A construction of hyperkähler metrics through Riemann-Hilbert problems II
César Garza










Department of Mathematics, IUPUI, Indianapolis, USA

cegarza@iu.edu









[2010]Primary 








We develop the theory of Riemann-Hilbert problems necessary for the results in <cit.>. In particular, we obtain solutions for a family of non-linear Riemann-Hilbert problems through classical contraction principles and saddle-point estimates. We use compactness arguments to obtain the required smoothness property on solutions. We also consider limit cases of these Riemann-Hilbert problems where the jump function develops discontinuities of the first kind together with zeroes of a specific order at isolated points in the contour. Solutions through Cauchy integrals are still possible and they have at worst a branch singularity at points where the jump function is discontinuous and a zero for points where the jump vanishes.


A construction of hyperkähler metrics through Riemann-Hilbert problems II
    C. Garza
    
=========================================================================





§ INTRODUCTION


This article presents the analytic results needed in <cit.>. As stated in said article, in order to construct complete hyperkähler metrics g on a special case of complex integrable systems (where moduli spaces of Higgs bundles constitute the prime example of this), one must obtain solutions to a particular infinite-dimensional, nonlinear family of Riemann-Hilbert problems. The analytical methods used to obtain these solutions and the smoothness results can be studied separately from the geometric motivations and so we present them in this article.

For limiting values of the parameter space, the Riemann-Hilbert problem degenerates in the sense that discontinuities appear in the jump function G(ζ) at ζ = 0 and ζ = ∞ in the contour Γ. Moreover, G(ζ) may vanish at isolated pairs of points in Γ. We study the behavior of the solutions to this boundary value problem near such singularities and we obtain their general form, proving that these functions do not develop singularities even in the presence of these pathologies, thus proving the existence of the hyperkähler metrics in <cit.>.

The paper is organized as follows:

In Section <ref> we state the Riemann-Hilbert problems to be considered. As shown in <cit.>, this arises from certain complex integrable systems satisfying a set of axioms motivated by the theory of moduli spaces of Higgs bundles, but we shall not be concerned about the geometric aspects in this paper.

In Section <ref> we solve the Riemann-Hilbert problem by iterations running estimates based on saddle-point analysis. Under the right Banach space, these estimates show that we have a contraction, proving that solutions exist and are unique. We then apply the Arzela-Ascoli theorem and uniform estimates to show that the solutions are smooth with respect to the parameter space. 

In Section <ref> we consider the special case when the parameter a approaches 0 yielding a Riemann-Hilbert problem whose jump function has discontinuities and zeroes along the contour. We apply Cauchy integral techniques to obtain the behavior of the solutions near the points on the contour with these singularities. We show that a discontinuity of the jump function induces a factor ζ^η in the solutions, where η is determined by the discontinuities of the jump function G. A zero of order k at ζ_0 induces a factor (ζ - ζ_0)^k on the left-side part of the solutions. The nature of these solutions are exploited in <cit.> to reconstruct a holomorphic symplectic form ϖ(ζ) and, ultimately, a hyperkähler metric.

Acknowledgment: The author would like to thank Professor Alexander Its for many illuminating conversations that greatly improved the manuscript.




§ FORMULATION OF THE RIEMANN-HILBERT PROBLEM




 §.§ Monodromy data


We state the monodromy data we will use in this paper. For a more geometric description of the assumptions we make, see <cit.>.

Since we will only consider the manifolds ℳ in <cit.> from a local point of view, we can consider it as a trivial torus fibration. With that in mind, here are the key ingredients we need in order to define our Riemann-Hilbert problem:



  * A neighborhood U of 0 in  with coordinate a. On U we have a trivial torus fibration U × T^2 := U × (S^1)^2 with θ_1, θ_2 the torus coordinates.


  * Γ≅^2 is a lattice equipped with an antisymmetric integer valued pairing ⟨ , ⟩. We also assume we can choose primitive elements γ_1, γ_2 in Γ forming a basis for the lattice and such that ⟨γ_1, γ_2⟩ = 1.


  * A homomorphism Z from Γ to the space of holomorphic functions on U.


  * A function Ω : Γ→ such that Ω(γ) = Ω(-γ), γ∈Γ and such that, for some K > 0,

    |Z_γ|/γ > K

for a positive definite norm on Γ and for all γ for which Ω(γ) ≠ 0.


For the first part of this paper we work with the extra assumption


  (5) Z_γ_1(a), Z_γ_2(a) ≠ 0 for any a in U.

Later in this paper we will relax this condition.

Observe that the torus coordinates θ_1, θ_2 induce a homomorphism θ from Γ to the space of functions on T^2 if we assign γ_k↦θ_k, k = 1, 2. We denote by θ_γ, γ∈Γ the result of this map.
  
We consider a different complex plane  with coordinate ζ. Let R > 0 be an extra real parameter that we consider. We define the “semiflat” functions 𝒳_γ : U × T^2 ×→ for any γ∈Γ as

    𝒳^sf_γ (a, θ_1, θ_2, ζ) = exp( π R Z_γ(a)/ζ + iθ_γ + π R ζZ_γ(a))

As in the case of the map θ, it suffices to define 𝒳^sf_γ_1 and 𝒳^sf_γ_2.

For each a ∈ U and γ∈Γ such that Ω(γ) ≠ 0, the function Z_γ defines a ray ℓ_γ(a) in  given by

    ℓ_γ(a) = {ζ∈ : ζ = -t Z_γ(a), t > 0 }


Given a pair of functions 𝒳_k :  U × T^2 ×→, k = 1, 2, we can extend this with the basis {γ_1, γ_2} as before to a collection of functions 𝒳_γ, γ∈Γ. Each element γ in the lattice also defines a transformation 𝒦_γ for these functions in the form

    𝒦_γ𝒳_γ' = 𝒳_γ' (1 - 𝒳_γ)^⟨γ', γ⟩


For each ray ℓ from 0 to ∞ in  we can define a transformation

    S_ℓ = ∏_γ : ℓ_γ(u) = ℓ𝒦_γ^Ω(γ)

Observe that all the γ's involved in this product are multiples of each other, so the 𝒦_γ commute and the order for the product is irrelevant.

We can now state the main type of Riemann-Hilbert problem we consider in this paper. We seek to obtain two functions 𝒳_k :  U × T^2 ×→, k = 1, 2 with the following properties:

	
  * Each 𝒳_k depends piecewise holomorphically on ζ, with discontinuities only at the rays ℓ_γ(a) for which Ω(γ) ≠ 0. The functions are smooth on U × T^2.
	
  * The limits 𝒳_k^± as ζ approaches any ray ℓ from both sides exist and are related by
	
    𝒳^+_k = S_ℓ^-1∘𝒳^-_k

	
  * 𝒳 obeys the reality condition
	
    𝒳_-γ(-1/ζ) = 𝒳_γ(ζ)

	
  * For any γ∈Γ, lim_ζ→ 0𝒳_γ(ζ) / 𝒳^sf_γ(ζ) exists and is real.




 §.§ Isomonodromic Deformation


It will be convenient for the geometric applications to move the rays to a contour that is independent of a. Even though the rays ℓ_γ defining the contour for the Riemann-Hilbert problem above depend on the parameter a, we can assume the open set U ⊂ is small enough so that there is a pair of rays r, -r such that for all a ∈ U, half of the rays  lie inside the half-plane ℍ_r of vectors making an acute angle with r; and the other half of the rays lie in ℍ_-r. We call such rays admissible rays. We are allowing the case that r is one of the rays ℓ_γ, as long as it satisfies the above condition.

For γ∈Γ, we define γ > 0 (resp. γ < 0) as ℓ_γ∈ℍ_r (resp. ℓ_γ∈ℍ_-r). Our Riemann-Hilbert problem will have only two anti-Stokes rays, namely r and -r.  In this case, the Stokes factors are the concatenation of all the Stokes factors S^-1_ℓ in (<ref>) in the counterclockwise direction:

    S_r     = ∏^_γ > 0𝒦^Ω(γ; a)_γ
    
    S_-r    = ∏^_γ < 0𝒦^Ω(γ; a)_γ

Thus, we reformulate the Riemann-Hilbert problem in terms of two functions 𝒴_k :  U × T^2 ×→, k = 1, 2 with discontinuities at the admissible rays r, -r by replacing condition <ref> above with

    [         𝒴^+_k = S_r ∘𝒴^-_k,       along r;         𝒴^+_k = S_-r∘𝒴^-_k,      along -r ]

The other conditions remain the same:

	
  * The functions 𝒴_k are smooth on U × T^2.
	
  * 𝒴 obeys the reality condition
	
    𝒴_-γ(-1/ζ) = 𝒴_γ(ζ)

	
  * For any γ∈Γ, lim_ζ→ 0𝒴_γ(ζ) / 𝒳^sf_γ(ζ) exists and is real. 


In the following section we will prove the main theorem of this paper:


There exists a pair of functions 𝒴_k :  U × T^2 ×→, k = 1, 2 satisfying (<ref>) and conditions (<ref>), (<ref>), (<ref>). These functions are unique up to multiplication by a real constant.







§ SOLUTIONS


We start working on a proof of Theorem <ref>. As in the classical scalar Riemann-Hilbert problems, we obtain the solutions 𝒴_k by solving the integral equation

    𝒴_k(a,ζ) = 𝒳_γ_k^sf(a,ζ) exp( 1/4π i{∫_r K(ζ, ζ') log(S_r 𝒴_k) + ∫_-r K(ζ, ζ') log(S_-r𝒴_k) }),    k = 1, 2

where we abbreviated  dζ'ζ'ζ'+ζζ'-ζ as K(ζ',ζ). The dependence of 𝒴_k on the torus coordinates θ_1, θ_2 has been omitted to simplify notation. We will write 𝒴_γ to denote the function resulting from the (multiplicative) homomorphism from Γ to nonzero functions on U × T^2 × induced by 𝒴_k, k = 1, 2. 

It will be convenient to write

    𝒴_γ(a, ζ, θ) = 𝒳_γ^sf(a, ζ, Θ),

for Θ_k : U ×  T^2 ×→, k = 1, 2. We abuse notation and write θ for (θ_1, θ_2), as we do with Θ.

If we take the power series expansion of log(S_r 𝒴_k), log(S_-r𝒴_k) and decompose the terms into their respective components in each γ∈Γ, we can rewrite the integral equation (<ref>) as

    𝒴_γ(a,ζ) = 𝒳_γ^sf(a,ζ)exp( 1/4π i{∑_γ' > 0 f^γ'∫_r K(ζ,ζ') 𝒴_γ'(a,ζ') +  ∑_γ' < 0 f^γ'∫_-r K(ζ,ζ') 𝒴_γ'(a,ζ')})

where 

    f^γ' = c_γ'⟨γ, γ' ⟩,

c_γ' a rational constant obtained by power series expansion. 

[The Pentagon case]

As our main example of this families of Riemann-Hilbert problems, we have the Pentagon case, studied in more detail in <cit.>. Here the jump functions S_r, S_-r are of the form

    . [                   𝒴_1           ↦𝒴_1(1-𝒴_2);                   𝒴_2 ↦𝒴_2(1-𝒴_1(1-𝒴_2))^-1 ]}    S_r 
    and, similarly. [                      𝒴_1        ↦𝒴_1(1-𝒴^-1_2)^-1;                      𝒴_2 ↦𝒴_2(1-𝒴^-1_1(1-𝒴^-1_2)) ]}    S_-r


If we expand log(S_r𝒴_k), k = 1, 2 etc. we obtain

    f^iγ_1+jγ_2= {[             -1j⟨γ, γ_2⟩                  if i=0;   (-1)^ji|i||j|⟨γ, γ_1⟩ if 0≤ j≤ i or i ≤ j ≤ 0;                       0              otherwise. ].



Back in the general case, our approach for a solution to (<ref>) is to work with iterations. For ν∈ℕ:

    𝒴^(ν+1)_γ(a,ζ) = 𝒳_γ^sf(a,ζ)exp( 1/4π i{∑_γ' > 0 f^γ'∫_r K(ζ,ζ') 𝒴^(ν)_γ'(a,ζ') +  ∑_γ' < 0 f^γ'∫_-r K(ζ,ζ') 𝒴^(ν)_γ'(a,ζ')})

Formula (<ref>) requires an explanation. Assuming 𝒴^(ν-1)_γ', γ' ∈Γ has been constructed, by definition, 𝒴^(ν)_γ' has jumps at r and -r. By abuse of notation, 𝒴^(ν)_γ' in (<ref>) denotes the analytic continuation to the ray r (resp. -r) along ℍ_r (resp. ℍ_-r) in the case of the first (resp. second) integral.

By using (<ref>), we can write (<ref>) as an additive Riemann-Hilbert problem where we solve the integral equation

    e^iΘ_γ = e^iθ_γexp( 1/4π i{∑_γ' > 0 f^γ'∫_r K(ζ,ζ') 𝒳^sf_γ'(a,ζ',Θ) + ∑_γ' < 0 f^γ'∫_-r K(ζ,ζ') 𝒳^sf_γ'(a,ζ',Θ)})


As in (<ref>), the solution of (<ref>) is obtained through iterations:

    Θ^(0)(ζ,θ) = θ,


    e^iΘ_γ^(ν+1) = e^iθ_γexp(1/4π i{∑_γ' > 0 f^γ'∫_r K(ζ,ζ') 𝒳^sf_γ'(a,ζ',Θ^(ν)) + ∑_γ' < 0 f^γ'∫_-r K(ζ,ζ') 𝒳^sf_γ'(a,ζ',Θ^(ν))})


We need to show that Θ^(ν) = (Θ_1^(ν), Θ_2^(ν)) converges uniformly in a to well defined functions Θ_k : U × T^2 ×→, k = 1, 2 with the right smooth properties on a and ζ. Define X as the completion of the space of bounded functions of the form Φ: U × T^2 ×→^2 that are smooth on U × T^2 under the norm

    Φ =  sup_ζ,θ, aΦ(ζ, θ, a) _^2,

where ^2 is assumed to have as norm the maximum of the Euclidean norm of its coordinates. Notice that we have not put any restriction on the functions Φ in the  slice, except that they must be bounded. Our strategy will be to solve the Riemann-Hilbert problem in  X and show that for sufficiently big (but finite) R, we can get uniform estimates on the iterations yielding such solutions and any derivative with respect to the parameters a, θ. The Arzela-Ascoli theorem will give us that the solution Φ not only lies in X, but it preserves all the smooth properties. The very nature of the integral equation will guarantee that its solution is piecewise holomorphic on ζ, as desired.

We're assuming as in <cit.> that Γ has a positive definite norm satisfying the Cauchy-Schwarz property

    |⟨γ, γ' ⟩| ≤γγ'

as well as the “Support property” (<ref>).




For any Φ∈X, let Φ_k denote the composition of Φ with the kth projection π_k : ^2→, k = 1, 2. Instead of working with the full Banach space X, let X^* be the collection of Φ∈X in the closed ball

    Φ - θ≤ϵ,

for an ϵ > 0 so small that

    sup_ζ,θ,a|  e^iΦ_k| ≤ 2,

for k =  1, 2. In particular, X^* is closed, hence complete. Note that by (<ref>), if Φ∈X^*, then e^iΦ∈X. Furthermore, by (<ref>), the transformation in ζ is only as an integral transformation, so Θ^(ν) is holomorphic in either of the half planes ℍ_r or ℍ_-r. 



 §.§ Saddle-point Estimates


We will prove the first of our uniform estimates on Θ^(ν).


Θ^(ν)∈ for all ν.


We follow <cit.>, using induction on  ν. The statement is clearly true for ν = 0 by (<ref>). Assuming Θ^(ν)∈X^*, take the log in both sides of (<ref>):

    Θ^(ν+1)_k - θ_k = -1/4π{∑_γ' > 0 f^γ'∫_r K(ζ,ζ') 𝒳^sf_γ'(a,ζ',Θ^(ν)) + ∑_γ' < 0 f^γ'∫_-r K(ζ,ζ') 𝒳^sf_γ'(a,ζ',Θ^(ν)) },    k = 1, 2

For general Φ∈X^*, Φ can be very badly behaved in the  slice, but by our inductive construction, Θ^(ν+1) is even holomorphic in ℍ_r and ℍ_-r. Consider the integral

    ∫_r K(ζ,ζ') 𝒳^sf_γ'(a,ζ',Θ^(ν))

The function Θ^(ν) can be analytically extended along the ray r so that it is holomorphic on the sector V bounded by r and ℓ_γ', γ' > 0 (see Figure <ref>). By Cauchy's theorem, we can move (<ref>) to one along the ray ℓ_γ', possibly at the expense of a residue of the form




    4π i exp[iΘ_γ'^(ν) + π R ( Z_γ'/ζ + Z_γ'ζ) ]

if ζ lies in V. This residue is in control. Indeed, by the induction hypothesis, | e^i Θ_γ'^(ν)| < 2^γ', independent of ν. Moreover, we pick a residue only if ζ lies in the sector S bounded by the first and last ℓ_γ_k, γ_k ∈{γ_1, γ_2} included in ℍ_r traveling in the counterclockwise direction. This sector is strictly smaller than ℍ_r (see Figure <ref>), so Z_γ' - ζ∈ (-π,π) and, since r makes an acute angle with all rays ℓ_γ', γ'>0:




    | Z_γ' - ζ| > const > π/2 for all γ' > 0, ζ∈ S.

In particular,

    cos( Z_γ' - ζ) < -const <0  for all γ' > 0, ζ∈ S.

Using the fact that inf (|ζ| + 1/|ζ|) = 2, the sum of residues of the form (<ref>) is bounded by:

    ∑_γ' >0| f^γ'| 2^γ'  e^-const R|Z_γ'|

Recall that γ' < const |Z_γ'|, so (<ref>) can be simplified to

    ∑_γ' >0|f^γ'|  e^(-const R + δ)|Z_γ'|

for a constant δ. We're assuming that Ω(γ') do not grow too quickly with γ', by the support property (<ref>), so |f^γ'| is dominated by the exponential term and the above sum can be made arbitrarily small if R is big enough. This bound can be chosen to be independent of ν, ζ and the basis element γ_k (by choosing the maximum among the γ_1, γ_2). The exact same argument can be used to show that the residues of the integrals along -r are in control. In fact, let ϵ > 0 be given. Choose R > 0 so that the total sum of residues Res(ζ) is less than ϵ/2.

Thus, we can assume the integrals are along ℓ_γ' and consider

    ∫_ℓ_γ' K(ζ,ζ') 𝒳^sf_γ'(a,ζ',Θ^(ν))

The next step is to do a saddle point analysis and obtain the asymptotics for large R. Since this type of analysis will be of independent interest to us, we leave these results to a separate Lemma at the end of this section.

By (<ref>), | exp( i Θ^(ν)_γ'(ζ_0)) | ≤ 2^γ'.  Thus, by Lemma <ref>, for ζ away from the saddle ζ_0, we can bound the contribution from the integral by

    const | f^γ'| 2^γ'e^-2π R|Z_γ'|/√(R |Z_γ'|)

if R is big enough.

The case of ζ = ζ_0 is, by Lemma <ref>, as in (<ref>) except without the √(R) term in the denominator. In any case, by (<ref>), and since exp( i Θ^(ν)_γ'(ζ_0)) ≤ 2^γ' by (<ref>) and by (<ref>),































































































    | ∑_γ' f^γ'∫_ℓ_γ' K(ζ,ζ') 𝒳^sf_γ'(a,ζ',Θ^(ν)) | <
    const ∑_γ'|
     f^γ'| e^(-2π R + δ)|Z_γ'|.

The δ constant is the same appearing in (<ref>). This sum is convergent by the tameness condition on the Ω(γ') coefficients, and can be made arbitrarily small if R is big enough. Putting everything together:

    sup_ζ,θ| Θ^(ν+1)_γ - θ_γ|     =const ∑_γ'|
     f^γ'| e^(-2π R + δ)|Z_γ'| + Res(ζ)
        < ϵ/2 + ϵ/2 = ϵ.

Therefore Θ^(ν+1) - θ < ϵ. In particular, Θ^(ν+1) < ∞, so Θ^(ν+1)∈X^*. Since ϵ was arbitrary, Θ^(ν+1) satisfies the side condition (<ref>) and thus Θ^(ν)∈X^* for all ν if R is big enough.


We finish this subsection with the proof of some saddle-point analysis results used in the previous lemma.


For every ν consider an integral of the form

    F(ζ) = ∫_ℓ_γ' K(ζ,ζ') 𝒳^sf_γ'(a,ζ',Θ^(ν))

Let ζ_0 = -e^i  Z_γ'. Then, for ζ≠ζ_0, we can estimate the above integral as

    F(ζ) = -ζ_0 + ζ/ζ_0 - ζexp( i Θ^(ν)(ζ_0)) 1/√(R|Z_γ'|) e^-2π R |Z_γ'| + O( e^-2π R |Z_γ'|/R),     as R →∞

For ζ = ζ_0,

    F(ζ_0) = O( e^-2π R |Z_γ'|/R),     as R →∞



Equation (<ref>) is of the type

    h(R) = ∫_ℓ_γ' g(ζ') e^π R f(ζ')

where

    g(ζ') = ζ'+ζ/ζ'(ζ'-ζ),  f(ζ') = Z_γ'/ζ' 
    + ζ' Z_γ'.

The function f has a saddle point ζ_0 = -e^i  Z_γ' at the intersection of the ray ℓ_γ' with the unit circle. Moreover, f(ζ_0) = -2|Z_γ'|. The ray ℓ_γ' and the unit circle are the locus of Im  f(ζ') = Im f(ζ_0) = 0. It's easy to see that in ℓ_γ' f(ζ') < f(ζ_0) if ζ' ≠ζ_0, so ℓ_γ' is the path of steepest descent (see Figure <ref>).




Introduce τ by


    1/2 (ζ' - ζ_0)^2 f”(ζ_0) + O((ζ' -ζ_0)^3) = -τ^2


and so


    ζ' - ζ_0 = {-2/f”(ζ_0)}^1/2τ + O(τ^2)


for an appropriate branch of {f”(ζ_0)}^1/2. Let α =  f”(ζ_0) = -2 Z_γ' + π. The branch of {f”(ζ_0)}^1/2 is chosen so that τ > 0 in the part of the steepest descent path outside the unit disk in Figure <ref>. That is, τ > 0 when (ζ' - ζ_0) = 1/2π - 1/2α, and so {f”(ζ_0)}^1/2 = i√(2|Z_γ'|)e^-i Z_γ'. Thus (<ref>) simplifies to


    ζ' - ζ_0 = -ζ_0/√(|Z_γ'|)τ + O(τ^2)


We expand g(ζ'(τ)) as a power series[In our case, g depends also on the parameter R, so this is an expansion on ζ']:


    g(ζ'(τ))  = g(ζ_0) + g'(ζ_0) {-2/f”(ζ_0)}^1/2τ + O(τ^2)


As in <cit.>, 


    h(R) ∼ e^Rf(ζ_0)g(ζ_0)  {-2/f”(ζ_0)}^1/2∫_-∞^∞ e^-R τ^2 dτ + …

and so

    h(R)     = √(2π/R|f”(ζ_0)|) g(ζ_0) e^R f(ζ_0) + (i/2)(π - α) + O( e^R f(ζ_0)/R) 
    in our case, and since ζ_0 = -e^i  Z_γ'    = -ζ_0 + ζ/ζ_0 - ζexp( i Θ^(ν)(ζ_0)) 1/√(R|Z_γ'|) e^-2π R |Z_γ'| + O( e^-2π R |Z_γ'|/R),     as R →∞
	
This shows (<ref>).







If ζ→ζ_0, we take a different path of integration, consisting of 3 parts ℓ_1, ℓ_2, ℓ_3 (see Figure <ref>).



If we parametrize the ℓ_γ' ray as ζ' = -e^t + i Z_γ' = - e^t ζ_0, -∞ < t < ∞, the ℓ_2 part is a semicircle around t = -ϵ and t = ϵ, for small ϵ. The contribution from ℓ_2 is clearly (up to a factor of 2 π i) half of the residue of the function in (<ref>). As in (<ref>), this residue is:

    2 π i exp( i Θ^(ν)(ζ_0) -2 π R |Z_γ'|).


If we denote by exp( i Θ^(ν)(t) ) the evaluation exp( i Θ^(ν)(-tζ_0) ), the contributions from ℓ_1 and ℓ_3 in the integral are of the form

    lim_ϵ→ 0{.    ∫_-∞^-ϵ dt -e^t  + 1/-e^t  - 1exp( i Θ^(ν)(t) ) exp( π R (e^t + e^-t)) 
       . + ∫_ϵ^∞ dt -e^t  + 1/-e^t  - 1exp( i Θ^(ν)(t)) exp( π R (e^t + e^-t)) }


If we do the change of variables t ↦ -t in the first integral, (<ref>) simplifies to

    ∫_0^∞ dt -e^t  + 1/-e^t  - 1[ exp( i Θ^(ν)(t)) - exp( i Θ^(ν)(-t)) ] exp( π R (e^t + e^-t))


(<ref>) is of the type (<ref>), with

    g(ζ') = ζ'+ζ_0/ζ'(ζ'-ζ_0)[ exp( i Θ^(ν)(ζ')) - exp( i Θ^(ν)(1/ζ')) ]


Since ζ_0 = 1/ζ_0, the apparent pole at ζ_0 of g(ζ') is removable and the integral can be estimated by the same steepest descent methods as in (<ref>). The only difference is that the saddlepoint now lies at one of the endpoints. This only introduces a factor of 1/2 in the estimates (see <cit.>). If g(ζ_0) ≠ 0 in this case, the integral is just

    g(ζ_0)/2√(R|Z_γ'|) e^-2π R |Z_γ'|+i  Z_γ' + O( e^-2π R |Z_γ'|/R)


If g(ζ_0) = 0, then the estimate is at least of the order O( e^-2π R |Z_γ'|/R). This finishes the proof of (<ref>).




 §.§ Uniform Estimates on Derivatives


Now let β = (β_1, β_2, β_3, β_4) be a multi-index in ℕ^4, and let D^β be a differential operator acting on the iterations Θ^(ν):

    D^βΘ^(ν)_γ = ∂/∂θ_1^β_1∂θ_2^β_2∂ a^β_3∂a^β_4Θ^(ν)_γ



We need to uniformly bound the partial derivatives of Θ^(ν) on compact subsets:


Let K be a compact subset of U × T^2. Then

    sup_× K D^βΘ^(ν) < C_β,K

for a constant C_β,K independent of ν.


Lemma <ref> is the case |β| := ∑β_i = 0, with ϵ as C_0,K. To simplify notation, we'll drop the K subindex in these constants. Assume by induction we already did this for |β| = k - 1 derivatives and for the first ν≥ 0 iterations, the case ν = 0 being trivial. Take partial derivatives with respect to θ_s, for s = 1, 2 in (<ref>). This introduces a factor of the form

    i∂/∂θ_sΘ^(ν)_γ'


By induction on ν, the above can be bounded by γ'  C_β', where β' = (1,0,0,0) or (0,1,0,0), depending on the index s. When we take the partial derivatives with respect to a in (<ref>), we add a factor of

    π R/ζ'∂/∂ a Z_γ'(a) + i ∂/∂ aΘ^(ν)_γ'

in the integrals (<ref>). Similarly, a partial derivative with respect to a adds a factor of

    π Rζ' ∂/∂aZ_γ'(a) + i ∂/∂aΘ^(ν)_γ'

As in (<ref>), the second term in (<ref>) and (<ref>) can be bounded by γ'  C_β' for |β'| = 1. Since Z_γ' is holomorphic on U ⊂, and since K ⊂ U × T^2 is compact,

    | ∂^k/∂a^k Z_γ'| ≤ k! γ' C

for all k and some constant C, independent of k and a. Likewise for a, Z_γ'. Thus if we take D^βΘ^(ν+1)_γ in (<ref>) for a multi-index β with |β| = k, the right side of (<ref>) becomes:

    -1/4π{∑_γ' > 0 f^γ'∫_r K(ζ,ζ') 𝒳^sf_γ'(a,ζ',Θ^(ν))P_γ'(a,ζ',θ) +  ∑_γ' < 0 f^γ'∫_-r K(ζ,ζ') 𝒳^sf_γ'(a,ζ',Θ^(ν))Q_γ'(a,ζ',θ) },

where each P_γ' or Q_γ' is a polynomial obtained as follows: 

Each 𝒳^sf_γ'(a,ζ',Θ^(ν)) is a function of the type e^g, for some g(a,a̅,θ_1, θ_2). If {x_1, …, x_k} denotes a choice of k of the variables a,a̅, θ_1, θ_2 (possibly with multiplicities), then by the Faà di Bruno Formula:

    ∂^k/∂ x_1 ⋯∂ x_k e^g = e^g ∑_π∈Π∏_B ∈π∂^|B|g/∏_j ∈ B∂ x_j:= e^g P_γ'

where

	
  * π runs through the set Π of all partitions of the set {1, …, k}.
	
  * B ∈π means the variable B runs through the list of all of the “blocks” of the partition π, and
  
  * |B| is the size of the block B.



 The resulting monomials in P_γ' (same thing holds for Q_γ') are products of the variables given by (<ref>), (<ref>), (<ref>) or their subsequent partial derivatives in θ, a, a. For each monomial, the sum of powers and total derivatives of terms must add up to k by (<ref>). For instance, when computing
 
    ∂^3/∂θ_1 ∂ a^2𝒳^sf_γ'(a,ζ',Θ^(ν)) = ∂^3/∂θ_1 ∂ a^2 e^g,

 a monomial that appears in the expansion is:
 
    ∂ g/∂θ_1[ ∂ g/∂ a]^2 = i∂/∂θ_1Θ^(ν)_γ'[ π R/ζ'∂/∂ a Z_γ'(a) + i ∂/∂ aΘ^(ν)_γ']^2

 There are a total of (possibly repeated) B_k monomials in P_γ', where B_k is the Bell number, the total number of partitions of the set {1, …, k} and B_k ≤ k!. We can assume, without loss of generality, that any constant C_β is considerably larger than any of the C_β' with |β'| < |β|, by a factor that will be made explicit. First notice that since there is only one partition of {1, …, k} consisting of 1 block, the Faà di Bruno Formula (<ref>) shows that P_γ' contains only one monomial with the factor D^βΘ^(ν). The other monomials have factors D^β'Θ^(ν) for |β'| < |β|. We can do a saddle point analysis for each integrand of the form
 
    ∫_r K(ζ,ζ') 𝒳^sf_γ'(a,ζ',Θ^(ν))P^i_γ'(a,ζ',θ),

 for P^i_γ' (or Q^i_γ') one of the monomials of P_γ' (Q_γ'). The saddle point analysis and the induction step for the previous Θ^(ν) give the estimate
 
    C_β·const∑_γ'| ⟨γ, f^γ'⟩| e^(-2π R + δ)|Z_γ'|

 for the only monomial with D^βΘ^(ν) on it. The estimates for the other monomials contain the same exponential decay term, along with powers s of C_β', C such that s · |β'| ≤ |β|, and constant terms. By making C_β significantly bigger than the previous C_β', we can bound the entire (<ref>) by C_β, completing the induction step
 




To see better the estimates we obtained in the previous proof, let's consider the particular case k = |β| = 3.  If k = 3, there are a total of 4+3-13 = 20 different third partial derivatives for each Θ^(ν + 1). There are a total of 5 different partitions of the set {1, 2, 3} and correspondingly

    ∂^3 /∂ x_1 ∂ x_2 ∂ x_3 e^g     = 
        e^g [ ∂^3 /∂ x_1 ∂ x_2 ∂ x_3g + (  ∂^2 /∂ x_1 ∂ x_2 g) ( ∂/∂ x_3g) + (  ∂^2 /∂ x_1 ∂ x_3 g) ( ∂/∂ x_2g) . 
       . + (  ∂^2 /∂ x_2 ∂ x_3 g) ( ∂/∂ x_1g) + ( ∂/∂ x_1 g ) ( ∂/∂ x_2 g ) ( ∂/∂ x_3 g ) ]


If x_1 = x_2 = x_3 = a, 

    ∂^3/∂ a^3𝒳^sf_γ'(a,ζ',Θ^(ν))     = 𝒳^sf_γ'(a,ζ',Θ^(ν)) [ π R/ζ'∂^3/∂ a^3 Z_γ' + i ∂^3/∂ a^3Θ^(ν)_γ'.
        + 3 (π R/ζ'∂^2/∂ a^2 Z_γ' + i ∂^2/∂ a^2Θ^(ν)_γ')( π R/ζ'∂/∂ a Z_γ' + i ∂/∂ aΘ^(ν)_γ') 
       . + ( π R/ζ'∂/∂ a Z_γ' + i ∂/∂ aΘ^(ν)_γ')^3 ]
        = 𝒳^sf_γ'(a,ζ',Θ^(ν))P(Θ^(ν)_γ')


There is one and only one term containing ∂^3/∂ a^3Θ^(ν)_γ'. By induction on ν, |∂^3/∂ a^3Θ^(ν)_γ'| < γ'  C_β. For the estimates of 

    i f^γ'∫_r K(ζ,ζ') 𝒳^sf_γ'(a,ζ',Θ^(ν))∂^3/∂ a^3Θ^(ν)_γ',

we do exactly the same as in the proof of Lemma <ref>. Namely, move the ray r to the corresponding BPS ray ℓ_γ', possibly at the expense of gaining a residue bounded by

    C_β·const|  f^γ'| e^(-2π R + δ)|Z_γ'|

The sum of all these residues over those γ' such that ⟨γ, γ'⟩≠ 0 is just a fraction of C_β. After moving the contour we estimate

    i f^γ'∫_ℓ_γ' K(ζ,ζ') 𝒳^sf_γ'(a,ζ',Θ^(ν))∂^3/∂ a^3Θ^(ν)_γ'

As in (<ref>), we run a saddle point analysis and obtain a similar estimate (<ref>) as in Lemma <ref>. The result is that the estimate for this monomial is an arbitrarily small fraction of C_β.

If we take other monomials, like say

    P^1_γ' = 3 ( π R/ζ')^2 ∂^2/∂ a^2 Z_γ'∂/∂ a Z_γ'

and estimate

    3 f^γ'∂^2/∂ a^2 Z_γ'∂/∂ a Z_γ'∫_r ( π R/ζ')^2 K(ζ,ζ') 𝒳^sf_γ'(a,ζ',Θ^(ν)),

we do as before, computing residues and doing saddle point analysis. The difference with these terms is that partial derivatives of Z_γ' are bounded by (<ref>), and at most second derivatives of Θ^(ν) (for this specific monomial, there are no such terms) appear. The extra powers of π R/ζ' that appear here don't affect the estimates, since 𝒳^sf_γ' has exponential decay on π R/ζ'. The end result is an estimate of the type

    C_β'_1^s_1⋯ C_β'_m^s_m C^j ·const| f^γ'| e^(-2π R + δ)|Z_γ'|

with all s_i · |β'_i| and j ≤ |β|. By induction on |β|, we can make C_β big enough so that (<ref>) are just a small fraction of C_β. This completes the illustration of the previous proof for β = (0,0,3,0) of the fact that sup |D^βΘ^(ν + 1)| < C_β on the compact set K.


Now we're ready to prove the main part of Theorem <ref>, that of the existence of solutions to the Riemann-Hilbert problem.

The sequence {Θ^(ν)} converges in X. Moreover, its limit Θ is piecewise holomorphic on ζ with jumps along the rays r, -r and continuous on the closed half-planes determined by these rays. Θ is C^∞ on a, a, θ_1,  θ_2.


We first show the contraction of the Θ^(ν) in the Banach space X thus proving convergence. We will use the fact that e^x is locally Lipschitz and the Θ^(ν) are arbitrarily close to θ if R is big. In particular,

    sup_ζ,θ,a| e^iΘ_γ^(ν) - e^iΘ_γ^(ν-1)| < const·sup_ζ,θ,a| Θ_γ^(ν) - Θ_γ^(ν-1)| ≤constΘ^(ν) - Θ^(ν-1),

for γ one of the basis elements γ_1,  γ_2. For arbitrary γ', recall that if γ' =
 c_1 γ_1 + c_2 γ_2, then Θ_γ'^(ν) = c_1 Θ_γ_1^(ν) +  
c_2Θ_γ_2^(ν). It follows from the last inequality that

    sup_ζ,θ| e^iΘ_γ'^(ν) - e^iΘ_γ'^(ν-1)| < const^γ' Θ^(ν) - Θ^(ν-1)


We estimate

    Θ^(ν+1) - Θ^(ν)  =    1/4π∑_γ'>0 f^γ'∫_r K(ζ,ζ') [ 𝒳^sf_γ'(a,ζ',Θ^(ν)) -
     𝒳^sf_γ'(a,ζ',Θ^(ν-1)) ].
       . + ∑_γ'<0 f^γ'∫_-r K(ζ,ζ') [ 𝒳^sf_γ'(a,ζ',Θ^(ν))-𝒳^sf_γ'(a,ζ',Θ^(ν-1)] 
       ≤1/4π∑_γ'>0 f^γ'∫_r K(ζ,ζ') | 𝒳^sf_γ'(a,ζ',θ)|| e^iΘ_γ'^(ν) -  e^iΘ_γ'^(ν-1)| 
        + 1/4π∑_γ'<0 f^γ'∫_r K(ζ,ζ') | 𝒳^sf_γ'(a,ζ',θ)|  | e^iΘ_γ'^(ν) -  e^iΘ_γ'^(ν-1)|

As in the proof of Lemma <ref>, we can move the integrals to the rays ℓ_γ' introducing an arbitrary small contribution from the residues. The differences of the form

    | e^iΘ_γ'^(ν) -  e^iΘ_γ'^(ν-1)|

can be expressed in terms of Θ^(ν) - Θ^(ν-1) by (<ref>).


The sum of the resulting integrals can be made arbitrarily small if R is big by a saddle point analysis as from (<ref>) onwards. By (<ref>):

    Θ^(ν+1) - Θ^(ν)    < const∑_γ' f^γ' e^(-2π R + δ)|Z_γ'|Θ^(ν) - Θ^(ν-1),

By making R big, we get the desired contraction in X and the convergence is proved.

The holomorphic properties of Θ on ζ are clear since Θ solves the integral equation (<ref>) and the right side of it is piecewise holomorphic, regardless of the integrand.

Finally, by Lemma <ref>, {D^βΘ^(ν)} is an equicontinuous and uniformly bounded family on compact sets K for any differential operator D^β as in (<ref>). By Arzela-Ascoli, a subsequence converges uniformly and hence its limit is of type C^k for any k. Since we just showed that Θ^(ν) converges, this has to be the limit of any subsequence. Thus such limit Θ must be of type C^∞ on U × T^2, as claimed.


By Theorem <ref>, the functions 𝒴_k(a, ζ, θ)  := 𝒳^sf_k(a, ζ, Θ), k = 1, 2 satisfy (<ref>) and condition (<ref>). It remains to show that the functions also satisfy the reality conditions.


For 𝒴_k(a, ζ, θ) defined as above and with γ = c_1 γ_1 + γ_2 ∈Γ, we define 𝒴_γ = 𝒴_1^c_1𝒴_2^c_2. Then

    𝒴_-γ(-1/ζ) = 𝒴_γ(ζ)



Ignoring the parameters a, θ_1, θ_2 for the moment, it suffices to show 

    Θ_k(-1/ζ) = Θ_k(ζ),      k = 1, 2

We show that this is true for all Θ^(ν) defined as in (<ref>) by induction on ν. For ν = 0, Θ^(0) = (θ_1, θ_2) which are real torus coordinates and independent of ζ, so (<ref>) is true. 

Assuming (<ref>) is true for ν, we obtain Θ^(ν+1) as in (<ref>). If we write ζ as te^iφ, t > 0 for some angle φ, and if we parametrize the admissible ray r as se^iρ, s > 0, then (<ref>) for ν+1 follows by induction and by rewriting the integrals in (<ref>) after the reparametrization s →1/s. An essential part of the proof is the form of the symmetric kernel

    K(ζ, ζ') = dζ'/ζ'ζ' + ζ/ζ' - ζ

which inverts the roles of 0 and ∞ after the reparametrization.


To verify the last property of 𝒴_k, we prove

 
For 𝒴_k(a, ζ, θ) defined as above

    lim_ζ→ 0𝒴_γ(ζ) / 𝒳^sf_γ(ζ)

 exists and is real.


Write Θ_k^0 for lim_ζ→ 0Θ_k. In a similar way we can define Θ_k^∞. It suffices to show that Θ_k^0 - θ_k is imaginary. This follows from Lemma <ref> by letting ζ→ 0.

Observe that this and the reality condition give

    Θ_k^0 = Θ_k^∞


To finish the proof of Theorem <ref>, we apply the classical arguments: given two solutions 𝒴_k, 𝒵_k satisfying the conditions of the theorem, the functions 𝒴_k 𝒵^-1_k are entire functions bounded at ∞, so this must be a constant. By the reality condition <ref>, this constant must be real. This finishes the proof of Theorem <ref>.








§ SPECIAL CASES


In our choice of admissible rays r, -r, observe that due to the exponential decay of 𝒳^sf_k, k = 1, 2 along these rays (see (<ref>)) and the rays ℓ_γ, the jumps S_ℓ or S_r, S_-r are asymptotic to the identity transformation as ζ→ 0 or ζ→∞ along these rays. Thus, one can define a Riemann-Hilbert problem whose contour is a single line composed of the rays r, -r, the latter with orientation opposite to the one in the previous section. The jump S along the contour decomposes as S_r, S_-r^-1 in the respective rays and we can proceed as in the previous section with a combined contour.



 §.§ Jump Discontinuities


In <cit.>, we will be dealing with a modification of the Riemann-Hilbert problem solved in <ref>. In particular, that paper deals with the new condition


  (5')  Z_γ_2(0) ≠ 0 for any a in U but Z_γ_1 attains its unique zero at a = 0.


Because of this condition, the jumps loose the exponential decay along those rays and they are no longer asymptotic to identity transformations. In fact, in <cit.> we show that this causes the jump function S(ζ) to develop a discontinuity of the first kind along ζ = 0 and ζ = ∞.

In this paper we obtain the necessary theory of scalar boundary-value problems to obtain solutions to this special case of Riemann-Hilbert problems appearing in <cit.>. We consider a general scalar boundary value problem consisting in finding a sectionally analytic function X(ζ) with discontinuities at an oriented line ℓ passing through 0. If X^+(t) (resp. X^-(t)) denotes the limit from the left-hand (resp. right-hand) side of ℓ, for t ∈ℓ, they must satisfy the boundary condition

    X^+(t) = G(t) X^-(t),      t ∈ℓ

for a function G(t) that is Hölder continuous on ℓ except for jump discontinuities at 0 and ∞. We require a symmetric condition on these singularities: if Δ_i, i = 0 or ∞ represents the jump of the function G near any of these points,

    Δ_0 = lim_t → 0^+ G(ζ) - lim_t → 0^- G(ζ),     etc.

Then we assume

    Δ_0 = -Δ_∞

Near 0 or ∞, we require for the analytic functions X^+(ζ), X^-(ζ) to have only one integrable singularity of the form

    |X^±(ξ)| < C/|ξ|^η,      (0 ≤η < 1)

For ξ a coordinate of  centered at either 0 or ∞. By (<ref>), each function X^± is asymptotic to 0 near the other point in the set {0, ∞}.


There exists functions X^+(ζ), X^-(ζ), analytic on opposite half-planes on  determined by the contour ℓ and continuous on the closed half-planes such that, along ℓ, the functions obey (<ref>) and (<ref>), with a Hölder continuous jump function G(t) satisfying (<ref>). The functions X^+(ζ), X^-(ζ) are unique up to multiplication by a constant.


We follow <cit.> for the solution of this exceptional case. As seen above, we only have jump discontinuities at 0 and ∞. For any point t_0 in the contour ℓ, and a function f with discontinuities of the first kind on ℓ at t_0, we denote by f(t_0 - 0) (resp. f(t_0+0)) the left (resp. right) limit of f at t_0, according to the given orientation of ℓ.

Let

    η_0 = 1/2π ilogG(0-0)/G(0+0)

Similarly, define

    η_∞ = 1/2π ilogG(∞-0)/G(∞+0)

 Since G obeys condition <ref>, η_0 = - η_∞. Observe that by definition, |η_0| < 1, and hence the same is true for η_∞.
 
Let D^+ be the region in  bounded by ℓ with the positive, counterclockwise orientation. Denote by D^- the region where ℓ as a boundary has the negative orientation. We look for solutions of the homogeneous boundary problem (<ref>) . To solve this, pick a point ζ_0 ∈ D^+ and introduce two analytic functions 

    (ζ - ζ_0)^η_0,     ζ^η_0

Make a cut in the ζ-plane from the point ζ_0 to ∞ through 0, with the segment of the cut from ζ_0 to 0 wholly in D^+. Consider the functions 

    ω^+(ζ) = ζ^η_0,     ω^- = ( ζ/ζ - ζ_0)^η_0

Due to our choice of cut, ω^+ is analytic in D^+ and ω^- is analytic in D^-. Introduce new unknown functions Y^± setting

    X^± (ζ) = ω^± (ζ) Y^± (ζ)

The boundary condition (<ref>) now takes the form

    Y^+(t) = G_1(t) Y^-(t),      t ∈ℓ

where

    G_1(t) = ω^-(t)/ω^+(t) G(t) = (t - ζ_0)^-η_0 G(t),      t ∈ℓ

By the monodromy of the function (ζ - ζ_0)^-η_0 around 0 and infinity and since η_∞ = -η_0, it follows that G_1 is continuous in the entire line ℓ. Hence, we reduced the problem (<ref>) to a problem (<ref>) with continuous coefficient, which can be solved with classical Cauchy integral methods.

By assumption, we seek solutions of (<ref>) with only one integrable singularity i.e. estimates of the form (<ref>). The notion of index (winding number) for G(t) in the contour ℓ is given by (see <cit.>) ϰ = ⌊η_0⌋ +  ⌊η_∞⌋  + 1 = 0, so the usual method of solution of (<ref>) as

    Y = exp( 1/2π i∫_ℓ K(ζ',ζ) log G_1(ζ') )

(for a suitable kernel K(ζ',ζ) that makes the integral along ℓ convergent) needs no modification. We can also see from (<ref>) that X^± has an integrable singularity at 0 (resp. ∞) if η_0 is negative (resp. positive).
 
 We need to show that for different choices of ζ_0 ∈ D^+ the solutions X only differ by a constant. To see this, by taking logarithms in (<ref>) it suffices to show uniqueness of solutions to the homogeneous additive boundary problem
 
    Φ^+(t) - Φ^-(t) = 0,      t ∈ℓ

 and with the assumption that Φ vanishes at a point and at the points of discontinuity of G(ζ), Φ^± satisfies an estimate as in (<ref>). The relation (<ref>) indicate that the functions Φ^+, Φ^- are analytically extendable through the contour ℓ and, consequently, constitute a unified analytic function in the whole plane. This function has, at worst, isolated singularities but according to the estimates (<ref>), these singularities cannot be poles or essential singularities, and hence they can only be branch points. But a single valued function with branch points must have lines of discontinuity, which contradicts the fact that Φ^+ = Φ^- is analytic (hence continuous) on the entire plane except possibly at isolated points. Therefore, the problem (<ref>) has only the trivial solution.




 §.§ Zeroes of the boundary function


Because of condition <ref>, yet another special kind of Riemann-Hilbert problem arises in <cit.>. We still want to find a
sectionally analytic function X(ζ) satisfying the conditions (<ref>) with G(t) having jump discontinuities at 0, ∞ with
the properties (<ref>) and (<ref>). In this subsection, we allow the case of G(t) having zeroes of integer order on finitely many
points α_1, …, α_μ along ℓ. Thus, we consider a Riemann-Hilbert problem of the form

    X^+(t) = ∏_j = 1^μ (t - α_j)^m_j G_1(t) X^-(t),      t ∈ℓ

where m_j are integers and G_1(t) is a non-vanishing function as in <ref>, still with discontinuities at 0 and ∞ as in (<ref>). 


For a scalar Riemann-Hilbert problem as in <ref> and with G_1(t) a non-vanishing function with discontinuities of the first kind at 0 and ∞ obeying (<ref>), there exist solutions X^±(ζ) unique up to multiplication by a constant. At all points α_j as above, both analytic functions X^+(ζ), X^-(ζ) are bounded and X^+ has a zero of order m_j.


By Lemma <ref>, there exists non-vanishing analytic functions Y^+(ζ), Y^-(ζ) on opposite half-planes D^+, D^- determined by ℓ and continuous along the boundary such that

    G_1(t) = Y^+(t)/Y^-(t),      t ∈ℓ

We can define 

    X^+(ζ)     =  ∏_j = 1^μ (ζ - α_j)^m_j Y^+(ζ) 
    
    X^-(ζ)     =   Y^-(ζ)

This clearly satisfies (<ref>) and, since Y^+ is non-vanishing on D^+, it shows that X^+ has a zero of order m_j at α_j ∈ℓ. To show uniqueness of solutions, note that if X^+, X^- are any solutions to the Riemann-Hilbert problem, we can write the boundary condition (<ref>) in the form

    X^+(t)Y^+(t) ∏_j = 1^μ (t - α_j)^m_j = X^-(t)/Y^-(t),      t ∈ℓ

The last relation indicates that the functions

    X^+(ζ)Y^+(ζ) ∏_j = 1^μ (ζ - α_j)^m_j, X^-(ζ)/Y^-(ζ)

are analytic in the domains D^+, D^- respectively and they constitute the analytical continuation of each other through the contour ℓ. The points α_j cannot be singular points of this unified analytic function, since this would contradict the assumption of boundedness of X^+ or X^-. The behavior of X^± at 0 or ∞ is that of Y^±, so by Liouville's Theorem,

    X^+(ζ)Y^+(ζ) ∏_j = 1^μ (ζ - α_j)^m_j = X^-(ζ)/Y^-(ζ) = C

for C a constant. This forces X^+(ζ), X^-(ζ) to be of the form (<ref>), (<ref>).






amsplain
        


