Lee-Yang-Fisher zeros

Lee-Yang-Fisher zeros for DHL
and 2D rational dynamics,
II. Global Pluripotential Interpretation.

Pavel Bleher, Mikhail Lyubich and Roland Roeder Pavel Bleher
IUPUI Department of Mathematical Sciences
402 N. Blackford St., LD270
Indianapolis, IN 46202-3267.
bleher@math.iupui.edu Mikhail Lyubich
Mathematics Department and IMS, Stony Brook University, Stony Brook, NY 11794.
mlyubich@math.sunysb.edu Roland Roeder
IUPUI Department of Mathematical Sciences
402 N. Blackford St., LD270
Indianapolis, IN 46202-3267.
July 14, 2019

In a classical work of the 1950’s, Lee and Yang proved that for fixed nonnegative temperature, the zeros of the partition functions of a ferromagnetic Ising model always lie on the unit circle in the complex magnetic field. Zeros of the partition function in the complex temperature were then considered by Fisher, when the magnetic field is set to zero. Limiting distributions of Lee-Yang and of Fisher zeros are physically important as they control phase transitions in the model. One can also consider the zeros of the partition function simultaneously in both complex magnetic field and complex temperature. They form an algebraic curve called the Lee-Yang-Fisher (LYF) zeros. In this paper we study their limiting distribution for the Diamond Hierarchical Lattice (DHL). In this case, it can be described in terms of the dynamics of an explicit rational function in two variables (the Migdal-Kadanoff renormalization transformation). We prove that the Lee-Yang-Fisher zeros are equidistributed with respect to a dynamical -current in the projective space. The free energy of the lattice gets interpreted as the pluripotential of this current. We also describe some of the properties of the Fatou and Julia sets of the renormalization transformation.

Stony Brook IMS Preprint #2011/3 July 2011

1. Introduction

1.1. Phenomenology of Lee-Yang-Fisher zeros

We will begin with a brief survey of the Ising model of magnetic matter. More background is given in Part I of the series [BLR1]; see also [Ba, R2].

The matter in a certain scale is represented by a graph . Let and stand respectively for the set of its vertices (representing atoms) and edges (representing magnetic bonds between the atoms). A magnetic state of the matter is represented by a spin configuration on . The spin represents a magnetic momentum of an atom .

The total energy of such a configuration is given by the Hamiltonian


Here, is the total magnetic moment of the configuration , is a constant describing the interaction between neighboring spins and is a constant describing an external magnetic field applied to the matter.

By the Gibbs Principle, a configuration occurs with probability proportional to its Gibbs weight , where is the temperature. The total Gibbs weight is called the partition function. It is a Laurent polynomial in two variables , where is a “field-like” variable and is “temperature-like”.

For a fixed , the complex zeros of in are called the Lee-Yang zeros. Their role comes from the fact that some important observable quantities can be calculated as electrostatic-like potentials of the equally charged particles located at the Lee-Yang zeros. A celebrated Lee-Yang Theorem [YL, LY] asserts that for the ferromagnetic111i.e., with Ising model on any graph, the zeros of the partition function lie on the unit circle in the complex plane (corresponding to purely imaginary magnetic field ).222 We will use either the -coordinate or the angular coordinate on , without a comment. .

Magnetic matter in various scales can be modeled by a hierarchy of graphs of increasing size (corresponding to finer and finer scales). For suitable models, zeros of the partition functions will have an asymptotic distribution on the unit circle. This distribution supports singularities of the magnetic observables (or rather, their thermodynamical limits), and hence it captures phase transitions in the model.

Instead of freezing temperature , one can freeze the external field , and study zeros of in the -variable. They are called Fisher zeros as they were first studied by Fisher for the regular two-dimensional lattice, see [F, BK]. Similarly to the Lee-Yang zeros, asymptotic distribution of the Fisher zeros is supported on the singularities of the magnetic observables, and is thus related to phase transitions in the model. However, Fisher zeros do not lie on the unit circle any more. For instance, for the regular 2D lattice at , the asymptotic distribution lies on the union of two Fisher circles, see Figure 1.1.

Figure 1.1. The Fisher circles: .

More generally, one can study the distribution of zeros for on other complex lines in . In order to organize the limiting distributions over all such lines into a single object, we use the theory of currents; see [dR, Le].

A -current on is a linear functional on the space of -forms that have compact support (see Appendix A.5). A basic example is the current of integration over an algebraic curve . A plurisubharmonic function is called a pluripotential of if , in the sense of distributions. (Informally, this means that for almost any complex line , so is an electrostatic potential of the charge distribution .)

For each , the zero locus of is an algebraic curve , which we call the Lee-Yang-Fisher (LYF) zeros. Let be the degree of . It is natural to ask whether there exists a current so that

It will describe the limiting distribution of Lee-Yang-Fisher zeros. Within almost any complex line , the limiting distribution of zeros can be obtained as the restriction .

In order to justify existence of , one considers the sequence of “free energies”

where is the polynomial obtained by clearing the denominators of . We will say that the sequence of graphs has a global thermodynamic limit if

in . In Proposition 2.1 we will show that this is sufficient for the limiting current to exist. More details, including the relation of this notion to the classical definition of thermodynamic limit, are given in §2.2.

The support of consists of the singularities of the magnetic observables of the model, thus describing global phase transitions in . Connected components of describe the distinct (complex) phases of the system.

This discussion can also be extended to the compactification of by considering lifts of the free energies to ; See §2.3.

1.2. Diamond hierarchical model

The Ising model on hierarchical lattices was introduced by Berker and Ostlund [BO] and further studied by Bleher & Žalys [BZ1, BZ2, BZ3] and Kaufman & Griffiths [KG1].

The diamond hierarchical lattice (DHL), illustrated on Figure 1.2, is a sequence of graphs with two marked vertices such that is an interval, is a diamond, and is obtained from by replacing each edge of with so that the marked vertices of match with the vertices of . We then mark two vertices in so that they match with the two marked vertices of . Part I of the series [BLR1] and the present paper are both fully devoted to study of this lattice.

Figure 1.2. Diamond hierarchical lattice.
Remark 1.1.

For the DHL we will use the following slightly different definition of magnetic momentum:


For a motivation, see Appendix F of [BLR1]. Also, we will use for the temperature-like variable, as it makes formulas nicer.

There is a general physical principle that the values of physical quantities depend on the scale where the measurement is taken. The corresponding quantities are called renormalized, and the (semi-group) of transformations relating them at various scales is called the renorm-group (RG). However, it is usually hard to justify rigorously the existence of the RG, let alone to find exact formulas for RG transformations. The beauty of hierarchical models is that all this can actually be accomplished.

The Migdal-Kadanoff RG equations [M1, M2, K] (see also [BO, BZ1, KG1]) for the DHL assume the form:


where and are the renormalized field-like and temperature-like variables on . The map that relates these quantities is also called the renormalization transformation.

To study the Fisher zeros, we consider the line in . This line is invariant under , and reduces to a fairly simple one-dimensional rational map

The Fisher zeros at level are obtained by pulling back the point under . As shown in [BL], the limiting distribution of the Fisher zeros in this case exists and it coincides with the measure of maximal entropy (see [Br, Ly]) of . The limiting support for this measure is the Julia set for , which is shown in Figure 1.3. It was studied by [DDI, DIL, BL, Ish] and others.

Figure 1.3. On the left is the Julia set for . On the right is a zoomed-in view of a boxed region around the critical point . The invariant interval corresponds to the states with real temperatures and vanishing field .

To study the Lee-Yang zeros for the DHL, one considers the Lee-Yang cylinder , which is invariant under . The Lee-Yang zeros for are the real algebraic curve . Equation (1.3) shows that is the pullback of under the -fold iterate of , i.e., . Part I of the series [BLR1] uses this dynamical approach to obtain detailed information about the limiting distribution of Lee-Yang zeros for the DHL.

In this paper, we will use the Migdal-Kadanoff RG equations to study the global limiting distribution of Lee-Yang-Fisher zeros for the DHL. Their extension to , which we will also denote by , is a curve of degree . Our main result is:

Theorem (Global Lee-Yang-Fisher Current).

The currents converge distributionally to some -current on whose pluripotential coincides with the (lifted) free energy of the system.

An important subtlety arises because the degrees of do not behave properly under iteration:

This algebraic instability333For the definition, see Appendix B.1. of has the consequence that

The issue is resolved by working with another rational mapping coming directly from the Migdal-Kadanoff RG Equations, without passing to the “physical” -coordinates. This map is semi-conjugate to by a degree two rational map . Moreover, is algebraically stable, satisfying . For each , we have:

where is an appropriate projective line.

Associated to any (dominant, algebraically stable) rational mapping is a canonically defined invariant current , called the Green current of . It satisfies = , where . If satisfies an additional (minor) technical hypothesis, then equals the Julia set of . (In our case, does not satisfy this additional hypothesis, but we still have that equals the Julia set for ; see Prop. 4.4).

Such invariant currents are a powerful tool of higher-dimensional holomorphic dynamics: see Bedford-Smillie [BS], Fornaess-Sibony [FS1], and others (see [Si] for an introductory survey to this subject). In Appendix B we provide some further background on complex dynamics in several variables.

We will show that the normalized sequence of currents converges distributionally to the Green current of . Pulling everything back under , this will imply the Global Lee-Yang-Fisher Theorem. In this way, the classical Lee-Yang-Fisher theory gets linked to the contemporary dynamical pluripotential theory.

Asymptotic distribution for pullbacks of algebraic curves under rational maps has been a focus of intense research in multidimensional holomorphic dynamics since the early 1990’s; see [BS, FS1, RuSh, FaJ, DS2, DDG] for a sample of papers on the subject. Our result above is very close in spirit to this work. However, the above theorem does not seem to be a consequence of any available results.

1.3. Structure of the paper

We begin in §2 by recalling the definitions of free energy and the classical notion of thermodynamic limit for the Ising model. We then discuss the notion of global thermodynamic limit, which is sufficient in order to guarantee that some lattice have a -current describing its limiting distribution of LYF zeros in . Following this, we extend the whole description from to by lifting the partition function and free energy to . We also give an alternative interpretation of the partition function as a section of (an appropriate tensor power of) the co-tautological line bundle over that will be central to the proof of the Global LYF Current Theorem. We conclude §2 by summarizing material on the Migdal-Kadanoff RG equations.

In §3 we summarize the global features of the mappings and on the complex projective space that were studied in [BLR1], including their critical and indeterminacy loci, superattracting fixed points and their separatrices.

In the next section, §4, we define the Fatou and Julia sets for and show that the Julia set coincides with the closure of preimages of the invariant complex line (corresponding to the vanishing external field). It is based on M. Green’s criteria for Kobayashi hyperbolicity of the complements of several algebraic curves in [G1, G2] that generalize the classical Montel Theorem. We then use this result to prove that points in the interior of the solid cylinder are attracted to a superattracting fixed point of .

We prove the Global LYF Current Theorem in §5, relying on estimates of how volume is transformed under a single iterate of that are completed in §6.

We finish with several Appendices. In Appendix A we collect needed background in complex geometry (normality, Kobayashi hyperbolicity, currents and their pluri-potentials, line bundles over , etc). In Appendix B we provide background on the complex dynamics in higher dimensions, including the notion of algebraic stability and information on the Green current. In Appendix C we provide background on Whitney Folds, a normal form for the simplest critical points of our mapping. In Appendix D we collect several open problems.

1.4. Basic notation and terminology

, , , , and . Given two variables and , means that for some constants .

Acknowledgment. We thank Jeffrey Diller, Han Peters, Robert Shrock, and Dror Varolin for interesting discussions and comments. The work of the first author is supported in part by the NSF grants DMS-0652005 and DMS-0969254. The work of the second author has been partially supported by NSF, NSERC and CRC funds. The work of the third author was partially supported by startup funds from the Department of Mathematics at IUPUI.

2. Description of the model

2.1. Free Energy

A configuration of spins has energy given by the Hamiltonian


The Ising model is called ferromagnetic if , and anti-ferromagnetic otherwise. In this paper (and Part I) we will assume that the model is ferromagnetic.

The partition function (or the statistical sum) is the total Gibbs weight of all of the configurations of spins:

It is a Laurent polynomial in and . Notice that each of the Gibbs weights is unchanged under . This basic symmetry of the Ising model makes invariant under the involution . Therefore, it has the form


Moreover, . Thus, for any given , has roots called the Lee-Yang zeros.

A configuration occurs with probability . Its entropy is defined as . The free energy of the system is defined as


It is independent of the configuration .

Equations (2.3) and (2.2) imply:


where the summation is taken over the Lee-Yang zeros of .

Remark 2.1.

It will be useful to consider the following variant of the free energy:

Here, is obtained by clearing the denominators of . The advantage of using , instead of , is that it extends as a plurisubharmonic function on all of . We will also refer to as the “free energy”.

2.2. Thermodynamic limit

Assume that we have a “lattice” given by a “hierarchy” of graphs of increasing size with partition functions , free energies and magnetizations . To pass to the thermodynamic limit we normalize these quantities per bond. One says that the hierarchy of graphs has a thermodynamic limit if


In this case, the function is called the free energy of the lattice. For many444Note that the DHL is not in this class—instead, dynamical techniques are used to justify its classical thermodynamic limit. lattices (e.g. ), existence of the thermodynamic limit can be justified by van Hove’s Theorem [vH, R2]. If the classical thermodynamic limit exists, then one can justify existence of the limiting distribution of Lee-Yang zeros and relate it to the limiting free energy; see [BLR1, Prop. 2.2].

When considering Fisher zeros, it is more convenient to work with the variant of the free energy that is defined in Remark 2.1. In order to prove existence of a limiting distribution for the Fisher zeros, one needs to prove existence of a limiting free energy:


For the lattice this is achieved by the Onsager solution, which provides an explicit formula the limiting free energy; See, for example, [Ba]. Similar techniques apply to the triangular, hexagonal, and various homopolygonal lattices (see [MaSh1, MaSh2] for suitable references and an investigation of the distribution of Fisher zeros for these lattices). For various hierarchical lattices, (2.6) can be proved by dynamical means. It seems to be an open question for other lattices, including , when .

The situation is similar for the Lee-Yang-Fisher zeros:

Proposition 2.1.

Let be a hierarchy of graphs and suppose that


Then, there is a closed positive -current on describing the limiting distribution of Lee-Yang-Fisher zeros. Its pluripotential coincides with the free energy .

For the DHL, we will prove existence of the limit (2.7) in the Global LYF Current Theorem. It is an open question whether such a limit exists for other lattices, including the classical lattices for ; see Problem D.1.


The locus of Lee-Yang-Fisher zeros are the zero set (counted with multiplicities) of the degree polynomial . The Poincaré-Lelong Formula describes its current of integration:

Here, is the pluri-Laplacian; see Appendix A.

Hypothesis (2.7) implies

2.3. Global consideration of partition functions and free energy on

It will be convenient for us to extend the partition functions and their associated free energies from to . We will use the homogeneous coordinates
on , with the “physical” copy of given by the affine coordinates .

For each , we clear the denominators of , obtaining a polynomial of degree . It lifts to a unique homogeneous polynomial of the same degree that satisfies The associated free energy becomes a plurisubharmonic function

on . It is related by the Poincaré-Lelong Formula to the current of integration over the Lee-Yang-Fisher zeros: .

Both of these extensions are defined on , rather than . In the proof of the Global LYF Current Theorem, it will be useful for us to interpret the partition function as an object defined on . Instead of being a function on , it gets interpreted as a section of an appropriate tensor power of the co-tautological line bundle; See Appendix A.4. The Lee-Yang-Fisher zeros are described as the zero locus of this section. (See Remark A.1 for an explanation of why a similar interpretation of the free energy as a section of a suitable real line bundle is not used.)

2.4. Migdal-Kadanoff renormalization for the DHL

Migdal-Kadanoff renormalization will allow us to write recursive formuli for the partition function . Restricting the spins at the marked vertices we obtain three conditional partition functions, , and as follows:


The total partition function is equal to

Migdal-Kadanoff RG Equations:

See Part I [BLR1] for a proof. They give a homogeneous degree polynomial map


called the Migdal Kadanoff Renormalization, with the property that . We will also call the induced mapping the Migdal Kadanoff Renormalization.

In order to express and in terms of and , one can use


as initial conditions for the iteration. According to (2.9), these coordinates are related to the “physical” -coordinates as follows:


Therefore, in the -coordinates, the renormalization transformation assumes the form:


Physically, the iterates are interpreted as the renormalized field-like and temperature-like variables.

By the basic symmetry of the Ising model, the change of sign of interchanges the conditional partition functions and keeping and the total sum invariant. Consequently, the RG transformation commutes with the involution , which is also obvious from the explicit formula (2.8), while commutes with .

Existence of the renormalization mapping makes discussion of the free energies especially clear for the DHL. Consider the linear form

which is chosen so that is the zero divisor of . For each ,


Here, and are expressed in terms of the initial values of . To rewrite (2.12) in terms of , we can pull back these expressions under .

3. Global properties of the RG transformation in .

We will now summarize (typically without proofs) results from [BLR1] about the global properties of the RG mappings.

3.1. Preliminaries

The renormalization mappings and are semi-conjugate by the degree two rational mapping given by (2.10).

Both mappings have topological degree (see Prop. 4.3 from Part I). However, as noted in the introduction, their algebraic degrees behave differently: is algebraically stable, while is not. Since , for any algebraic curve of degree , the pullback is a divisor of degree . (For background on divisors, see Appendix A.) For this reason, we will focus most of our attention on the dynamics of .

The semiconjugacy sends Lee-Yang cylinder to a Mobius band that is invariant under . It is obtained as the closure in of the topological annulus


Let be the “top” circle of , while be the slice of at infinity. In fact, is a conjugacy, except that it maps the bottom of by a -to- mapping to (see Prop. 3.1 from Part I).

3.2. Indeterminacy points for

In homogeneous coordinates on , the map has the form:


One can see that has precisely two points of indeterminacy and . Resolving all of the indeterminacies of by blowing-up the two points (see Appendix A.3), one obtains a holomorphic mapping .

In coordinates and near , we obtain the following expression for the map near :


(Similar formulas hold near .) The exceptional divisor is mapped by to the conic

3.3. Superattracting fixed points and their separatrices

We will often refer to as the line at infinity. It contains two symmetric superattracting fixed points, and . Let and stand for the attracting basins of these points. It will be useful to consider local coordinates near .

The line at infinity is -invariant, and the restriction is the power map . Thus, points in the disk in are attracted to , points in the disk are attracted to , and these two basins are separated by the unit circle . We will also call the fast separatrix of and .

Let us also consider the conic


passing through points and . It is an embedded copy of that is invariant under , with , where . Thus, points in the disk in are attracted to , points in the disk are attracted to , and these two basins are separated by the unit circle (see §3.1 from Part I). We will call the slow separatrix of and .

If a point near (resp. ) does not belong to the fast separatrix , then its orbit is “pulled” towards the slow separatrix at rate , with some , and converges to (resp. ) along at rate , with some .

The strong separatrix is transversally superattracting: all nearby points are pulled towards uniformly at rate (see also the proof of Lemma 3.3). It follows that these points either converge to one of the fixed points, or , or converge to the circle .

Given a neighborhood of , let


(where is implicit in the notation, and an assertion involving means that it holds for arbitrary small suitable neighborhoods of ). It is shown in Part I (§9.2) that has the topology of a -manifold that is laminated by the union of holomorphic stable manifolds of points .

We conclude:

Lemma 3.1.

fills in some neighborhood of .

3.4. Regularity of

For a diffeomorphism the existence and regularity of the local stable manifold for a hyperbolic invariant manifold has been studied extensively in [HPS]. In order to guarantee a local stable manifold , a strong form of hyperbolicity known as normal hyperbolicity is assumed. Essentially, is normally hyperbolic for if the expansion of in the unstable direction dominates the maximal expansion of tangent to and the contraction of in the stable direction dominates the maximal contraction of tangent to . See [HPS, Theorem 1.1]. If, furthermore, the expansion in the unstable direction dominates the -th power of the maximal expansion tangent to and the contraction in the stable direction dominates the -th power of the maximal contraction tangent to , this guarantees that the stable manifold is of class . The corresponding theory for endomorphisms is less developed, although note that other aspects of [HPS], related to persistence of normally hyperbolic invariant laminations, have been generalized to endomorphisms in [Be].

In our situation, is not normally hyperbolic because it lies within the invariant line and is holomorphic. This forces the expansion rates tangent to and transverse to (within this line) to coincide. Therefore, the following result does not seem to be part of the standard hyperbolic theory:

Lemma 3.2.

is a manifold and the stable foliation is a foliation by complex analytic discs.


In Proposition 9.11 from Part I, we showed that within the cylinder the stable foliation of has regularity and that the stable curve of each point is real analytic. Mapping forward under , we obtain the same properties for the stable foliation of within .

Let us work in the local coordinates and . In these coordinates, . The stable curve of some can be given by expressing as the graph of a holomorphic function of :


The right hand side is a convergent power series with coefficients depending on , having a uniform radius of convergence over every . The series is uniquely determined by its values on the real slice , in which the leaves depend with regularity on . Therefore, each of the coefficients is in . This gives that is a manifold.

Remark 3.1.

The technique from the proof of Lemma 3.2 applies to a more general situation: Suppose that is a real analytic manifold and is a real analytic map. Let be a compact real analytic invariant submanifold for , with expanding and with transversally attracting under . Then, will have a stable foliation of regularity , for some (see the beginning of this subsection), with the stable manifold of each point being real-analytic. The stable manifold for the extension of to the complexification of will then also have regularity.

3.5. Critical locus

We showed in Part I [BLR1] that the critical locus of consists of 6 complex lines and one conic:

(Here the curves are written in the homogeneous coordinates and in the affine ones, .) The critical locus is schematically depicted on Figure 3.1, while its image, the critical value locus, is depicted on Figure 3.2.

SeparatrixSeparatrixFixed point Fixed point collapsingline
Figure 3.1. Critical locus for shown with the separatrix at infinity.
Figure 3.2. Critical values locus of .

It will be helpful to also consider the critical locus for the lift . Each of the critical curves lifts by proper transform (see Appendix A.3) to a critical curve for . Moreover, any critical point for is either one of these proper transforms or lies within the exceptional divisors of .

By symmetry, it is enough to consider the blow-up of . We saw in Part I that there are four critical points on the exceptional divisor occurring at and , where . They correspond to intersections of with the collapsing line , the , and the critical lines and , respectively.

Whitney Folds are a normal form for the simplest type of critical points of a mapping (see Appendix C). We have:

Lemma 3.3.

All critical points of except the fixed points , the collapsing line , and two points , are Whitney folds.

The only critical values obtained as images of non-Whitney folds are: and .

4. Fatou and Julia sets and the measure of maximal entropy

4.1. Julia set

For a rational map , the Fatou set is defined to be the maximal open set on which the iterates are well-defined and form a normal family. The complement of the Fatou set is the Julia set .

If is dominant and has no collapsing varieties, Lemma A.1 gives that is locally surjective (except at indeterminate points), so that the Fatou set is forward invariant and consequently, the Julia set is backward invariant.

If has indeterminate points, then, according to this definition they are in . In this case, and are not typically totally invariant. One can see this by blowing up an indeterminate point and observing that the image of the resulting exceptional divisor typically intersects . Moreover, if is not algebraically stable, then any curve that is mapped by some iterate to the indeterminacy points is in .

The Migdal-Kadinoff renormalization is not locally surjective at any . More specifically, if is a small neighborhood of that avoids , then

since any point of only has preimages in . However, we still have the desired invariance:

Lemma 4.1.

The Migdal-Kadinoff renormalization has forward invariant Fatou set and, consequently, backward invariant Julia set.


It suffices to show that , since is locally surjective at any other point, by Lemma A.1. By definition, , so we consider . Let be any small neighborhood of . Note that is a fixed point of saddle-type, with one-dimensional stable and unstable manifolds. Therefore, in order for the iterates to form a normal family on , we must have . However, this is impossible, since there are plenty of regular points for in . ∎

Lemmas 3.1 and 3.2 give a clear picture of in a neighborhood of the line at infinity .

Proposition 4.2.

Within some neighborhood of we have that . Within this neighborhood, is a -dimensional manifold.

Let us consider the locus of vanishing magnetic field in for the DHL. In the affine coordinates, it an -invariant line ; in the physical coordinates, it is an -invariant line . The two maps are conjugate by the Möbius transformation , . Dynamics of on was studied in [BL]. In particular, it is shown in that paper that the Fatou set for consists entirely of the basins of attraction of the fixed points and which are superattracting within this line: see Figure 1.3.

Proposition 4.3.



Since is conformally conjugate to , every point in the Fatou set of is in the basin of attraction of either or . Since these points are of saddle-type in , the family of iterates cannot be normal at any point on . Thus . It follows that since is closed and backward invariant.

We will now show that is dense in . Consider a configuration of five algebraic curves

We will use the results of M. Green to check that the complement of these curves, , is a complete Kobayashi hyperbolic manifold hyperbolically embedded in (see Appendix A.6). We will first check that is Brody hyperbolic, i.e., there are no non-constant holomorphic maps . To this end we will apply Green’s Theorem A.5. It implies that the image of must lie in a line