Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 18, 1993, 355–367

NON-UNIQUENESS OF GEODESICS IN INFINITE-DIMENSIONAL TEICHMÜLLER SPACES (II)

Li Zhong

Peking University, Department of Mathematics Beijing, 100871, People’s Republic of China

Abstract. The non-uniqueness of geodesics joining two given points in universal Teichmüller space is proved in the previous paper [7]. The purpose of the present paper is to discuss the nonuniqueness of geodesics in any infinite-dimensional Teichmüller space. It is proved that if μ_{1} and μ_{2} are two extremal Beltrami differentials belonging to a point [μ] of a Teichmüller space and μ_{1} − μ_{2} does not belong to N -class, the paths [tμ_{1}] and [tμ_{2}] (0 ≤ t ≤ 1) are different geodesics joining [0] and [μ]. Making use of this theorem, the result of [7] is generalized to hold for any infinite-dimensional Teichmüller space. This is a complete answer to a problem posed by F.P. Gardiner [1].

1. Introduction

There are some essential differences in the geometry between the finite-di-mensional Teichmüller spaces and the infinite-dimensional Teichmüller spaces. It is well known that a finite-dimensional Teichmüller space is a straight geodesic space in the sense of Buseman (S. Kravetz [4]), that is, for any pair of points in a finite-dimensional Teichmüller space there is a unique geodesic line through them. But we do not know whether or not the same is true for the infinite-dimensional case. We further explain the question as follows.

Let T (S) be a Teichmüller space of a Riemann surface S. T (S) is defined as a quotient space of the Beltrami differentials on S. If dim T (S) < ∞, each point [μ] ∈ T (S) contains a unique extremal Beltrami differential, say μ_{0} , and the path [tμ_{0}] (0 ≤ t ≤ 1) is the unique geodesic joining [0] and [μ]. But in the infinite-dimensional case the situation is different. When dim T (S) = ∞, a point [μ] of T (S) may contain more than one extremal differential. The first example of such a point in the universal Teichmüller space was given by K. Strebel, known as the Strebel chimney (K. Strebel [11] or see O. Lehto [6]). Suppose the point [μ] contains two extremal differentials μ_{1} and μ_{2} . The question is how to determine whether [tμ_{1}] is the same as [tμ_{2}] (0 ≤ t ≤ 1). This question was proposed by F.P. Gardiner [1].

In [7] the author answers this question by constructing two extremal differentials μ_{1} and μ_{2} such that they are in the same point of the universal Teichmüller space but the geodesic paths [tμ_{1}] and [tμ_{2}] (0 ≤ t ≤ 1) are different. This example shows that the geodesics joining two points in the universal Teichmüller space are not unique and hence the universal Teichmüller space is not a straight geodesic space.

1991 Mathematics Subject Classification: Primary 32G15; Secondary 30G60, 30C70.

This study was supported by the grant of NSF and DPF.

After this result, a natural question that follows concerns the general infinitedimensional Teichmüller space except the universal Teichmüller space. The purpose of this paper is to investigate the geodesic problem for any infinite-dimensional Teichmüller space.

The main result is Theorem 3.1, which states that if a point [μ] ∈ T (S) contains two extremal differentials μ_{1} and μ_{2} such that μ_{1} − μ_{2} does not belong to the N -class of Ahlfors, then [tμ_{1}] and [tμ_{2}] (0 ≤ t ≤ 1) are different geodesics joining [0] and [μ].

As a consequence of this main result, we get a criterion for the non-uniqueness of the geodesics in any infinite-dimensional Teichmüller space: If μ is an extremal Beltrami differential on S and the set {p ∈ S | |μ(p)| < ∥μ∥_{∞} − ε} (ε > 0) has an interior point, there are infinitely many geodesic lines through [0] and [μ] (see Theorem 3.2).

As as application of Theorem 3.2, we shall construct in Section 4 a pair of points in any infinite-dimensional Teichmüller space such that they have infinitely many geodesics.

Acknowledgement. This study was carried out when the author was a visiting research fellow of the United College of Chinese University of Hong Kong. The author would like to thank Professor Y.C. Wong for his invitation and discussions. The author also would like to thank Professor M.T. Cheng, Dr. G.Z. Cui and Dr. Y.L. Shen for their reading the manuscript and their suggestions.

Remark. Having prepared this paper, the author received a preprint of a paper [12] by Harumi Tanigawa, who shows a similar result to Theorem 3.2 of this paper. Her result is as follows: If μ is an extremal Beltrami differential which vanishes on a Jordan domain U and does not vanish identically on the whole Riemann surface, there exists a family of geodesic discs through [0] and [μ] with a complex analytic parameter. She constructs an extremal Beltrami differential which satisfies the conditions in her theorem. The method of her proof is completely different from ours.

From the proofs of Theorem 3.1 and 3.2 of this paper one may find that these theorems still hold if the “geodesics” in the statements of Theorem 3.1 and 3.2 is replaced by “geodesic discs”.

2. Preliminaries

Let S be a Riemann surface with a holomorphic universal covering map: π: Δ → S, where Δ is the unit disc. Then S can be expressed as a quotient space Δ/Γ, where Γ is a Fuchsian group acting on Δ. Denote by Bel(S) the space of the Beltrami differentials on S with the supremum norm smaller than 1. For each element μ ∈ Bel(S) there exists a Riemann surface S^{μ} and a quasiconformal mapping f^{μ}: S → S^{μ} such that the complex dilatation of f^{μ} is μ. We denote by f̃^{μ}: Δ → Δ the lift of f^{μ} with the points 1, i and −1 fixed. Then f̃^{μ} is uniquely determined by μ. An element μ_{1} ∈ Bel(S) is said to be equivalent to μ_{2} ∈ Bel(S) if and only if f̃^{μ_{1}} |_{∂Δ}= f̃^{μ_{2}} |_{∂Δ} . Then the Teichmüller space of S, denoted by T (S), can be defined as the quotient space of Bel(S) under the equivalence relations. The Teichmüller metric between two points [μ] and [ν] is defined as follows:

d([μ], [ν]) = \frac{1}{2} inf_{f} log K[f],

where f runs through all quasiconforml mappings of S^{μ} onto S^{ν} in the homotopy class [f^{ν} ∘ (f^{μ})^{−1}] and K[f] is the maximal dilatation of f.

An element μ ∈ Bel(S) is said to be extremal if and only if its norm is the smallest among the elements of [μ]. It is known that the extremal Beltrami differential always exists for any point [μ] ∈ T (S). Moreover, if μ is extremal, then tμ is also extremal for t (0 ≤ t ≤ 1) and the path [tμ] is a geodesic (in the Teichmüller metric) joining the points [0] and [μ]. If dim T (S) < +∞, the extremal differential of [μ] is unique for each point [μ] ∈ T (S) and must be of the following form: k\overline{φ}/|φ|, where φ is a holomorphic quadratic differential with finite norm and k is a constant non-negative and smaller than 1.

Let Q(S) be the space of quadratic differentials on S with L_{1} -norms finite. The dimension of Q(S) is finite if and only if dim T (S) is finite.

An element μ ∈ Bel(S) is extremal if and only if

(2.1)  sup_{{φ∈Q(S) \atop ∥φ∥=1}} \{| ∫∫_{S} μφ dx dy|\} = ∥μ∥_{∞} .

This result is due to R.S. Hamilton [3] and S. Krushkal [5] (for necessity), and E. Reich and K. Strebel [8] (for sufficiency). Condition (2.1) is called Hamilton– Krushkal condition.

If μ is extremal and φ_{n} ∈ Q(S) such that ∥φ_{n}∥ = 1 and

(2.2)  lim_{n→∞} | ∫∫_{S} μφ_{n} dx dy| = ∥μ∥_{∞} ,

the sequence {φ_{n}} is called a Hamilton sequence.

Another known result we need is the main inequality of Reich and Strebel (see [9]). If μ ∈ Bel(S) is equivalent to 0, i.e., f̃^{μ} |_{∂Δ}= id, we have

(2.3)  ∥φ∥ ≤ ∫∫_{S} |φ|\frac{|1 − μφ/|φ||^{2}}{1 − |μ|^{2}} dx dy

for all φ ∈ Q(S).

A Beltrami differential μ ∈ Bel(S) is said to be an element of N(S), known as the N -class of Beltrami differentials, if and only if

(2.4)  ∫∫_{S} μφ dx dy = 0,  for all φ ∈ Q(S).

3. Main results

In this section we shall give some sufficient conditions of a point [μ] ∈ T (S) having more than 1 geodesics joining [0] and [μ].

Lemma 3.1. Let μ ∈ Bel(Δ) and F_{t}: Δ → Δ be a quasiconformal mapping of the unit disc Δ onto itself with the points 1, i and −1 fixed and with the complex dilatation tμ (0 ≤ t ≤ 1). Then we have a sequence {t_{n}}, 0 < t_{n} < 1, such that

(3.1)  ∂_{z}F_{t_{n}}(z) → 1,  as t_{n} → 0,

almost everywhere in Δ.

Proof. It is known that the Beltrami equation

∂z̄w = tμ∂_{z}w (0 < t < 1)

has a solution of the following form:

w_{t}(z) = z − \frac{1}{π} ∫∫_{Δ} \frac{ω_{t}(ζ)}{ζ − z} dξ dη (ζ = ξ + iη),

where ω_{t} is a solution of the equation

(3.2)  ω − tμS(ω) = tμ,

with S a singular integration operator:

S(ω) = −\frac{1}{π} ∫∫_{Δ} \frac{ω(ζ)}{(ζ − z)^{2}} dξ dη.

Let Λ_{p} be the norm of the operator S: L_{p} → L_{p}. Then Λ_{2} = 1 and Λ_{p} is a continuous function of p > 1 (see [13]). Suppose

|μ(z)| ≤ k < 1, a.e. z ∈ Δ,

where k is a constant. We choose p_{0} > 2 such that Λ_{p_{0}}k < 1. Then (3.2) has a solution ω_{t} ∈ L_{p_{0}} satisfying

∥ω_{t}∥_{L_{p_{0}}} ≤ tk/(1 − Λ_{p_{0}}k).

Hence we get ∥ω_{t}∥_{L_{p_{0}}} → 0 (as t → 0) and

(3.3)  w_{t}(z) → z  (uniformly for z ∈ Δ)

as t → 0. On the other hand, we see

∂_{z}w_{t} = 1 + S(ω_{t}).

It follows that

∥∂_{z}w_{t} − 1∥_{L_{p_{0}}} ≤ Λ_{p_{0}} ∥ω_{t}∥_{L_{p_{0}}} → 0 (as t → 1),

which implies that there exists a sequence {t_{n}} such that

(3.4)  ∂_{z}w_{t_{n}} → 1  (a.e.) as t_{n} → 0.

By the assumption of the lemma, F_{t} can be expressed as φ_{t}(w_{t}), where φ_{t} is a conformal mapping of w_{t}(Δ) onto Δ with the following conditions:

φ_{t}(w_{t}(1)) = 1,  φ_{t}(w_{t}(i)),  φ_{t}(w_{t}(−1)) = −1.

By (3.3) and the Caratheodory theorem, it follows from the above conditions that φ is locally uniformly convergent to z and hence φ′_{t} is locally convergent to 1 as t → 0. The lemma is proved by (3.4). QED.

Theorem 3.1. Let μ_{1} and μ_{2} be two extremal Beltrami differentials in [μ] ∈ T (S). If μ_{1} − μ_{2} does not belong to the N -class of Beltrami differentials on S, then [tμ_{1}] and [tμ_{2}] (0 ≤ t ≤ 1) are two different geodesics joining [0] and [μ].

Proof. Suppose [tμ_{1}] = [tμ_{2}] for every t ∈ [0, 1]. The theorem will be proved by obtaining a contradiction with the assumption that μ_{1} − μ_{2} ∉ N(S).

Let S = Δ/Γ, where Γ is a Fuchsian group. For each t ∈ [0, 1] there are a Riemann surface S_{t} and a quasiconformal mapping f_{t,j} = f^{tμ_{j}}: S → S_{t} (j = 1, 2) such that f_{t,1} is homotopic to f_{t,2} modulo the boundary. Suppose F_{t,j}: Δ → Δ is the lift of f_{t,j} with the points 1, i and −1 fixed (j = 1, 2). Then we have

F_{t,1} |_{∂Δ}= F_{t,2} |_{∂Δ} .

Let S_{t} = Δ/Γ_{t}, where Γ_{t} is a Fuchsian group. It is easy to see that we can assume that

Γ_{t} = {F_{t,j} ∘ γ ∘ F_{t,j}^{−1} | γ ∈ Γ},

where j = 1 or 2.

Let φ be an arbitrary fixed element of Q(S) and φ̃(z) dz^{2} be the lift of φ. Then φ̃ satisfies

φ̃(γ(z))[γ′(z)]^{2} = φ̃(z), z ∈ Δ.

There is a holomorphic function ψ(z) in Δ with the condition

∫∫_{Δ} |ψ(z)| dx dy < +∞,

such that the Poincaré series of ψ (see [2, Chapter 4, Theorem 3]).

Θψ(z) = ∑_{γ∈Γ} ψ(γ(z))[γ′(z)]^{2}

is equal to φ̃. Using this function ψ, we define

φ̃_{t}(z) = ∑_{γ_{t}∈Γ_{t}} ψ(γ_{t}(z))[γ′_{t}(z)]^{2}.

Then we get

φ̃_{t}(γ_{t}(z))[γ′_{t}(z)]^{2} = φ̃_{t}(z)

for z ∈ Δ and γ_{t} ∈ Γ_{t}. This means that φ̃_{t}(z) dz^{2} is a lift of a holomorphic quadratic differential φ_{t} on S_{t} with finite norm.

Let g_{t} be the composition mapping of f_{t,1} and f_{t,2}^{−1} , i.e., g_{t} = f_{t,1}∘f_{t,2}^{−1}: S_{t} → S_{t}. Denote by σ_{t} the complex dilatation of g_{t}. Then it follows from the assumption that [tμ_{1}] = [tμ_{2}], that σ_{t} is equivalent to 0. Hence by the main inequality we have

∥φ_{t}∥ ≤ ∫∫_{S_{t}} |φ_{t}|\frac{|1 − σ_{t}φ_{t}/|φ_{t}||^{2}}{1 − |σ_{t}|^{2}} dξ dη,

which can be rewritten in the following form:

(3.5)  Re ∫∫_{S_{t}} \frac{σ_{t}φ_{t}}{1 − |σ_{t}|^{2}} dξ dη ≤ ∫∫_{S_{t}} \frac{|σ_{t}|^{2}|φ_{t}|}{1 − |σ_{t}|^{2}} dξ dη.

Let Ω be a fundamental domain of S. Then Ω_{t} = F_{t,2}(Ω) is a fundamental domain of S_{t}. Let μ̃_{1} , μ̃_{2} and σ̃_{t} be the lifts of μ_{1} , μ_{2} and σ_{t}, respectively. Then we have

σ̃_{t}(ζ) = \frac{t(μ̃_{1} − μ̃_{2})}{1 − t^{2}μ̃_{1}\overline{μ̃_{2}}} · \frac{∂_{z}F_{t,2}}{\overline{∂_{z}F_{t,2}}} ∘ F_{t,2}^{−1}(ζ).

From inequality (3.5) it follows that

Re ∫∫_{Ω_{t}} σ̃_{t}φ̃_{t} dξ dη ≤ ∫∫_{Ω_{t}} \frac{|σ̃_{t}|^{2}|φ̃_{t}|}{1 − |σ̃_{t}|^{2}} dξ dη + O(t^{2})

and hence

lim sup_{t→0} Re ∫∫_{Ω_{t}} \frac{μ̃_{1} − μ̃_{2}}{1 − t^{2}μ̃_{1}\overline{μ̃_{2}}} · \frac{∂_{z}F_{t,2}}{\overline{∂_{z}F_{t,2}}} ∘ F_{t,2}^{−1}(ζ)φ̃_{t}(ζ) dξ dη ≤ 0

or

lim sup_{t→0} Re ∫∫_{Ω_{t}} \frac{μ̃_{1} − μ̃_{2}}{1 − t^{2}μ̃_{1}\overline{μ̃_{2}}} · \frac{∂_{z}F_{t,2}}{\overline{∂_{z}F_{t,2}}} φ̃_{t}(F_{t,2}) dx dy ≤ 0.

It is easy to see that φ̃_{t}(z) → φ̃(z). By Lemma 3.1 there is a sequence {t_{n}} (0 < t_{n} < 1) such that ∂_{z}F_{t_{n},2} → z as t_{n} → 0. Then we have

Re ∫∫_{Ω} (μ̃_{1} − μ̃_{2})φ̃ dx dy ≤ 0

or

Re ∫∫_{S} (μ_{1} − μ_{2})φ dx dy ≤ 0.

Since φ is an arbitrary element of Q(S), this inequality implies

∫∫_{S} (μ_{1} − μ_{2})φ dx dy = 0, for all φ ∈ Q(S),

which means μ_{1} − μ_{2} ∈ N(S). This is a contradiction with the assumption of the theorem. QED.

Theorem 3.1 gives a sufficient condition of the non-uniqueness of geodesics. But the condition that μ_{1} − μ_{2} ∉ N(S) is not very explicit for practical use. Below there follows a more explicit condition which can be regarded as a criterion of the non-uniqueness of geodesics.

Theorem 3.2. Let μ be an extremal Beltrami differential on a Riemann surface S. If the set

A_{μ} = {p ∈ S | |μ|(p) < ∥μ∥_{∞} − ε} (for some ε > 0)

has an interior point, there are infinitely many geodesics joining the points [0] and [μ] in T (S).

Proof. Suppose p_{0} ∈ S is an interior point of A_{μ} and U ⊂ A_{μ} is a neighbourhood of p_{0} . Without any loss of generality we assume that μ is C^{2} smooth in U (otherwise one may make a small perturbation of μ|_{U} such that the resulting differential is C^{2} smooth and its absolute values are smaller than ∥μ∥_{∞} − ε′ for another ε′ > 0).

Let φ ∈ Q(S) \ {0} and φ|_{p_{0}} ≠ 0. We choose U so small that φ|_{p} ≠ 0 for every p in U. Suppose z is a natural parameter of φ such that z(p_{0}) = 0 and φ|_{U} = dz^{2} . Let f^{μ}: S → S^{μ} be a quasiconformal mapping of S onto another Riemann surface S^{μ} with the Beltrami differential μ. Let ζ be a local parameter of a neighbourhood V of q_{0} = f^{μ}(p_{0}) with ζ(q_{0}) = 0. Suppose f^{μ}(V) ⊂ U and the local expression of f^{μ}|_{U} is ζ = f(z), the complex dilatation of which is μ(z). Without any loss of generality we assume

(3.6)  ∂_{ζ}[∂_{z}f(f^{−1}(ζ))] ≢ 0  in V.

(Otherwise one can change the parameter ζ or take a small perturbation of f|_{U}). We look at a disc Δ_{r} = {ζ | |ζ| < r}, where r > 0 is sufficiently small to allow Δ_{r} ⊂ V . Define a mapping of Δ_{r} onto itself:

h_{α}(ζ) = ζ + αη(ζ),

where α is a complex parameter and η ∈ C^{∞} has a compact supporting set in Δ. Then we have

(3.7)  h_{α}(ζ) ≡ ζ,  for all ζ ∈ ∂Δ_{r}.

Let ν_{α} = ∂ζ̄h_{α}/∂_{ζ}h_{α}. It is easy to see that ∥ν_{α}∥_{∞} is less than 1 when |α| is sufficiently small. By the argument principle for quasiregular functions and condition (3.7), h_{α} is an 1 − 1 mapping and hence a quasiconformal mapping of Δ_{r} onto itself.

Let us look at the composed mapping g_{α} = h_{α} ∘ f. It is easy to see that the complex dilatation of g_{α} is

(3.8)  σ_{α} = \frac{μ(z) + ν_{α}(ζ)θ(z)}{1 + \overline{μ(z)}ν_{α}(ζ)θ(z)},

where ζ = f(z) and θ(z) = ∂_{z}f/\overline{∂_{z}f}.

Let α_{1} and α_{2} be two sufficiently small parameters and α_{1} ≠ α_{2} . Then we have

σ_{α_{1}}(z) − σ_{α_{2}}(z) = \frac{[ν_{α_{1}}(ζ) − ν_{α_{2}}(ζ)]θ(z)[1 − |μ(z)|^{2}]}{[1 + μ̄(z)ν_{α_{1}}(ζ)θ(z)][1 + μ̄(z)ν_{α_{2}}(ζ)θ(z)]}.

Non-uniqueness of geodesics in Teichmüller spaces 363

Noting the fact that ν_{α}(ζ) = O(|α|), we get

\begin{aligned}σ_{α_{1}}(z) − σ_{α_{2}}(z) & = [ν_{α_{1}}(ζ) − ν_{α_{2}}(ζ)]θ(z)[1 − |μ(z)|^{2}] \\ & · [1 + O(|α_{1}| + |α_{2}|)], ζ = f(z). \end{aligned}

It follows from the definition of ν_{α} that

ν_{α_{1}}(ζ) − ν_{α_{2}}(ζ) = (α_{1} − α_{2})∂ζ̄η[1 + O(|α_{1}| + |α_{2}|)],

and hence

(3.9)  \begin{aligned}σ_{α_{1}}(z) − σ_{α_{2}}(z) & = (α_{1} − α_{2})∂ζ̄η(ζ)θ(z)[1 − |μ(z)|^{2}] \\ & · [1 + O(|α_{1}| + |α_{2}|)]. \end{aligned}

We are now going to show that

(3.10)  ∫∫_{D_{r}} [σ_{α_{1}}(z) − σ_{α_{2}}(z)] dx dy ≠ 0

for sufficiently small α_{1} and α_{2} (α_{1} ≠ α_{2} ), where D_{r} = f^{−1}(Δ_{r}). By (3.9) it is easy to see that, to show (3.10), it is sufficient to prove

∫∫_{D_{r}} ∂ζ̄η(f(z))θ(z)[1 − |μ(z)|^{2}] dx dy ≠ 0

or

(3.11)  ∫∫_{Δ_{r}} ∂ζ̄η(ζ)/\overline{[∂_{z}f(f^{−1}(ζ))]^{2}} dξ dη ≠ 0.

There exists a function η ∈ C_{0}^{∞}(Δ_{r}) such that (3.11) holds; otherwise we have

∫∫_{Δ_{r}} ∂ζ̄η(ζ)/\overline{[∂_{z}f(f^{−1}(ζ))]^{2}} dξ dη = 0,

for all η ∈ C_{0}^{∞}(Δ_{r}), which implies

∂ζ̄(1/\overline{[∂_{z}f(f^{−1}(ζ))]^{2}} ) ≡ 0 (in Δ_{r})

or

∂_{ζ}[∂_{z}f(f^{−1}(ζ))] ≡ 0 (in Δ_{r})

being a contradiction with (3.6). From now on we assume η to be chosen such that (3.11) holds and hence (3.10) holds.

We identify the points in U with their parameters. Then D_{r} are identified with the domain of S which D_{r} represent. Now we define a Beltrami differential on S:

σ_{α}^{∗} = \{ \begin{matrix}μ, & in S \ D_{r}; \\ σ_{α}, & in D_{r}. \end{matrix}

Obviously, σ_{α}^{∗} ∈ [μ] and σ_{α}^{∗} is extremal when |α| is sufficiently small. By the definition we have

∫∫_{S} (σ_{α_{1}}^{∗} − σ_{α_{2}}^{∗})φ dx dy = ∫∫_{D_{r}} (σ_{α_{1}} − σ_{α_{2}})φ dx dy,

where φ is the holomorphic quadratic differential on S which is given at the beginning of the proof. Recalling the fact that the local expression of φ|_{U} is dz^{2} , we get

∫∫_{S} (σ_{α_{1}}^{∗} − σ_{α_{2}}^{∗})φ dx dy = ∫∫_{D_{r}} (σ_{α_{1}} − σ_{α_{2}}) dx dy.

It follows from (3.10) that

∫∫_{S} (σ_{α_{1}}^{∗} − σ_{α_{2}}^{∗})φ dx dy ≠ 0,

which implies that (σ_{α_{1}}^{∗} − σ_{α_{2}}^{∗}) does not belong to the N -class. By Theorem 3.1 we see that [tσ_{α_{1}}^{∗}] and [tσ_{α_{2}}^{∗}] (0 ≤ t ≤ 1) are different geodesics joining [0] and [μ], provided α_{1} ≠ α_{2} and both are sufficiently small. QED.

Remark. It is easy to prove that the mapping t ↦ [tμ/ ∥μ∥_{∞}] (μ is extremal) is a holomorphic isometry of Δ = {t : |t| < 1} into T (S) with the Poincaré metric and Teichmüller metric. So one can replace the geodesic lines in Theorem 3.2 with the geodesic discs [12].

In her paper [12], H. Tanigawa investigates the boundary behaviour of a holomorphic mapping of Δ into T (S) and gives a sufficient condition for holomorphic mappings into T (S) to be rigid.

4. Existence

Now we want to construct an extremal Beltrami differential which satisfies the conditions in Theorem 3.2. The main point is to construct an extremal Beltrami differential which is not of the Teichmüller form or the Teichmüller form while the associated quadratic differential has infinite norm.

In [12], H. Tanigawa gives a construction of such an extremal Beltrami differential. Here we shall give another construction.

Let T (S) be an arbitrary infinite dimensional Teichmüller space of a Riemann surface S. Suppose that S = Δ/Γ, where Γ is a Fuchsian group acting on Δ. Then Γ is of the second kind or of the first kind with infinite generators.

First, we assume that Γ is a Fuchsian group of the second kind (finitely or infinitely generated). Let z_{0} ∈ ∂Δ be a point of ∂Δ \ ∧(Γ), where ∧(Γ) is the limit set of Γ. Then z_{0} is a point on a free side of a fundamental polygon G of Γ. Take a function φ which is holomorphic on C \ {z_{0}} and has a pole of the second order at the point z_{0} . We look at its Poincaré series

ψ̃ = Θφ = ∑_{γ∈Γ} φ(γ(z))γ′(z)^{2}.

It is easy to see that z_{0} is a pole of ψ̃ of the second order and

∫∫_{G} |ψ̃| dx dy = ∞.

Obviously, ψ̃ induces a holomorphic quadratic differential on S, which is denoted by ψ. Then we have

∫∫_{S} |ψ| dx dy = ∞.

Define a Beltrami differential on S:

μ = kψ̃/|ψ|,

where k ∈ (0, 1) is a constant. Since ψ̃ has a pole of the second order on the boundary of Δ, the Beltrami differential μ̃ = k\overline{ψ̃}/|ψ̃| with respect to Γ is extremal ([10]). Hence μ is an extremal Beltrami differential on S. By the Hamilton theorem there is a sequence of quadratic differentials, {ψ_{n}}, such that ∥ψ_{n}∥ = 1 and

lim_{n→∞} | ∫∫_{S} μψ_{n} dx dy| = k.

It is easy to see that there is a subsequence of {ψ_{n}} which is uniformly convergent to a holomorphic quadratic differential ψ_{0} in any compact set of S. Without any loss of generality we may assume that the subsequence is the sequence {ψ_{n}} itself. Then we have

∥ψ_{n} − ψ_{0}∥ ↛ 0 (n → ∞).

Otherwise, ∥ψ_{n} − ψ_{0}∥ → 0 (n → ∞) implies ∥ψ_{0}∥ = 1 and

| ∫∫_{S} μψ_{0} dx dy| = k,

which means μ = e^{iθ}kψ̄_{0}/|ψ_{0}|. This is a contradiction with the definition of μ.

Without loss of generality we assume that ∥ψ_{n} − ψ_{0}∥ ≠ 0 for every integer n. Define η_{n} = (ψ_{n} −ψ_{0})/ ∥ψ_{n} − ψ_{0}∥. Then η_{n} has norm 1 and {η_{n}} is a degenerate Hamilton sequence of μ, namely,

lim_{n→∞} | ∫∫_{S} μη_{n} dx dy| = k

and η_{n} locally tends to zero.

Now we get the desired Beltrami differential as follows. We define

μ^{∗} = \{ \begin{matrix}μ, & in S \ U; \\ k′, & in U, \end{matrix}

where k′ ∈ (0, 1) is a constant smaller than k and U is a small compact neighbourhood of a point. Obviously, μ^{∗} is also an extremal Beltrami differential and {η_{n}} is also a Hamilton sequence of μ^{∗}. μ^{∗} satisfies the conditions in Theorem 3.2.

Now we look at the case where Γ is infinitely generated and of the first kind, namely, the fundamental polygon of Γ has infinitely many of the vertexes and has no free side. In this case, we take the point z_{0} as a limit point of the vertexes of a fundamental polygon of Γ. We define the quadratic differentials ψ̃ and ψ as above. By a result in [10], we see that the Beltrami differential kψ̄/|ψ| is extremal and ∥ψ∥ = ∞. All of the above arguments hold.

We have proved

Theorem 4.1. In any infinite dimensional Teichmüller space T (S), there is a pair of points such that there exist infinitely many of geodesic lines through them.

5. Problems.

The following problems seem naturally to present themselves:

(i) Does the uniqueness of the extremal differentials in a point [μ] always imply the uniqueness of the geodesics joining [0] and [μ]?

(ii) Does the uniqueness of the geodesics joining [0] and [μ] always imply the uniqueness of the extremal differentials in [μ]?

Non-uniqueness of geodesics in Teichmüller spaces 367

References

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[3] Hamilton, R.S.: Extremal quasiconformal mappings with prescribed boundary values. - Trans. Amer. Math. Soc. 138, 1969, 399–406.

[4] Kravetz, S.: On the geometry of Teichmüller spaces and the structure of their modular groups. - Ann. Acad. Sci. Fenn. Ser. A I Math. 278, 1959, 1–35.

[5] Krushkal, S.L.: On Teichmüller theorem on extremal quasiconformal mappings. - Math. UUSR-Sb. 8, 1967, 313–332.

[6] Lehto, O.: Univalent functions and Teichmüller spaces. - Springer-Verlag, 1985.

[7] Li, Z.: Non-uniqueness of geodesics in infinite dimensional Teichmüller spaces. - Complex Variables Theory Appl. 16, 1991, 261–372.

[8] Reich, E., and K. Strebel: Extremal quasiconformal mappings with given boundary values. - In: Contribution to Analysis, Academic Press, New York, 1974, 375–392.

[9] Reich, E., and K. Strebel: On quasiconformal mappings which keep the boundary points fixed. - Trans. Amer. Math. Soc. 137, 1969, 211–221.

[10] Shen, Y.: The uniqueness of extremal quasiconformal mappings. - To appear.

[11] Strebel, K.: Zur Frage der Eindeutigkeit extremaler quasikonformer Abbildungen des Einheitskreises. - Comm. Math. Helv. 36, 1961–1962, 306–343.

[12] Tanigawa, H.: Holomorphic family of geodesic discs in infinite dimensional Teichmüller spaces. - To appear.

[13] Vekua, Y.N.: Generalized analytic functions. - Physic-Math. Press, Moscow, 1959.

Received 13 May 1992