MONODROMIES OF SCHWARZIAN DIFFERENTIAL EQUATIONS OF A CERTAIN TYPE

TOSHIYUKI SUGAWA 須川敏幸 HIROSHIMA UNIVERSITY 広島大学大学院理学研究科

Abstract. We consider a second-order linear homogenous ordinary differential equations in connection with the Teichmüller space of a four-times punctured sphere. Interests will be focused on the mysterious relation between the shape of the Teichmüller space and the Fibonacci sequence. The monodromy homomorphism induced by the differential equations plays a decisive role in our framework. This note is based on the author’s paper [7] and the joint paper [3] with Y. Komori.

1. An ordinary differential equation

Let Y be a four-times punctured sphere ℂ̂\{0, 1, ∞, λ} with hyperbolic metric ρ_{Y} (z)|dz| of Gaussian curvature −4. It is known that the space B_{2}(Y) of holomorphic quadratic differentials ψ(z)dz^{2} with finite norm

∥ψ∥_{Y} = sup_{z∈Y} ρ_{Y} (z)^{2}|ψ(z)|

is one dimensional vector space over ℂ. We may, for instance, take

ψ_{0}(z)dz^{2} = \frac{dz^{2}}{z(z − 1)(z − λ)}

as a basis of B_{2}(Y). Let π : 𝔻 → Y be a holomorphic universal covering map of the unit disk 𝔻 = {z ∈ ℂ : |z| < 1} onto Y and let Γ be the covering transformation group. Then Γ is a Fuchsian group of the first kind acting on 𝔻. It can be seen that the Schwarzian derivative S_{π^{−1}} of a local inverse of π is independent of the choice of branch and even of the particular choice of π. Hence it defines a (single-valued) analytic function ν_{Y} on Y. Here the Schwarzian derivative S_{f} of f is defined by

S_{f} = (\frac{f′′}{f′} )′ − \frac{1}{2} (\frac{f′′}{f′} )^{2} .

The function ν_{Y} is sometimes called the uniformizing connection of Y. Then it is known that ν_{Y} has the form

ν_{Y} (z) = \frac{1}{2z^{2}(z − 1)^{2}} + \frac{1}{2(z − λ)^{2}} + \frac{c(λ)}{z(z − 1)(z − λ)},

Date: 20 March, 2004.

1991 Mathematics Subject Classification. Primary 30F60; Secondary 30F40, 34A20.

2 T. SUGAWA

where c(λ) is a constant determined by λ (see, for instance, [1, Ch. X, p. 492]). This constant is known as an accessory parameter. It is generally difficult to compute c(λ). A method of numerical computation of c(λ) is given in [3] and the method will be indicated in Remark 2.

We now consider the second-order homogeneous linear ordinary differential equation 2y′′ + (ν_{Y} + tψ_{0})y = 0, (1.1)

where t is a complex parameter. We fix a base point z_{0} in Y and let y_{0} and y_{1} are fundamental (local) solutions of (1.1) around z_{0}, namely, they are determined by the initial conditions y_{0}(z_{0}) = 1, y′_{0}(z_{0}) = 0, y_{1}(z_{0}) = 0 and y′_{1}(z_{0}) = 1. It is easy to see that the Wronskian y_{0}y′_{1} − y′_{0}y_{1} is identically 1. For a curve γ with initial and terminal points at z_{0}, the solutions y_{0} and y_{1} can be analytically continued along γ to, say, ỹ_{0} and ỹ_{1}, respectively. These are written as linear combinations of y_{0} and y_{1}, say,

ỹ_{1} = ay_{1} + by_{0}
ỹ_{0} = cy_{1} + dy_{0},

where a, b, c, d are complex constants. Note that ad − bc = 1 holds. By the monodromy theorem, these constants depend only on the homotopy class [γ] of γ. In this way, we obtain a homomorphism χ_{t} : π_{1}(Y, z_{0}) → SL(2, ℂ) satisfying

χ_{t} : [γ] ↦ (\begin{matrix}a & b \\ c & d \end{matrix}) .

The homomorphism is called the monodromy homomorphism or holonomy representation of π_{1}(Y, z_{0}) with respect to tψ_{0}. We write G_{t} = χ_{t}(π_{1}(Y, z_{0})).

In particular, when t = 0, the quotient f = y_{1}/y_{0} of fundamental solutions to (1.1) has Schwarzian derivative S_{f} = ν_{Y} , and thus, f = L ∘ π^{−1} locally, where L is a Möbius transformation. Therefore, the image χ_{0}(π_{1}(Y, z_{0})) is Möbius conjugate with the uniformizing Fuchsian group Γ of Y.

Remark 1. Up to Möbius conjugate, the group G_{0} may be regarded as a lift of Γ ⊂ PSU(1, 1) to SU(1, 1). This lift is determined by the condition tr χ_{0}(γ) = −2 for every simple closed curve rounding about a puncture of Y.

Let T̃(Y) be the set of those elements in B_{2}(Y) of the form tψ_{0} for which G_{t} is a quasiconformal deformation of G_{0}. The connected component of T̃(Y) which contains the origin is called (the Bers embedding of) the Teichmüller space of Y and will be denoted by T (Y). (Rigorously, our T (Y) is the Teichmüller space of the mirror image Y^{∗} of Y. To avoid confusion, we adopt a different definition here from the standard one.)

It is known that T (Y) is a Jordan domain with {ψ ∈ B_{2}(Y) : ∥ψ∥_{Y} < 2} ⊂ T (Y) ⊂ {ψ ∈ B_{2}(Y) : ∥ψ∥_{Y} < 6} (see [3]). The quantities

ι(T (Y)) = inf{∥ψ∥_{Y} : ψ ∈ ∂T (Y)} and o(T (Y)) = sup{∥ψ∥_{Y} : ψ ∈ ∂T (Y)}

are called the inner radius and the outer radius of T (Y), respectively. Therefore, we have 2 ≤ ι(T (Y)) ≤ o(T (Y)) ≤ 6. (In fact, these inequalities are all strict.)

2. Pinching deformation along a simple closed curve

Let τ be a complex number with Im τ > 0. Then T = ℂ/⟨1, τ⟩ becomes a complex torus, where ⟨1, τ⟩ denotes the lattice group generated by 1 and τ over ℤ. We denote by [z] the equivalence class represented by z ∈ ℂ with respect to the action of the lattice. Let ℘ be the Weierstrass ℘-function with period lattice ⟨1, τ⟩ and set e_{1} = ℘(1/2), e_{2} = ℘(τ/2), e_{3} = ℘((1 + τ)/2). As is well known, the quantity

λ(τ) = \frac{e_{3} − e_{2}}{e_{1} − e_{2}}

is an elliptic modular function. We now choose τ so that λ(τ) = λ. Then the mapping p : T → ℂ̂ defined by p([z]) = (℘(z) − e_{2})/(e_{1} − e_{2}) is a two-sheeted branched covering map of T onto ℂ̂ with branch point of order 2 at [0], [1/2], [τ/2], [(1 + τ)/2]. Therefore, the four-times punctured torus Z = T \ {[0], [1/2], [τ/2], [(1 + τ)/2]} is a two-sheeted unbranched covering space over Y.

A simple closed curve is called peripheral if it is homotopic to either a point or a puncture. Let γ : [0, 1] → Y be a non-peripheral simple closed curve in Y and γ̃ be a lift of γ via the composition of covering maps ℂ \ ⟨1/2, τ/2⟩ → Z → Y. The difference γ̃(1) − γ̃(0) of the terminal point and the initial point of γ̃ has the form m + nτ for relatively prime integers m and n. The ratio r = n/m ∈ ℚ̂ = ℚ ∪ {∞} is called the slope of γ. It is known that the ratio determines the homotopy class of γ and that any number in ℚ̂ is realized as the ratio of a non-peripheral simple closed curve in Y.

Let [γ_{r}] be an element of π_{1}(Y, z_{0}) with slope r ∈ ℚ̂ and define the function σ_{r} to be σ_{r}(t) = tr^{2}χ_{t}([γ_{r}]). Note that σ_{r} is an entire function and σ_{r}(0) > 4.

Remark 2. The origin is a special point where the entire functions σ_{r} take real values simultaneously. In turn, this property can be used to compute the accessory parameter c(λ). See [3] for details.

We next recall fundamental facts about Farey triangles (cf. [6]). For a more detailed explanation, see [3]. The reader also finds a good account for Farey sequences in [2] as well as an interesting historical remark.

For three points a, b, c in ℝ̂, we denote by Δ(a, b, c) the hyperbolic triangle formed by three hyperbolic geodesics in the upper half plane ℍ connecting two of the three points a, b and c. Let Δ = Δ(0, 1, ∞). Then ℍ is tessellated by Δ and its conjugates by the modular group PSL(2, ℤ). Note that the stabilizer of Δ in PSL(2, ℤ) consists of three elements and permutes the vertices of Δ cyclically. Each triangle which is conjugate with Δ by the action of PSL(2, ℤ) is called a Farey triangle. The initial Farey triangle Δ and its reflection Δ′ = Δ(0, −1, ∞) in the imaginary axis form a fundamental domain of the

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modular group Γ_{2} = {±C ∈ PSL(2, ℤ) : C ≡ I mod 2} of level 2. We will say that both Δ and Δ′ are of level 0. A Farey triangle which shares a side with that of level 0 will be called of level 1 unless it is of level 0. Similarly, a Farey triangle which shares a side with that of level n will be called of level n + 1 unless it is of level ≤ n. It is important to note that the corresponding graph to the above tessellation is a tree, namely, there is no closed circuit.

It is well known that the orbit of 0 under the action of PSL(2, ℤ) coincides with ℚ̂. We denote by F̃(n) the set of rationals which appear as vertices of Farey triangles of level ≤ n. Set F(n) = F̃(n) \ F̃(n − 1) for n = 0, 1, …. For instance, F(0) = {−1, 0, 1, ∞}, F(1) = {−2, −1/2, 1/2, 2}, F(2) = {−3, −3/2, −2/3, −1/3, 1/3, 2/3, 3/2, 3} and so on. We note that #F(n) = 2^{n+1} for n ≥ 1. An element r in F(n) will be called of level n and designated by level(r) = n. Note that if p_{1}/q_{1} and p_{2}/q_{2} are of level ≤ n and if p_{1}q_{2} − q_{1}p_{2} = ±1, then (p_{1} + p_{2})/(q_{1} + q_{2}) is of level ≤ n + 1.

Note that the rationals ±r_{n} and ±1/r_{n} belong to F(n), where r_{n} = b_{n+1}/b_{n} and b_{n} is the n-th Fibonacci number, namely, b_{n} is determined by b_{0} = 1, b_{1} = 1, b_{n} = b_{n−1} + b_{n−2} (n = 2, 3, 4, …).

It is shown in [3] that the connected component P_{r} of the inverse image σ_{r}^{−1}([4, σ_{r}(0)]) containing 0 is a closed analytic Jordan arc for each rational r. The arc P_{r} is called the pleating ray with slope r. The the other end point of P_{r}, denoted by β(r), lies on the boundary of the Teichmüller space T (Y) and corresponds to a cusp group. The deformation along P_{r} can be regarded as a pinching deformation of the Riemann surface Y along the hyperbolic geodesic with slope r.

When λ = 1/2, the following observation was made in [7] by the aid of Mathematica for, at least, small n’s.

Conjecture 1. When λ = 1/2, then the following relation holds for each n :

max_{r∈F̃(n)} ∥β(r)∥ = ∥β(r_{n})∥.

Note also that ∥β(r_{n})∥ = ∥β(−r_{n})∥ = ∥β(1/r_{n})∥ = ∥β(−1/r_{n})∥ by symmetry in the case λ = 1/2. It is well known that r_{n} converges to the golden ratio (\sqrt{5}−1)/2. Since cusps are dense in the boundary [5], if the above conjecture is valid, then the next conjecture is valid, too.

Conjecture 2 ([7]). If λ = 1/2, then

o(T (Y)) = ∥β((\sqrt{5} − 1)/2)∥

holds, where β((\sqrt{5} − 1)/2) is the end point of the pleating ray with irrational slope (\sqrt{5} − 1)/2.

Based on the last conjecture, we numerically obtained o(T (Y)) ≈ 2.9386 when λ = 1/2. It is naturally expected that, for a general λ, a similar statement would hold for some irrational number r whose continued fraction expansion has the same tail as that of the golden ratio.

At present, we have no idea to prove the above conjectures in a rigorous way. It is also interesting to see the asymptotic behaviour of the monodromy homomorphism χ_{t} in connection with the configuration of exotic projective structures (see [4]). We end this note with the specialized problem.

Problem 3. Investigate the asymptotic behaviour of the entire function σ_{r}(t) as t → ∞

for an extended rational number r ∈ ℚ̂.

References

1. A. R. Forsyth, Theory of Differential Equations Part III, Vol. IV,, Cambridge University Press, 1902.

2. G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, fifth ed., The Clarendon Press, Oxford University Press, New York, 1979.

3. Y. Komori and T. Sugawa, Bers embedding of the Teichmüller space of a once-punctured torus, to appear in Conform. Geom. Dyn.

4. Y. Komori, T. Sugawa, M. Wada, and Y. Yamashita, Drawing Bers embeddings of the Teichmüller space of once-punctured tori, preprint.

5. C. McMullen, Cusps are dense, Ann. of Math. (2) 133 (1991), 217–247.

6. Y. Minsky, The classification of punctured-torus groups, Ann. of Math. (2) 149 (1999), 559–626.

7. T. Sugawa, Estimates of hyperbolic metric with applications to Teichmüller spaces, Kyungpook Math. J. 42 (2002), 51–60.

Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, 739-8526 Japan

  E-mail address: sugawa@math.sci.hiroshima-u.ac.jp