Fuchsian Equations with Three Non-Apparent Singularities
Alexandre Eremenko Vitaly Tarasov
math.CA math-ph math.MP 34M03, 34M35, 57M50









Fuchsian Equations with Three Non-Apparent Singularities

Fuchsian Equations with Three Non-Apparent
 Singularities

Alexandre EREMENKO ^† and Vitaly TARASOV ^
A. Eremenko and V. Tarasov
^† Purdue University, West Lafayette, IN 47907, USA
mailto:eremenkos@math.purdue.edueremenko@math.purdue.edu
<http://www.math.purdue.edu/ eremenko/>
^ Indiana University – Purdue University Indianapolis, Indianapolis, IN 46202, USA
mailto:vtarasov@iupui.eduvtarasov@iupui.edu
^ St. Petersburg Branch of Steklov Mathematical Institute, St. Petersburg, 191023, Russia
mailto:vt@pdmi.ras.ruvt@pdmi.ras.ru

Received February 02, 2018, in final form June 10, 2018; Published online June 15, 2018

We show that for every second order Fuchsian linear differential equation E with n singularities of which n-3 are apparent there exists a hypergeometric equation H and a linear differential operator with polynomial coefficients which maps the space of solutions of H into the space of solutions of E. This map is surjective for generic parameters. This justifies one statement of Klein (1905). We also count the number of such equations E with prescribed singularities and exponents. We apply these results to the description of conformal metrics of curvature 1 on the punctured sphere with conic singularities, all but three of them having integer angles.

Fuchsian equations; hypergeometric equation; difference equations; apparent singularities; bispectral duality; positive curvature; conic singularities

34M03; 34M35; 57M50



§ INTRODUCTION


We consider Fuchsian linear differential equations of second order such that all but three singularities are apparent. A singularity is called apparent if the ratio of two linearly independent solutions is meromorphic at the singularity. Our main result says that all such equations can be obtained from hypergeometric equations by certain linear differential operators. The result was hinted by Klein at the end of his lecture on November 29, 1905 <cit.>, without a proof or a precise formulation. F. Shilling <cit.> investigated the case of one apparent singularity with exponent difference 2.

Our motivation is the following. In <cit.>, we studied conformal Riemannian metrics of constant curvature 1 on the punctured sphere with conic singularities at the punctures. The question is how many such metrics with prescribed singularities and prescribed conic angles at the singularities exist, and how to describe all of them. Unlike the similar problem with non-positive curvature where a complete answer is known <cit.>, this problem is wide open, see, for example the survey in the introduction to <cit.>. The developing map of such a metric is a ratio of two linearly independent solutions of a Fuchsian differential equation (see Section <ref>). Singularities of this equation are the conic singularities of the metric, and the exponent difference α at a singularity corresponds to the conic angle 2πα of the metric. Another condition on this equation is that the projective monodromy group is conjugate to a subgroup of PSU(2)= SU(2)/{± I}. We call such projective monodromy groups unitarizable.

So the question about the metrics is equivalent to the following:

For given a_j∈ and positive α_j, 1≤ j≤ n, how many Fuchsian differential equations of the form

    w”+(∑_j=1^n-11-α_j/z-a_j)w'+Az^n-3+⋯/(z-a_1)⋯(z-a_n-1)w=0

with unitarizable projective monodromy exist?

Here we assume without loss of generality that a_n=∞ and the prescribed exponent difference at ∞ is α_n; together with the Fuchs relation this determines the constant A. The dots in the numerator of the third summand of (<ref>) stand for a polynomial of degree n-4 whose n-3 coefficients are called the accessory parameters. These accessory parameters must be determined from the condition that the projective monodromy is unitarizable. This gives a system of equations on these parameters, and the question is how many solutions it has.

The difficulty of this problem strongly depends on the number of non-integer[Here and in what follows “integer angle” means an integer multiple of 2π radians.] angles α_j. The condition of unitary projective monodromy implies that at a singular point with integer exponent difference the projective monodromy must be trivial, that is this singular point must be apparent.

A complete solution of this problem is only known when n=3, <cit.>. The case when n is arbitrary and all angles are integers was studied in <cit.>, and the case where there are only two non-integer
angles in <cit.>. The case when n=4 and there are three non-integer angles was investigated in <cit.> under the additional assumption that equation (<ref>) is real. In terms of the metric this means that the metric is symmetric with respect to some circle which contains all singularities. The problem of description of symmetric metrics is equivalent to the problem of classification of spherical n-gons of prescribed conformal type with prescribed angles, and this was the main subject of Klein's lectures <cit.>.

Theorem <ref> of this paper gives a classification of metrics of constant positive curvature with conic singularities on the sphere in the case when only three singularities have non-integer angles, for generic singularities and generic non-integer angles.

The word generic is used in this paper in the sense of Zariski topology: a set in ^n is generic if its complement is contained is a proper algebraic subset.



§ STATEMENT OF RESULTS


Let a_j be distinct complex numbers, a_j∉{0,1} for 1≤ j≤ k. Let m_j be positive integers, and α, β, γ, δ complex numbers. Consider the differential equation

    y”-( α/z+β/z-1+∑_i=1^k m_i/z-a_i) y'+γδ z^k+⋯/z(z-1)∏_i=1^k(z-a_i)y = 0,

where γ+δ=α+β+1+∑_j=1^km_j (Fuchs's condition). The Riemann scheme of this equation is

    (
    0   1   a_1   …   a_k   ∞
    
    0   0   0   …   0   -γ
    α+1   β+1   m_1+1   …   m_k+1   -δ).

We recall that the first line of the Riemann scheme contains the singularities, and the other two the exponents at each singularity <cit.>. Set d=∑_j=1^km_j. Assume that the singularities at a_1 a_k are all apparent and

    γ, δ, γ-α-1, δ-α-1 ∉{0,1 d-1}.



Under these conditions, there exists a polynomial Q of degree d=∑_j=1^km_j such that every solution of (<ref>) has the form Q(z d/ dz)F(z), where F is a solution of the hypergeometric equation

    F”-( α/z+β+d/z-1) F'+γδ/z(z-1) F = 0.

In particular, the monodromy group of equation (<ref>) coincides with that of the hypergeometric equation (<ref>).


 Clearly, conditions (<ref>) imply that for any non-zero polynomial p of degree at most d-1, the functions p(z) and z^α+1p(z) are not solutions of the hypergeometric equation (<ref>).


 Under conditions (<ref>), for any non-zero polynomial Q of degree at most d and any non-zero solution F of (<ref>), the function Q(z d/ dz)F(z) is non-zero. Indeed, a solution F of (<ref>) such that Q(z d/ dz)F(z)≡ 0, has the following form: there are polynomials p_1, p_2 of degrees at most d-1 and constants A,B, such that

    F(z) = p_1(z) + z^αp_2(z)           if α is not an integer,   

    F(z) = p_1(z) + z^αp_2(z) (A+Blogz)           if α is a non-negative integer,   

    F(z) = z^αp_1(z) + p_2(z) (A+Blogz)           if α is a negative integer.    

Then in the first case, both p_1(z) and z^α p_2(z) are solutions of equation (<ref>) in contradiction with Remark <ref>. In the second case, z^α p_2(z) is a solution of (<ref>) if B 0, while for B=0, F(z) is a polynomial and F<d since Q(z d/ dz)F(z)=0. Both options contradict Remark <ref>. The third case can be worked out similarly.



For every α, β, γ, δ, m_1 m_k, a_1 a_k, there exist at least one and at most ∏_j=1^k(m_j+1) equations (<ref>) with apparent singularities at all points a_j. For generic α, β, γ, δ, a_1 a_k, the equality holds.


 Let positive numbers (α_1α_n) be given, where α_1, α_2, α_3 are not integers, and all combinations

    α_1±α_2±α_3

for any choice of signs, while α_4α_n are integers, α_j≥ 2. Let the points (z_1 z_n) in  be given. Then the necessary and sufficient condition of existence of a conformal metric of curvature 1 with conic singularities at z_j with angles 2πα_j is

    cos^2 πα_1+cos^2 πα_2+cos^2 πα_3+2(-1)^σcosπα_1 cosπα_2 cosπα_3 < 1,

where

    σ =∑_j=4^n(α_j-1).

The number of such distinct metrics does not exceed the product α_4⋯α_n, with equality for generic α_1, α_2, α_3 and z_j.


The necessary and sufficient condition in this theorem was found in <cit.> in the special case of symmetric metrics. The case when (<ref>) does not hold was called exceptional in <cit.>. In this case the unitarizable projective monodromy must be co-axial, that is conjugate to a subgroup of the unit circle <cit.>, and generic singularities z_1,…,z_n cannot be assigned in the co-axial case <cit.>. Moreover, in the coaxial case, existence of one metric implies the existence of a continuous family of them.

Condition (<ref>) is a special case of the Mondello and Panov necessary condition on the angles of arbitrary metric of positive curvature with conic singularities and non-coaxial monodromy.



§ PROOF OF THEOREM <REF>


It will be convenient to use a different normalization for our differential equation. By a linear-fractional transformation of the independent variable in (<ref>) we place the singular points in , so that they become

    (z_1,…,z_n,∞),

where n=k+3, and infinity is an apparent singularity with exponent difference 1. Then we multiply the dependent variable on some function to shift the exponents and obtain the Riemann scheme

    (z_1   z_2   …   z_n   ∞
    
    1+α_1/2   1+α_2/2   …   1+α_n/2    0
    
    1-α_1/2   1-α_2/2   …   1-α_n/2   -1),

where α_1=α+1, α_2=β+1, α_3=m_1+1α_n-1=m_k+1, α_n=γ-δ. This transformation changes neither the exponent differences nor the projective monodromy group, and the new equation has the form

    w”+∑_j=1^n(1-α_j^2/4(z-z_j)^2 +β_j/z-z_j)w=0.

Regularity of the point ∞ implies

    ∑_j=1^nβ_j=0,
    ∑_j=1^nβ_jz_j=∑_j=1^∞(1-α_j^2)/4,

and the condition that the formal solution at ∞ contains no logarithm is

    ∑_j=1^nβ_jz_j^2=∑_j=1^n z_j(1-α_j^2)/2.

When α_j is an integer, the singularity at z_j may be apparent. The test for the apparent singularity can be found, for example in <cit.>. Write our equation near z_j as

    (z-z_j)^2w”+(∑_k=0^∞ x_k,j(z-z_j)^k)w=0.

The singularity at z_j is apparent if and only if
α_j=ℓ_j is an integer,

    x_0,j=(1-ℓ_j^2)/4,

and

    Y_ℓ_j(x_1,j x_ℓ_j,j)=0,

where for any ℓ, the polynomial Y_ℓ is defined by

    Y_ℓ(x_1 x_ℓ)=
    |x_1   1·(1-ℓ)   0   …   0
    
    x_2   x_1   2·(2-ℓ)   …   0
    ⋮   ⋮   ⋮   ⋱   ⋮
    
    x_ℓ-1   x_ℓ-2   x_ℓ-3   …   (ℓ-1)·(-1)
    
    x_ℓ   x_ℓ-1   x_ℓ-2   …   x_1
    |.

The matrix has x_1 on the main diagonal, x_2 on the next diagonal below the main one, etc., and the given numbers on the next diagonal above the main one.

Notice that the polynomial Y_ℓ has the form

    Y_ℓ(x_1 x_ℓ)=x_1^ℓ+Y^*_ℓ(x_1 x_ℓ),     Y^*_ℓ< ℓ.


Suppose that all singularities z_j and exponents α_j are fixed, and that α_3α_n-1 are integers, α_j=ℓ_j. Then the number of such equations with apparent singularities at z_3 z_n-1 is finite.


 We take β_3β_n-1 as variables, and express β_1, β_2, β_n in terms of these variables, using (<ref>)–(<ref>). (The determinant of this 3×3 system with respect to β_1, β_2, β_n is the Vandermonde determinant, so it is not zero.) Then for each j∈[3,n-1] the variables x_k,j become linear functions of β_3β_n-1. Noticing that x_1,j=β_j, we obtain that in terms of β_j, 3≤β_j≤ n-1, our equations (<ref>), (<ref>) have the form

    β_j^ℓ_j=S_j(β_3β_n-1),     3≤ j≤ n-1,

where S_j are polynomials of degrees at most ℓ_j-1. Thus

    |β_j|≤ C(1+β^1-1/ℓ_j)≤ C_1(1+β^1-1/ℓ),    ℓ=max_jℓ_j,

where β^2:=β_3^2+⋯+β_n-1^2. Hence, β≤ C_2. As a bounded algebraic set must be finite, this proves the lemma.

So the condition that singularities z_3 z_n-1 of equation (<ref>) are apparent is expressed as n-3 equations (<ref>) with n-3 unknowns β_3β_n-1. Adding one extra variable β_0, we make these equations homogeneous:

    β_j^ℓ_j=β_0 S_j(β_0,β_3β_n-1),     3≤ j≤ n-1,

with some homogeneous polynomials S of degree ℓ_j-1. Evidently no non-zero solution can have β_0=0, so Bezout's theorem <cit.> implies that the number of solutions
of (<ref>), counting multiplicity, equals the product of degrees ℓ_3⋯ℓ_n-1=α_3⋯α_n-1.

Now we show that for generic α_1, α_2, α_n, solutions of (<ref>) are simple. To do this, we specify α_1, α_2, α_n to be large integers, and show that in this case, there are at least α_3⋯α_n-1 solutions. It will follow that for generic α_1, α_2, α_n, solutions are of multiplicity 1, because the set of all large integers is Zariski dense.

Solutions of (<ref>) correspond to equations (<ref>) with trivial local projective monodromy around z_3 z_n-1. The number of such equa­tions (<ref>) is at least the number of equations (<ref>) with trivial projective monodromy around all singularities, and ratios of solutions of such equations are rational functions with critical points at z_1 z_n of multiplicities α_j-1 at z_j. Rational functions obtained from the same differential equation are equivalent in the sense that they are obtained from each other by linear-fractional transformations.


So we prove


Suppose that

    α_1+α_n≥ (1/2)∑_j=1^n(α_j-1)+2,    α_2≥∑_j=3^n-1(α_j-1)+1.

Then the number of equivalence classes of rational functions with critical points at generic points z_j of multiplicities α_j-1, for 1≤ j≤ n is the product α_3⋯α_n-1.


As a corollary we obtain that trivial local monodromy around z_4 z_n implies that the whole projective monodromy group is trivial, assuming that α_1, α_2, α_3 are large enough in comparison with
other α_j.

It is known <cit.> that classes of rational functions with prescribed generic critical points are enumerated by the Young tableaux of shape 2× (d-1), where

    d=1/2(∑_j=1^n(α_j-1)+2)

is the degree of the rational function.

The diagrams are filled with α_j-1 numbers j by the usual rules: the entries do not decrease in rows (left to right) and strictly increase in columns. It follows from this rule that 1's can be only in the first row on the left, while n's in the second row in the right. Condition (<ref>) implies that two entries other than 1, n cannot stand in the same column: they occupy the place in the first row on the right, and in the second row on the left. There is no other restriction on placement of these entries (from 2 to n-1), except that they must be non-decreasing in their rows. So the tableau is completely determined by specifying the list of entries from 2 to n-1 in the first row. The possible number of k's (3≤ k≤ n-1) in the first row is α_k, and the 2's occupy the remaining places. The second assumption of the lemma ensures that there are always enough of 2's to fill the remaining places. So we have α_3⋯α_n-1 possibilities.

This proves Lemma <ref> and completes the proof of Theorem <ref>.




§ PRELIMINARIES ON DIFFERENCE EQUATIONS


In the proof of Theorem <ref> we use the tools developed in <cit.> which are explained in this section. Our exposition is formally independent of <cit.>.

Denote by τ be the shift operator, (τ f)(x)=f(x+1) acting on meromorphic functions on the real line. For functions u_1(x) u_k(x), define their Casorati determinant (discrete Wronskian) by

    _k[u_1 u_k] = (τ^j-1u_i)_i,j=1^k.

By convention, _0=1.

For given functions u_1(x) u_k(x) such that _k[u_1 u_k] 0 and a non-zero function F(x), there exists a unique difference operator of the form

    D_u_1 u_k;F:=τ^k+1+∑_j=1^k K_j(x)τ^j+F(x)

such that D_u_1 u_k;Fu_i=0 for every i=1 k.


 For given u_1 u_k and F, consider a system of linear equations

    ∑_j=1^kK_j τ^ju_i=-τ^k+1u_i-Fu_i,     i=1 k,

for the coefficients K_1 K_k of D_u_1 u_k;F. The determinant of this linear system equals τ(_k[u_1 u_k]) 0, so the system has a unique solution.



For every function f(x), we have

    D_u_1 u_k; F f = τ(_k+1[u_1 u_k,f])+ (-1)^kF_k+1[u_1 u_k,f]/τ(_k[u_1 u_k]).



 Denote by D the difference operator whose action on f is given by the right-hand side of (<ref>). Then Du_i=0 for every i=1 k, and D is of the form (<ref>) with some coefficients K_1 K_k. Hence, D=D_u_1 u_k; F, by Lemma <ref>.


 D_u_1 u_k; F=(τ-g_k+1) (τ-g_k) ⋯ (τ-g_1), where

    g_i = τ f_i/f_i,     f_i = _i[u_1 u_i]/_i-1[u_1 u_i-1],
         i=1 k,

and

    g_k+1 = (-1)^k+1F_k[u_1 u_k]/τ(_k[u_1 u_k]).


Set F_i=(-1)^i τ(_i[u_1 u_i])/ _i[u_1 u_i] for i=1 k, and F_k+1=F. Let D_i=(τ-g_i) … (τ-g_1). We will prove by induction on i, that D_i=D_u_1 u_i-1; F_i for any i=1 k+1.

The base of induction for i=1 is immediate. Clearly, D_i=τ^i+∑_j=1^i-1 K_i,jτ^j+F_i for some coefficients K_i,1 K_i,i-1. Thus by Lemma <ref>, it remains to show that D_iu_j=0 for any j=1 i-1. Since D_i=(τ-g_i) D_i-1, it suffices to show that D_i-1u_j=0 for any j=1 i-1. This follows from Lemma <ref> since D_i-1=D_u_1 u_i-2; F_i-1 by the induction assumption.


 Let u_1(x) u_k(x) be solutions of the difference equation

    ∑_i=0^k+1 A_i(x) u(x+i) = 0,     A_0 0,     A_k+1 0.

Let w(x) be a non-zero solution of the difference equation A_0(x) w(x)=(-1)^k+1A_k+1(x) w(x+1). Then the function

    v(x) = _k[u_1 u_k](x+1)/A_0(x) w(x)

satisfies the difference equation

    ∑_i=0^k+1 A_i(x-i) v(x-i) = 0.



 If _k[u_1 u_k]=0, the claim is trivial. If _k[u_1 u_k] 0, consider the difference operator D=D_u_1 u_k; K_0 of the form (<ref>) with K_i=A_i/A_k+1, i=0 k. Then the difference equation (<ref>) reads A_k+1 Du=0.

Define the difference operator D^†=τ^-k-1+∑_i=0^kK_i(x-i) τ^-i. Then the difference equa­tion (<ref>) reads D^†(A_k+1v)=0.

The operator D^† is obtained from D by applying the formal conjugation antiautomorphism that sends τ to τ^-1 and the multiplication by x to itself. Thus Lemma <ref> yields

    D^†= (τ^-1-g_1) ⋯(τ^-1-g_k) (τ^-1-g_k+1),

where one takes F=A_0/A_k+1. Straightforwardly, (τ^-1-g_k+1) (A_k+1v)=0, which proves Lemma <ref>.
 

Now we consider our equation (<ref>). Define the polynomial A and numbers b_i,2 by

    A(z)=(z-1)∏_i=1^k(z-a_i)= z^k+1+∑_i=1^k b_i,2z^i+(-1)^k+1a_1⋯ a_k,

and

    W(z)=z^α(z-1)^β∏_i=1^k(z-a_i)^m_i.

Multiply equation (<ref>) by z^2A(z), so that it becomes

    A(z) (z d/ dz)^ 2y-B(z) z dy/ dz+C(z)y=0,

where

    B(z) = A(z)(1+zW'(z)/W(z))
     = (γ+δ)z^k+1-∑_i=1^k b_i,1z^i+(-1)^k+1(α+1) a_1⋯ a_k,

with some numbers b_i,1, and C(z) is a polynomial of degree k+1 such that C(0)=0.

Let x be a new variable and τ be the shift: (τ u)(x)=u(x+1). The difference equation

    (x^2A(τ)-xB(τ)+C(τ)) u(x)=0

is called the bispectral dual of equation (<ref>). Bispectral dual equations are discussed in Theorems 4.1 and 4.2 of <cit.>.

Write (<ref>) in the form

    (-1)^k+1a_1⋯ a_k x(x-α-1) u(x) +∑_i=1^k b_i(x) u(x+i) 
         +(x-γ)(x-δ) u(x+k+1) = 0,

where b_1 b_k are quadratic polynomials

    b_i(x)=b_i,2x^2+b_i,1x+b_i,0,

the constants b_i,1 and b_i,2 being defined in (<ref>) and (<ref>), respectively.

The equation

    (A(τ^-1) x^2-B(τ^-1) x+C(τ^-1))v(x)=0

is called the formal conjugate to equation (<ref>). Equivalently, (<ref>) can be written as

    (-1)^k+1a_1⋯ a_k x(x-α-1) v(x) +∑_i=1^k b_i(x-i) v(x-i)
         + (x-k-1-γ)(x-k-1-δ) v(x-k-1) = 0.

If u_1 u_k are solutions of equation (<ref>), then by Lemma <ref>

    v(x) = (a_1⋯ a_k)^-x-1Γ(x-γ) Γ(x-δ)/Γ(x+1) Γ(x-α)(u_i(x+j))_i,j=1^k

is a solution of equation (<ref>).

Equations (<ref>) and (<ref>) are related as follows: if v(x) solves (<ref>) and the series

    f(z) = ∑_n=-∞^∞ v(n+ξ) z^n+ξ

converges for some ξ, then f(z) is a solution of equation (<ref>).



§ PROOF OF THEOREM <REF>


Let y(z) be a solution of equation (<ref>). Define functions

    u_i(x) = ∮_a_iy(z) z^x-α-2 dz/(z-1)^β+1∏_j=1^k(z-a_j)^m_j+1
    = 2π√(-1)_z=a_iy(z) z^x-α-2/(z-1)^β+1∏_j=1^k(z-a_j)^m_j+1,

where the integral is over a small circle around a_i counterclockwise. Clearly, u_i(x)=a_i^xp_i(x), where p_i(x) is a polynomial of degree m_i. Since y(z) solves equation (<ref>), the function

    (z) =
    y(z)/z^α+1(z-1)^β+1∏_j=1^k(z-a_j)^m_j+1 =y(z)/zA(z)W(z)

satisfies the differential equation

    (z d/ dz)^ 2( A(z) )+ z d/ dz(B(z) )+C(z)  = 0.

By formulas (<ref>), (<ref>),
    u_i(x)=∮_a_i(z) z^x-1 dz,

and equation (<ref>) yields that u_1 u_k are solutions of equation (<ref>).

Use functions (<ref>) in formula (<ref>) for a solution v(x) of equation (<ref>). Then

    v(x) = Γ(x-γ) Γ(x-δ)/Γ(x+1) Γ(x-α) Q(x),

where

    Q(x) = (a_1⋯ a_k)^-x-1(u_i(x+j))_i,j=1^k=
     (a_i^j-1p_i(x+j))_i,j=1^k

is a non-zero polynomial of degree d=m_1+…+m_k.

To prove Theorem <ref>, it suffices to show that for any solution F(z) of (<ref>), the function Q(z d/ dz)F(z) solves (<ref>) because under conditions (<ref>) the function Q(z d/ dz)F(z) is non-zero, see Remark <ref> after Theorem <ref>. Moreover, it is enough to prove this statement for generic values of α, γ, δ. Indeed, by the proof of Theorem <ref>, equation (<ref>) can be deformed to generic values of α, β, γ, δ without moving the singularities at a_1 a_k and keeping all of them apparent. To make F and Q change continuously under the deformation, one can parametrize F and a solution y(z) of (<ref>) defining Q by their data at a fixed regular point.

For generic α, γ, δ, the functions

    F_1(z) = ∑_n=0^∞Γ(n-γ) Γ(n-δ)/Γ(n+1) Γ(n-α) z^n = Γ(-α)/Γ(-γ) Γ(-δ) _2F_1(-γ,-δ;-α;z)

and

    F_2(z)= ∑_n=0^∞Γ(n+1+α-γ) Γ(n+1+α-δ)/Γ(n+2+α) Γ(n+1) z^n+1+α
     = Γ(2+α)/Γ(1+α-γ) Γ(1+α-δ) z^1+α _2F_1(1+α-γ,1+α-δ;2+α;z)

are independent solutions of the hypergeometric equation (<ref>), see for example <cit.>. The functions

    f_1(z) = Q(z d/ dz) F_1(z) = ∑_n=0^∞Γ(n-γ) Γ(n-δ)/Γ(n+1) Γ(n-α) Q(n) z^n,
    
    f_2(z) = Q(z d/ dz) F_2(z) = ∑_n=0^∞Γ(n+1+α-γ) Γ(n+1+α-δ)/Γ(n+2+α) Γ(n+1) Q(n+1+α) z^n+1+α,

coincide with the series (<ref>) for function (<ref>) and ξ=0 or ξ=1+α. Since the series converge for |z|<1, both f_1(z) and f_2(z) are solutions of (<ref>).

This proves Theorem <ref>.



§ APPLICATION TO METRICS OF POSITIVE CURVATURE


Derivation of Theorem <ref> from Theorems <ref> and <ref> consists of standard arguments which we outline here.

Suppose that we have a Riemannian metric of curvature 1 on the punctured sphere S\{ z_1 z_n}, and conic singularities at z_j. Conic singularity at z_j means that a neighborhood of z_j is isometric to a cone with intrinsic angle 2πα_j>0 at the vertex, and curvature 1 at all other points. It is well-known that a surface of constant curvature 1 is locally isometric to the standard unit sphere. Consider this local isometry f in a neighborhood of a non-singular point. As f is an isometry, it is conformal, thus analytic, and admits an analytic continuation along any curve which does not pass through the singularities. We obtain a multi-valued developing map

    f S\{ z_1 z_n}→.

Developing map extends continuously (in the sense of radial limits) to the singularities. As the sphere has only one conformal structure, S is also the Riemann sphere. The monodromy of our map consists of rotations of the sphere which can be represented by fractional linear transformations in PSU(2). The Schwarzian derivative

    { f,z}:=f”'/f'-3/2(f”/f')^2

remains unchanged when f undergoes a fractional-linear transformation, so it is a single-valued analytic map from S\{z_1 z_n} to . Consideration near a singularity shows that f(z)=(c+o(1))z^α in the local coordinate with angle 2πα. Computing the Schwarzian we obtain that it has a double pole with the principal term

    1-α^2/2z^2,

so we have a third order differential equation for f:

    { f,z}=∑_j=1^n1-α_j^2/2(z-z_j)^2+2β_j/z-z_j.

When f is a developing map, the monodromy of this differential equation must be unitarizable, which means that there is at least one solution with monodromy in PSU(2). Developing maps correspond to the
same metric if they are related by f_1=L∘ f_2 where L∈ PSU(2). The general solution of (<ref>) consists of fractional-linear transformations of one solution.

So different solutions of the same equation can define different metrics if they are related by f_1=L∘ f_2 where L satisfies the condition that ϕ^-1Lϕ is unitary for all ϕ from the projective monodromy group. This is only possible when L itself is in PSU(2) or when the monodromy is coaxial.

Finally we mention that the general solution of (<ref>) is the ratio of two linearly independent solutions of (<ref>).

 Suppose that condition (<ref>) holds. Then Theorem <ref> is applicable. By Theorem <ref>, equations (<ref>) and (<ref>) have the same monodromy. Monodromy of the hypergeometric equation (<ref>) depends only on the exponents, and under the assumption (<ref>), it is unitarizable if and only if the exponents satisfy (<ref>), as it is proved in <cit.>. So condition (<ref>) is necessary and sufficient for unitarizability
of the monodromy when (<ref>) is satisfied.

Now if (<ref>) holds, the monodromy is not co-axial, so every
equation (<ref>) with apparent singularities and unitary monodromy defines exactly one metric. So the statement on the number of the metrics follows from Theorem <ref>.



A referee suggested another proof of Theorem <ref> without using Theorem <ref>. Suppose that A_1, A_2, A_3 are matrices in SL(2) which generate a non-commutative group and satisfy A_1A_2A_3=I and t_j:= A_j∈(-2,2). Then a necessary and sufficient condition of simultaneous unitarizability of these matrices is
    t_1^2+t_2^2+t_3^2-t_1t_2t_3<4.

This statement is well-known and easy to prove, see, for example <cit.>.

Consider now an equation with the Riemann scheme (<ref>) where α_1, α_2, α_3 are not integers while all other α_j are integer and the singularities at z_j, 4≤ j≤ n, and infinity are apparent. Let M_j be the monodromy matrices. Then it is easy to see that M_j=1 for all singularities, M_j=(-I)^α_j-1 for all apparent singularities including infinity, where α_∞=1, and M_i=-cosπα_i for i=1,2,3. Thus M_1M_2M_3=(-I)^σ-1, and applying condition (<ref>) to the matrices A_1=M_1, A_2=M_2, A_3=(-1)^σ-1M_3, we obtain a necessary and sufficient condition of unitarizability, which is nothing but (<ref>).




 §.§ Acknowledgments


A. Eremenko was supported by NSF grant DMS-1665115. V. Tarasov was supported in part by Simons Foundation grant 430235. We thank Andrei Gabrielov for illuminating discussions of this paper and the referees whose remarks improved the exposition.

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