1_\Omega(\mu)$ is projectively induced for all positive real number $\alpha\geq \frac{(r-1)a}{2\mu}$, exhibing the first example of complete, noncompact, nonhomogeneous and projectively induced K\"ahler-Einstein metric. 
In \cite{ijgmmp} the  author of the present paper proved that for $d_0=1$, $g(\mu)$ is extremal (in the sense of Calabi \cite{Calabi82}) if and only if it is K\"ahler--Einstein and that the coefficient $a_2$ of Engli\v{s} expansion of the $\varepsilon$-function associated to a Cartan--Hartogs domain $(M^1_\Omega(\mu),g(\mu))$ is constant, then $(M^1_\Omega(\mu), g(\mu))$ is K\"ahler--Einstein, conjecturing also that the converse was true. In \cite{fengtu}, Z. Feng and Z. Tu generalize that theorem to generic $d_0$ and proved that conjecture. More precisely, they prove the following:

\begin{theor}[Z. Feng, Z. Tu {\cite[Th. 1.3]{fengtu}}]\label{fengzu}
The coefficient $a_2$ of the Rawnsley's $\epsilon$-function expansion is a constant on $M^{d_0}_\Omega(\mu)$ if and only if $\left(M^{d_0}_{\Omega}(\mu),g(\mu)\right)$ is biholomorphically isometric to the complex hyperbolic space $(\mathds{C}H^{d+d_0},g_{\rm hyp})$.
\end{theor}
Notice that  $g_{\rm hyp}$ denotes the hyperbolic metric on $\mathds{C}H^{d+d_0}$ and $$\left(\mathds{C}H^{d+d_0},g_{\rm hyp}\right)=\left(M^{d_0}_{\C H^{d}}(1),g(1)\right).$$ 


The prove of the previous theorem  is based on the  explicit formula for the $\epsilon$-function $\epsilon_{\alpha g(\mu)}$ of Cartan--Hartogs domains:
\begin{equation}\label{epsilonCH}
\epsilon_{\alpha g(\mu)}(z,w)=\frac 1{\mu^d}\sum_{k=0}^d\frac{D^k\tilde \chi(d)}{k!}\left(1-\frac{||w||^2}{N_\Omega(z, z)^\mu}\right)^{d-k}\frac{\Gamma(\alpha-d+k)}{\Gamma(\alpha-d-d_0)},
\end{equation}
for
$$
D^k\tilde \chi(d)=\sum_{j=0}^k{ k\choose j}(-1)^j\tilde \chi(d-j)
$$
and
$$
\tilde \chi(d-j)= \prod_{j=1}^r\frac{\Gamma(\mu (d-j)-\gamma-(j+1)\frac a2+2+b+ra)}{\Gamma(\mu (d-j)-\gamma+1+(j-1)\frac a2)},
$$
where $\Gamma$ is the usual $\Gamma$-function. Observe that formula \eqref{epsilonCH} shows that Engli\v{s} expansion of the $\epsilon$-function of Cartan--Hartogs domains is finite. In \cite{berezinCH} the author of this paper uses this formula to prove the existence of a Berezin-Engli\v{s} quantization for Cartan--Hartogs domains.



The aim of this paper is to generalize Theorem \ref{fengzu}  above (see next section) by proving the following: 
\begin{theor}\label{main}
For all $j=2,\dots, d+d_0$, any coefficient $a_j$ of the Rawnsley's $\epsilon$-function expansion is a constant on $M^{d_0}_\Omega(\mu)$ if and only if $\left(M^{d_0}_{\Omega}(\mu),g(\mu)\right)$ is biholomorphically isometric to the complex hyperbolic space $(\mathds{C}H^{d+d_0},g_{\rm hyp})$.
\end{theor}


The author would like to thank prof. Andrea Loi for the interesting and encouraging conversations.

\section{Proof of Theorem \ref{main}}

Due to Theorem \ref{fengzu} we need only to prove that if $a_j$ is constant for some $j=3,4,\dots, d+d_0$, then $a_2$ is.


Consider first the polynomial $P(\alpha)$ in the variable $\alpha$:
$$
P(\alpha)=\frac{\Gamma(\alpha-d+k)}{\Gamma(\alpha-d-d_0)},
$$
and observe that for $k=d$, ($d\geq 1$):
$$
P(\alpha)=\prod_{j=1}^{d+d_0}(\alpha-j),\quad \deg(P(\alpha))=d+d_0,%coefficienti dall'$a_0$ in poi});
$$
for $k=d-1$, ($d\geq 1$):
$$
P(\alpha)= \prod_{j=2}^{d+d_0}(\alpha-j),\quad  \deg(P(\alpha))=d+d_0-1,%coefficienti dall'$a_1$ in poi});
$$
for $k=d-2$, ($d\geq 2$):
$$
P(\alpha)=\prod_{j=3}^{d+d_0}(\alpha-j),\quad \deg(P(\alpha))=d+d_0-2,%, coefficienti dall'$a_2$ in poi});
$$
and so on.
Thus, from \eqref{epsilonCH}
we get that the factor with $k=d$ contributes to all the coefficients $a_0, a_1,\dots, a_{d+d_0}$ ($d\geq 1$), the factor with $k=d-1$ to all from $a_1$ to $a_{d+d_0}$ ($d\geq 1$), the factor with $k=d-2$ to all from $a_2$ ($d\geq 2$), and so on. Obviously the $j$-th coefficient is constant iff each one of its factors (except the $k=d$ one) vanishes, in fact the term $\left(1-\frac{||w||^2}{N_\Omega(z,\bar z)^\mu}\right)$ in each factor has a different power.

In particular, the coefficient $a_i$, $i=1,\dots, d+d_0,$ contains the factor:
$$
 \frac 1{\mu^d}\frac{D^{d-1}\tilde \chi(d)}{(d-1)!}\left(1-\frac{||w||^2}{N_\Omega(z,  z)^\mu}\right)A_{i}^2,\qquad (d\geq 1),
$$
and 
 the coefficient $a_i$, $i=2,\dots, d+d_0$, contains the factor:
$$
\frac 1{\mu^d}\frac{D^{d-2}\tilde \chi(d)}{(d-2)!}\left(1-\frac{||w||^2}{N_\Omega(z,  z)^\mu}\right)^{2}A_{i-1}^3,\qquad (d\geq 2),
$$
where we denote by $A_p^q$ the $p$-th coefficient of the polynomial in $\alpha$: 
$$
\prod_{j=q}^{d+d_0}(\alpha-j).
$$
Observe that $A_i^2$ and $A_{i-1}^3$ do not vanish. In fact we have:
$$\prod_{j=2}^{d+d_0}=\alpha^{d+d_0}+e_1(2,\dots,d+d_0)\alpha^{d+d_0-1}+\cdots+e_{d+d_0}(2,\dots,d+d_0),$$
$$\prod_{j=3}^{d+d_0}=\alpha^{d+d_0-1}+e_1(3,\dots,d+d_0)\alpha^{d+d_0-2}+\cdots+e_{d+d_0-1}(3,\dots,d+d_0)$$
where $e_j(x_1,\dots,x_n)$ is the elementary symmetric polynomial in the variables $(x_1,\dots,x_n)$, i.e.:
$$
e_j(x_1,\dots, x_n)=\sum_{1\leq k_1<k_2<\dots<k_j\leq n}x_{k_1}\dots x_{k_j}.
$$ 
Since in our case $x_j$ are positive integers, $A_i^2=e_i(2,\dots, d+d_0)$ and $A_{i-1}^3=e_{i-1}(3,\dots, d+d_0)$ do not vanish.

Thus we have that for $d\geq 2$ and for each $i=3,\dots, d$, if $a_i$ is constant then $D^{d-2}\tilde \chi(d)=D^{d-1}\tilde \chi(d)=0,$ and conclusion follows by \cite{fengtu}, where in the proof of Theorem 1.3 it is pointed out that when $d>1$ we have
$$D^{d-2}\tilde \chi(d)=D^{d-1}\tilde \chi(d)=0,$$
if and only if $a_2$ is constant.

If $d=1$, then $r=1$ and $\Omega=\mathds{C}H^1$, thus we need only to prove that if $a_j$ is constant for some $j=3,4,\dots, d_0+1$, then $\mu=1$. By the discussion above, if $a_j$ is constant for some $j=3,4,\dots, d_0+1$ then $D^{0}\tilde \chi(1)=0$, which by \cite[Lemma 3.5]{fengtu} directly implies $\mu=1$, concluding the proof.



\begin{thebibliography}{99}

\bibitem{arlquant} C. Arezzo, A. Loi,
{\em Quantization of K\"{a}hler
manifolds and the asymptotic
expansion of Tian--Yau--Zelditch},
J. Geom. Phys. 47  (2003), 87-99.

%\bibitem{arezzoloi}
%C. Arezzo, A. Loi,
%{\em Moment maps, scalar curvature and quantization of K\"ahler manifolds},
%Comm. Math.  Phys. 243 (2004), 543-559.

%\bibitem{Ber1} F. A. Berezin, {\em
%Quantization},
%Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 1116--1175 (Russian).

%\bibitem{BoCo} D. Boichu, G. Coeur\'{e}, {\em Sur le noyau de Bergman des domaines de Reinhardt}, Invent. Math. 72 (1983), 131--152.
%
%\bibitem{Bout} L. Boutet de Monvel, {\em Le noyau de Bergman en dimension 2}, S\'{e}minaire sur les Equations aux D\'{e}riv\'{e}es partielles 1987-1988, Exp. no. XXII, 13pp., Ecole Polytech., Palaiseau 1988.
%
%
%\bibitem{cgr1} M. Cahen, S. Gutt, J. H. Rawnsley,
%{\em Quantization of K\"{a}hler manifolds I: Geometric
%interpretation of Berezin's quantization}, JGP. 7 (1990), 45-62.
%
%\bibitem{cgr2} M. Cahen, S. Gutt, J. H. Rawnsley,
%{\em Quantization of K\"{a}hler manifolds II}, Trans. Amer. Math.
%Soc. 337 (1993), 73-98.

\bibitem{CGR} M. Cahen, S. Gutt, J. Rawnsley, \emph{Quantization of K\"ahler manifolds. I: Geometric interpretation of Berezin's quantization}, J. Geom. Physics 7 (1990), 45--62.

%\bibitem{cgr3} M. Cahen, S. Gutt, J. H. Rawnsley,
%{\em Quantization of K\"{a}hler manifolds  III}, Lett. Math. Phys.
%30 (1994), 291-305.

%\bibitem{cgr4} M. Cahen, S. Gutt, J. H. Rawnsley,
%{\em Quantization of K\"{a}hler manifolds IV}, Lett. Math. Phys. 34
%(1995), 159-168.



\bibitem{Calabi82} E. Calabi, \emph{Extremal K\"ahler metrics}, In Seminar on Differential Geometry vol. 16 of 102 (1982), Ann. of Math. Stud., Princeton University Press, 259-290.


%\bibitem{globsymp} F. Cuccu, A. Loi,
%{\em Global symplectic coordinates on complex domains},
%J. Geom. Phys. 56 (2006), 247-259.

%\bibitem{balancedC} F. Cuccu,  A. Loi,
%{\em Balanced metrics on ${\C}^n$},
%J. Geom. Phys. 57 (2007), 1115-1123.


%\bibitem{donaldson} S. Donaldson, {\em Scalar Curvature and Projective Embeddings, I}, J. Diff. Geometry 59 (2001),  479-522.
%
%\bibitem{do2} S. Donaldson,
%{\em Scalar Curvature and Projective Embeddings, II},
%Q. J. Math. 56 (2005),  345--356.

\bibitem{englisasymp} M. Engli\v{s}, {\em The asymptotics of a Laplace integral on a K\"ahler manifold}, J. Reine Angew. Math. 528 (2000) 1--39. 


%\bibitem{englisweigh} M.  Engli$\rm \check{s}$, \emph{Weighted Bergman kernels and balanced metrics}, RIMS Kokyuroku 1487 (2006), 40--54.



\bibitem{fengtu} Z. Feng, Z. Tu, {\em On canonical metrics on Cartan-Hartogs domains}, Math. Zeit. 278 (2014), Issue 1-2, 301--320.


%\bibitem{grecoloi} A. Greco, A. Loi, {\em Radial balanced metrics on the unit disk} J. Geom. Phys. 60  (2010), 53-59.
%


%\bibitem{regcov}
%A. Loi, \emph{Regular quantizations of K\"ahler manifolds and constant scalar curvature metrics},  J. Geom. Phys.  53 (2005), 354--364.

%\bibitem{loianal} A. Loi, \emph{The Tian--Yau--Zelditch asymptotic expansion for real analytic K\"ahler metrics}, Int. J. Geom. Methods in Modern Phys. 1 (2004) 253--263.


%\bibitem{loismooth} A. Loi, {\em A Laplace integral, the TYZ expansion and Berezin's transform on a K\"ahler manifold}, Int. J. Geom. Methods in Modern Physics 2 (2005), 359--371.

\bibitem{articwall} A. Loi, M. Zedda, \emph{K\"ahler--Einstein  submanifolds of  the infinite dimensional  projective space}, Math. Ann. 350 (2011), 145--154.



%\bibitem{hartogsbalanced} A. Loi, M. Zedda, \emph{Balanced metrics on Hartogs domains}, Abh. Math. Semin. Univ. Hambg. 81 (2011), no. 1, 69--77.


%\bibitem{balancedCH} A. Loi, M. Zedda, {\em Balanced metrics on Cartan and Cartan--Hartogs domains}, Math. Z. 270 (2012), no. 3-4, 1077--1087.
%
%

%\bibitem{lu} Z. Lu,
%{\em On the lower terms of the asymptotic expansion of Tian--Yau--Zelditch},
%Amer. J. Math. 122 (2000),  235-273.


\bibitem{rawnsley} J. Rawnsley, \emph{Coherent states and K\"ahler manifolds}, Quart. J. Math. Oxford (2), n. 28 (1977), 403--415.

\bibitem{roos} G. Roos, A. Wang, W. Yin, L. Zhang,
\emph{The K\"ahler-Einstein metric for some Hartogs
domains over bounded symmetric domains}, Science in China 49 (September
2006).

\bibitem{ze} S. Zelditch,
{\em Szeg\"{o} Kernels and a Theorem of Tian},
Internat. Math. Res. Notices  6 (1998), 317--331.

\bibitem{ijgmmp} M. Zedda, {\em Canonical metrics on Cartan-Hartogs domains},
Int. J. Geom. Methods Mod. Phys. 9 (2012), no. 1, 1250011, 13 pp..

\bibitem{berezinCH} M. Zedda, {\em Berezin--Engli\v{s}' quantization of Cartan--Hartogs domains}, arXiv:1404.1749 [math.DG] (preprint 2014)

\end{thebibliography}






\end{document}
1409.8660.extracted_bib
{arlquant C. Arezzo, A. Loi, {\em Quantization of K\"{ahler manifolds and the asymptotic expansion of Tian--Yau--Zelditch, J. Geom. Phys. 47 (2003), 87-99. %
1409.8660.extracted_bib
{arezzoloi %C. Arezzo, A. Loi, %{\em Moment maps, scalar curvature and quantization of K\"ahler manifolds, %Comm. Math. Phys. 243 (2004), 543-559. %
1409.8660.extracted_bib
{Ber1 F. A. Berezin, {\em %Quantization, %Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 1116--1175 (Russian). %
1409.8660.extracted_bib
{BoCo D. Boichu, G. Coeur\'{e, {\em Sur le noyau de Bergman des domaines de Reinhardt, Invent. Math. 72 (1983), 131--152. % %
1409.8660.extracted_bib
{Bout L. Boutet de Monvel, {\em Le noyau de Bergman en dimension 2, S\'{eminaire sur les Equations aux D\'{eriv\'{ees partielles 1987-1988, Exp. no. XXII, 13pp., Ecole Polytech., Palaiseau 1988. % % %
1409.8660.extracted_bib
{cgr1 M. Cahen, S. Gutt, J. H. Rawnsley, %{\em Quantization of K\"{ahler manifolds I: Geometric %interpretation of Berezin's quantization, JGP. 7 (1990), 45-62. % %
1409.8660.extracted_bib
{cgr2 M. Cahen, S. Gutt, J. H. Rawnsley, %{\em Quantization of K\"{ahler manifolds II, Trans. Amer. Math. %Soc. 337 (1993), 73-98.
1409.8660.extracted_bib
{CGR M. Cahen, S. Gutt, J. Rawnsley, Quantization of K\"ahler manifolds. I: Geometric interpretation of Berezin's quantization, J. Geom. Physics 7 (1990), 45--62. %
1409.8660.extracted_bib
{cgr3 M. Cahen, S. Gutt, J. H. Rawnsley, %{\em Quantization of K\"{ahler manifolds III, Lett. Math. Phys. %30 (1994), 291-305. %
1409.8660.extracted_bib
{cgr4 M. Cahen, S. Gutt, J. H. Rawnsley, %{\em Quantization of K\"{ahler manifolds IV, Lett. Math. Phys. 34 %(1995), 159-168.
1409.8660.extracted_bib
{Calabi82 E. Calabi, Extremal K\"ahler metrics, In Seminar on Differential Geometry vol. 16 of 102 (1982), Ann. of Math. Stud., Princeton University Press, 259-290. %
1409.8660.extracted_bib
{globsymp F. Cuccu, A. Loi, %{\em Global symplectic coordinates on complex domains, %J. Geom. Phys. 56 (2006), 247-259. %
1409.8660.extracted_bib
{balancedC F. Cuccu, A. Loi, %{\em Balanced metrics on ${\C^n$, %J. Geom. Phys. 57 (2007), 1115-1123. %
1409.8660.extracted_bib
{donaldson S. Donaldson, {\em Scalar Curvature and Projective Embeddings, I, J. Diff. Geometry 59 (2001), 479-522. % %
1409.8660.extracted_bib
{do2 S. Donaldson, %{\em Scalar Curvature and Projective Embeddings, II, %Q. J. Math. 56 (2005), 345--356.
1409.8660.extracted_bib
{englisasymp M. Engli\v{s, {\em The asymptotics of a Laplace integral on a K\"ahler manifold, J. Reine Angew. Math. 528 (2000) 1--39. %
1409.8660.extracted_bib
{englisweigh M. Engli$\rm \check{s$, Weighted Bergman kernels and balanced metrics, RIMS Kokyuroku 1487 (2006), 40--54.
1409.8660.extracted_bib
{fengtu Z. Feng, Z. Tu, {\em On canonical metrics on Cartan-Hartogs domains, Math. Zeit. 278 (2014), Issue 1-2, 301--320. %
1409.8660.extracted_bib
{grecoloi A. Greco, A. Loi, {\em Radial balanced metrics on the unit disk J. Geom. Phys. 60 (2010), 53-59. % %
1409.8660.extracted_bib
{regcov %A. Loi, Regular quantizations of K\"ahler manifolds and constant scalar curvature metrics, J. Geom. Phys. 53 (2005), 354--364. %
1409.8660.extracted_bib
{loianal A. Loi, The Tian--Yau--Zelditch asymptotic expansion for real analytic K\"ahler metrics, Int. J. Geom. Methods in Modern Phys. 1 (2004) 253--263. %
1409.8660.extracted_bib
{loismooth A. Loi, {\em A Laplace integral, the TYZ expansion and Berezin's transform on a K\"ahler manifold, Int. J. Geom. Methods in Modern Physics 2 (2005), 359--371.
1409.8660.extracted_bib
{articwall A. Loi, M. Zedda, K\"ahler--Einstein submanifolds of the infinite dimensional projective space, Math. Ann. 350 (2011), 145--154. %
1409.8660.extracted_bib
{hartogsbalanced A. Loi, M. Zedda, Balanced metrics on Hartogs domains, Abh. Math. Semin. Univ. Hambg. 81 (2011), no. 1, 69--77. %
1409.8660.extracted_bib
{balancedCH A. Loi, M. Zedda, {\em Balanced metrics on Cartan and Cartan--Hartogs domains, Math. Z. 270 (2012), no. 3-4, 1077--1087. % % %
1409.8660.extracted_bib
{lu Z. Lu, %{\em On the lower terms of the asymptotic expansion of Tian--Yau--Zelditch, %Amer. J. Math. 122 (2000), 235-273.
1409.8660.extracted_bib
{rawnsley J. Rawnsley, Coherent states and K\"ahler manifolds, Quart. J. Math. Oxford (2), n. 28 (1977), 403--415.
1409.8660.extracted_bib
{roos G. Roos, A. Wang, W. Yin, L. Zhang, The K\"ahler-Einstein metric for some Hartogs domains over bounded symmetric domains, Science in China 49 (September 2006).
1409.8660.extracted_bib
{ze S. Zelditch, {\em Szeg\"{o Kernels and a Theorem of Tian, Internat. Math. Res. Notices 6 (1998), 317--331.
1409.8660.extracted_bib
{ijgmmp M. Zedda, {\em Canonical metrics on Cartan-Hartogs domains, Int. J. Geom. Methods Mod. Phys. 9 (2012), no. 1, 1250011, 13 pp..
1409.8660.extracted_bib
{berezinCH M. Zedda, {\em Berezin--Engli\v{s' quantization of Cartan--Hartogs domains, arXiv:1404.1749 [math.DG] (preprint 2014)