GENERALIZED OBSTACLE PROBLEM

RIE SASAI

Abstract. Fehlmann and Gardiner [3] considered the obstacle problem which asks what embedding of a Riemann surface S of finite topological type minus an obstacle E into another surface R of the same type which induces the isomorphisms π_{1}(S) → π_{1}(R) of the fundamental groups does maximize the L^{1}-norm of the holomorphic quadratic differential on R corresponding to a given one on S under the heights mapping. In this paper we consider obstacles with arbitrarily many connected components while they considered the case where the obstacle has finitely many components. As an application we give a slit mapping theorem of an open Riemann surface of finite genus.

1. Introduction.

Since early 20th century, through the works of Koebe and de Possel and others, it has been turned out that a solution to an extremal problem often gives a conformal mapping with distinguished properties, such as parallel slit mappings. In this regard, it is important to find a good extremal problem for the investigations on complex analysis. Fehlman and Gardiner [3] studied such an extremal problem, which is called the obstacle problem. In this paper, we extend their results to the case where an obstacle may have uncountably many connected component. In order to formulate the problem, we give some explanations for necessary notions.

For a holomorphic quadratic differential φ = φ(z)dz^{2} on a Riemann surface S, we define the L^{1}-norm ∥φ∥_{L^{1}(S)} of it by

∥φ∥_{L^{1}(S)} = ∫∫ _{S} |φ(z)|dx dy.

If ∥φ∥_{L^{1}(S)} < ∞, the quadratic differential φ is called integrable. We denote by A(S) the set of integrable holomorphic quadratic differentials on S.

1991 Mathematics Subject Classification. Primary 30F60; Secondary 30C75, 32G15.

Suppose that S is a hyperbolic Riemann surface. Then there is a holomorphic universal covering projection P of 𝔻 = {z ∈ ℂ | |z| < 1} onto S. Then

Γ = {γ ∈ Aut𝔻 | P ∘ γ = P }

is called a Fuchsian model of S. When Γ is of the second kind, namely, when the limit set Λ(Γ) of Γ does not coincide with ∂𝔻, the Riemann surface S^{d} = (ℂ̄ \ Λ(Γ))/Γ is called the (Schottky) double of S, and ∂S = (∂𝔻 \ Λ(Γ))/Γ is called the border of S. When Γ is of the first kind, we set S^{d} = S and ∂S = ∅.

Let S be a Riemann surface of finite topological type (κ, m, l), that is, S is obtained by removing mutually disjoint l topological compact disks and m points from a compact Riemann surface of genus κ. When l = 0, R is called of finite analytic type (κ, m). When l > 0, the double S^{d} is of finite analytic type (2κ+l −1, 2m). We assume that the number 6κ − 6 + 3l + 2m is positive in the sequel so that dim_{ℂ} A(S^{d}) = 6κ − 6 + 3l + 2m > 0.

A quadratic differential φ = φ(z)dz^{2} in A(S) is called symmetric in ∂S if there is a quadratic differential φ^{d} ∈ A(S^{d}) with φ^{d}|_{S} = φ for which \overline{j^{∗}φ^{d}} = φ^{d} holds. Here j : S^{d} → S^{d} is the canonical anti-conformal involution of S^{d} induced by conjugation

z ↦ \frac{1}{z̄}, in other words, the lift φ̃ of φ^{d} to ℂ̄ \ Λ(Γ) is symmetric in ∂𝔻 : \overline{φ̃(\frac{1}{z̄}) · \frac{1}{z̄^{4}}} = φ̃(z).

We denote by A_{s}(S) the set of quadratic differentials φ ∈ A(S) symmetric in ∂S. Note that dim_{ℝ} A_{s}(S) = 6κ − 6 + 3l + 2m.

For a Riemann surface S of finite analytic type let 𝔖(S) be the set of simple closed curves γ on S which are homotopic neither to a point in S nor to a puncture of S, and let 𝔖[S] be the set of the free homotopy classes [γ] of γ ∈ 𝔖(S).

For a given φ ∈ A(S) \ {0}, we set

height_{φ}(γ) := ∫_{γ} |Im (\sqrt{φ(z)}dz)|

and

height_{φ}[γ] := inf_{β} height_{φ}(β),

where the infimum is taken over all closed curves β ∈ 𝔖(S) freely homotopic to γ. Conversely, such a height vector h ∈ ℝ^{𝔖[S]} together with a conformal structure on S determines a quadratic differential φ ∈ A(S) with height_{φ}[γ] = h[γ] (see Section 2 or [5, Chap. 12] for details).

We are now ready to state the obstacle problem in the sense of Fehlmann and Gardiner [3]. They thought of a “simply connected” compact subset with finitely many connected components as an obstacle. We will consider a more general set as an obstacle.

Definition 1.1. A subset E of a Riemann surface S is said to be allowable if E is compact and contractible in S and if S \ E is connected. For an element φ ∈ A(S), if each component of an allowable subset E is either a horizontal arc of φ or the union of a finite number of horizontal arcs and critical points of φ, then E is called an allowable slit with respect to φ.

Note that an allowable set E may have uncountably many components. For detailed properties of an allowable set, see Section 2.

Let E be an allowable subset of a Riemann surface S of finite topological type. We denote by 𝔉(S, E) the set of pairs (g, S_{g}), where S_{g} is a Riemann surface of the same type as S and g is a conformal embedding of S \ E into S_{g} such that g maps the border ∂S of S to that of S_{g} and the same applies to the punctures. We remark that E^{d} = E ∪ j(E) and E_{g} = S_{g} \ g(S \ E) are both allowable (see Lemma 2.3). For every (g, S_{g}) ∈ 𝔉(S, E), the mapping g naturally extends to a conformal map g^{d} from S^{d} \ E^{d} into the double S_{g}^{d} of S_{g}. Then g^{d} induces an isomorphism ι_{g} of the fundamental group π_{1}(S^{d}) of S^{d} onto that of S_{g}^{d} (see Lemma 2.4). For each φ ∈ A_{s}(S) \ {0}, we assign the new height vector [γ] ↦ height_{φ^{d}}(ι_{g}^{−1}[γ]), [γ] ∈ 𝔖[S_{g}^{d}]. Thus there is the unique holomorphic quadratic differential φ_{g}^{d} ∈ A(S_{g}^{d}) \ {0} such that

height_{φ_{g}^{d}}[γ] = height_{φ^{d}}(ι_{g}^{−1}[γ])

for every [γ] ∈ 𝔖[S_{g}^{d}]. Since φ_{g}^{d} is symmetric in ∂S_{g}, φ_{g} = φ_{g}^{d}|_{S_{g}} belongs to A_{s}(S_{g}). (See the next section for a more detailed account.)

The obstacle problem for (S, E, φ) is to find an element (g, S_{g}) in 𝔉(S, E) which maximizes the quantity

M_{g} = ∥φ_{g}∥_{L^{1}(S_{g})} = ∫∫ _{S_{g}} |φ_{g}|.

We show existence and uniqueness of a solution to the problem and, as an application, we deduce a slit mapping theorem (see Section 5). Our first main theorem is stated as in the following, which is a generalization of a result of Fehlmann and Gardiner [3].

Theorem 1.2 (existence). Suppose that S is a Riemann surface of finite topological type and that φ ∈ A_{s}(S) \ {0}. Let E be an allowable subset of S. Then there is an element (g, S_{g}) ∈ 𝔉(S, E) with

M_{g} = sup_{(h,S_{h})∈𝔉(S,E)}M_{h}

such that E_{g} = S_{g} \ g(S \ E) is an allowable slit with respect to φ_{g}, and E_{g} is of zero area.

We call such an element (g, S_{g}) ∈ 𝔉(S, E) as in Theorem 1.2 extremal for (S, E, φ) and φ_{g} ∈ A(S_{g}) \ {0} the extremal differential associated with (g, S_{g}) ∈ 𝔉(S, E).

In [3], Fehlmann and Gardiner showed the above result under the additional assumption that E consists of finitely many components. They also asserted in the paper that the extremal pair (g, S_{g}) is unique in the sense that, if (u, S_{u}) ∈ 𝔉(S, E) is also extremal for (S, E, φ), then g ∘ u^{−1} extends to a conformal map from S_{u} onto S_{g}. The uniqueness, however, does not necessarily hold in this sense (see [9]).

In this paper we show a uniqueness result for the obstacle problem in the following form. Theorem 1.3 (uniqueness). Under the same hypotheses as in Theorem 1.2, the extremal differential φ_{g} ∈ A(S_{g}) \ {0} is uniquely determined. Namely, if an element (u, S_{u}) ∈ 𝔉(S, E) is also extremal for (S, E, φ), then the extremal differential φ_{u} associated with (u, S_{u}) satisfies

φ_{u} = (φ_{g} ∘ w)(w′)^{2} on u(S \ E),

where w = g ∘ u^{−1}.

We remark that, if we consider the extremal problem of finding an element minimizing ∥φ_{g}∥_{L^{1}(S_{g})} instead of maximizing, then the similar result can be obtained just by replacing the term “horizontal” by “vertical”. Acknowledgement. The author wishes to express her deepest gratitude to Professors Takehiko Morita, Masakazu Shiba, Mitsuhiro Shishikura, Toshiyuki Sugawa and Masahiko Taniguchi for many valuable suggestions and ceaseless encouragement.

2. Preliminaries.

Let S be a Riemann surface of finite analytic type. The following variant of the second minimal norm property will play an important role in this note. The assertion can be seen by analyzing the proof of Theorem 9 in [4, p. 54].

Proposition 2.1. Let S be a Riemann surface of finite analytic type. Suppose that φ ∈ A(S) and ψ is an integrable quadratic differential on S such that

height_{φ}[γ] ≤ height_{ψ}(γ)

for almost every φ-polygonal curve γ ∈ 𝔖(S). Then

∥φ∥_{L^{1}(S)} ≤ ∫∫ _{S} |\sqrt{φ}\sqrt{ψ}|

and, in particular, ∥φ∥_{L^{1}(S)} ≤ ∥ψ∥_{L^{1}(S)}. Moreover ∥φ∥_{L^{1}(S)} = ∥ψ∥_{L^{1}(S)} only if φ = ψ a.e.

We now recall a definition of the Teichmüller space T (S) of a Riemann surface S of finite analytic type (see [6] for the compact case). Let (R, ι) be a marked Riemann surface of the same type as S, that is, a pair of a Riemann surface R of the same type as S and an orientation-preserving isomorphism ι of the fundamental group π_{1}(S) onto π_{1}(R). (More rigorously, we should consider the fundamental group with base point. Though we do not refer to the base point to avoid complexity, the reader can formulate it in an obvious way.) Two pairs (R_{1}, ι_{1}) and (R_{2}, ι_{2}) are called (Teichmüller) equivalent if there exists a conformal mapping u of R_{1} onto R_{2} such that (u)_{∗} ∘ ι_{1} = ι_{2}. The set of such equivalent classes [R, ι] is called the Teichmüller space of S and denoted by T (S). Every point in T (S) is represented as [R, f_{∗}] by a (smooth) quasiconformal map f : S → R. For the existence of such a quasiconformal map, see [7].

We next recall heights mappings (cf. [4, §11.7], [5, §12.6]). Let |dv| be a measured foliation on a Riemann surface S of finite analytic type, namely, there are finitely many singuralities p_{1}, …, p_{t} in Ŝ, where Ŝ is the completion of S and an open cover {U_{j}} of S \ {p_{1}, …, p_{t}} and real-valued continuous funtions v_{j} on U_{j} with locally L^{2} derivatives in such a way that v_{j} = ±v_{k} + const. on U_{j} ∩ U_{k} and |dv| behaves around each p_{s} like the pull-back of |Im (z^{n/2}dz)| under a quasiconformal map with certain restriction on the integer n.

Let φ be a non-zero integrable holomorphic quadratic differential on a Riemann surface S of finite analytic type. We say that a curve γ ∈ 𝔖(S) is φ-polygonal if γ is an union of finitely many horizontal and vertical arcs of φ. For a measured foliation |dv| on S, by abuse of language, we define the height of γ relative to |dv| by

height_{v}(γ) = ∫_{γ} |dv|.

Note that the height is defined for almost every φ-polygonal curves in 𝔖(S). We denote by height_{v}[γ] the essential infimum of height_{v}(γ′) over γ′ ∈ [γ]. Measured foliations |dv_{1}| and |dv_{2}| on S are called measure equivalent if height_{v_{1}}[γ] = height_{v_{2}}[γ] holds for all [γ] ∈ 𝔖[S]. Let 𝔐𝔉(S) be the set of measure equivalence classes of measured foliations on S. Every measure equivalence class [|dv|] ∈ 𝔐𝔉(S) induces a real-valued function

height_{v} : 𝔖[S] → ℝ ([γ] ↦ height_{v}[γ]).

In this way, we obtain an embedding

𝔐𝔉(S) → ℝ^{𝔖[S]} ([|dv|] ↦ height_{v}).

Then the product topology of ℝ^{𝔖[S]} induces a topology of 𝔐𝔉(S). It is known that the mapping

Ψ : A(S) \ \{0\} → 𝔐𝔉(S) (φ ↦ |Im (\sqrt{φ(z)}dz)|)

is a homeomorphism (see, for example, [5, p. 227]).

For a given φ ∈ A(S)\{0} and a quasiconformal map f : S → R, the measured foliation |dv| = |Im (\sqrt{φ(z)}dz)| on S induces a measured foliation f_{∗}(|dv|) = |d(v ∘ f^{−1})| on R. The relation

height_{φ}[γ] = height_{f_{∗}(|dv|)}[f_{∗}(γ)], [γ] ∈ 𝔖[S],

implies that the measure equivalent class [f_{∗}(|dv|)] ∈ 𝔐𝔉(R) depends only on the Teichmüller equivalence class τ = [R, f_{∗}]. Then we obtain the unique holomorphic quadratic differential τ_{∗}φ ∈ A(R) \ {0} as Ψ^{−1}[f_{∗}(|dv|)], namely,

height_{τ_{∗}φ}[γ] = height_{(f)_{∗}(|dv|)}[γ]
= height_{φ}[f^{−1}(γ)]

for every [γ] ∈ 𝔖[R]. The mapping τ_{∗} : A(S) \ {0} → A(R) \ {0} is called the heights mapping.

We collect here basic properties of allowable subsets of a Riemann surface. First we give another characterization of allowable sets.

Lemma 2.2. Let E be a compact subset of a Riemann surface S. Then E is contractible in S if and only if E is contained in a compact topological disk.

Proof. The “if ” part is trivial. We show the “only if ” part when S is hyperbolic. The other cases can be treated similarly. Suppose that E is contractible in S, namely, there is a continuous map H : E × [0, 1] → S such that H(p, 1) = p for all p ∈ E and that H(p, 0) = p_{0} is independent of p ∈ E. Let P be the universal covering projection of 𝔻 onto S such that P (0) = p_{0} and Γ be the covering transformation group of P. Let H̃ be a lift of the homotopy H via P such that H̃(p, 0) = 0 for all p ∈ E. The set Ẽ = {H̃(p, 1) : p ∈ E} is compact. For every p̃ ∈ Ẽ we can take a positive number rp̃ with the property that a closed disk Vp̃ centered at p̃ with radius rp̃ satisfies

dist(A(Vp̃), Ẽ) ≥ 2rp̃ for every A ∈ Γ, A ≠ id.

By compactness of Ẽ we may assume that there exist finitely many of points p̃_{1}, …, p̃_{k} ∈ Ẽ so that Ẽ is contained in the interior of the closed set Ṽ = ∪_{i=1}^{k}Vp̃_{i}. We may assume that by replacing radii for every i ≠ j, Vp̃_{i} is not tangent to Vp̃_{j}. We can see that A(Ṽ) ∩ Ṽ = ∅ for every A ∈ Γ, A ≠ id. Indeed, if A(Ṽ) ∩ Ṽ ≠ ∅ for some A ∈ Γ, A ≠ id, there exist i, j ∈ {1, …, k} and q̃_{i} ∈ Vp̃_{i} and q̃_{j} ∈ Vp̃_{j} such that A(q̃_{i}) = q̃_{j}. Because of dist(A(Vp̃_{i}), Ẽ) ≥ 2rp̃_{i}, we have rp̃_{j} ≥ dist(q̃_{j}, p̃_{j}) ≥ dist(q̃_{j}, Ẽ) ≥ dist(A(Vp̃_{i}), Ẽ) ≥ 2rp̃_{i} holds. On the other hand by considering A^{−1} we have rp̃_{i} ≥ 2rp̃_{j}. This cannot occur. Hence A(Ṽ) ∩ Ṽ = ∅ for every A ∈ Γ, A ≠ id. Let W̃ be a convex hall of Ṽ, that is W̃ ⊂ 𝔻 is the closure of the set of point p̃ such that there exist a Jordan curve γ in Ṽ whoes interior contains p̃. Then W̃ is a disjoint union of finitely many closed topological disks whose boundaries are contained in Ṽ. Then we can see that for every components W̃_{1}, W̃_{2} of W̃ and for every A ∈ Γ, A ≠ id intersection A(W̃_{1})∩W̃_{2} is empty or contained in either A(W̃_{1}) or W̃_{2}, because by the above argument A(∂W̃_{1}) ∩ ∂W̃_{2} = ϕ holds. Therefore the projection P (W̃) also consists of mutually disjoint closed topological disks in S whose union contains E in its interior. Consequently, by joining these disks with suitable canals we obtain a topological closed disk in S containing E.

By using the above lemma, we can also show the following. Lemma 2.3. Let E be an allowable subset of a Riemann surface S of finite topological type. Then E^{d} is an allowable subset of the double S^{d}. Also, for any (g, S_{g}) ∈ 𝔉(S, E), the set E_{g} = S_{g} \ g(S \ E) is allowable in S_{g}. Proof. By joining a compact topological disk Δ containing E and its reflection in ∂S with a suitable canal, we obtain a compact topological disk Δ^{d} containing E^{d}. Thus E^{d} is allowable by Lemma 2.2. We may take the compact topological disk Δ so that E is contained in the interior of Δ. If S is of finite topological type (κ, m, l), then S \ Δ and g(S \ Δ) are of type (κ, m, l + 1). The set Δ_{g} = S_{g} \ g(S \ Δ) ⊂ S_{g} is of finite topological type, say, (κ′, m′, l′). Since S_{g} is reconstructed from g(S \ Δ) and Δ_{g} by glewing along a single Joldan curve, (κ, m, l) = (κ, m, l+1)+(κ′, m′, l′)−(0, 0, 2), i.e., (κ′, m′, l′) = (0, 0, 1). Then the set Δ_{g} which contain E_{g} in the interior is a compact topological disk in S_{g}. Thus the latter assertion has been proved.

The following lemma now easily follows. Lemma 2.4. Let E be an allowable subset of a Riemann surface S of finite topological type and (g, S_{g}) ∈ 𝔉(S, E). Then the natural isomorphism g_{∗}^{d} : π_{1}(S^{d} \ E^{d}) → π_{1}(S_{g}^{d} \ E_{g}^{d}) induces the isomorphism ι_{g} : π_{1}(S^{d}) → π_{1}(S_{g}^{d}).

Proof. It is sufficient to show that if analytic closed curves γ_{1}, γ_{2} ⊂ S^{d} \ E^{d} are homotopic in S^{d}, then g^{d}(γ_{1}) and g^{d}(γ_{2}) are also homotopic in S_{g}^{d}. Let Δ^{d} be a compact topological closed disk which contains E^{d} in the interior. Since the number of components of γ_{i} ∩ ∂Δ^{d}(i = 1, 2) is finite, we can decompose Δ^{d} to finitely many components each of which does not intersect to γ_{1}∪γ_{2}. Then by suitable composition of components of the boundary ∂Δ^{d} of the decomposed topological disk Δ^{d} we can modify γ_{1} to γ′_{1} so that γ′_{1} is homotopic to γ_{2} in S^{d} \ E^{d}. Then g^{d}(γ′_{1}) is homotopic to g^{d}(γ_{2}) in S_{g}^{d} \ E_{g}^{d}. We know by Lemma 2.3 g^{d}(γ′_{1}) is homotopic to g^{d}(γ_{1}) in S_{g}^{d}. Hence g^{d}(γ_{2}) is homotopic to g^{d}(γ_{1}) in S_{g}^{d}.

Following [3], we now introduce a few lemmas. For a simple closed curve β on a Riemann surface S, we denote by λ([β], S) the extremal length of the family [β] of all closed curves on S freely homotopic to β. For a point τ = [R, ι] in T (S), let K_{0}(τ) be the dilatation of the unique extremal quasiconformal map from S onto R which induces the isomorphism ι : π_{1}(S) → π_{1}(R). In other words, log K_{0}(τ) is the Teichmüller distance between τ and the base point [S, id].

Lemma 2.5 ([5, p. 247]). Let S be a Riemann surface of finite analytic type. Then there exists a positive constant c and finitely many simple closed curves β_{1}, …, β_{N} on S such that the inequality

K_{0}(τ) ≤ c max_{1≤l≤N} λ(ι[β_{l}], R)

holds for every point τ = [R, ι] in T (S).

Lemma 2.6 ([4, p. 218]). Let S be a Riemann surface of finite analytic type and φ ∈ A(S) \ {0}. Then for every τ ∈ T (S)

K_{0}(τ)^{−1}∥φ∥_{L^{1}(S)} ≤ ∥τ_{∗}φ∥_{L^{1}(S_{τ})} ≤ K_{0}(τ)∥φ∥_{L^{1}(S)}.

Let E be an allowable subset of a Riemann surface S of finite topological type. By Lemma 2.4, every pair (g, S_{g}) ∈ 𝔉(S, E) corresponds to a point τ = [S_{g}^{d}, ι_{g}] ∈ T (S^{d}). Note that the point τ is represented as [S_{g}^{d}, f_{∗}] by a quasiconformal map f of S^{d} onto S_{g}^{d} which is symmetric in the sense that f ∘ j = j_{g} ∘ f holds, where j : S^{d} → S^{d} and j_{g} : S_{g}^{d} → S_{g}^{d} are the canonical anti-conformal involutions of S^{d} and S_{g}^{d}, respectively. Recall that the holomorphic quadratic differential τ_{∗}φ ∈ A(S_{g}^{d}) \ {0} is determined by

height_{τ_{∗}φ}[γ] = height_{φ^{d}}[f^{−1}(γ)]

for every [γ] ∈ 𝔖[S_{g}^{d}]. We now see that the quadratic differential φ_{g} = τ_{∗}φ|_{S_{g}} is symmetric in ∂S_{g}. Indeed,

height\overline{j_{g}^{∗}τ_{∗}φ}[γ] = height_{τ_{∗}φ}[j_{g}(γ)]
= height_{φ^{d}}[f^{−1} ∘ j_{g}(γ)]
= height_{φ^{d}}[j ∘ f^{−1}(γ)]
= height_{φ^{d}}[f^{−1}(γ)]
= height_{τ_{∗}φ}[γ]

for every [γ] ∈ 𝔖[S_{g}^{d}] and Proposition 2.1 implies

\overline{j_{g}^{∗}τ_{∗}φ} = τ_{∗}φ,

that is, φ_{g} is symmetric in ∂S_{g}.

Let S be a Riemann surface of finite topological type, φ ∈ A_{s}(S) \ {0} and E be an allowable subset of S. We now show that

{M_{h} : (h, S_{h}) ∈ 𝔉(S, E)}

is bounded above and away from zero. Moreover we see that the set

𝔗(S, E) = {[S_{g}^{d}, ι_{g}] : (g, S_{g}) ∈ 𝔉(S, E)}

is relatively compact in T (S^{d}).

For the double S^{d} we take a family of curves β_{1}, …, β_{N} and a positive constant c as in Lemma 2.5. We may assume that each β_{k} is contained in S^{d} \ E^{d}. For every (h, S_{h}) ∈ 𝔉(S, E) set τ = [S_{h}^{d}, ι_{h}] ∈ T (S^{d}). Lemma 2.6, together with ∥φ^{d}∥_{L^{1}(S^{d})} = 2∥φ∥_{L^{1}(S)} and ∥φ_{h}^{d}∥_{L^{1}(S_{h}^{d})} = 2∥φ_{h}∥_{L^{1}(S_{h})}, implies

\frac{∥φ∥_{L^{1}(S)}}{K_{0}(τ)} ≤ ∥φ_{h}∥_{L^{1}(S_{h})} ≤ K_{0}(τ)∥φ∥_{L^{1}(S)}.

Moreover from Lemma 2.5, we obtain

K_{0}(τ) ≤ c max_{1≤l≤N} λ(ι_{h}[β_{l}]), S_{h}^{d})
= c max_{1≤l≤N} λ([h^{d}(β_{l})], S_{h}^{d})
≤ c max_{1≤l≤N} λ([h^{d}(β_{l})], h^{d}(S^{d} \ E^{d}))
= c max_{1≤l≤N} λ([β_{l}], S^{d} \ E^{d}) < ∞.

The last constant is independent of (h, S_{h}) ∈ 𝔉(S, E). This means that 𝔗(S, E) is relatively compact in T (S^{d}). Moreover, letting

c_{0} = c max_{1≤l≤N} λ([β_{l}], S^{d} \ E^{d}),

we have

c_{0}^{−1}∥φ∥_{L^{1}(S)} ≤ M_{h} = ∥φ_{h}∥_{L^{1}(S_{h})} ≤ c_{0}∥φ∥_{L^{1}(S)}

for every (h, S_{h}) ∈ 𝔉(S, E). We have shown

Lemma 2.7. Let E be an allowable subset of a Riemann surface S of finite topological type and φ ∈ A_{s}(S) \ {0}. Then, the set 𝔗(S, E) is relatively compact in T (S^{d}) and there is a positive constant c_{0} such that

c_{0}^{−1}∥φ∥_{L^{1}(S)} ≤ M_{h} ≤ c_{0}∥φ∥_{L^{1}(S)}

for every (h, S_{h}) ∈ 𝔉(S, E).

3. Proof of Theorem 1.2

Take a sequence {(g_{n}, S_{g_{n}})}_{n} ⊂ 𝔉(S, E) such that

M_{g_{n}} ↗ sup_{(h,S_{h})∈𝔉(S,E)}M_{h}.

Set τ_{n} = [S_{g_{n}}^{d}, ι_{g_{n}}] ∈ T (S^{d}). Let Γ and Γ_{n} (n ∈ ℕ) be the normalized Fuchsian models ([6, p. 59]) of S and S_{g_{n}}, respectively. Since {[S_{g_{n}}^{d}, ι_{g_{n}}]}_{n} is relatively compact in T (S^{d}) by Lemma 2.7, we may assume that [S_{g_{n}}^{d}, ι_{g_{n}}] converge in T (S^{d}). Then the isomorphisms χ_{n} : Γ → Γ_{n} induced by ι_{g_{n}} algebraically converge to an isomorphism χ_{∞} : Γ → Γ_{∞}. We denote by S_{∞} the quotient space 𝔻/Γ_{∞}. Let G be a Fuchsian model of S \ E and ρ : G → Γ be the surjective homomorphism induced by the natural homomorphism π_{1}(S \ E) → π_{1}(S). For each n ∈ ℕ, set

ρ_{n} = χ_{n} ∘ ρ : G → Γ_{n}.

Since χ_{n} converge to χ_{∞}, the homomorphisms ρ_{n} converge to the surjective homomorphism

ρ_{∞} = χ_{∞} ∘ ρ : G → Γ_{∞}.

Let g̃_{n} : 𝔻 → 𝔻 (n ∈ ℕ) be the lift of g_{n} associated with ρ_{n}. Then

g̃_{n} ∘ A = ρ_{n}(A) ∘ g̃_{n}

for every A ∈ G. Since {g̃_{n}}_{n} is normal, there is a subsequence which converges to a holomorphic map g̃ on 𝔻 uniformly on any compact subset of 𝔻. Then we obtain

g̃ ∘ A = ρ_{∞}(A) ∘ g̃

for every A ∈ G. If g̃ is a constant c, then Γ_{∞} has the common fixed point c, and thus Γ_{∞} is a cyclic group, which is not the case. So g̃ is not constant and thus a holomorphic map into 𝔻. Therefore g̃ descends to an injective holomorphic map g of S \ E into S_{∞} such that the homomorphism (g)_{∗} corresponds to ρ_{∞}. We denote S_{∞} by S_{g}. Then it is easy to see that (g, S_{g}) ∈ 𝔉(S, E).

Let Γ̂, Γ̂_{n} (n ∈ ℕ) and Γ̂_{∞} be the normalized Fuchsian models of the compact Riemann surface Ŝ, Ŝ_{g_{n}} and Ŝ_{g}, which are the completion of the doubles S^{d}, S_{g_{n}}^{d} and S_{g}^{d}, respectively. (In the case when Ŝ is not hyperbolic, the proof will be simpler.) Let f_{n} be a quasiconformal map of S^{d} onto S_{g_{n}}^{d} which represents [S_{g_{n}}^{d}, ι_{g_{n}}] ∈ T (S^{d}) and f be a quasiconformal map of S^{d} onto S_{g}^{d} representing the element [S_{g}^{d}, ι_{g}] ∈ T (S^{d}). We denote by f̂_{n} and f̂ the lifts of f_{n} and f fixing ±1, −i to the covering space 𝔻 over Ŝ_{g_{n}} and Ŝ_{g}, respectively. Let Ω̂ be a relatively compact fundamental domain of Γ̂. Set Ω̂_{n} = f̂_{n}(Ω̂) (n ∈ ℕ) and Ω̂_{∞} = f̂(Ω̂). Then Ω̂_{n} and Ω̂_{∞} are the fundamental domains of Γ̂_{n} and Γ̂_{∞}, respectively. Let φ̂_{g_{n}} be a lift of φ_{g_{n}}^{d} to the universal covering space 𝔻 over Ŝ_{g_{n}}. For each n ∈ ℕ, φ̂_{g_{n}} is a meromorphic quadratic differential on 𝔻 with at most simple poles. Let D be a subdomain of 𝔻, which is obtained from 𝔻 by removing the points corresponding to the punctures of S_{g}^{d}.

Lemma 3.1. The family {φ̂_{g_{n}}}_{n} is normal on D.

Proof. It is sufficient to show that {∥φ̂_{g_{n}}∥_{L^{1}(K)}}_{n} is bounded for any compact subset K of 𝔻. Let

{A ∈ Γ̂_{∞} : A(Ω̂_{∞} ∪ ∂Ω̂_{∞}) ∩ K ≠ ∅} = {A_{1}, …, A_{m_{0}}}.

Then, the euclidean distance d_{∞} between K and (∪_{i=1}^{m_{0}}A_{i}(Ω̂_{∞}))^{c} is positive. Since f̂_{n} converge to f̂ uniformly on any compact subset of 𝔻, for each A_{i} (1 ≤ i ≤ m_{0}) an element

A_{i,n} = (f̂_{n} ∘ f̂^{−1}) ∘ A_{i} ∘ (f̂_{n} ∘ f̂^{−1})^{−1} ∈ Γ̂_{n}

converge to A_{i} as n → ∞. Then the distance d_{n} between K and (∪_{i=1}^{m_{0}}A_{i,n}(Ω̂_{n}))^{c} converge to d_{∞}. Hence d_{n} > 0 for sufficiently large n. This implies

K ⊂ ⋃_{i=1}^{m_{0}} A_{i,n}(Ω̂_{n}).

Then

#{A ∈ Γ̂_{n} : A(Ω̂_{n}) ∩ K ≠ ∅} ≤ m_{0}

for sufficiently large n. This result together with boundness of {M_{g_{n}}}_{n} (Lemma 2.7) implies that {∥φ̂_{g_{n}}∥_{L^{1}(K)}}_{n} is bounded.

Lemma 3.2. M_{g} = sup{M_{h} : (h, S_{h}) ∈ 𝔉(S, E)}.

Proof. By Lemma 3.1, there is a subsequence of {φ̂_{g_{n}}}_{n} which uniformly converges on any compact subset of D to a holomorphic function φ̂_{∞} . For each n ∈ ℕ, φ̂_{g_{n}} is Γ̂_{n}-invariant, that is,

(φ̂_{g_{n}} ∘ A_{n})(A′_{n})^{2} = φ̂_{g_{n}}

for every A_{n} ∈ Γ̂_{n}. Then we see that φ̂_{∞} is Γ̂_{∞}-invariant, so we can project φ̂_{∞} to a holomorphic quadratic differential φ_{∞}^{∗} on S_{g}^{d}. Moreover for any compact subset K of D,

∥φ̂_{∞}∥_{L^{1}(K)} = lim_{n→∞} ∥φ̂_{g_{n}}∥_{L^{1}(K)} ≤ 2c_{0}∥φ∥_{L^{1}(S)}

holds, where c_{0} is the positive number obtained in Lemma 2.7. Hence we can see that φ̂_{∞} is integrable on Ω̂_{∞}.

Next we show that the integrable holomorphic quadratic differential φ_{∞}^{∗} on S_{g}^{d} is symmetric in ∂S_{g}. Let Λ_{n}(n ∈ ℕ) and Λ_{∞} be the limit sets of Γ_{n} and Γ_{∞} respectively. Let φ̃_{g_{n}}(n ∈ ℕ) be the lift of φ_{g_{n}}^{d} to the covering space ℂ̄ \ Λ_{n} over S_{g_{n}}^{d} and let φ̃_{∞} be the representation of φ_{∞}^{∗} on the double ℂ̄ \ Λ_{∞}. By assumption, for each n ∈ ℕ, φ̃_{g_{n}} satisfies

\overline{φ̃_{g_{n}}(\frac{1}{z̄^{4}} )} · \frac{1}{z^{4}} = φ̃_{g_{n}} on ℂ̄ \ Λ_{n}.

Since φ̃_{g_{n}} converge to φ̃_{∞} uniformly on any compact subset of ℂ̄ \ Λ_{∞}, we have

\overline{φ̃_{∞}(\frac{1}{z̄^{4}} )} · \frac{1}{z^{4}} = φ̃_{∞} on ℂ̄ \ Λ_{∞}.

Hence φ̃_{∞} is symmetric for ∂𝔻, that is, the projection φ_{∞}^{∗} of φ̃_{∞} to S_{g}^{d} is symmetric in ∂S_{g}. We denote by φ_{∞} the restriction of φ_{∞}^{∗} to S_{g}.

Now we show

lim_{n→∞} ∥φ_{g_{n}}∥_{L^{1}(S_{gn})} = ∥φ_{∞}∥_{L^{1}(S_{g})}.

Let p_{1}, …, p_{k} ∈ Ω̂_{∞} be the points which correspond to the punctures of S_{g}^{d}. We may assume that all p_{1}, …, p_{k} are interior points of Ω̂_{∞} ∪ ∂Ω̂_{∞}. Set

p_{i,n} = f̂_{n} ∘ f̂^{−1}(p_{i}) (n ∈ ℕ, 1 ≤ i ≤ k).

Then p_{i,n} → p_{i} as n → ∞.

For sufficiently small ε > 0, let V_{i} ⊂ Ω̂_{∞} (1 ≤ i ≤ k) be the open disk of radius ε centered at p_{i} such that V_{i} ∩ V_{j} = ϕ if i ≠ j. Since holomorphic functions (z − p_{i,n})φ̂_{g_{n}} (n ∈ ℕ) on V_{i} converge to (z − p_{i})φ̂_{∞} uniformly on V_{i}, the residues c_{i,n} = Res(φ̂_{g_{n}}, p_{i,n}) converge to c_{i} = Res(φ̂_{∞}, p_{i}) as n → ∞. Moreover we can also see that holomorphic functions

φ̂_{g_{n}} − \frac{c_{i,n}}{z − p_{i,n}}

converge to

φ̂_{∞} − \frac{c_{i}}{z − p_{i}}

uniformly on V_{i}. We take the union ⋃_{n≥n_{0}} Ω̂_{n} ∪ ∂Ω̂_{n} for sufficiently large n_{0} ∈ ℕ. Then φ̂_{g_{n}} converge to φ̂_{∞} uniformly on the compact set

( ⋃_{n≥n_{0}}(Ω̂_{n} ∪ ∂Ω̂_{n})) ⋃(Ω̂_{∞} ∪ ∂Ω̂_{∞}) \ ⋃_{i=1}^{k} V_{i}.

Set

C = max_{1≤i≤k} |c_{i}|.

Then we have

lim sup_{n→∞} |∥φ̂_{g_{n}}∥_{L^{1}(Ω̂_{n})} − ∥φ̂_{∞}∥_{L^{1}(Ω̂_{∞})}|
≤ lim sup ∑_{i=1}^{k} (∥\frac{c_{i}}{z − p_{i}} ∥_{L^{1}(V_{i})} + ∥\frac{c_{i,n}}{z − p_{i,n}} ∥_{L^{1}(V_{i})})

+ lim sup ∑_{i=1}^{k} ∥(φ̂_{g_{n}} − \frac{c_{i,n}}{z − p_{i,n}} ) − (φ̂_{∞} − \frac{c_{i}}{z − p_{i}} )∥_{L^{1}(V_{i})}

+ lim sup |∥φ̂_{g_{n}}∥_{L^{1}(Ω̂_{n}\∪_{i=1}^{k}V_{i})} − ∥φ̂_{∞}∥_{L^{1}(Ω̂_{∞}\∪_{i=1}^{k}V_{i})}|

≤ Ck6πε.

Thus

lim_{n→∞} ∥φ_{g_{n}}∥_{L^{1}(S_{gn})} = ∥φ_{∞}∥_{L^{1}(S_{g})},

and we obtain ∥φ_{∞}∥_{L^{1}(S_{g})} = sup_{(h,S_{h})∈𝔉(S,E)}M_{h}. Moreover, by Lemma 2.7, we have

∥φ_{∞}∥_{L^{1}(S_{g})} = lim_{n→∞} ∥φ_{g_{n}}∥_{L^{1}(S_{gn})} ≥ c_{0}^{−1}∥φ∥_{L^{1}(S)}.

Therefore φ_{∞} ∈ A_{s}(S_{g}) \ {0}.

Next we show φ_{∞} = φ_{g} on S_{g}. Take any curve γ ∈ 𝔖(S_{g}^{d}) and sufficiently small ε > 0. Set γ_{n} = f_{n} ∘ f^{−1}(γ) (n ∈ ℕ). Assume that A_{∞} ∈ Γ̂_{∞} corresponds to the homotopy class [γ] on Ŝ_{g}. Then

A_{n} = (f̂_{n} ∘ f̂^{−1}) ∘ A_{∞} ∘ (f̂_{n} ∘ f^{−1})^{−1}

corresponds to the homotopy class [γ_{n}] on Ŝ_{g_{n}}.

Let α_{n} ∈ 𝔖(S_{g_{n}}^{d}) be a φ_{g_{n}}^{d}-geodesic curve freely homotopic on S_{g_{n}}^{d} to γ_{n}, that is, α_{n} satisfies [α_{n}] = [γ_{n}] on S_{g_{n}}^{d} and ℓ_{φ_{gn}^{d}}[γ_{n}] = ℓ_{φ_{gn}^{d}}(α_{n}), where ℓ_{φ_{gn}^{d}}(α_{n}) is the length

∫_{α_{n}} |\sqrt{φ_{g_{n}}^{d}(z)}dz|

of α_{n} associated with φ_{g_{n}}^{d} and ℓ_{φ_{gn}^{d}}[γ_{n}] is that of homotopy class [γ_{n}] defined by

inf_{β} ℓ_{φ_{gn}^{d}}(β),

where the infimum is taken over all closed curves β freely homotopic to γ_{n} on S_{g_{n}}^{d}. It is known [11, Theorem 24.1] that each α_{n} satisfies height_{φ_{gn}^{d}}[γ_{n}] = height_{φ_{gn}^{d}}(α_{n}). Let α̂_{n} (n ∈ ℕ) be a lift of α_{n} to the universal covering space 𝔻 over Ŝ_{g_{n}} such that α̂_{n} is a closed arc starting from a point on ∂Ω̂_{n}. We parametrize each α̂_{n} by the φ̂_{g_{n}}-length s and we set l(n) = ℓφ̂_{gn}(α̂_{n}). Then {l(n)}_{n} is bounded because the set of {[S_{g_{n}}^{d}, ι_{g_{n}}]}_{n} is relatively compact in T (S^{d}) by Lemma 2.7, so we may assume that l(n) converge to a positive constant l_{∞}. For each n ∈ ℕ by re-parametrizing α̂_{n} by α̂_{n}(l(n)s), we may assume that for every n ∈ ℕ, α̂_{n} is defined on [0, 1]. The curve α_{n} (n ∈ ℕ) is analytic except for finitely many points where φ_{g_{n}}^{d} vanishes. Since the number of zeros of φ_{g_{n}}^{d} on S_{g_{n}}^{d} is uniformly bounded for n ∈ ℕ and {l(n)}_{n} is also bounded, the orders of singularities of α̂_{n} are uniformly bounded. Then we can assume that there exists a natural number N_{0} and real numbers t_{1}, t_{2}, …, t_{N_{0}} ∈ [0, 1] (t_{1} ≤ t_{2} ≤ … ≤ t_{N_{0}}) such that the number of critical points of α̂_{n} is N_{0} for all n ∈ ℕ and for each i = 1, …, N_{0} critical point t_{n,i} of α̂_{n} converge to t_{i} as n → ∞. We can see by reduction to absurdity that the family {α̂_{n}}_{n∈ℕ} is equicontinuous. Then, together with uniform boundedness of {α̂_{n}}_{n∈ℕ}, we may assume that α̂_{n} uniformly converge to a continuous function α̂_{∞} on [0, 1]. The image α̂_{∞}[0, 1] is a lift of a closed curve on Ŝ_{g}.

Now we can show that heightφ̂_{gn}(α̂_{n}) converge to heightφ̂_{∞}(α̂_{∞}). Let s_{0} be any point on (0, 1) \ {t_{1}, …, t_{N_{0}}} such that φ̂_{∞}(α̂_{∞}(s_{0})) ≠ 0 and α̂_{∞}(s_{0}) ∈ D. We can take a neighborhood U of s_{0} in (0, 1) such that for every n ∈ ℕ φ̂_{g_{n}}(α̂_{n}(s)) ≠ 0 on U ∪ ∂U and α̂_{n}(U ∪∂U) does not contain any points corresponding to punctures of S_{g_{n}}^{d}. Let ζ̂_{n} (n ∈ ℕ) and ζ̂_{∞} be the natural parameter near α̂_{n}(s_{0}) and α̂_{∞}(s_{0}), which are represented by

ζ̂_{n} = ∫_{α̂_{n}(s_{0})}^{z} \sqrt{φ̂_{g_{n}}(z)}dz

and

ζ̂_{∞} = ∫_{α̂_{∞}(s_{0})}^{z} \sqrt{φ̂_{∞}(z)}dz,

respectively. For every n ∈ ℕ, α_{n} is geodesic relative to φ_{g_{n}}^{d}-length, so we see that

\frac{dζ̂_{n}}{|dζ̂_{n}|} = e^{iθ_{n}} on U

for a constant θ_{n} ∈ [0, 2π). For any subsequence {θ_{n_{k}}}_{k} which is converging to some θ_{∞} ∈ [0, 2π]

\frac{dα̂_{n_{k}}}{ds} = \frac{l(n_{k})dα̂_{n_{k}}}{|dζ̂_{n_{k}}|}
= \frac{l(n_{k})dα̂_{n_{k}}}{dζ̂_{n_{k}}} · \frac{dζ̂_{n_{k}}}{|dζ̂_{n_{k}}|}
= \frac{l(n_{k})}{\sqrt{φ̂_{g_{n_{k}}} (α̂_{n_{k}})}} · e^{iθ_{n_{k}}}
⟶^{k→∞} \frac{l_{∞}}{\sqrt{φ̂_{∞}(α̂_{∞})}} · e^{iθ_{∞}}.

The convergence is uniform on U, so we can see that α̂_{∞} is of C^{1}-class on U and

\frac{dα̂_{∞}}{ds} = lim_{k→∞} \frac{dα̂_{n_{k}}}{ds} = \frac{l_{∞}}{\sqrt{φ̂_{∞}(α̂_{∞})}} · e^{iθ_{∞}}.

Since the argument θ_{∞} is independent of the choice of subsequence, the sequence {θ_{n}}_{n} is converging itself and we obtain

\frac{dα̂_{∞}}{ds} = lim_{n→∞} \frac{dα̂_{n}}{ds} on U.

Consequently the curve α̂_{∞} satisfies

\frac{dα̂_{∞}}{ds} = lim_{n→∞} \frac{dα̂_{n}}{ds} on (0, 1)

except for finitely many of points. Then we obtain lim_{n→∞} heightφ̂_{gn}(α̂_{n}) = heightφ̂_{∞}(α̂_{∞})

because of

|Im (\sqrt{φ̂_{g_{n}}(α̂_{n})}\frac{dα̂_{n}}{ds} )| ≤ |\sqrt{φ̂_{g_{n}}(α̂_{n})}\frac{dα̂_{n}}{ds} | = l(n) ≤ sup_{n∈ℕ}l(n) < ∞

for every n ∈ ℕ and

lim_{n→∞} |Im (\sqrt{φ̂_{g_{n}}(α̂_{n}(s))}\frac{dα̂_{n}}{ds} )| = |Im (\sqrt{φ̂_{∞}(α̂_{∞}(s))}\frac{dα̂_{∞}}{ds} )| a.e. on [0, 1].

If there exists s_{1} ∈ [0, 1] such that α̂_{∞}(s_{1}) corresponds to a puncture of S_{g}^{d}, we modify the curve α̂_{∞} locally in a neighborhood of s_{1} to a curve α̂′_{∞} so that the projection on S_{g}^{d} is homotopic to γ on S_{g}^{d}. It is possible to choose α̂′_{∞} so that ℓφ̂_{∞}(α̂′_{∞}) is as close to ℓφ̂_{∞}(α̂_{∞}) as we want. Therefore, for any ε > 0, there is a curve α̂′_{∞} so that

height_{φ_{∞}^{d}}[γ] ≤ heightφ̂_{∞}(α̂′_{∞})
< heightφ̂_{∞}(α̂_{∞}) + ε
= lim_{n→∞} heightφ̂_{gn}(α̂_{n}) + ε
= lim_{n→∞} height_{φ_{gn}^{d}}[γ_{n}] + ε
= height_{φ_{g}^{d}}[γ] + ε.

Then we have the inequality height_{φ_{∞}^{d}}[γ] ≤ height_{φ_{g}^{d}}[γ]. Particularly, by Proposition 2.1,

∥φ_{∞}^{d}∥_{L^{1}(S_{g}^{d})} ≤ ∥φ_{g}^{d}∥_{L^{1}(S_{g}^{d})}.

On the other hand, from maximality of ∥φ_{∞}∥_{L^{1}(S_{g})} in {M_{h}; (h, S_{h}) ∈ 𝔉(S, E)} we have ∥φ_{g}^{d}∥_{L^{1}(S_{g}^{d})} ≤ ∥φ_{∞}^{d}∥_{L^{1}(S_{g}^{d})}.

Therefore ∥φ_{∞}^{d}∥_{L^{1}(S_{g}^{d})} = ∥φ_{g}^{d}∥_{L^{1}(S_{g}^{d})} holds, and by Proposition 2.1, we have

φ_{∞}^{d} = φ_{g}^{d} on S_{g}^{d}.

Lemma 3.3. E_{g} = S_{g} \ g(S \ E) has measure zero.

Proof. If E_{g} has positive measure, we can define a Beltrami coefficient ν on S_{g}^{d} so that

ν = 0 on S_{g}^{d} \ E_{g}^{d} and ∫∫ _{S_{g}^{d}} νφ_{g}^{d} > 0.

For each t ∈ (0, 1), let f^{tν} be a quasiconformal map of S_{g}^{d} onto the double S_{tν}^{d} of a Riemann surface S_{tν} with Beltrami coefficient tν. Let τ(t) = [S_{tν}^{d}, (f^{tν})_{∗}] ∈ T (S_{g}^{d}). For φ_{g} ∈ A_{s}(S_{g}) \ {0} we set φ_{tν} = τ(t)_{∗}φ_{g}|_{S_{tν}} ∈ A_{s}(S_{tν}) \ {0} whose height on S_{tν}^{d} is equal to that of φ_{g}^{d} on S_{g}^{d}. By the variational formura [4, p. 217], we have

log ∥φ_{tν}^{d}∥_{L^{1}(S_{tν}^{d})} = log ∥φ_{g}^{d}∥_{L^{1}(S_{g}^{d})} + \frac{2}{∥φ_{g}^{d}∥_{L^{1}(S_{g}^{d})}} Re ∫∫ _{S_{g}^{d}} tνφ_{g}^{d} dxdy + o(∥tν∥_{∞}).

Then we can see

∥φ_{tν}∥_{L^{1}(S_{tν})} > ∥φ_{g}∥_{L^{1}(S_{g})}

for sufficiently small t > 0. On the other hand, (f^{tν} ∘ g, S_{tν}) ∈ 𝔉(S, E) for every t ∈ (0, 1), and the maximality of ∥φ_{g}∥_{L^{1}(S_{g})} yields the inequality

∥φ_{tν}∥_{L^{1}(S_{tν})} ≤ ∥φ_{g}∥_{L^{1}(S_{g})},

which is a contradiction. Lemma 3.4. A component of E_{g} is either

(i) a horizontal arc of φ_{g} or, (ii) a connected union of finitely many horizontal arcs and critical points of φ_{g}. In particular, E_{g} is an allowable slit with respect to φ_{g}.

Proof. Fix any component J of E_{g}. We consider the obstacle problem for (S_{g}, J, φ_{g}). Then there is a solution (h, S_{h}) ∈ 𝔉(S_{g}, J) and φ_{h} ∈ A_{s}(S_{h}) \ {0} satisfying

∥φ_{h}∥_{L^{1}(S_{h})} = sup{∥φ_{f}∥_{L^{1}(S_{f})} : (f, S_{f}) ∈ 𝔉(S_{g}, J)}.

Since (h∘g, S_{h}) ∈ 𝔉(S, E), we have ∥φ_{h}∥_{L^{1}(S_{h})} ≤ ∥φ_{g}∥_{L^{1}(S_{g})}. On the other hand, (id, S_{g}) ∈ 𝔉(S_{g}, J) yields the equality ∥φ_{g}∥_{L^{1}(S_{g})} ≤ ∥φ_{h}∥_{L^{1}(S_{h})}. Thus (id, S_{g}) ∈ 𝔉(S_{g}, J) attains the extremum. By the same argument as in [3], we conclude that the component J satisfies either (i) or (ii) in the lemma.

We have proved Theorem 1.2 completely.

4. Proof of Theorem 1.3

Let E_{n} ⊂ S (n ∈ ℕ) be an allowable subset of S such that each E_{n} is a disjoint union of finitely many of closed analytic disks and satisfying E_{n} ⊃ E_{n+1} ⊃ ⋯ and ∩_{n=1}^{∞}E_{n} = E. Set U_{n} = S \E_{n} (n ∈ ℕ). (Consider a regular exhaustion {R_{n}}_{n∈ℕ} of the Riemann surface Ŝ\E^{d} and set U_{n} = S ∩R_{n} and E_{n} = S \U_{n} for sufficiently large n’s; cf. [1, p. 144].) First we show that we can obtain a solution (h_{∞}, S_{h_{∞}}) ∈ 𝔉(S, E) of the obstacle problem for (S, E, φ) from the family of solutions (h_{n}, S_{h_{n}}) ∈ 𝔉(S, E_{n}) for (S, E_{n}, φ) (n ∈ ℕ). Next we show that any other extremal element (u, S_{u}) ∈ 𝔉(S, E) for (S, E, φ) satisfies

φ_{u} = (φ_{h_{∞}} ∘ w_{∞})(w′_{∞})^{2} on u(S \ E),

where w_{∞} = h_{∞} ∘ u^{−1}. The solution (g, S_{g}) ∈ 𝔉(S, E) in the Theorem 1.2 also satisfies

φ_{g} = (φ_{h_{∞}} ∘ w_{0})(w′_{0})^{2} on g(S \ E),

where w_{0} = h_{∞} ∘ g^{−1}, so we have

φ_{u} = (φ_{g} ∘ w)(w′)^{2} on u(S \ E),

where w = g ∘ u^{−1}.

We now proceed to the proof. For each n ∈ ℕ, by considering the obstacle problem for (S, E_{n}, φ), we obtain a solution (h_{n}, S_{h_{n}}) ∈ 𝔉(S, E_{n}). Let τ_{n} = [S_{h_{n}}^{d}, ι_{h_{n}}] ∈ T (S^{d}). We can see in a similar manner to Section 2 that the set {τ_{n}}_{n} ⊂ T (S^{d}) is relatively compact in T (S^{d}). In fact, we may assume that the curve family β_{1}, …, β_{N} in Lemma 2.5 is contained in S^{d} \ U_{1}^{d}. Then the dilatation K_{0}(τ_{n}) of the extremal quasiconformal map of the Teichmüller class τ_{n} satisfies

K_{0}(τ_{n}) ≤ c max_{1≤l≤N} λ([β_{l}], U_{1}^{d}) =: c_{1}

for every n ∈ ℕ. Since the constant c_{1} is independent of n ∈ ℕ, {τ_{n}}_{n} is relatively compact in T (S^{d}). We can also see

c_{1}^{−1}∥φ∥_{L^{1}(S)} ≤ ∥φ_{h_{n}}∥_{L^{1}(S_{hn})} ≤ c_{1}∥φ∥_{L^{1}(S)}

for every n ∈ ℕ. By the same argument as in the proof of Theorem 1.2, we obtain a subsequence {τ_{n_{k}}}_{k} and a point τ_{∞} = [S_{h_{∞}}, ι_{h_{∞}}] ∈ T (S^{d}) associated with an element (h_{∞}, S_{h_{∞}}) ∈ 𝔉(S, E) such that τ_{n_{k}} converge to τ_{∞} in T (S^{d}), and the lift of h_{n_{k}}^{d} to the covering space 𝔻 over S^{d} \ E^{d} converge to that of h_{∞}^{d} uniformly on any compact subset of 𝔻. For brevity, we renumber n_{k} by k. For every (h, S_{h}) ∈ 𝔉(S, E) and every n ∈ ℕ, by considering the restriction of h to U_{n}, we have

∥φ_{h}∥_{L^{1}(S_{h})} ≤ ∥φ_{h_{n}}∥_{L^{1}(S_{hn})}.

Since ∥φ_{h_{n}}∥_{L^{1}(S_{hn})} converge to ∥φ_{h_{∞}}∥_{L^{1}(S_{h∞})}, we have

∥φ_{h}∥_{L^{1}(S_{h})} ≤ ∥φ_{h_{∞}}∥_{L^{1}(S_{h∞})}

for every (h, S_{h}) ∈ 𝔉(S, E). Hence, (h_{∞}, S_{h_{∞}}) ∈ 𝔉(S, E) is extremal for (S, E, φ) ;

M_{h_{∞}} = sup_{(h,S_{h})∈𝔉(S,E)}M_{h}.

Next we assume that an element (u, S_{u}) ∈ 𝔉(S, E) also attains the supremum. Set

w_{n} = h_{n} ∘ u^{−1} , ψ_{n} = (φ_{h_{n}} ∘ w_{n})(w′_{n})^{2} (n ∈ ℕ)
w_{∞} = h_{∞} ∘ u^{−1} , ψ_{∞} = (φ_{h_{∞}} ∘ w_{∞})(w′_{∞})^{2}.

We show that φ_{u} = ψ_{∞} on u(S \ E). For the sake of convenience, we extend ψ_{n} to S_{u} so that ψ_{n} = 0 on S_{u} \ u(U_{n}), similarly ψ_{∞} = 0 on S_{u} \ u(S \ E). Then ψ_{n} (n ∈ ℕ) and ψ_{∞} are integrable quadratic differentials on S_{u}.

Fix any point p_{0} ∈ S_{u}^{d} with φ_{u}^{d}(p_{0}) ≠ 0. Let ζ_{u} = ξ_{u} + iη_{u} be the natural parameter of φ_{u}^{d}, which is defined about p_{0} by

ζ_{u} = ∫_{z_{0}}^{z} \sqrt{φ_{u}^{d}(z)}dz,

where z is a local chart near p_{0} and z(p_{0}) = z_{0}. Let N_{p_{0}} be a sufficiently small closed neighborhood of p_{0} such that N_{p_{0}} corresponds to, by the parameter ζ_{u}, a square centered at p_{0} and each side of which is parallel to the axis. Since for each n ∈ ℕ ψ_{n}^{d} is integrable on S_{u}^{d}, the height relative to ψ_{n}^{d} of almost every vertical segment of φ_{u}^{d} on S_{u}^{d}, particularly on N_{p_{0}}, is well defined and the same holds also to that relative to ψ_{∞}^{d}. Then we can see that for almost every vertical segment of φ_{u}^{d} on S_{u}^{d}, particularly on N_{p_{0}}, the heights relative to ψ_{n}^{d} (n ∈ ℕ) and ψ_{∞}^{d} are all well defined. We denote by V_{p_{0}} the set of such vertical segments of φ_{u}^{d} in N_{p_{0}}.

For each β ∈ V_{p_{0}} we set β_{n} = β ∩ u^{d}(U_{n}^{d}) (n ∈ ℕ). Fix any ε > 0 and any 1dimensional measurable set A ⊂ N_{p_{0}} on a horizontal trajectory of φ_{u}^{d}. Since φ^{d} and φ_{u}^{d} are integrable on S^{d} and S_{u}^{d}, respectively, by Fubini’s theorem and Schwarz’s inequality, for every n, m ∈ ℕ (n ≥ m) we have

∫_{A} |height_{ψ_{n}^{d}}(β) − height_{ψ_{n}^{d}}(β_{m})|dξ_{u} ≤ ∫_{A}∫_{β_{n}\β_{m}} |\sqrt{ψ_{n}^{d}(ζ_{u})}|dη_{u} dξ_{u}
= ∫∫_{V (n,m,A)} |\sqrt{ψ_{n}^{d}(ζ_{u})}|dξ_{u} dη_{u}
≤ ∥ψ_{n}^{d}∥L^{1}(V (n,m,A))^{\frac{1}{2}} · ∥φ_{u}^{d}∥L^{1}(V (n,m,A))^{\frac{1}{2}}
≤ (c_{1}∥φ^{d}∥_{L^{1}(S^{d})})^{\frac{1}{2}} · ∥φ_{u}^{d}∥L^{1}(V (n,m,A))^{\frac{1}{2}}
⟶^{m→∞} 0,

where V (n, m, A) = {p ∈ N_{p_{0}} ∩ u^{d}(U_{n}^{d} \ U_{m}^{d}) | Re ζ_{u}(p) ∈ A}. Then there exists a natural number N_{1} such that for every n ≥ N_{1}

∫_{A} |height_{ψ_{n}^{d}}(β) − height_{ψ_{n}^{d}}(β_{N_{1}})|dξ_{u} < ε.

Furthermore we can see that

∫_{A} |height_{ψ_{n}^{d}}(β_{N_{1}}) − height_{ψ_{∞}^{d}}(β_{N_{1}})|dξ_{u} → 0 (n → ∞),

because ψ_{n}^{d} uniformly converge to ψ_{∞}^{d} on N_{p_{0}} ∩ u^{d}(U_{N_{1}}^{d}). Then, by Fatou’s lemma, we have

∫_{A} lim inf_{n→∞} height_{ψ_{n}^{d}}(β)dξ_{u}
≤ lim inf_{n→∞} ∫_{A} height_{ψ_{n}^{d}}(β)dξ_{u}
≤ lim inf_{n→∞} {∫_{A} (height_{ψ_{n}^{d}}(β) − height_{ψ_{n}^{d}}(β_{N_{1}})) dξ_{u}
+ ∫_{A} (height_{ψ_{n}^{d}}(β_{N_{1}}) − height_{ψ_{∞}^{d}}(β_{N_{1}})) dξ_{u} + ∫_{A} height_{ψ_{∞}^{d}}(β_{N_{1}})dξ_{u}}
= lim inf_{n→∞} ∫_{A} (height_{ψ_{n}^{d}}(β) − height_{ψ_{n}^{d}}(β_{N_{1}})) dξ_{u}
+ lim_{n→∞} ∫_{A} (height_{ψ_{n}^{d}}(β_{N_{1}}) − height_{ψ_{∞}^{d}}(β_{N_{1}})) dξ_{u} + ∫_{A} height_{ψ_{∞}^{d}}(β_{N_{1}})dξ_{u}
≤ ε + ∫_{A} height_{ψ_{∞}^{d}}(β)dξ_{u}.

Therefore almost every β ∈ V_{p_{0}} satisfies

lim inf_{n→∞} height_{ψ_{n}^{d}}(β) ≤ height_{ψ_{∞}^{d}}(β).

The same holds for almost every vertical segments on S_{u}^{d}. Then we can see that almost every φ_{u}^{d}-polygonal curve γ, which is a union of finitely many horizontal and vertical segments, satisfies

lim inf_{n→∞} height_{ψ_{n}^{d}}(γ) ≤ height_{ψ_{∞}^{d}}(γ).

Let w̃_{n}(γ) be a simple closed curve on S_{h_{n}}^{d} which is obtained from w_{n}(γ) by supplying finitely many horizontal segments of φ_{h_{n}}^{d}. We have

height_{φ_{u}^{d}}[γ] = height_{φ_{hn}^{d}}[w̃_{n}(γ)]
≤ height_{φ_{hn}^{d}} (w̃_{n}(γ))
= height_{φ_{hn}^{d}}(w_{n}(γ))
= height_{ψ_{n}^{d}}(γ)

for every n ∈ ℕ. Therefore height_{φ_{u}^{d}}[γ] ≤ lim inf_{n→∞} height_{ψ_{n}^{d}}(γ) holds, and we obtain height_{φ_{u}^{d}}[γ] ≤ height_{ψ_{∞}^{d}}(γ) and, in particular, ∥φ_{u}^{d}∥_{L^{1}(S_{u}^{d})} ≤ ∥ψ_{∞}^{d}∥_{L^{1}(S_{u}^{d})}. Then the maximality of M_{u} = ∥φ_{u}^{d}∥_{L^{1}(S_{u}^{d})} implies, together with Proposition 2.1,

φ_{u}^{d} = ψ_{∞}^{d} on u^{d}(S^{d} \ E^{d}).

Consequently we have

φ_{u} = (φ_{h_{∞}} ∘ w_{∞})(w′_{∞})^{2} on u(S \ E).

We have proved Theorem 1.3.

5. Slit mapping theorem of open Riemann surface of finite genus. Let R be an open Riemann surface of finite genus. As was observed by Bochner, R

can be embedded into a compact Riemann surface S of the same genus. Let w be a conformal embedding of R into S such that (w)_{∗} : π_{1}(R) → π_{1}(S) is surjective. We fix such a compact Riemann surface S, an embedding w and an element φ ∈ A(S) \ {0}. We remark that E = S \ w(R) is allowable in S. Let 𝔐 be the family of pairs (g, S_{g}), where g is a conformal map from R into a compact Riemann surface S_{g} such that (g)_{∗} is surjective. For every (g, S_{g}) ∈ 𝔐, g ∘ w^{−1} : S \ E ↪ S_{g} induces an isomorphism from π_{1}(S) onto π_{1}(S_{g}). We denote by τ = [S_{g}, (g ∘ w^{−1})_{∗}] the Teichmüller class in T (S). Then for this τ ∈ T (S) and for φ ∈ A(S) \ {0} we obtain a unique element φ_{g} ∈ A(S_{g}) \ {0} whose height on S_{g} is equal to that of φ on S. Consider the extremal problem to find an element (g, S_{g}) ∈ 𝔐 maximizing ∥φ_{g}∥_{L^{1}(S_{g})}. By Theorem 1.2, we obtain an extremal element (g, S_{g}) ∈ 𝔐. The set S_{g} \ g(R) is of zero area on S_{g} and each component of S_{g} \ g(R) is either

(i) a horizontal arc of φ_{g} or, (ii) a connected union of finitely many horizontal arcs and critical points of φ_{g}. We have shown Corollary 5.1. Let R be an open Riemann surface of finite genus. Then there are a compact Riemann surface S and a conformal embedding g : R → S and a holomorphic quadratic differential φ_{g} on S such that S \ g(R) has measure zero and each component of S \ g(R) is a possibly branched arc on horizontal trajectories of φ_{g}.

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

 E-mail address: sasair@jeans.ocn.ne.jp