Abstract
We provide a characterization of isometries in the sense of the Carathéodory–Reiffen metric in the symmetrized bidisc.
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1 Introduction
Let \(D\subset {{\mathbb {C}}}^n\) be a domain. We say that a function \(F_D:D\times {{\mathbb {C}}}^n\rightarrow [0,+\infty )\) is a Finsler metric if it is an upper semi-continuous function and
Assume that \(D_1\subset {{\mathbb {C}}}^{n_1}\) and \(D_2\subset {{\mathbb {C}}}^{n_2}\) are domains with Finsler metrics \(F_{D_1}\) and \(F_{D_2}\), respectively. We say that a \(C^1\) mapping \(\phi :D_1\rightarrow D_2\) is an isometry if
where \(d_{\lambda } \phi \) denotes the \({{\mathbb {R}}}\)-differential of \(\phi \) at \(\lambda \). In case \(n_1=1\) the equation (1) is equivalent with
where \({{\mathbb {T}}}=\{t\in {{\mathbb {C}}}: |t|=1\}\) denotes the unit circle.
The paper is motivated by the question whether an isometry is holomorphic or anti-holomorphic. This problem was considered in many papers and, it seems, that in case of invariant metrics (Kobayashi metric, Carathéodory metric, etc) is very difficult (see [2, 8, 10]).
We denote by \({{\mathbb {D}}}=\{t\in {{\mathbb {C}}}: |t|<1\}\) the unit disc. For a domain \(D\subset {{\mathbb {C}}}^n\) we define a biholomorphically invariant Finsler metric as follows
for any \(\lambda \in D\) and any \(X\in {{\mathbb {C}}}^n\). We call \(\gamma _D\) the Carathéodory-Reiffen (pseudo)metric for D (see e.g. [6], Chapter 2). According to the above definition a \(C^1\)-mapping \(\phi :D_1\rightarrow D_2\) is a \(\gamma \)-isometry if
for any \(\lambda \in D_1\) and any \(X\in {{\mathbb {C}}}^{n_1}\).
Put
and \({{\mathbb {G}}}_2=\pi ({{\mathbb {D}}}^2)\). We call \({{\mathbb {G}}}_2\) the symmetrized bidisc (see e.g. [1]). One of the main results of our paper is the following.
Theorem 1
Let \(\phi :{{\mathbb {D}}}\rightarrow {{\mathbb {G}}}_2\) be a mapping of class \(C^1\). Assume that \(\phi \) is a \(\gamma \)-isometry. Then \(\phi \) is holomorphic or anti-holomorphic.
As a corollary, we obtain the following result.
Corollary 2
Let \(F:{{\mathbb {G}}}_2\rightarrow {{\mathbb {G}}}_2\) be a mapping of class \(C^1\) and a \(\gamma \)-isometry. Then F is holomorphic or anti-holomorphic.
2 Proof of Theorem 1
For any \(\lambda \in {{\mathbb {G}}}_2\), there exists an automorphism \(\Phi \) of \({{\mathbb {G}}}_2\) such that \(\Phi (\lambda )=(0,p)\), where \(p\in [0,1)\). Define a set \(\Sigma =\{(2t,t^2): t\in {{\mathbb {D}}}\}\) called the royal set. We have
Recall the following result (see [6], Chapter 7)
where \(\lambda =(s,p)\in {{\mathbb {G}}}_2\) and \(F_{\eta }(s,p)=\frac{2\eta p-s}{2-\eta s}\), \(\eta \in {{\mathbb {T}}}\). Note that
Hence,
In particular, we have
where \(p\in [0,1)\) and \((X,Y)\in {{\mathbb {C}}}^2\).
The main result of this section is the following.
Theorem 3
Let \((X_1,Y_1), (X_2,Y_2)\in {{\mathbb {C}}}^2\) be fixed vectors and let \(p\in (0,1)\) be a fixed number. Assume that there exists a constant \(C>0\) such that
Then \((X_1,Y_1)=0\) or \((X_2,Y_2)=0\).
We assume that \((X_1,Y_1)\not =0\) and \((X_2,Y_2)\not =0\). Consider polynomials
and their duals,
We have \(P_j(z)Q_j(z)=z^2 |P_j(z)|^2\) for any \(z\in {{\mathbb {T}}}\). Note that the equality (3) means that
It is easy to see that
We put
and
We have divided the proof of Theorem 3 into a sequence of lemmas.
Lemma 4
Assume that \(z_0\in E\). Then \(P_1(z_0)P_2(z_0)\not =0\).
Proof
Assume that \(P_1(z_0)=0\) and that \(X_1X_2\not =0\). Then
Put \(Q(z)=z^2|P_2(z_0)|^2-P_2(z)Q_2(z)\). Then Q is a holomorphic polynomial of degree \(\le 4\) such that
Then by the Fejér–Riesz theorem (see [4, 7], see also [1, 3]) there exists a polynomial R of degree \(\le 2\) such that
Hence, \(P_1\) has a double zero at \(z_0\). We get
and, therefore, \(pX_1z_0^2=-X_1\). So, \(X_1=0\). A contradiction. \(\square \)
Lemma 5
We have \(E(\xi )\subset E\) for any \(\xi \in {{\mathbb {T}}}\). Moreover, \(E(\xi _1)\cap E(\xi _2)=\varnothing \) when \(\xi _1\not =\xi _2\).
In particular, the set E is infinite.
Proof
It suffices to note that the equality \(|a+\xi b|=|a|+|b|\) where \(a,b\in {{\mathbb {C}}}{\setminus }\{0\}\) and \(\xi \in {{\mathbb {T}}}\) is equivalent with \(\xi =|ab|/ (b{\overline{a}})\). Now we use the previous Lemma and get that \(z_0\in E\) is such that \(z_0\in E(\xi )\) if and only if
\(\square \)
Lemma 6
Assume that the equation
has at least 9 solutions in \({{\mathbb {T}}}\). Then at least one of the following conditions hold:
-
(1)
\(X_1=X_2=0\);
-
(2)
\(P_j(z)=pX_j(z-\zeta _j)^2\), \(j=1,2\), where \(|X_1|=|X_2|=\frac{1}{2p+2}\), \(\zeta _1=-\zeta _2=\frac{\epsilon i}{\sqrt{p}}\), and \(\epsilon \in \{-1,1\}\).
Proof
From the equality \(|P_1(z)|+|P_2(z)|=1\) we get \(|P_2(z)|^2=(1-|P_1(z)|)^2\) and, therefore,
We have
has at least 9 different solutions. Hence, it holds for any \(z\in {{\mathbb {C}}}\). Using that
we have \(|X_1|=|X_2|\). Moreover, there exists a polynomial \(R_1\) such that \(P_1Q_1=R_1^2\).
Let \(P_1(z)=pX_1(z-\zeta _1)(z-\xi _1)\). Note that \(pX_1\zeta _1\xi _1=-X_1\). Hence, \(X_1=0\) or \(\zeta _1\xi _1=-\frac{1}{p}\). Assume that \(X_1\not =0\). Note that
So, \(P_1Q_1=R_1^2\) if and only if \(\zeta _1=\xi _1\) or \({\bar{\zeta }}_1=1/\xi _1\). Since \(\zeta _1\xi _1=-\frac{1}{p}\), we get \(\zeta _1=\xi _1\) and, therefore, \(\zeta _1^2=-\frac{1}{p}\) and \(P_1(z)=pX_1(z-\zeta _1)^2\). By similar arguments, we get \(P_2(z)=pX_2(z-\zeta _2)^2\). Putting these equalities to (4) we get \(\zeta _2=-\zeta _1\) and \(|X_1|=|X_2|=\frac{1}{2p+2}\). \(\square \)
Proof of Theorem 3
For the proof, we apply Lemma 6 to the polynomials \({\tilde{P}}_j=P_j/(2C(1-p^2))\) with \({\tilde{X}}_j=X_j/(2C(1-p^2))\), \({\tilde{Y}}_j=Y_j/(2C(1-p^2))\), where \(j=1,2\). Then we have \({\tilde{P}}_1(z)=p{\tilde{X}}_1(z-\zeta _0)^2\) and \({\tilde{P}}_2(z)=p{\tilde{X}}_2(z+\zeta _0)^2\), where \(|\zeta _0|=\frac{1}{\sqrt{p}}\) and, \(|{\tilde{X}}_1|=|{\tilde{X}}_2|=\frac{1}{2+2p}\).
There exists \(\xi _0\in {{\mathbb {T}}}\) such that \(\xi _0 {\tilde{X}}_2=-{\tilde{X}}_1\). Then
And, therefore, we have
Hence, we get \(p=1\). A contradiction with the condition \(p\in [0;1)\). \(\square \)
Lemma 7
Let \(f:{{\mathbb {D}}}\rightarrow {{\mathbb {C}}}\) be a \(C^1\) function such that for any \(z\in {{\mathbb {D}}}\) we have \(\frac{\partial f}{\partial z}(z)=0\) or \(\frac{\partial f}{\partial {\bar{z}}}(z)=0\). Then f is holomorphic or anti-holomorphic in \({{\mathbb {D}}}\).
Proof
Put \(h=\frac{\partial f}{\partial z}\). Then h is a continuous function on \({{\mathbb {D}}}\) and on a set \(\{z\in {{\mathbb {D}}}: h(z)\not =0\}\) we have \(\frac{\partial f}{\partial {\bar{z}}}\equiv 0\). Hence, h is holomorphic on \({{\mathbb {D}}}{\setminus } h^{-1}(0)\). By Radó’s theorem, h is holomorphic in \({{\mathbb {D}}}\). If \(\{z\in {{\mathbb {D}}}: h(z)=0\}\) is a discrete, locally finite set, then f is holomorphic in \({{\mathbb {D}}}\). For otherwise it is equal to \({{\mathbb {D}}}\) and, therefore, f is an anti-holomorphic function. \(\square \)
Proof of Theorem 1
It suffices to show that for any \((X_1,Y_1), (X_2,Y_2)\in {{\mathbb {C}}}^2\), any \((s,p)\in {{\mathbb {G}}}_2\), and any constant \(C>0\) such that
we have \((X_1,Y_1)=0\) or \((X_2,Y_2)=0\). There exists an automorphism \(\Phi \) of \({{\mathbb {G}}}_2\) such that \(\Phi (s,p)=(0,{\tilde{p}})\), where \({\tilde{p}}\ge 0\). Then
Since \(\det \Phi '\not =0\), we get \((X_1,Y_1)=0\) or \((X_2,Y_2)=0\). \(\square \)
3 Carathéodory isometries
For a domain \(D\subset {{\mathbb {C}}}^n\), we define another biholomorphically invariant function. For any \(\lambda _1,\lambda _2\in D\) we put
We say that a mapping \(\Phi :D_1\rightarrow D_2\) between domains \(D_1,D_2\) is a Carathéodory isometry if
We say that \(\Phi \) is local c-isometry if for any \(\lambda _0\in D_1\) there exists a neighborhood \(U_0\subset D_1\) of \(\lambda _0\) such that
Note that any c-isometry is a local c-isometry and any local c-isometry is a \(\gamma \)-isometry (see e.g. [10]). In particular, we have
Theorem 8
Let \(\phi :{{\mathbb {D}}}\rightarrow {{\mathbb {G}}}_2\) be a local c-isometry of class \(C^1\). Then \(\phi \) is holomorphic or anti-holomorphic.
4 Applications
In [5] the authors gave a proof of the description of automorphisms of the symmetrized bidisc. One of the main step in the proof is to show that for any automorphism \(\Phi :{{\mathbb {G}}}_2\rightarrow {{\mathbb {G}}}_2\) we have \(\Phi (\Sigma )\subset \Sigma \). We show this property by using our approach.
Corollary 9
Let \(F:{{\mathbb {G}}}_2\rightarrow {{\mathbb {G}}}_2\) be a holomorphic mapping such that F is a \(\gamma \)-isometry at 0. Then \(F(0)\in \Sigma \). In particular, if F is a \(\gamma \)-isometry on \(\Sigma \), then \(F(\Sigma )\subset \Sigma \).
Proof
Assume that \(F(0)=(0,p)\), where \(p\in (0,1)\). Then
where \(a_1=\frac{\partial F_1}{\partial s}(0)\), \(b_1=\frac{\partial F_1}{\partial p}(0)\), \(a_2=\frac{\partial F_2}{\partial s}(0)\), \(b_2=\frac{\partial F_2}{\partial p}(0)\). Take pairs \((X,Y)=(1,\xi )\), where \(\xi \in {{\mathbb {T}}}\). Recall that
In this way we get a contradiction. \(\square \)
Now we can prove a Vigué type result (see e.g. [9]).
Corollary 10
Let \(F:{{\mathbb {G}}}_2\rightarrow {{\mathbb {G}}}_2\) be a holomorphic mapping such that F is a \(\gamma \)-isometry at 0. Then F is an automorphism of \({{\mathbb {G}}}_2\).
Proof
By the above Corollary, we may assume that \(F(0)=0\). Then
where \(a_1=\frac{\partial F_1}{\partial s}(0)\), \(b_1=\frac{\partial F_1}{\partial p}(0)\), \(a_2=\frac{\partial F_2}{\partial s}(0)\), \(b_2=\frac{\partial F_2}{\partial p}(0)\).
By taking \(X=0\) and later \(Y=0\) we get \(\frac{|b_1|}{2}+|b_2|=1\) and \(\frac{|a_1|}{2}+|a_2|=\frac{1}{2}\). So,
We get \(a_1b_1=0\) and \(a_2b_2=0\). Therefore, \(a_1=0\), \(|a_2|=\frac{1}{2}\), \(b_2=0\), \(|b_1|=2\) or \(b_1=0\), \(|b_2|=1\), \(a_2=0\), \(|a_1|=1\). We have \(|\det F'(0)|=1\). From the Cartan theorem we get that F is an automorphism. \(\square \)
Remark 11
The author thanks the anonymous referee for her/his helpful comments that improved the presentation of the results.
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Edigarian, A. Isometries in the symmetrized bidisc. Math. Z. 306, 51 (2024). https://doi.org/10.1007/s00209-024-03449-0
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DOI: https://doi.org/10.1007/s00209-024-03449-0