1 Introduction

Let (L, h) be a positive Hermitian line bundle over a Kähler manifold (M, g) of dimension n such that \({\text {Ric}}(h)=\omega _g\). Here, \({\text {Ric}}(h)\) denotes the two-form on M whose local expression is given by

$$\begin{aligned} {\text {Ric}}(h)=-\frac{i}{2}\partial {\overline{\partial }}\log h(\sigma (z),\sigma (z)), \end{aligned}$$

for a trivializing holomorphic section \(\sigma :U\subset M\rightarrow L{\setminus } \{0\}\). In the quantum mechanics terminology, the pair (L, h) is also called a geometric quantization of the Kähler manifold (M, g).

For all integers \(\alpha >0\), we can define a complex Hilbert space \({\mathcal {H}}^{\alpha }\) consisting of the global holomorphic sections s of the line bundle \((L^\alpha ,h_\alpha )\) over M with \({\text {Ric}}\;h_\alpha =\alpha \omega _g\) which are bounded with respect to the following norm:

$$\begin{aligned} \langle s,s\rangle _\alpha :=\int _{M}h_{\alpha }(s(z),s(z))\frac{\omega _{g}^n}{n!}<+\infty . \end{aligned}$$

Let \(\{s_j\}\) be an orthonormal basis of \({\mathcal {H}}^\alpha \) with respect to \(\langle ,\rangle _\alpha \). Then, one can define a smooth real-valued function on M, called Rawnsley’s \(\varepsilon \)-function:

$$\begin{aligned} \varepsilon _{(\alpha ,g)}(z)=\sum \limits _{j=1}^{d_\alpha }h_\alpha (s_j(z),s_j(z)). \end{aligned}$$

One can check that this function depends only on the Kähler metric \(\omega _g\) and not on the orthonormal basis chosen. It is well known that Rawnsley’s \(\varepsilon \)-function \(\varepsilon _{(\alpha ,g)}\) has a asymptotic expansion in terms of the parameter \(\alpha \) (e.g., [8, 33]). There are two important branches of research on Rawnsley’s \(\varepsilon \)-function. The first one is the existence of balanced metrics on complex manifolds.

Definition 1.1

The metric g on M is balanced if the Rawnsley’s \(\varepsilon \)-function \(\varepsilon _{(1,g)}(z)\;(z\in M)\) is a positive constant on M.

The definition of balanced metrics was originally given by Donaldson (cf. [16]) in the case of a compact polarized Kähler manifold in 2001. Later on, it was generalized by Arezzo–Loi [1] and Englis̆ [20] to the noncompact case. Furthermore, balanced metrics had been widely used to study the quantization of a Kähler manifold, the expansion of the Bergman kernel function and the stability of the projective algebraic varieties. The reader is referred to Cahen–Gutt–Rawnsley [6], Englis̆ [20], Zhang [34] and references therein.

In fact, by Donaldson’s results we know that there exist balanced metrics on compact manifold with finite automorphism group. In the noncompact case, the existence and uniqueness of balanced metrics is still an open problem. Therefore, it makes sense to study the existence and uniqueness of balanced metrics on some special noncompact manifolds.

Unfortunately, despite the extensive studies of the compact case, very little seems to be known about the existence of balanced metrics on noncompact manifolds and even on the domains in \({\mathbb {C}}^n\).

We want to start with the simplest situation, namely (L, h) is the trivial positive holomorphic line bundle over a domain \(M\subset {\mathbb {C}}^n\) equipped with a Kähler metric g. In this case, the metric g can be described by a strictly plurisubharmonic real-valued function \(\varphi \), called a Kähler potential for g, that is \(\omega _g=\frac{\sqrt{-1}}{2\pi }\partial {{\bar{\partial }}}\varphi \). It is not hard to see that in this case, the Hilbert space \({\mathcal {H}}^\alpha \) equals the weighted Hilbert space \(H_{\alpha \varphi }\) of square integrable holomorphic functions on (M, g) with the weight \(\exp \{-\,\alpha \varphi \}\) defined by

$$\begin{aligned} H_{\alpha \varphi }(M):=\left\{ f\in {\text {Hol}}(M): \int _M\vert f\vert ^2\exp \{-\,\alpha \varphi \}\frac{\omega ^n}{n!}<+\infty \right\} , \end{aligned}$$

where \({\text {Hol}}(M)\) is the space of holomorphic functions on M. If \(H_{\alpha \varphi }(M)\ne \{0\}\), let \(K_{\alpha }(z,{\overline{z}})\) be its weighted Bergman kernel. Then, it is not difficult to see that the Rawnsley’s \(\varepsilon \)-function in this case can be expressed as

$$\begin{aligned} \varepsilon _{(\alpha ,g)}(z):=\exp \{-\,\alpha \varphi (z)\}K_{\alpha }(z,{\overline{z}}),\;\; z\in M. \end{aligned}$$

It can be easily verified that this function depends only on the metric g and not on the choice of the Kähler potential \(\varphi \) (which is defined up to the sum with the real part of a holomorphic function on M).

Some progress had been made in this simplest case. In 2012, Loi–Zedda [28] proved the existence of balanced metrics on bounded symmetric domains. Note that bounded symmetric domains are homogeneous domains. Inspired by this, similar results were recently generalized by Loi–Mossa [26] to all bounded homogeneous (not necessarily symmetric) domains.

Recently, Feng–Tu [21] firstly found the existence of balanced metrics on a class of nonhomogeneous domains called generalized Cartan–Hartogs domains. Later on, Bi–Feng–Tu [5] proved that balanced metric can also exist on Fock–Bargmann–Hartogs domains. For the study of the balanced metrics, see Hélène–Englis̆–Youssfi [22], Loi [25], Loi–Zedda [27], and Zedda [31].

In this paper, we study the canonical metric on the Hartogs domains called n-dimensional Hartogs triangles which generalize the classical Hartogs triangles defined by

$$\begin{aligned} \varOmega _n:=\{(z_1,\ldots ,z_n)\in {\mathbb {C}}^n:\vert z_1\vert<\vert z_2\vert<\cdots<\vert z_n\vert <1\}, \ \ \ (n\ge 2). \end{aligned}$$

The Hartogs triangles have attracted many attentions and been deeply investigated by many authors from different views. In 2013, Chakrabarti–Shaw [9] focused on Sobolev regularity of the \({\overline{\partial }}\)-equation over the Hartogs triangle. In 2016, Edholm [17] obtained the explicit form for the Bergman kernel for the generalized Hartogs triangle of exponent \(\gamma >0\), that is \(H_\gamma :=\{(z_1,z_2)\in {\mathbb {C}}^2: \vert z_1\vert ^\gamma<\vert z_2\vert <1\}\). By using the close form for the Bergman kernel, Edholm–McNeal [18] studied \(L^p\) boundedness of the Bergman projection on \(H_\gamma \). Inspired by this, Chen [10] obtained the necessary and sufficient condition for the Bergman projection on \(L^p\) space of more general bounded Hartogs domains to be bounded. For the reference of the theories of Bergman kernel, see also Park [29].

Recently, Zapałowski [35] gave the rigidity of proper holomorphic self-mappings between generalized Hartogs triangle and obtained automorphism group of the generalized Hartogs triangle. The reader is also referred to [11,12,13,14, 23] for the studies of rigidity of the proper holomorphic mappings between Hartogs triangles. Moreover, we can see that the n-dimensional Hartogs triangles are nonhomogeneous pseudoconvex domains with nonsmooth boundary. More importantly, much less seems to be known about the geometric properties of Hartogs triangles. Thus, all the above inspire us to study the canonical metrics on n-dimensional Hartogs triangles.

Firstly, let us introduce a new Kähler metric \(g(\nu )\) on \(\varOmega _n\). Define the strictly plurisubharmonic function \(\varPhi _n(z)\) on the Hartogs triangles \(\varOmega _n\) as follows

$$\begin{aligned} \varPhi _n(z):=-\sum _{k=1}^{n-1}\nu _k\ln (\vert z_{k+1}\vert ^2-\vert z_k\vert ^2)-\nu _n\ln (1-\vert z_n\vert ^2), \end{aligned}$$
(1.1)

where \(\nu =(\nu _1,\ldots ,\nu _n)\) with \(\nu _k>0\), \(1\le k\le n\). The Kähler form \(\omega \) on \(\varOmega _n\) is given by

$$\begin{aligned} \omega :=\frac{\sqrt{-1}}{2\pi }\partial {{\bar{\partial }}}\varPhi _n. \end{aligned}$$

Hence, the Kähler metric \(g(\nu )\) on \(\varOmega _n\) associated with \(\omega \) can be expressed by

$$\begin{aligned} g(\nu )_{i{{\bar{j}}}}=\frac{\partial ^2 \varPhi _n}{\partial z_i\partial {{\bar{z}}}_j},\quad (1\le i,j\le n). \end{aligned}$$

Thus, we can define the weighted Hilbert space \(H_{\varPhi _n}(\varOmega _n)\) as follows:

$$\begin{aligned} H_{\varPhi _n}(\varOmega _n):=\left\{ f\in {\text {Hol}}(\varOmega _n): \int _{\varOmega _n}\vert f\vert ^2\exp \{-\,\varPhi _n\}\frac{\omega ^n}{n!}<+\infty \right\} . \end{aligned}$$
(1.2)

One of the main results of our paper is the following.

Theorem 1.2

The Kähler metric \(g(\nu )\) on \(\varOmega _n\) is balanced if and only if \(\nu _k\ge 2\) is an integer for all \(k=1,\ldots ,n-1\), and \(\nu _n>1\).

Remark 1.3

Notice that if \(\nu _1=\cdots =\nu _n=2\), the Kähler metric \(g(\nu )\) is exactly the Bergman metric for \(\varOmega _n\).

Another important application of Rawnsley’s \(\varepsilon \)-function is to study whether a Berezin quantization can be established on some Kähler manifolds. In recent years, Berezin quantization has attracted a lot of attention and has been deeply studied by mathematicians and physicists, see, e.g., Cahen–Gutt–Rawnsley [6], Engliš [19], Loi–Mossa [26] and Zedda [32]. Roughly, a quantization is a construction of a quantum system from the classical mechanics of a system. In 1927, for seek of finding the purely mathematical significance of quantization, Weyl made an attempt at a quantization known as Weyl quantization. He associated a self-adjoint operators on a separable Hilbert space with functions on a symplectic manifold and some certain commutations are fulfilled. Later on, Berezin [3] raised a new quantization procedure, i.e., Berezin quantization. A Berezin quantization on a Kähler manifold \((\varOmega ,\omega )\) is given by a family of associative algebra \({\mathcal {A}}_{h}\) where the parameter h runs through a set E of the positive reals with 0 in its closure, and moreover, there exists a subalgebra \({\mathcal {A}}\) of \(\bigoplus \{{\mathcal {A}}_{h};\;h\in E\}\) such that some properties are satisfied (refer to Berezin [3]). More precisely, we call an associative algebra with involution \({\mathcal {A}}\) a quantization of \((\varOmega ,\omega )\) if the following properties are satisfied.

  1. (i)

    There exist a family of associative algebras \({\mathcal {A}}_{h}\) of functions on \(\varOmega \) where the parameter h runs through a set E of the positive reals with 0 in its closure. Moreover, \({\mathcal {A}}\) is a subalgebra of \(\bigoplus \{{\mathcal {A}}_{h};\;h\in E\}\).

  2. (ii)

    For each \(f\in {\mathcal {A}}\) which will be written f(h, x) (\(h\in E\), \(x\in \varOmega \)) such that \(f(h,\cdot )\in {\mathcal {A}}_{h}\), the limit

    $$\begin{aligned} \lim _{h\rightarrow 0+}f(h,x)=\varphi (f)(x) \end{aligned}$$

    exists.

  3. (iii)

    \(\varphi (f*g)=\varphi (f)\cdot \varphi (g)\), \(\varphi (h^{-1}(f*g-g*f))=\frac{1}{i}\{\varphi (f),\varphi (g)\}\) for \(f,g\in {\mathcal {A}}\). Here, \(*\) and \(\{,\}\) denote the product of \({\mathcal {A}}\) and the Poisson bracket.

  4. (iv)

    For any two points \(x_{1},x_{2}\in \varOmega \), there exists \(f\in {\mathcal {A}}\) such that \(\varphi (f)(x_{1})\ne \varphi (f)(x_{2})\).

For a given Kähler manifold \(\varOmega \) endowed with a Kähler metric g associated with a Kähler form \(\omega \), suppose that there exists a global Kähler potential \(\varphi (z):\varOmega \rightarrow {\mathbb {R}}\) which can extend to a sesquianalytic function \(\varphi (z,{\overline{w}})\) on \(\varOmega \times \varOmega \) such that \(\varphi (z,{\overline{z}})=\varphi (z)\). Then, the Calabi’s diastasis function is defined by (see Calabi [7])

$$\begin{aligned} D_{g}(z,w):=\varphi (z,{\overline{z}})+\varphi (w,{\overline{w}}) -\varphi (z,{\overline{w}})-\varphi (w,{\overline{z}}),\;(z,w)\in \varOmega \times \varOmega . \end{aligned}$$
(1.3)

It is not hard to see that the Calabi’s diastasis function \(D_{g}(z,w)\) is symmetric in z and w and is uniquely defined up to the real part of a holomorphic function.

Moreover, the Calabi’s diastasis function has played a crucial rule in studying balanced metric, Berezin quantization and Kähler immersions (i.e., holomorphic and isometric immersions). For more details, please see [2, 7, 24].

In fact, by using the Rawnsley’s \(\varepsilon \)-function and the Calabi’s diastasis function, Englis̆ [19] gave a sufficient condition for a Kähler manifold \((\varOmega ,g)\) to admit a Berezin quantization.

Theorem 1.4

(see [19]) Let \(\varOmega \) be a Kähler manifold endowed with a Kähler metric g associated with Kähler form \(\omega \). If

  1. (I)

    The function \(\exp \{-\,D_{g}(z,w)\}\) is globally defined on \(\varOmega \times \varOmega \), \(\exp \{-\,D_{g}(z,w)\}\le 1\) and \(\exp \{-\,D_{g}(z,w)\}=1\) if and only if \(z=w\), where \(D_{g}(z,w)\) denotes the Calabi’s diastasis function.

  2. (II)

    There exists a subset \(E\subset {\mathbb {R}}^{+}\) which has \(+\infty \) in its closure such that the Rawnsley’s \(\varepsilon \)-function \(\varepsilon _{(\alpha ,g)}(z)\) is a positive constant for \(\alpha \in E\).

Then, \((\varOmega ,g)\) admits a Berezin quantization.

As far as we know, the above conditions are satisfied by homogeneous Kähler manifold, a contractible homogeneous Kähler manifold (i.e., all the products \((\varOmega , g)\times ({\mathbb {C}}^m, g_0)\), where \((\varOmega , g)\) is an homogeneous bounded domain and \(g_0\) is the standard flat metric) and some special pseudoconvex domains (cf. [4, 19, 26, 30]). So some experts are dedicated to find more noncompact Kähler manifolds which a Berezin quantization can be carried out.

In this paper, by using Theorems 1.2 and 1.4, we will show that the conditions (I) and (II) can be satisfied by the Hartogs triangles \((\varOmega _n,g(\nu ))\), that is

Theorem 1.5

Let \(\varOmega _n\) be the Hartogs triangle endowed with the Kähler metric \(g(\nu )\). If \(\nu _k\) for all \(k=1,\ldots ,n-1\) are positive rational numbers and \(\nu _n>0\), then \((\varOmega _n,g(\nu ))\) admits a Berezin quantization.

The paper is organized as follows. In Sect. 2, we give an explicit formula for the Bergman kernel of the weighted Hilbert space of square integrable holomorphic functions on \((\varOmega _n, g(\nu ))\) with the weight \(\exp \{-\,\varPhi _n\}\) for some special \(\nu _k\). By using the expression of the Rawnsley’s \(\varepsilon \)-function, we give the proof of Theorem 1.2. In Sect. 3, using the Calabi’s diastasis function, Theorems 1.2 and 1.4, we prove Theorem 1.5.

2 Weighted Bergman kernel and balanced metrics on Hartogs triangles

In the following lemma, we describe the volume form of the Kähler metric \(g(\nu )\). The proof is omitted since it can be obtained by a straightforward induction argument of n.

Lemma 2.1

For \(n\ge 2\), let \(\varPhi _n\) be defined by (1.1). Then, we have

$$\begin{aligned} \det \left( \frac{\partial ^2\varPhi _n}{\partial z^t\partial {{\bar{z}}}}\right) (z)=\frac{\prod _ {j=1}^{n}\nu _j\prod _ {k=1}^{n-1}\vert z_{k+1}\vert ^2}{(1-\vert z_n\vert ^2)^2\prod _{k=1}^{n-1} (\vert z_{k+1}\vert ^2-\vert z_k\vert ^2)^2}, \end{aligned}$$
(2.1)

where \(z=(z_1,\ldots ,z_n)\in \varOmega _n\).

Since \(\varOmega _n\) is a Reinhardt domain, we are going to compute the squared \(L_{\varPhi _n}^2\)-norms for some holomorphic monomials in \(\varOmega _n\).

Lemma 2.2

Let \(z=(z_1,\ldots ,z_n)\in \varOmega _n\), and \(p=(p_1,\ldots ,p_n)\in {\mathbb {Z}}^n\), we have

$$\begin{aligned} \Vert z^p\Vert ^2_{L_{\varPhi _n}^2}=\prod _{k=1}^{n}(\pi \nu _k)\prod _{k=1}^{n} B\left( \sum _{j=1}^k (p_j+ \nu _j)-\nu _k+1, \nu _k-1\right) , \end{aligned}$$
(2.2)

where \(B(p,q)=\int _0^1 x^{p-1}(1-x)^{q-1}dx\) is the beta function.

Proof

Combining (1.1) and (2.1), we get

$$\begin{aligned} \Vert z^p\Vert ^2_{L_{\varPhi _n}^2}&=\int _{\varOmega _n}\vert z\vert ^{2p}\exp \{-\,\varPhi _n\}\frac{\omega ^n}{n!}\\&=\int _{\varOmega _n}\vert z\vert ^{2p}(1-\vert z_n\vert ^2)^{\nu _n}\prod _{k=1}^{n-1}(\vert z_{k+1}\vert ^2-\vert z_k\vert ^2)^{\nu _k}\\&\quad \times \frac{\prod _{k=1}^n\nu _k\prod _{k=1}^{n-1}\vert z_{k+1}\vert ^2}{(1-\vert z_n\vert ^2)^2\prod _{k=1}^{n-1}(\vert z_{k+1}\vert ^2-\vert z_k\vert ^2)^2}dm(z), \end{aligned}$$
(2.3)

where dm(z) is the Euclidean measure. We introduce polar coordinates in each variable by putting \(z_k=t_k {\hbox {e}}^{i\theta _k}, 1\le k\le n\). After doing so, and integrating out the angular variables, (2.3) becomes

$$\begin{aligned} \prod _{k=1}^n(2\pi \nu _k)\int _{0\le t_1<\cdots<t_n<1}t_1^{2p_1+1}(1-t_n^2)^{\nu _n-2}\prod _{k=1}^{n-1} t_{k+1}^{2p_{k+1}+3}(t_{k+1}^2-t_k^2)^{\nu _k-2}dt_1\cdots dt_n. \end{aligned}$$

Next, we set \(s_k=t_k^2, 1\le k\le n\) and then change variables again. We can obtain

$$\begin{aligned} \prod _{k=1}^n(\pi \nu _k)\int _{0\le s_1<\cdots<s_n<1}s_1^{p_1}(1-s_n)^{\nu _n-2}\prod _{k=1}^{n-1} s_{k+1}^{p_{k+1}+1}(s_{k+1}-s_{k})^{\nu _k-2}ds_1\cdots ds_n. \end{aligned}$$
(2.4)

Claim that

$$\begin{aligned} \Vert z^p\Vert ^2_{L_{\varPhi _n}^2}=\prod _{k=1}^n(\pi \nu _k)\prod _{k=1}^n B\left( \sum _{j=1}^k (p_j+ \nu _j)-\nu _k+1, \nu _k-1\right) . \end{aligned}$$

We will prove this claim by induction for n. For \(n=2\), by (2.4), we learn that

$$\begin{aligned} \Vert z^p\Vert ^2_{L_{\varPhi _n}^2}&=\pi ^2\nu _1\nu _2\int _{0\le s_1<s_2<1}(1-s_2)^{\nu _2-2}s_{2}^{p_{2}+1}s_1^{p_1}(s_{2}-s_{1})^{\nu _1-2}ds_1ds_2\\&=\pi ^2\nu _1\nu _2\int _0^1(1-s_2)^{\nu _2-2}s_{2}^{p_{2}+1}ds_2\int _{0}^{s_2}s_1^{p_1}(s_{2}-s_{1})^{\nu _1-2}ds_1\\&=\pi ^2\nu _1\nu _2B(p_1+1, \nu _1-1)\int _0^1(1-s_2)^{\nu _2-2}s_{2}^{p_{2}+1}s_2^{p_1+\nu _1-1}ds_2\\&=\pi ^2\nu _1\nu _2B(p_1+1, \nu _1-1)B(p_1+p_2+\nu _1+1,\nu _2-1). \end{aligned}$$

This means that the claim holds for \(n=2\). Thus, assume that the claim holds for \(n=\ell \); then, for \(n=\ell +1\), by (2.4), we obtain

$$\begin{aligned} \Vert z^p\Vert ^2_{L_{\varPhi _n}^2}&= \prod _{k=1}^{\ell +1}(\pi \nu _k)\int _{0\le s_1<\cdots<s_{\ell +1}<1}s_1^{p_1}(1-s_{\ell +1})^{\nu _{\ell +1}-2}\prod _{k=1}^{\ell }s_{k+1}^{p_{k+1}+1}(s_{k+1}-s_{k})^{\nu _k-2}ds_1\cdots ds_{\ell +1}\\&= \prod _{k=1}^{\ell +1}(\pi \nu _k)\int _0^1(1-s_{\ell +1})^{\nu _{\ell +1}-2}s_{\ell +1}^{p_{\ell +1}+1}ds_{\ell +1}\\&\quad \times \int _{0\le s_1<\cdots<s_\ell<s_{\ell +1}}s_1^{p_1}(s_{\ell +1}-s_\ell )^{\nu _\ell -2}\prod _{k=1}^{\ell -1}s_{k+1}^{p_{k+1}+1}(s_{k+1}-s_{k})^{\nu _k-2}ds_1\cdots ds_{\ell }\\&= \prod _{k=1}^{\ell +1}(\pi \nu _k)\int _0^1(1-s_{\ell +1})^{\nu _{\ell +1}-2}s_{\ell +1}^{p_{\ell +1}+1} s_{\ell +1}^{\sum _{k=1}^{\ell }(p_k+\nu _k)-1}ds_{\ell +1}\\&\times \int _{0\le {{\hat{s}}}_1<\cdots<{{\hat{s}}}_\ell <1}{{\hat{s}}}_1^{p_1}(1-{{\hat{s}}}_\ell )^{\nu _\ell -2}\prod _{k=1}^{\ell -1}{{\hat{s}}}_{k+1}^{p_{k+1}+1}({{\hat{s}}}_{k+1}-{{\hat{s}}}_{k})^{\nu _k-2}d{{\hat{s}}}_1\cdots d{{\hat{s}}}_{\ell }.\\ \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert z^p\Vert ^2_{L_{\varPhi _n}^2}&= \prod _{k=1}^{\ell +1}(\pi \nu _k)\prod _{k=1}^\ell B\left( \sum _{j=1}^k (p_j+ \nu _j)-\nu _k+1, \nu _k-1\right) \\&\times \int _0^1(1-s_{\ell +1})^{\nu _{\ell +1}-2}s_{\ell +1}^{\sum _{k=1}^{\ell +1}(p_k+\nu _k)-\nu _{\ell +1}} ds_{\ell +1}\\&= \prod _{k=1}^{\ell +1}(\pi \nu _k)\prod _{k=1}^\ell B\left( \sum _{j=1}^k (p_j+\nu _j)-\nu _k+1, \nu _k-1\right) \\&\times B\left( \sum _{k=1}^{\ell +1}(p_k+\nu _k)-\nu _{\ell +1}+1,\nu _{\ell +1}-1\right) \\&= \prod _{k=1}^{\ell +1}(\pi \nu _k)\prod _{k=1}^{\ell +1} B\left( \sum _{j=1}^k (p_j+\nu _j)-\nu _k+1,\nu _k-1\right) . \end{aligned}$$

The proof is completed. □

Hence, by the definition of the beta function, we can easily obtain the following property.

Proposition 2.3

\(H_{\varPhi _n}(\varOmega _n)\ne \{0\}\) if and only if \(\nu _k>1\) for all \(k=1,\ldots , n\).

Now, we give an elementary lemma for the gamma function.

Lemma 2.4

(see D’Angelo [15] Lemma 2) Let \(x=(x_1,\ldots ,x_m)\in {\mathbb {R}}^m\) with \(\Vert x\Vert <1\) and \(s\in {\mathbb {R}}\) with \(s>0\). Then,

$$\begin{aligned} \sum _{q\in {\mathbb {N}}^m}\frac{\varGamma (\vert q\vert +s)}{\varGamma (s)\prod _{i=1}^m\varGamma (q_i+1)}x^{2q}=\frac{1}{(1-\Vert x\Vert ^2)^s}. \end{aligned}$$

Theorem 2.5

Suppose that \((\varOmega _n,g(\nu ))\) is the n-dimensional Hartogs triangle endowed with the Kähler metric \(g(\nu )\). Let \(\nu _k\ge 2\) be integers for all \(k=1,\ldots , n-1\), and let \(\nu _n>1\). Let \(H_{\varPhi _n}(\varOmega _n)\) be the weighted Hilbert space of square integrable holomorphic functions on \((\varOmega _n,g(\nu ))\) with the weight \(\exp \{-\,\varPhi _n\}\) (see (1.2)). Then, \(H_{\varPhi _n}(\varOmega _n)\ne \{0\}\), and the Bergman kernel of \(H_{\varPhi _n}(\varOmega _n)\) is given by

$$\begin{aligned} K_{\varPhi _n}(z,{{\bar{z}}})=\frac{\nu _n-1}{\pi ^n\nu _n(1-\vert z_n\vert ^2)^{\nu _n}}\prod _{k=1}^{n-1}\frac{\nu _k-1}{\nu _k(\vert z_{k+1}\vert ^2-\vert z_{k}\vert ^2)^{\nu _k}}. \end{aligned}$$
(2.5)

Proof

Since \(\varOmega _n\) is a Reinhardt domain, together with Lemma 2.2 and the definition of the beta function, we can obtain that \(\{\frac{z^p}{\Vert z^p\Vert _{L_{\varPhi _n}^2}}\}\) forms a complete orthonormal basis of \(H_{\varPhi _n}(\varOmega _n)\), where the multi-index \(p=(p_1,\ldots ,p_n)\) ranges all integers that satisfy the following inequalities for all \(k=1,\ldots , n\),

$$\begin{aligned} \sum _{j=1}^k(p_j+\nu _j)-\nu _k\ge 0. \end{aligned}$$

Let N denote the set of all the multi-index \(p=(p_1,\ldots ,p_n)\) satisfying such inequalities. Hence, Formula (2.2) implies that

$$\begin{aligned} K_{\varPhi _n}(z,{{\bar{z}}})&= \sum _{p\in N}\frac{\vert z^p\vert ^{2}}{\Vert z^p\Vert _{L_{\varPhi _n}^2}^2}\\&= \frac{1}{\prod _{k=1}^n(\pi \nu _k)}\sum _{p_1=0}^{+\infty }\frac{\vert z_1\vert ^{2p_1}}{B(p_1+1,\nu _1-1)}\sum _{p_2=-p_1-\nu _1}^{+\infty }\frac{\vert z_2\vert ^{2p_2}}{B(p_1+p_2+\nu _1+1,\nu _2-1)}\\&\cdots \sum _{p_n=-\sum _{k=1}^{n-1}(p_k+\nu _k)}^{+\infty }\frac{\vert z_n\vert ^{2p_n}}{B(\sum _{k=1}^{n}(p_k+\nu _k)-\nu _n+1,\nu _n-1)}. \end{aligned}$$

Notice that by Lemma 2.4, we can learn that

$$\begin{aligned}&\sum _{p_n=-\sum _{k=1}^{n-1}(p_k+\nu _k)}^{+\infty }\frac{\vert z_n\vert ^{2p_n}}{B(\sum _{k=1}^{n}(p_k+\nu _k)-\nu _n+1,\nu _n-1)}\\&\quad =\vert z_n\vert ^{-\sum _{k=1}^{n-1}2(p_k+\nu _k)}\sum _{m=0}^{+\infty }\frac{\vert z_{n}\vert ^{2m} }{B(m+1,\nu _n-1)}\\&\quad =\vert z_n\vert ^{-\sum _{k=1}^{n-1}2(p_k+\nu _k)}\frac{\varGamma (\nu _n)}{\varGamma (\nu _n-1)}\sum _{m=0}^{+\infty }\frac{\varGamma (m+\nu _n)}{\varGamma (m+1)\varGamma (\nu _n)}\vert z_n\vert ^{2m}\\&\quad =(\nu _n-1)\vert z_n\vert ^{-\sum _{k=1}^{n-1}2(p_k+\nu _k)}\frac{1}{(1-\vert z_n\vert ^2)^{\nu _n}}. \end{aligned}$$

Thus, we obtain

$$\begin{aligned} K_{\varPhi _n}(z,{{\bar{z}}})&= \frac{1}{\prod _{k=1}^n(\pi \nu _k)}\frac{\nu _n-1}{\vert z_n\vert ^{2\nu _{n-1}}(1-\vert z_n\vert ^2)^{\nu _n}}\sum _{p_1=0}^{+\infty }\frac{\vert z_1\vert ^{2p_1}}{B(p_1+1,\nu _1-1)}\\&\cdots \sum _{p_{n-1}=-\sum _{k=1}^{n-2}(p_k+\nu _k)}^{+\infty }\frac{\vert z_{n-1}\vert ^{2p_{n-1}}\vert z_{n}\vert ^{-2p_{n-1}-\sum _{k=1}^{n-2}2(p_k+\nu _k)}}{B(\sum _{k=1}^{n-1}(p_k+\nu _k)-\nu _{n-1}+1,\nu _{n-1}-1)}. \end{aligned}$$

Similarly, we can see that

$$\begin{aligned}&\sum _{p_{n-1}=-\sum _{k=1}^{n-2}(p_k+\nu _k)}^{+\infty }\frac{\vert z_{n-1}\vert ^{2p_{n-1}}\vert z_{n}\vert ^{-2p_{n-1}-\sum _{k=1}^{n-2}2(p_k+\nu _k)}}{ B(\sum _{k=1}^{n-1}(p_k+\nu _k)-\nu _{n-1}+1,\nu _{n-1}-1)}\\&\quad =\vert z_{n-1}\vert ^{-\sum _{k=1}^{n-2}2(p_k+\nu _k)}\sum _{m=0}^{+\infty }\frac{\vert z_{n-1}/z_n\vert ^{2m}}{B(m+1,\nu _{n-1}-1)}\\&\quad =(\nu _{n-1}-1)\vert z_{n-1}\vert ^{-\sum _{k=1}^{n-2}2(p_k+\nu _k)}\frac{1}{(1-\vert \frac{z_{n-1}}{z_n}\vert ^2)^{\nu _{n-1}}}. \end{aligned}$$

Hence,

$$\begin{aligned} K_{\varPhi _n}(z,{{\bar{z}}})&= \frac{1}{\prod _{k=1}^n(\pi \nu _k)} \frac{(\nu _n-1)(\nu _{n-1}-1)}{\vert z_{n-1}\vert ^{2\nu _{n-2}}(1-\vert z_n\vert ^2)^{\nu _n}(\vert z_n\vert ^2-\vert z_{n-1}\vert ^2)^{\nu _{n-1}}}\\&\times \sum _{p_1=0}^{+\infty }\frac{\vert z_1\vert ^{2p_1}}{B(p_1+1,\nu _1-1)} \cdots \sum _{p_{n-2}}^{+\infty }\frac{\vert z_{n-2} \vert ^{2p_{n-2}}\vert z_{n-1}\vert ^{-2p_{n-2}-\sum _{k=1}^{n-3}2(p_k+\nu _k)}}{B(\sum _{k=1}^{n-2}(p_k+\nu _k)-\nu _{n-2}+1,\nu _{n-2}-1)}. \end{aligned}$$

Therefore, by induction, we conclude that

$$\begin{aligned} K_{\varPhi _n}(z,{{\bar{z}}})=\frac{\nu _n-1}{\pi ^n\nu _n(1-\vert z_n\vert ^2)^{\nu _n}}\prod _{k=1}^{n-1}\frac{\nu _k-1}{\nu _k(\vert z_{k+1}\vert ^2-\vert z_{k}\vert ^2)^{\nu _k}}. \end{aligned}$$

The proof is completed. □

Now, we are able to prove Theorem 1.2.

Proof of Theorem 1.2

By the definition of balanced metric and Theorem 2.5, we see that

$$\begin{aligned} \varepsilon _{(1,g(\nu ))}(z)&=\exp \{-\,\varPhi _n(z)\}K_{\varPhi _n}(z,{{\bar{z}}})\\&=\frac{1}{\pi ^n}\prod _{k=1}^n\frac{\nu _k-1}{\nu _k}. \end{aligned}$$

Thus, the metric \(g(\nu )\) is balanced. On the other hand, now assume that \(g(\nu )\) is balanced. This means that there exists a constant \(C>0\) such that

$$\begin{aligned} K_{\varPhi _n}(z,{{\bar{z}}})&= C\exp \{\varPhi _n(z)\}\\&= C(1-\vert z_n\vert ^2)^{-\nu _n}\prod _{k=1}^{n-1}(\vert z_{k+1}\vert ^2-\vert z_k\vert ^2)^{-\nu _k}. \end{aligned}$$

Notice that by Lemma 2.4, we get

$$\begin{aligned} (\vert z_{k+1}\vert ^2-\vert z_k\vert ^2)^{-\nu _k}=\sum _{p_k=0}^{+\infty } \frac{\varGamma (p_k+\nu _k)}{\varGamma (\nu _k)\varGamma (p_k+1)}\vert z_k\vert ^{2p_k}\vert z_{k+1}\vert ^{-2(\nu _k+p_k)}. \end{aligned}$$

Thus, for any \(p_1\in {\mathbb {N}}\), consider the coefficient of \(\vert z_1\vert ^{2p_1}\) in the series expansion of \(K_{\varPhi _n}(z,{{\bar{z}}})\), and then, one can see that

$$\begin{aligned} {\text {the coefficient of}}\,\vert z_1\vert ^{2p_1}={\widetilde{C}} \sum _{k=2}^n\sum _{p_k=0}^{+\infty }\frac{\varGamma (p_k+\nu _k)}{\varGamma (p_k+1)}\vert z_k\vert ^{2(p_k-p_{k-1}-\nu _{k-1})}. \end{aligned}$$
(2.6)

where \({\widetilde{C}}\) is a constant which is independent of z. Since \(z_1^{p_1}\) belongs to the basis of \(H_{\varPhi _n}(\varOmega _n)\), we can conclude that the right hand of (2.6) must contain a positive constant term. This means that we can find some term in (2.6) which is independent of \(z_k\), for all \(k=2,\ldots ,n\). Thus, for any \(1\le k\le n-1\), there exist \(p_k\) and \(p_{k+1}\) such that

$$\begin{aligned} \nu _k=p_{k+1}-p_k. \end{aligned}$$

Notice that for any \(1\le k\le n\), \(p_k\) is an integer, and thus, \(\nu _1,\ldots ,\nu _{n-1}\) are forced to be integers. Thus, the proof follows by Proposition 2.3. □

In 2016, Edholm [17] introduced a new domain named the generalized Hartogs triangle of exponent \(\gamma >0\) and obtained the closed form of Bergman kernel for this domain with some special \(\gamma \). And then Park [29] extended Edholm’s result to three-dimensional case. The method can even be applied to n-dimensional case as well. Inspired by their work, we state the following open problem:

Problem 2.6

Consider the generalized Hartogs triangle

$$\begin{aligned} {\mathbb {H}}_{p}:=\{z\in {\mathbb {C}}^n :\vert z_1\vert ^{p_1}<\vert z_2\vert ^{p_2}<\cdots<\vert z_n\vert ^{p_n}<1\} \end{aligned}$$

where \(p_1,\ldots ,p_n\) are any positive integers. Can we give some conditions of p to find balanced metrics on \({\mathbb {H}}_{p}\), even for the case \(n=2\)?

3 Berezin quantization of Hartogs triangles

Now, we consider the Berezin quantization on \((\varOmega _n,g(\nu ))\). At first, we give some useful lemma.

Lemma 3.1

(see Lemma 3.2 in [30]) Assume that \(\varOmega \) is a domain in \({\mathbb {C}}^{n}\). Let g be a Kähler metric on \(\varOmega \) associated to the Kähler form \(\omega =\frac{\sqrt{-1}}{2\pi }\partial {\overline{\partial }}\varphi \). Then, the following formula is established

$$\begin{aligned} \varepsilon _{(\alpha \beta , g)}(z)={\beta }^{n}\varepsilon _{(\alpha , \beta g)}(z). \end{aligned}$$

Now, we can give the proof of Theorem 1.5.

Proof of Theorem 1.5

Firstly, we prove that \((\varOmega _n,g(\nu ))\) satisfies condition (I) in Theorem 1.4. In fact, it is easy to see that

$$\begin{aligned}&\exp \{-\,D_{g(\nu )}(z,w)\}\\&\quad =\frac{\prod \limits _{k=1}^{n-1}\vert z_{k+1}\overline{w_{k+1}}-z_{k}\overline{w_{k}}\vert ^{-2\nu _k}}{\prod \limits _{k=1}^{n-1}\bigg ((\vert z_{k+1}\vert ^2-\vert z_k\vert ^2)(\vert w_{k+1}\vert ^2-\vert w_k\vert ^2)\bigg )^{-\nu _k}}\times \frac{\vert 1- z_n\overline{w_n}\vert ^{-2\nu _n}}{\bigg ((1-\vert z_n\vert ^2)(1-\vert w_n\vert ^2)\bigg )^{-\nu _n}}\\&\quad =\frac{\prod \limits _{k=1}^{n-1}\bigg \vert 1-\frac{z_{k}}{z_{k+1}}\overline{\frac{w_{k}}{w_{k+1}}}\bigg \vert ^{-2\nu _k}}{\prod \limits _{k=1}^{n-1}\bigg ((1-\vert \frac{z_k}{z_{k+1}}\vert ^2)(1-\vert \frac{w_k}{w_{k+1}}\vert ^2)\bigg )^{-\nu _k}}\times \frac{\vert 1- z_n\overline{w_n}\vert ^{-2\nu _n}}{\bigg ((1-\vert z_n\vert ^2)(1-\vert w_n\vert ^2)\bigg )^{-\nu _n}}.\\ \end{aligned}$$

By Taylor expansion, we know that

$$\begin{aligned} \bigg (1-\frac{z_{k}}{z_{k+1}}\overline{\frac{w_{k}}{w_{k+1}}}\bigg )^{-\nu _k}=\sum \limits _{\alpha =0} c_{\alpha }(\nu _k)\bigg (\frac{z_{k}}{z_{k+1}}\bigg )^\alpha \bigg (\overline{\frac{w_{k}}{w_{k+1}}}\bigg )^{\alpha }, \end{aligned}$$

where \(c_{\alpha }(\nu _k)\) are the constants depending on \(\alpha \) and \(\nu _k\). By Cauchy–Schwarz inequality, we get

$$\begin{aligned} \left| 1-\frac{z_{k}}{z_{k+1}}\overline{\frac{w_{k}}{w_{k+1}}}\right| ^{-2\nu _k}\le \left( \left( 1-\left| \frac{z_k}{z_{k+1}}\right| ^2\right) \left( 1-\left| \frac{w_k}{w_{k+1}}\right| ^2\right) \right) ^{-\nu _k} \end{aligned}$$
(3.1)

for \(1\le k \le n-1\). Similarly, we also have

$$\begin{aligned} \vert 1- z_n\overline{w_n}\vert ^{-2\nu _n}\le \bigg ((1-\vert z_n\vert ^2)(1-\vert w_n\vert ^2)\bigg )^{-\nu _n}. \end{aligned}$$
(3.2)

Hence, we must have

$$\begin{aligned} \exp \{-\,D_{g(\nu )}(z,w)\}\le 1,\;(z,w)\in \varOmega _n\times \varOmega _n. \end{aligned}$$

Furthermore, by (3.1) and (3.2), we get \(\exp \{-\,D_{g(\nu )}(z,w)\}=1\) if and only if for \(1\le k\le n-1\), we have

$$\begin{aligned} \frac{z_k}{z_{k+1}}=\frac{w_k}{w_{k+1}}\quad {\text {and}}\quad z_n=w_n. \end{aligned}$$

It follows that \(z=w\).

Now, we are in position to check the condition (II) in Theorem 1.4. Let \(E\subset {\mathbb {R}}^{+}\) be a set defined by

$$\begin{aligned} E:=\{\alpha \in {\mathbb {N}}^+;\alpha \nu _k \,{\text {are integers for}}\,1\le k\le n-1, \quad {\text {and}}\quad \alpha \nu _n>1\}. \end{aligned}$$

Since \(\nu _k\,(1\le k\le n-1)\) are positive rational numbers and \(\nu _n>0\), we can learn that \(+\infty \) is in the closure of the subset E. Then, we want to prove that this subset E satisfies condition (II) in Theorem 1.4. Actually, since \(\alpha \in E\), this means that \(\alpha \nu _k\) are integers for all \(1\le k\le n-1\), and \(\alpha \nu _n>1\); thus, by Theorem 1.2, we can conclude that \(\alpha g(\nu )\) is the balanced metric on \(\varOmega _n\), i.e., \(\varepsilon _{(1,\alpha g(\nu ))}(z)\) is a positive constant for all \(\alpha \in E\). Then, by Lemma 3.1, we can obtain that \(\varepsilon _{(\alpha , g(\nu ))}(z)\) is a positive constant for all \(\alpha \in E\). This follows that E satisfies condition (II) in Theorem 1.4. Therefore, we conclude that \((\varOmega _n,g(\nu ))\) admit Berezin quantization by Theorem 1.4. The proof is complete. □