Abstract
In Ghosh and Zwonek (2-proper holomorphic images of classical Cartan domains. https://doi.org/10.48550/arXiv.2303.11940) introduced a new class of domains \({{\mathbb {L}}}_n\), \(n\ge 1\), which are 2-proper holomorphic images of the Cartan domains of type four. This family contains biholomorphic images of the symmetrized bidisc and the tetrablock. It is well-known, that symmetrized bidisc and tetrablock are Lempert domains. In our paper we show that the whole family of domains \({{\mathbb {L}}}_n\) are Lempert domains.
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1 Introduction
Let X be a complex manifold. In many applications, it is natural to consider on X a distance \(d_X\) related to the complex structure and consider \((X,d_X)\) as a metric space. It turns out that there are many natural ways to introduce these distances. For a good reference on this subject, see e.g. [12]. The largest one is the Lempert function, related to the Kobayashi pseudodistance. We denote by \({{\mathbb {D}}}\) the unit disc and by \({{\mathbb {T}}}\) the unit circle in the complex plane \({{\mathbb {C}}}\). For simplicity, we restrict ourselves to domains in the space \({{\mathbb {C}}}^n\). Let \(D\subset {{\mathbb {C}}}^n\) be a domain and let \(z,w\in D\) be any points. The Lempert function is defined as
where \(\rho \) is the Poincaré distance in the disc (see [12], Chapter III).
On the other hand, the smallest one is the Carathéodory pseudodistance (see [12], Chapter II). We put
It is well-known that for the unit ball, products of the unit balls, and some modifications, these two functions agree, i.e., there is only one natural way to introduce a distance on these domains. More generally, it is true on classical Cartan domains,i.e., bounded symmetric homogeneous domains in \({{\mathbb {C}}}^n\) (see [11]). The proof uses that these domains are homogeneous and the Schwarz lemma type results at the origin.
In 1981 L. Lempert [13] proved a fundamental result that for a convex domain \(D\subset {{\mathbb {C}}}^n\) there is an equality
Later it was extended by L. Lempert to bounded strongly linearly convex pseudoconvex domains (see [14]). We say that a taut domain \(D\subset {{\mathbb {C}}}^n\) is a Lempert domain if the identity \(c_D=\ell _D\) holds.
For more than 20 years it was an open question, whether there are other types of Lempert domains, besides biholomorphic images of convex ones and some simple modifications. In a series of papers [2,3,4] J. Agler and N.J. Young introduced and analysed the symmetrized bidisc, which was the first known bounded hyperconvex domain for which we have the equality of the Carathéodory distance and Lempert function but which cannot be exhausted by domains biholomorphic to convex domains. The symmetrized bidisc \({{\mathbb {G}}}_2\) is the image of the bidisc \({{\mathbb {D}}}^2\) under the symmetrization mapping
In [5] the authors introduced another domain, tetrablock in \({{\mathbb {C}}}^3\), which has the same properties (for the definition and the properties of the tetrablock see below).
Recently G. Ghosh and W. Zwonek (see [10]) proposed a new class of domains, which includes symmetrized bidisc and tetrablock. Our main aim is to show that for domains in this class equality (1) holds, so they are Lempert domains.
Let \(\langle z,w\rangle =\sum _{j=1}^n z_j{\bar{w}}_j\) for \(z,w\in \mathbb {C}^n\) be the Hermitian inner product in \({{\mathbb {C}}}^n\) and let \(\Vert z\Vert ^2=\langle z,z\rangle \) be the corresponding norm. We define another norm in \({{\mathbb {C}}}^nC\), as follows
The unit ball in this norm we denote by \(L_n=\{z\in {{\mathbb {C}}}^n: p(z)<1\}\) and call it the Cartan domain of type four (sometimes it is called the Lie ball of dimension \(n\ge 1\)). It is a bounded symmetric homogeneous domain (see [6, 7]).
Following [10] we define a 2-proper holomorphic mapping
and put \({{\mathbb {L}}}_n=\Lambda _n(L_n)\).
The main result of the paper is the following.
Theorem 1
For any \(z,w\in {{\mathbb {L}}}_n\) we have
As we mentioned above, Theorem 1 was known for \(n=2\) (symmetrized bidisc) and for \(n=3\) (tetrablock, see [8]). So, for any \(n\ge 4\) it provides us with example of non-trivial (i.e., product of domains biholomorphic to convex ones) non-convex domains so that the equality (1) holds. Moreover, our proof shows that these domains, being a modification of the homogeneous symmetric domains, in a sense, are quite natural in relation with the invariant distances and metrics.
2 The Group of Automorphisms
In this part we describe automorphisms of the Cartan domain of type four in \({{\mathbb {C}}}^n\) and give some useful properties. For more information see [9, 11, 15].
For \(z\in {{\mathbb {C}}}^n\) we put
In [1], page 278, the numbers a(z) and b(z) are called the modules related to the Cartan domain of type four. Note that \(p(z)=a(z)+b(z)\) for any \(z\in {{\mathbb {C}}}^n\). For a point \(z\in {{\mathbb {C}}}^n\) we have \(p(z)<1\) if and only if
From this inequality we have.
Lemma 2
Let \(z=(z_1,z')\in {{\mathbb {C}}}\times {{\mathbb {C}}}^{n-1}\). Then \(z\in {{\mathbb {L}}}_n\) if and only if
Note that if \((z_1,\dots ,z_{n-1})\in L_{n-1}\) then \((z_1,\dots ,z_{n-1},0)\in L_{n}\). Let us show the following simple inequality.
Lemma 3
Let \(z=(z_1,z')\in {{\mathbb {C}}}\times {{\mathbb {C}}}^{n-1}\). Assume that \(\Vert z\Vert <1\). Then
Moreover, the equality holds if and only if \(z_1=0\).
Proof
We have to show \(|\langle z,{\bar{z}}\rangle |^2- |\langle z',\bar{z}'\rangle |^2\le 2|z_1|^2\). This is equivalent with \( |z_1|^4+2\Re \big ({\bar{z}}_1^2\langle z',{\bar{z}}')\big )\le 2|z_1|^2\). It suffices to show \(|z_1|^4+2|z_1|^2|\langle z',{\bar{z}}'\rangle |\le 2|z_1|^2\). But we have \(|z_1|^2+2|\langle z',{\bar{z}}'\rangle |\le |z_1|^2+2\Vert z'\Vert ^2<2\). \(\square \)
Assume that \(z=(z_1,\dots ,z_n)\in L_n\). Then from the above Lemma we have \((z_1,\dots ,z_{n-1})\in L_{n-1}\).
We denote \({{\,\textrm{SO}\,}}_n({{\mathbb {R}}})\) as the set of real matrices A satisfying \(A^T A = A A^T = {{\,\textrm{I}\,}}_n\) and \(\det A = 1\). Throughout our considerations, this set plays a crucial role, owing to its significant properties: \(\langle Az, Aw \rangle = \langle z, w \rangle \) and \(\langle Az, \overline{Aw} \rangle = \langle z, \bar{w} \rangle \) for any \(z, w \in {{\mathbb {C}}}^n\) and any \(A \in {{\,\textrm{SO}\,}}_n({{\mathbb {R}}})\). Notably, \(p(Az) = p(z)\) holds for any \(z \in {{\mathbb {C}}}^n\) and any \(A \in {{\,\textrm{SO}\,}}_n({{\mathbb {R}}})\).
Let \(z\in {{\mathbb {C}}}^n\). Then there exists an \(\eta \in {{\mathbb {T}}}\) such that \(\eta ^2\langle z,{\bar{z}}\rangle =|\langle z,{\bar{z}}\rangle |\). It is worth noting that \(\eta ^2\langle z,{\bar{z}}\rangle =\langle \eta z,\overline{\eta z}\rangle \). If \(\eta z=u+iv\), where \(u,v\in {{\mathbb {R}}}^n\), then the vectors u and v are orthogonal in \({{\mathbb {R}}}^n\) and \(\Vert u\Vert \ge \Vert v\Vert \). Simple calculations reveal that \(\Vert u\Vert =a(z)\) and \(\Vert v\Vert =b(z)\). Thus, there exists a matrix \(A\in {{\,\textrm{SO}\,}}_n({{\mathbb {R}}})\) (a ’rotation’ of \({{\mathbb {R}}}^n\)) such that \(Au=(a(z),0,\dots 0)\) and \(Av=(0,b(z),\dots ,0)\). Consequently, \(\eta Az=A(\eta z)=(a(z),b(z)i,\dots ,0)\). We formulate the above considerations as a separate result.
Lemma 4
For any \(z\in {{\mathbb {C}}}^n\), \(n\ge 2\), there exists a matrix \(A\in {{\,\textrm{SO}\,}}_n({{\mathbb {R}}})\) and a number \(\eta \in {{\mathbb {T}}}\) such that
Fixing first k-coordinates and "rotating" last \(n-k\) coordinates we have
Lemma 5
For any \(z\in {{\mathbb {C}}}^n\), \(n\ge 3\), and any \(k\in {{\mathbb {N}}}\) with \(k\le n\), there exists a matrix \(A\in {{\,\textrm{SO}\,}}_n({{\mathbb {R}}})\) and a number \(\eta \in {{\mathbb {T}}}\) such that \(Aw\in \{(w_1,\dots ,w_k)\}\times {{\mathbb {C}}}^{n-k}\) for any \(w=(w_1,\dots ,w_n)\in {{\mathbb {C}}}^n\) and
where \(z'=(z_{k+1},\dots ,z_n)\in {{\mathbb {C}}}^{n-k}\).
Note that for any \(a,b\in {{\mathbb {R}}}\) we have \((a,bi,0,\dots ,0)\in L_n\) if and only if \(|a|+|b|<1\).
Recall the following well-known result.
Theorem 6
Let \(\Phi :L_n\rightarrow L_n\) be a biholomorphic mapping such that \(\Phi (0)=0\). Then there exist a matrix \(A\in {{\,\textrm{SO}\,}}_n({{\mathbb {R}}})\) and a number \(\eta \in {{\mathbb {T}}}\) such that \(\Phi (z)=\eta Az\) for any \(z\in L_n\).
Automorphisms of \(L_n\) described in the above Theorem are called linear automorphisms. Now we are going to describe all automorphisms of the irreducible classical Cartan domain of type four (see [9, 11, 15]). We define first a group of matrices.
For any \(g\in G\) we define the following \({{\mathbb {C}}}^n\)-valued holomorphic function
where \(W(z)=\begin{bmatrix} \frac{1}{2}(\langle z,{\bar{z}}\rangle +1) \\ \frac{i}{2}(\langle z,z\rangle -1) \end{bmatrix}\). One can show (see [9]) that for any \(g\in G\) we have
Thus, \(\Psi _g\) is well-defined on \(\overline{L_n}\). Note that we have a homomorphism of groups \(\kappa :G(n)\rightarrow G(n+1)\) defined by
where \(g=\begin{bmatrix} A &{} B\\ C &{} D \end{bmatrix}\). As a simple corollary we get.
Corollary 7
For any \(\Psi \in {{\,\textrm{Aut}\,}}(L_n)\) there exists a \({\tilde{\Psi }}\in {{\,\textrm{Aut}\,}}(L_{n+1})\) such that \({\tilde{\Psi }}(0,z)=(0,\Psi (z))\) for any \(z\in L_n\).
Proof
Assume that \(\Psi =\Psi _g\) for some \(g\in G(n)\). Put \({\tilde{\Psi }}=\Psi _{\kappa (g)}\). \(\square \)
Remark 8
By permuting the coordinates we may take \({\tilde{\Psi }}(z,0)=(\Psi (z),0)\).
In [10] the authors gave a description of automorphisms of the domain \({{\mathbb {L}}}_n\). Using this description and results above we get.
Theorem 9
Let \(\Phi \in {{\,\textrm{Aut}\,}}({{\mathbb {L}}}_{n})\). Then there exists a \(g\in G(n-1)\) such that
Moreover, for any \(z\in L_{n-1}\) we have \(\Phi (0,z)=(0,\Psi _g(z))\).
From Remark 8 we get (cf. [10], Theorem 5.3).
Theorem 10
Let \(\Phi \in {{\,\textrm{Aut}\,}}({{\mathbb {L}}}_n)\). Then there exists a \({\tilde{\Phi }}\in {{\,\textrm{Aut}\,}}({{\mathbb {L}}}_{n+1})\) such that \({\tilde{\Phi }}(z,0)=(\Phi (z),0)\) for any \(z\in {{\mathbb {L}}}_n\).
From the definition and similar properties for \(L_n\) we have.
Lemma 11
Let \(\Pi _n:{{\mathbb {L}}}_n\ni (z_1,\dots ,z_n)\rightarrow (z_1,\dots ,z_{n-1})\in {{\mathbb {L}}}_{n-1}\) and \(Q_n:{{\mathbb {L}}}_{n-1}\ni (z_1,\dots ,z_{n-1})\rightarrow (z_1,\dots ,z_{n-1},0)\in {{\mathbb {L}}}_{n}\). Both \(\Pi _n\) and \(Q_n\) are well-defined holomorphic mappings. Furthermore, it holds that \(\Pi _n\circ Q_n={{\,\textrm{id}\,}}_{{{\mathbb {L}}}_{n-1}}\).
Corollary 12
Let \(n\ge 4\). Take \({\tilde{\Pi }}_n=\Pi _4\circ \Pi _{5}\circ \dots \circ \Pi _{n}\) and \({\tilde{Q}}_n=Q_{n}\circ Q_{n-1}\circ \dots \circ Q_4\). Then \({\tilde{\Pi }}_n:{{\mathbb {L}}}_n\rightarrow {{\mathbb {L}}}_3\) and \({\tilde{Q}}_n:{{\mathbb {L}}}_3\rightarrow {{\mathbb {L}}}_n\). Moreover, \({\tilde{\Pi }}_n\circ {\tilde{Q}}_n={{\,\textrm{id}\,}}_{{{\mathbb {L}}}_3}\). In particular, for any \(z,w\in {\tilde{Q}}_n({{\mathbb {L}}}_3)\) we have
3 The Properties of the Tetrablock and of the Domain \({{\mathbb {L}}}_3\)
In our paper properties of the tetrablock and a domain \({{\mathbb {L}}}_3\) are crucial. First recall the following definition (see [5], Definition 1.1).
Definition 13
The tetrablock is the domain
In [5] the authors gave several equivalent characterizations of the tetrablock. One of them is the following (compare with [5], Theorem 2.2).
Proposition 14
For \(z\in {{\mathbb {C}}}^3\) we have: \(z\in {{\mathbb {E}}}\) if and only if
From this Proposition, definition of the Cartan domain, and the domain \({{\mathbb {L}}}_n\) we have the following biholomorphism (see [10], Corollary 3.5).
Lemma 15
Put
Then \(\Psi :{{\mathbb {L}}}_3\rightarrow {{\mathbb {E}}}\) is a biholomorphic mapping such that \(\Psi (r,0,0)=(0,0,r)\) for any \(r\in [0,1)\).
Recall the following result (see [16], Theorem 5.2).
Theorem 16
Let \(z\in {{\mathbb {E}}}\) be any point. Then there exist an automorphism \(\Phi \) of \({{\mathbb {E}}}\) and a number \(r\in [0,1)\) such that \(\Phi (z)=(0,0,r)\).
From Theorem 16 and Lemma 15 we get.
Corollary 17
For any \(z\in {{\mathbb {L}}}_3\) there exists an automorphism \(\Phi \) of \({{\mathbb {L}}}_3\) such that \(\Phi (z)=(r,0,0)\), where \(r\in [0,1)\).
Proof
Fix \(z\in {{\mathbb {L}}}_3\). Then \(\Psi (z)\in {{\mathbb {E}}}\). By Theorem 16 there exist an automorphism \({\tilde{\Phi }}\) of \({{\mathbb {E}}}\) and a number \(r\in [0,1)\) such that \({\tilde{\Phi }}(\Psi (z))=(0,0,r)\). It suffice to put \(\Phi =\Psi ^{-1}\circ {\tilde{\Phi }}\circ \Psi \). \(\square \)
From this we get the following important result.
Corollary 18
For any \(z\in {{\mathbb {L}}}_n\) there exists an automorphism \(\Phi \) of \({{\mathbb {L}}}_n\) such that \(\Phi (z)=(\rho ,0,\dots ,0)\), where \(\rho \in [0;1)\).
All these analyses and remarks imply the following crucial result.
Theorem 19
For any \(z,w\in {{\mathbb {L}}}_n\) there exists an automorphism \(\Phi \) of \({{\mathbb {L}}}_n\) such that \(\Phi (z)=(\rho ,0,\dots ,0)\) and \(\Phi (w)\in {{\mathbb {L}}}_3\times \{0\}_{n-3}\).
Proof of Theorem 1
The proof follows from Theorem 19 and Corollary 12. \(\square \)
4 Invariant Distences and Metrics in \({{\mathbb {L}}}_n\)
Let \(D\subset {{\mathbb {C}}}^n\) be a domain. For a point \(z\in D\) and a vector \(X\in {{\mathbb {C}}}^n\), we recall that the infinitesimal Kobayashi metric at z in the direction X is defined to be
We show the following.
Theorem 20
Let \(n\ge 3\).
where \(X=(X_1,X')\in {{\mathbb {C}}}\times {{\mathbb {C}}}^{n-1}\).
Proof
We know that (see [16], Theorem 2.1)
From the biholomorphicity between \({{\mathbb {E}}}\) and \({{\mathbb {L}}}_3\) we infer
For the general case, by Lemma 4 there exists an \(\eta \in {{\mathbb {T}}}\) and a matrix \(A\in {{\,\textrm{SO}\,}}_{n-1}({{\mathbb {R}}})\) such that
Take an automorphism \(\Phi _A\) of \(L_{n-1}\) defined as \(\Phi _A(z)=Az\). By Theorem 5.3 in [10] there exists an automorphism \(\Phi \) of \({{\mathbb {L}}}_n\) induced by \(\Phi _A\). Moreover, \(\Phi (0)=0\) and
Then
and, therefore,
\(\square \)
Recall also the following result (see [5], Corollary 3.7).
Theorem 21
For any \(z=(z_1,z_2,z_3)\in {{\mathbb {E}}}\) with \(|z_1|\le |z_2|\), we have
As a corollary of this we get.
Corollary 22
Let \(z=(z_1,z')\in {{\mathbb {L}}}_n\), where \(z_1\in {{\mathbb {C}}}\) and \(z'\in {{\mathbb {C}}}^{n-1}\). If \(z'=0\) then \(c_{{{\mathbb {L}}}_n}(0;z)=\tanh ^{-1}(|z_1|)\). If \(z'\not =0\) then we have
Proof
In a similar way as above, if \(|z_2+iz_3|\le |z_2-iz_3|\) then
If \(z\in {{\mathbb {L}}}_n\), then using appropriate automorphism, we may assume that \(z=(z_1,{\bar{\eta }} a(z'),{\bar{\eta }} b(z')i,\dots ,0)\) and, therefore,
where \(\eta \in {{\mathbb {T}}}\) is such that \(\eta ^2\langle z',{\bar{z}}'\rangle = |\langle z',{\bar{z}}'\rangle |\). From this we obtain the formula. \(\square \)
Remark 23
The author thanks the anonymous referee for her/his helpful comments that improved the presentation of the results.
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Edigarian, A. A Family of Lempert Domains. J Geom Anal 34, 176 (2024). https://doi.org/10.1007/s12220-024-01640-7
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DOI: https://doi.org/10.1007/s12220-024-01640-7