A Kobayashi and Bergman complete domain without bounded representations

We construct an unbounded strictly pseudoconvex Kobayashi hyperbolic and complete domain in C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}^2$$\end{document}, which also possesses complete Bergman metric, but has no nonconstant bounded holomorphic functions.

A question of existence of the (complete) Bergman metric on a given complex manifold X is one of the important problems of complex analysis. When X = is a bounded domain in ℂ n , it is well known that the Bergman metric on exists. If is further assumed to be bounded pseudoconvex with C 1 -smooth boundary, then the Bergman completeness of was proved by Ohsawa in [25] using the celebrated Kobayashi's criterion [23]. (Here the Bergman completeness means that is a complete metric space with respect to the Bergman metric.) Later on, several articles appeared concerning the Bergman completeness for bounded pseudoconvex domains in ℂ n (see [10,30] and references therein). Among them, it is worth to mention the papers of Blocki-Pflug [5] and Herbort [21] were independently a long-standing question of Bergman completeness for any bounded hyperconvex domain has been solved. (By a hyperconvex domain we mean here that it has a bounded continuous plurisubharmonic exhaustion function.) A few years later, this result was further generalized by Chen [6] to the case of hyperconvex manifolds.
When X is an unbounded domain, or, more generally, a complex manifold, then not so much is known for the existence of the (complete) Bergman metric except the early work of Greene-Wu [17], which asserts that a simply connected complex manifold possesses a complete Bergman metric if it carries a complete Kähler metric with holomorphic sectional curvature which is negatively pinched. It is a bit surprising that such lower negative bound was dropped by Chen and Zhang in [9], where the main ingredient used in the proof is the pluricomplex Green function (see Sect. 4 for the definition) and the L 2 -method for the ̄equation. In particular, they proved that a Stein manifold X possesses a Bergman metric, provided that X carries a bounded continuous strictly plurisubharmonic function. Recently, a new characterization for the existence of the Bergman metric on unbounded domains has been given by Gallagher et al. [13]. It was proved there that a pseudoconvex domain with empty core (see Sect. 3 for the definition) possesses a Bergman metric.
Some other conditions (for certain unbounded X) which are sufficient for possessing a (complete) Bergman metric are also scattered in the literature, see for examples [2,8,26,28] et al.
A complex manifold X is called (Kobayashi) hyperbolic if its Kobayashi pseudodistance X is a distance (see Sect. 5 for definitions). We say that X is complete hyperbolic, if (X, X ) is a complete metric space. For example, all bounded domains in ℂ n are hyperbolic, while complex manifolds containing entire curves are not. The question of hyperbolicity has been intensively studied in the case of compact complex manifolds. Nevertheless, there are many interesting and quite long-standing conjectures in that context which are still open in their complete form, such as the Kobayashi conjecture and Green-Griffiths-Lang conjecture [12].
In the case of open complex manifolds, the questions related to the Kobayashi hyperbolicity are rather different. There are various characterizations of the hyperbolicity. For examples, an analytic description due to Sibony [29] says that a complex manifold X carrying a bounded strictly plurisubharmonic function is Kobayashi hyperbolic. In [1], Abate proved that a complex manifold X is Kobayashi hyperbolic if and only if the space of holomorphic maps from the unit disc Δ to X is relatively compact (with respect to the compact-open topology) in the space of continuous maps from Δ into the one point compactification X * of X. Later, Gaussier [14] gave sufficient conditions for hyperbolicity of an unbounded domain in terms of the existence of peak and antipeak functions at infinity. In [24], Nikolov and Pflug found some conditions at infinity which guarantee the hyperbolicity of unbounded domains. They also obtained a characterization of hyperbolicity in terms of asymptotic behavior of the Lempert function. Recently, Gaussier and Shcherbina [16] have given a new sufficient condition for Kobayashi hyperbolicity of unbounded domains in ℂ n using a concept of strong antipeak plurisubharmonic function at infinity. For more information on the progress in this direction (conditions which characterize (complete) Kobayashi hyperbolic (open) manifolds), we refer the reader to the survey paper of Gaussier [15].
In the present paper, in contrast to the situation treated by Chen in [6] and Sibony in [29] (when the existence of a bounded continuous strictly plurisubharmonic function was assumed), we consider both the Bergman completeness and the Kobayashi hyperbolicity for unbounded domains in ℂ n which have neither bounded smooth strictly plurisubharmonic functions nor nonconstant bounded holomorphic functions. More precisely, we prove here the following result.
Main Theorem There exists an unbounded strictly pseudoconvex domain ⊂ ℂ 2 with smooth boundary which has the following properties: is not Carathéodory hyperbolic. 5. The core ( ) of is nonempty, but the core � ( ) is empty.
The construction of is motivated by [16], where a Kobayashi hyperbolic model domain with a nonempty core has been constructed. Using a characterization of the Bergman space in terms of the core (see [13,Remark 7(b)]), we show the existence of the Bergman metric on . For the completeness of this metric, we apply a criterion given by Chen [

Construction of a special Wermer type set in ℂ 2
Let {a n } be the enumeration of points running through the set ℤ + iℤ ⊂ ℂ z as follows: Let { n } be a decreasing sequence of positive numbers converging to zero very fast that will be further specified later. Then, as in [18], for each n ∈ ℕ , we consider the set By definition, ∑ n k=1 k √ z − a k is a multi-valued function that takes 2 n values at each point z ∈ ℂ (counted with multiplicities). Therefore, for each z ∈ ℂ⧵(ℤ + iℤ) we can locally choose single-valued functions w (n) For every n ∈ ℕ , we define a function P n ∶ ℂ 2 → ℂ as Then each P n is a well-defined holomorphic polynomial in z and w (see for details [18]). Moreover, provided that { n } is decreasing to zero fast enough, the sets E n = {P n = 0} converge to a nonempty unbounded connected closed set E ⊂ ℂ 2 , where the convergence

3
is understood with respect to the Hausdorff metric on each compact subset of ℂ 2 . More precisely,

Construction of the domain
Consider first a plurisubharmonic function defined on ℂ 2 by where is the convex function constructed in the Section 2.2 of [16], and observe, that for each t > 0 the domain coincides with the domain F d of the inclusion (3.2) in [16] for d = 1 2 e t . The structure of the domains F d , which were systematically studied in [16], will be essential for proving in Sect. 5 the Kobayashi completeness of our domain . The crucial technical tool for this proof is the following property established in [16]: Property (F) For each d > 0 there exists r 0 ∶= r 0 (d) > 0 such that the domain F d contains no holomorphic disks of radius r > r 0 (the last part of the statement means, more precisely, that for every Let us now pick one of these domains, for example U −1 , and denote it (to simplify our notations) by U, then U will be a neighborhood of the Wermer type set E in ℂ 2 which is defined by is a function with the following properties: 1. ̃ is smooth strictly increasing and convex, Here = (z, w) ∈ ℂ 2 and ‖ ‖ 2 ∶= �z� 2 + �w� 2 is the Euclidean norm. Observe that, in view of strict monotonicity and convexity of the functions t → − log(−t) and ̃ , the function ̃ is strictly plurisubharmonic on U. Then, since ̃> 0 on U , and since ̃= −∞ on the set E , we conclude that, maybe after a small perturbation of the function ̃ , the domain will be a smoothly bounded strongly pseudoconvex neighborhood of E such that ⊂ U.
Note that, by Theorem 6.1 below, we know that any bounded from above continuous plurisubharmonic function u in the neighborhood of E is constant on E . Therefore, the core ( ) of contains E . Observe also that, for each constant c > u | |E , the sublevel set { ∈ ∶ u( ) ⩽ c} is not relatively compact in . This shows that the domain is not hyperconvex. Now we will study the existence of the Bergman metric as well as the Bergman completeness of . Note first that, since < −1 on U, the function − log(− ) is well defined and, moreover, it is plurisubharmonic, since is plurisubharmonic. A straightforward calculation yields Thus one can make ī̃ arbitrary large whenever ‖ ‖ 2 is large and ̃ grows sufficiently fast. Moreover, we can also force the volume of to be finite if ̃ goes to +∞ fast enough. This insures that the holomorphic polynomials are L 2 -integrable. In particular, the Bergman kernel of is non-degenerated.

Existence of the Bergman metric
A notion of the core ( ) of a domain ⊂ ℂ n (or, more general, of a domain in a complex manifold M ) was introduced and intensively studied in [19,20,27]. It can be defined as follows. Similar definition can also be given for plurisubharmonic functions from different smoothness classes.
Later on in [13] a slightly larger class of functions defined on a given domain ⊂ ℂ n , was considered: ( ) ∶= ∈ ∶ every smooth plurisubharmonic function on that is bounded from above fails to be strictly plurisubharmonic in where PSH( ) denotes the family of plurisubharmonic functions in and ( , 0 ) denotes the Lelong number of at 0 , i.e.
In order to formulate a general sufficient condition for the infinite dimensionality of the Bergman space of , they introduced the following notion of the core � ( ) of (which is a slight modification of the notion of the core ( ) given above).  If we now consider the domain defined in (4) and observe that the function ̃ defined by (3) is negative strictly plurisubharmonic on this domain and has zero Lelong numbers (this easily follows from the fact that h(t) ∶= −(1∕t) log(−t) is an increasing convex function in t and the fact that lim t→−∞ h(t) = 0 , see for details [13, Remark 7(c)]), then, applying Theorem 3.1 to the domain on the place of , we conclude that the Bergman metric on exists.

Remark 1 Since the restriction of ī to each vertical line
is spread over the Cantor set E z 0 ∶= E ∩ ℂ z 0 ,w , one can prove that the Lelong numbers of the strictly plurisubharmonic function +̃(‖ ‖ 2 ) + C‖ ‖ 2 , C > 0 , will also be identically equal to zero. Hence, we can use this function instead of the function ̃ in the definition of the domain and in all the arguments later on. But the proof of the fact that the Lelong numbers of ̃ are equal to zero is more elementary, that is why we have decided to use it here.

Completeness of the Bergman metric
The main argument which we will use to prove Bergman completeness of the domain (see the Case 2 below) follows closely the arguments presented in the proof of Theorem 1.2 in [7]. For the reader's convenience, we give them here in detail.
First we recall the notion of the pluricomplex Green function on an open set ⊂ ℂ n with logarithmic pole at 0 ∈ : where the supremum is taken over all negative plurisubharmonic functions u on such above fails to be strictly plurisubharmonic at }.
For each a > 0 , set The following criterion for the Bergman completeness is given in [ Using this criterion we will now prove that Bergman metric on is complete. Indeed, let { k } be an infinite sequence of points in without accumulation points in . Then we have two possibilities: Case 1 { k } admits an accumulation point on .
In this case, in view of strict pseudoconvexity of , a standard localization argument shows that the Bergman metric is complete. This is definitely known for experts, but for completeness of the presentation we include the sketch of the proof here.
Let p ∈ be an accumulation point of { k } and let U be a neighborhood of p. Since is strictly pseudoconvex and, hence, locally has a strictly plurisubharmonic defining function, there is a neighborhood V ⋐ U of the point p and a constant a > 0 , such that for every point 0 ∈ V , one has A ( 0 , a) ⊂ U ∩ . We fix an arbitrary point 0 ∈ V and consider a function f ∈ O(U ∩ ) which has the properties f ( 0 ) = 0 and ‖f ‖ 2 L 2 (U∩ ) = 1 . Then we set where g is the pluricomplex Green function of with a pole at 0 (written in what follows as g for simplicity) and ∶ (−∞, +∞) → [0, 1] is a smooth cut-off function such that (t) ≡ 1 for t ⩽ −2a and (t) ≡ 0 for t ⩾ −a . Observe that is a closed (0, 1)-form which can be extended by 0 to the whole of .
Next we consider the functions and Since g is a negative plurisubharmonic function on , a direct calculation (see also the estimate (5) above with ̃≡ 0 ) easily shows that Since, by (6), =̄( •g) ⋅ f = � •g ⋅̄g ⋅ f , it follows from (9) that where H ∶= | � •g| 2 ⋅ g 2 ⋅ |f | 2 . Then, by the celebrated Donnelly-Fefferman's existence theorem (see for example [4, Theorem 3.1]), and in view of (7) and (10), there exists u ∈ L 2 loc ( ) such that ̄u = and the following estimate holds From the definition of H and the fact that ‖f ‖ 2 L 2 (U∩ ) = 1 , we can conclude by (11) that where C(a) and C (a) are some constants that depend on a only (for , U and V being fixed). It follows now from the fact that g( , 0 ) − log | − 0 | is locally bounded and, hence, in view of (8), from the fact that the integrability of |u| 2 e − = |u| 2 e −(2n+2)g is the same as the integrability of |u| 2 | − 0 | −(2n+2) , that u( 0 ) = 0 and u ( 0 ) = 0 . Therefore, if we set we will get a function F ∈ O( ) such that By the estimate (11), we have that Recall that the Bergman metric is defined by where X is a nonzero tangent vector at . If we assume that the function f achieves the above supremum on U ∩ , we will get by (12) a function F ∈ O( ) such that, in view of (13), for any ∈ V ⋐ U the following estimate holds We can conclude now from the trivial inequality K ( ) ⩽ K U∩ ( ) that for all ∈ V . This completes the proof of the localization property for the Bergman metric and shows the completeness of the Bergman metric at the finite points of .
Consider a smooth cut-off function on ℂ 2 such that ( ) = 1 , when ‖ ‖ ⩽ 1 2 and ( ) = 0 , when ‖ ‖ ≥ 1, and then for every > 0 define the auxiliary function Notice that there exists a constant C 1 > 0 such that Let K be any compact subset in . Observe that for each > 0 there exists a positive integer k such that for every k > k one has that B( k ) ∩ K = � (here B( k ) denotes the ball of radius 1 with center at k ) and, moreover, that the function u ,k is a negative and plurisubharmonic on . We can insure plurisubharmonicity of u ,k ( ) for large enough k in the following way: • On the set ⧵B( k ) plurisubharmonicity of u ,k ( ) is clear, since on this set ( − k ) ≡ 0.

• On the set B( k ) one has
This, in view of our assumptions that lim t→∞̃� (t) = ∞ and that k → ∞ as k → ∞ , implies plurisubharmonicity of u ,k ( ) on B( k ) for all large enough k . Observe now that, since u ,k is a negative plurisubharmonic function with (at least) a logarithmic pole at k , it is included into the family of functions which defines the pluricomplex Green function g ( , k ) . By the definition of the pluricomplex Green function, for large enough k we have and, hence, also Since for every > 0 there exists > 0 so small that the volume of the set { ∈ K ∶̃< −a∕ } is less than , we conclude by the above arguments, that there exists k such that for all k > k . By Theorem 4.1, the Bergman metric is complete.

Kobayashi completeness
Let be an arbitrary domain in ℂ n . Recall that the Kobayashi pseudometric at a point ( , v) ∈ × ℂ n is defined as where Δ ∶= {z ∈ ℂ ∶ |z| < 1} is the unit disc in ℂ and O(Δ, ) denotes the set of all holomorphic maps from Δ to .
For any two point 1 , 2 ∈ , the Kobayashi pseudodistance is defined by where the infimum is taken over all piecewise C 1 curve ∶ [0, 1] → connecting 1 and 2 .
The domain is called Kobayashi hyperbolic if is a distance. is said to be Kobayashi complete (abbr. -complete) if any -Cauchy sequence { j } j∈ℕ converges to a point 0 ∈ with respect to the Euclidean topology of .
When is a bounded domain in ℂ n , it is well known that is -complete if and only if is locally -complete, i.e., for any boundary point a ∈ , there exists a bounded neighborhood U of a such that every connected component of ∩ U is -complete (see, for example, [22,Theorem 7.5.5]).
For unbounded domains which are (possibly) not biholomorphic to bounded domains in ℂ n , Nikolov and Pflug gave a criterion of the -completeness, by introducing the following notions of -points and ′ -points.

Definition 5.1 A point a ∈
is called a -point for if lim →a ( , ) = ∞ for any fixed ∈ .

Definition 5.2 A point a ∈
is called a ′ -point for if there is no -Cauchy sequence converging to a.
It is clear that any -point is a ′ -point. Nikolov and Pflug (see [24,Proposition 3.6]) have proved the following theorem for the -completeness.

Theorem 5.1 Let be an open set in
Then the following conditions are equivalent:

1.
is -complete. 2. Any finite boundary point of admits a neighborhood U such that ∩ U is -complete. 3. Any finite boundary point of is a ′ -point. 4. Any boundary point of is a -point.
We will now use the above theorem to prove that the domain is -complete. Observe first that, by our definition of the domain (which is given in Sect. 2), one has that with d = 1 2 e −1 . By Property ( F ) stated in Remark 3 of [16] and restated in Sect. 2 above, we know that the domain U −1 = F 1 2 e −1 , and hence also a smaller domain , contains only discs of uniformly bounded size, say r 0 . This implies that is Kobayashi hyperbolic. Since the domain is strictly pseudoconvex at each finite boundary point, it follows that is -complete at these points (see, for example, the arguments of Lemma 2.1.1 and Lemma 2.1.3 in [14]). Hence, for proving that is -complete we only need to check that ∞ is a ′ -point of . Now suppose that { j } is a -Cauchy sequence converging to ∞ . Then, by the definitions (14), (15) of the Kobayashi pseudodistance, and in view of the boundedness by r 0 of the size of holomorphic disks contained in , we see that for all k, l ∈ ℕ , where d E ( , ) denotes the Euclidean distance between the points , ∈ ℂ 2 . The last inequality obviously implies that { j } is also a Cauchy sequence with respect to the Euclidean distance, and hence cannot converge to ∞ . This shows that ∞ is a ′ -point of , and, therefore, proves the Kobayashi completeness of .

Nonexistence of nonconstant bounded holomorphic functions
In this section, we will show that the only bounded holomorphic functions defined in are constants. For proving this, we will need the following version of the Liouville-type theorem which was proved in [20, Theorem 2.2] for a slightly different Wermer type set. The argument provided there can also be applied to the set E considered in our paper. But, since for this set there is a bit simpler proof, we will present it here for the reader convenience.

Theorem 6.1 Let be a continuous plurisubharmonic function defined on an open
neighborhood V ⊂ ℂ 2 of the Wermer type set E. If is bounded from above, then ≡ C on E for some constant C ∈ ℝ.
Proof By construction of the Wermer type set (see Sect. 1), E is locally the limit in the Hausdorff distance of analytic sets E n and, therefore, the complement of E is pseudoconvex. Due to a theorem of Slodkowski, we know that E is an analytic multivalued function (see the definition of analytic multivalued functions and Slodkowski's theorem in [3, pages 15-16]). Now, for a given set F ⊂ ℂ 2 z,w and any point z 0 ∈ ℂ z , let us define the vertical slice F(z 0 ) of the set F by Then for any function , which is continuous and plurisubharmonic in a neighborhood of the set E , we can define the function It follows from part (ii) of the definition of analytic multivalued functions that ̃ is a subharmonic function in the complex plane ℂ z . If the function is further assumed to be bounded from above (without loss of generality we may assume that on the set E one has ⩽ 0 and sup = 0 ), the standard Liouville theorem asserts that ̃≡ 0 on ℂ z . The rest of the proof will be devoted to showing that even for the initial function one also has that ≡ 0 . In order to do this, we first define the set where z ∶ ℂ 2 z,w → ℂ z is the canonical projection. Then we observe that for every point p = (p 1 , p 2 ) ∈ A and every analytic disc D r (p) ∶= {(z, f (z)) ∶ z ∈ Δ r (p 1 )} ⊂ E around p such that Δ r (p 1 ) ⊂ ℂ z ⧵{ℤ + iℤ} , p = (p 1 , f (p 1 )) and f ∶ Δ r (p 1 ) → ℂ w holomorphic, the function (z, f (z)) is subharmonic on Δ r (p 1 ) ⊂ ℂ z . Here Δ r (p 1 ) denotes the disc of radius r centered at p 1 . Since | E ⩽ 0 and (p) = 0 , we conclude from the maximum principle that (z, f (z)) is identically equal to zero on Δ r (p 1 ) and, therefore, the set A is open in the topology defined along the leaves of E.
The claim easily follows if we cover the curve ([0, 1]) by suitable finite family of analytic discs D 1 , D 2 , … , D l ⊂ E and apply the maximum principle to the restriction of to each of these discs. Now, in analogy to the definitions (1) and (2) of the sets E n and E , respectively, for each m ∈ ℕ and each n ≥ m we can consider the set and then define the set It is easy to see from these definitions that for each m ∈ ℕ we have that and then, from the construction of the set E (more precisely, from the choice of the sequence { n } in this construction) we also conclude that if for every compact set K ⊂ ℂ z and every m ∈ ℕ we define then Next, for a fixed m ∈ ℕ , we define the notion of the lifting ̃ of a curve ⊂ ℂ z to the set E m . Let p be a point of E m such that z (p) ∉ ℤ + iℤ . Then for some r > 0 we have that Δ r ( z (p)) ⊂ ℂ z ⧵{ℤ + iℤ} and, hence, the set E m ∩ (Δ r ( z (p)) × ℂ w ) can be represented as the union of the graphs { (f )} ∈B of holomorphic functions {f ∶ Δ r ( z (p)) → ℂ w } ∈B . Let f (p) be a function of this family such that its graph (f ( ) ) contains the point p and let ∶ [0, 1] → ℂ z ⧵{ℤ + iℤ} be a continuous curve such that z (p) = (0) . Then we define the lifting of the curve with the initial data f (p) as follows: We divide the segment [0, 1] into small enough segments by points 0 = t 0 < t 1 < ⋯ < t k = 1 and then consider a family of disks {Δ r s ( (t s ))} 0≤s≤k such that for each s = 0, 1, … , k one has Δ r s ( (t s )) ⊂ ℂ z ⧵{ℤ + iℤ} and each s = 1, 2, … , k one also has ([t s−1 , t s ]) ⊂ Δ r s ( (t s )) (in particular, {Δ r s ( (t s )} 1≤s≤k is a covering of ([0, 1]) ). Observe that, as above, for each s = 1, 2, … , k the set E m ∩ (Δ r s ( (t s )) × ℂ w ) can be represented as the union of the graphs { (f s )} ∈B of holomorphic functions {f s ∶ Δ r s ( (t s )) → ℂ w } ∈B . Note now that, in view of the construction of the set E m and the unicity theorem for holomorphic functions, there exists exactly one function f 1 (p) in the family {f 1 ∶ Δ r 1 ( (t 1 )) → ℂ w } ∈B which coincides with f (p) on the set Δ r 0 ( (t 0 )) ∩ Δ r 1 ( (t 1 )) (without loss of generality we can assume here that r 0 < r , and hence the function f (p) is defined on the disk Δ r 0 ( (t 0 )) ). Then we proceed inductively and, if for some 2 ≤ s ≤ k the function {f s−1 (p) ∶ Δ r s−1 ( (t s−1 )) → ℂ w } is already chosen, we consider the (uniquely defined) function from the family {f s ∶ Δ r s ( (t s )) → ℂ w } ∈B which coincides with f s−1 (p) on the set Δ r s−1 ( (t s−1 )) ∩ Δ r s ( (t s )) and denote it by f s (p) . Now we can finally define the lifting ̃ of the curve to the set E m . Namely, for each s = 1, 2, … , k and each ∈ [t s−1 , t s ] we set To finish the proof of the theorem, we need to show that for an arbitrary point q ∈ E one has (q) = 0 . In view of continuity of the function , it is enough to prove this claim for points q such that z (q) ∉ ℤ + iℤ . Let us fix a point q with these properties, and let p be a point of the set A , i.e. (p) = 0 and z (p) ∉ ℤ + iℤ . Fix R > 0 so large that z (p), z (q) ∈ Δ R (0) and set K ∶= Δ R (0) . Then, by property (17) above, for each n ∈ ℕ there is m n ∈ ℕ such that for all m ≥ m n . Further, in view of property (16), we can find points p n , q n ∈ E m n such that z (p n ) = z (p) , z (q n ) = z (q) and where w ∶ ℂ 2 z,w → ℂ w is the canonial projection. Since the set E m n ⧵({ℤ + iℤ} × ℂ w ) is obviously connected, there is a continuous curve n ∶ [0, 1] → E m n such that n (0) = p n , n (1) = q n and z ( n (t)) ∉ ℤ + iℤ for all t ∈ [0, 1] . Let n ∶ [0, 1] → ℂ z ⧵{ℤ + iℤ} be a curve defined by n (t) ∶= z ( n (t)) . Consider now some initial data f (p n ) for the lifting of the curve n to the set E m n +1 , that is, consider a number r > 0 and a holomorphic function f (p n ) ∶ Δ r ( z (p n )) → ℂ w such that its graph (f (p n ) ) contains the point p n and is contained in the set E m n +1 . In the case when such initial data is not unique, one can just choose an arbitrary one. Then, in view of the construction above of the lifting of a curve, we will get a lifting ̃n of the curve n to the set E m n +1 with the initial point ̃n(0) =p n and an endpoint ̃n(1) =∶q n ∈ E m n +1 ( z (q)) . Now we can finally define the curve * n ∶ [0, 1] → E⧵({ℤ + iℤ} × ℂ w ) by and observe that, by our construction and property (19), one has that and, by property (20), one also has that Since, by properties (21) and (22), * n is a curve in E⧵({ℤ + iℤ} × ℂ w ) connecting the points p and q * n , and since, by the choice of p, one has that (p) = 0 , we can conclude from Claim 1 that (q * n ) = 0 . But properties (18), (20), (22) and the fact that q n ∈ E m n +1 ( z (q)) imply that̃( ) ∶= ( ( ), f s (p) ( ( ))).
Hence, by continuity of , we finally have that (q) = lim n→∞ (q * n ) = 0 . This concludes the proof of our Liouville-type theorem. ◻ Let us now consider an arbitrary bounded holomorphic function f on the domain . Without loss of generality, we can assume that |f | < 1 on . Then the following statement holds true.
Claim 2 The restriction f | E of the function f to the Wermer type set E ⊂ is constant.
Proof Due to the last theorem, we know that |f | ≡ c 0 on E for some constant c 0 < 1 . Hence, we can write f (z, w) = c 0 exp(i ), where ∈ [0, 2 ] , in principle, might be different for different values z and w. Consider now the function g ∶= |1 − f | . Then g is also bounded continuous and plurisubharmonic on some neighborhood of E , hence, by the last theorem, |g| is also constant (say C * ) on E . A simple calculation shows that there are at most two solutions 0 and 2 − 0 to the equation |1 − c 0 exp(i )| = C * . It follows then from holomorphicity (and, hence, continuity) of f and connectedness of E that the restriction f | E of f to E is constant. This completes the proof of Claim 2. ◻ Now we fix z = z 0 and consider the slice z 0 ∶= ∩ {z = z 0 } . Since, by construction, the core E intersected with z 0 is a Cantor set, it follows that there is a point w 0 and a sequence of points w j converging to w 0 such that j = (z 0 , w j ) ∈ E ∩ z 0 for each j ∈ ℕ . Then, by Claim 2, there is a constant C such that f | E ≡ C . In particular, we have that f ( j ) = C for each j ∈ ℕ . The standard one-dimensional identity theorem for holomorphic functions tells us now that f = C on the whole slice z 0 . Since the same argument holds true for each z 0 , we finally conclude that f ≡ C on , which completes the proof of the main statement of this section.
The next statement follows directly from nonexistence of bounded holomorphic functions.

Corollary 6.1
The domain has the following properties:

1.
cannot be biholomorphic to a bounded domain.

The Carathéodory metric on is identically equal to zero.
Acknowledgements Open Access funding provided by Projekt DEAL. Part of this work was done while the first author was a visitor at the Capital Normal University (Beijing). It is his pleasure to thank this institution for its hospitality and good working conditions. The work of the second author was partially supported by NFSC (Grant No. 11671270).
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/. ‖q − q * n ‖ = � w (q n ) − w (q n )� < � w (q n )� + � w (q n )� < 1 2 n−1 .