ON UNIQUENESS OF OBSTACLE PROBLEM ON FINITE RIEMANN SURFACE

RIE SASAI

Abstract. In [1], R. Fehlmann and F. P. Gardiner studied an extremal problem for a topologically finite Riemann surface and established a slit mapping theorem. In this article, we give a condition for non-uniqueness of such slit mappings, by using a deformation of a Riemann surface.

1. Introduction

Let S be a Riemann surface of finite analytic type. Let (S_{ι}, ι) be a pair of a Riemann surface S_{ι} of the same type as S and an isomorphism ι of the fundamental group π_{1}(S) of S onto π_{1}(S_{ι}). We say that two pairs (S_{ι_{1}}, ι_{1}) and (S_{ι_{2}}, ι_{2}) are equivalent if there exists a conformal map u of S_{ι_{1}} onto S_{ι_{2}} such that

(u)_{∗} ∘ ι_{1} = ι_{2}.

The family of such equivalence classes is said to be the Teichmüller space of S and denoted by T (S).

Let S be a finite bordered Riemann surface with border Γ. In other words, the border Γ consists of finitely many mutually disjoint simple closed curves, and the double S^{d} of S with respect to the border Γ is of finite analytic type. Note that the border Γ may be empty. In that case, the double S^{d} will be interpreted as S itself. Let A(S) be the set of integrable holomorphic quadratic differentials φ on S with the property that φ = φ(z)dz^{2} is real along the border Γ (cf. [2]). Every φ ∈ A(S) extends to a symmetric holomorphic quadratic differential φ^{d} on S^{d}.

Let 𝔖(S^{d}) be the family of simple closed curves on the double S^{d}, which are homotopic neither to a point of S^{d} nor to a puncture of S^{d}. Let 𝔖[S^{d}] be the set of free homotopy classes of elements of 𝔖(S^{d}). For φ ∈ A(S) and γ ∈ 𝔖(S^{d}), we denote the height of γ with respect to φ^{d} by

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

and the height of the homotopy class [γ] by

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

where the infimum is taken over all closed curves β ∈ 𝔖(S^{d}) freely homotopic to γ in S^{d}.

Now we state the obstacle problem in the sense of Fehlmann and Gardiner [1]. 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.

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

Definition 1.1. We say that E is allowable if E is a compact subset of the interior S^{∘} of S such that S^{∘} \ E is connected and E is contractible in S.

We further say that E is an allowable slit with respect to φ ∈ A(S)\{0} if E is allowable and if each component of E is a horizontal arc of φ or the union of a finite number of horizontal arcs and critical points of φ.

We remark that a compact subset E ⊂ S^{∘} is contractible if and only if there is a topological closed disk in S which contains E in its interior (see [4, Lemma 2.3]). Let E be an allowable subset of S. Set E^{d} = E ∪ j(E), where j : S^{d} → S^{d} is the canonical anticonformal involution. Let 𝔉(S, E) be the family of pairs (g, S_{g}), where g is a conformal map of S \ E into another Riemann surface S_{g} of the same type as S in such a way that g maps the border Γ onto the border of S_{g} and the same applies to the punctures. For every (g, S_{g}) ∈ 𝔉(S, E), g extends to a conformal map g^{d} of S^{d} \E^{d} into S_{g}^{d} symmetrically. Then (g, S_{g}) ∈ 𝔉(S, E) induces an isomorphism ι_{g} of the fundamental group π_{1}(S^{d}) of S^{d} onto π_{1}(S_{g}^{d}) (cf. [4, Lemma 2.5] ). We denote by [S_{g}^{d}, ι_{g}] the Teichmüller (equivalence) class of (S_{g}^{d}, ι_{g}) in T (S^{d}).

It is known (cf. [3]) that, for every (f, S_{f}) ∈ 𝔉(S, E) and φ ∈ A(S) \ {0}, there exists the unique holomorphic quadratic differential φ_{f} ∈ A(S_{f}) \ {0} such that

height_{φ_{f}^{d}}[γ] = height_{φ^{d}}(ι_{f}^{−1}[γ]) for every [γ] ∈ 𝔖[S_{f}^{d}].

Fehlmann and Gardiner [1] posed an obstacle problem for (S, E, φ) which asks the existence of (f, S_{f}) ∈ 𝔉(S, E) maximizing the quantity

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

in 𝔉(S, E), and showed the following result.

Theorem 1.2 (Fehlmann-Gardiner). Suppose that S is a finite bordered Riemann surface, and that φ ∈ A(S) \ {0}. Let E be an allowable subset of S with finitely many components. Then there exists an element (g, S_{g}) ∈ 𝔉(S, E) such that M_{g} attains the supremum

M_{g} = sup_{(f,S_{f})∈𝔉(S,E)} M_{f}.

Moreover, for this point (g, S_{g}) ∈ 𝔉(S, E), E_{g} = S_{g} \ g(S \ E) is an allowable slit with respect to φ_{g}.

The point (g, S_{g}) ∈ 𝔉(S, E) in Theorem 1.2 is called extremal for (S, E, φ), and the associated differential φ_{g} is called the extremal differential.

Fehlmann and Gardiner also asserted in the paper [1] 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 of S_{u} onto S_{g}. The uniqueness, however, does not necessarily hold in their sense.

We show in this note the following theorem which gives a condition for extremal, and hence extremal slit mappings, not to be unique.

Definition 1.3. Let E be an allowable slit in a finite bordered Riemann surface S with respect to a holomorphic quadratic differential φ ∈ A(S) \ {0}. We will call p_{0} ∈ E a refolding point of order m for (S, E, φ) if p_{0} is a zero of φ of order m and if E contains two horizontal arcs ℓ_{1} and ℓ_{2} with common end point p_{0} such that the angle formed by them at p_{0} is greater than 2π/(m + 2).

Theorem 1.4. Let R be a finite bordered Riemann surface, and ψ ∈ A(R) \ {0}. Let E_{ψ} be an allowable slit of R with respect to ψ. Suppose that E has a refolding point p_{0} of order m ≥ 3 for (R, E_{ψ}, ψ). Then, there exist (ũ, R̃) ∈ 𝔉(R, E_{ψ}) and ψ̃ ∈ A(R̃) \ {0} such that

(i) Eψ̃ = R̃ \ ũ(R \ E_{ψ}) is an allowable slit of R̃ with respect to ψ̃, (ii) heightψ̃^{d}[γ] = height_{ψ^{d}}(ι−1 ; ũ[γ]) for every [γ] ∈ 𝔖[R̃^{d}], and (iii) [R̃^{d}, ιũ] ≠ [R^{d}, id] in T (R^{d}).

Corollary 1.5. Suppose that S is a finite bordered Riemann surface and that φ ∈ A(S) \ {0}. Let E be an allowable subset of S, and (g, S_{g}) ∈ 𝔉(S, E) be extremal for (S, E, φ). If the allowable slit E_{g} of S_{g} with respect to the extremal differential φ_{g} has a refolding point of order at least three, then there exists another extremal element for (S, E, φ) which induces a point in T (S^{d}) different from [S_{g}^{d}, ι_{g}].

Proof. Take the triple (S_{g}, E_{g}, φ_{g}) as the triple (R, E_{ψ}, ψ) in Theorem 1.4. Then we obtain (ũ, R̃) ∈ 𝔉(S_{g}, E_{g}), and ψ̃ ∈ A(R̃) \ {0} satisfying (i) (ii) (iii) in the theorem. Then from [1] we see that (i) and (ii) implies that the point (ũ ∘ g, R̃) ∈ 𝔉(S, E) is extremal for (S, E, φ). Moreover we can see, by (iii), [R̃^{d}, ι_{ũ∘g}] ≠ [S_{g}^{d}, ι_{g}] in T (S^{d}). Thus we have the assertion.

Remark In the proof, we will actually construct a continuous family of extremal elements (ũ_{t}, R̃_{t}) ∈ 𝔉(R, E_{ψ}) for the same obstacle problem for (R, E_{ψ}, ψ) in such a way that the marked Riemann surface τ_{t} = [R̃_{t}^{d}, ιũ_{t}] varies continuously in T (R^{d}) \ {[R^{d}, id]} and that τ_{t} approaches [R^{d}, id] as t → 0. Since the Teichmüller modular group acts on T (R^{d}) discontinuously, this implies that the Riemann surface R̃_{t}^{d} is not conformally equivalent to R̃^{d} for sufficiently small t > 0.

In [4], the auther showed a uniqueness result in the weaker form: Let (g, S_{g}) and (u, S_{u}) be both extremal for (S, E, φ). Then the extremal differentials φ_{g} and φ_{u} satisfy the relation φ_{u} = (φ_{g} ∘ w)(w′)^{2} on u(S \ E), where w = g ∘ u^{−1} .

2. Example

In this section we give an example of the triple (S, E, φ) which satisfies the assumptions of Corollary 1.5.

First make three copies M_{1}, M_{2}, M_{3} of the rectangle

M = {z = x + iy ∈ ℂ | |x| ≤ 2, |y| ≤ 1},

and let z_{j} be the coordinate corresponding to z on each M_{j}. Next on each M_{j}, identify the two pairs of parallel sides under the translations

z_{j} → z_{j} + 4, z_{j} → z_{j} + 2i.

Then we obtain three copies T_{1}, T_{2}, T_{3} of a torus T. The quadratic differential dz^{2} on M induces a holomorphic quadatic differential φ_{0} on T.

Cut T_{j} along the segment

I_{j} = {z_{j} = x_{j} + iy_{j} | −1 ≤ x_{j} ≤ 0, y_{j} = 0},

and glue them cyclically. More precisely, we paste the upper edge I_{1}^{+} of the slit I_{1} to the lower edge I_{2}^{−} of the slit I_{2}, the upper edge I_{2}^{+} of the slit I_{2} to the lower edge I_{3}^{−} of the slit I_{3}, and the upper edge I_{3}^{+} of the slit I_{3} to the lower edge I_{1}^{−} of the slit I_{1}. Then we obtain a compact Riemann surface S of genus three.

Now let Π be the natural projection of S onto the torus T, and φ be the pull-back of φ_{0} by Π. Finally, let E be the subset of S consisting of ℓ_{1} and ℓ_{2}, where ℓ_{i} is the arc on T_{i} corresponding to {z | 0 ≤ x ≤ 1, y = 0}.

Now we consider the obstacle problem for (S, E, φ). Then the set E is an allowable slit of S with respect to φ. Hence we know that the identy mapping of S gives an extremal slit map associated with the extremal problem for this triple. Moreover, we can easily see that {p_{0}} = Π^{−1}(0) ⊂ S consists of the refolding point for (S, E, φ).

Thus the assumptions in Corollary 1.5 are satisfied and, as a consequence, the points in T (S^{d}) which are induced by the extremals for (S, E, φ) are not uniquely determined.

3. Proof of Theorem 1.4

Assume that a component J of E_{ψ} contains a refolding point p_{0} of ψ of order m ≥ 3 and horizontal arcs ℓ_{1} and ℓ_{2} with common end point p_{0} and that the angle formed by ℓ_{1} and ℓ_{2} at p_{0} is

\frac{2kπ}{m + 2} (2 ≤ k ≤ \frac{m + 2}{2}) .

Note that the arcs ℓ_{1}, ℓ_{2} are segments on the real axis with endpoint at the origin with respect to the natural parameter

ζ_{ψ} = ∫_{z_{0}}^{z} \sqrt{ψ(z)}dz,

where z is a local chart near p_{0} and z_{0} = z(p_{0}).

We take closed subarcs κ_{j} ⊂ ℓ_{j}(j = 1, 2) with the same ψ-length such that p_{0} is an endpoint of each κ_{j} and that ψ has no zeros on κ_{j} \ {p_{0}}. Let p_{j} be the other endpoint of κ_{j} for each j. Also set K = κ_{1} ∪ κ_{2}.

Now, cut R along κ_{1} and κ_{2}. For each j, let κ_{j}^{+} and κ_{j}^{−}, respectively, be the right-side and the left-side edges of the slit κ_{j}, with respect to the orientation which corresponds to the move along the slit from p_{0} to p_{j}. Assume that κ_{1}^{−} and κ_{2}^{+}, κ_{1}^{+} and κ_{2}^{−} form the angles

\frac{2kπ}{m + 2} and \frac{2π(m + 2 − k)}{m + 2}.

at p_{0}, respectively.

Paste κ_{1}^{−} and κ_{2}^{+} so that points having the same absolute value with respect to ζ_{ψ} are identified. In the same way, paste κ_{1}^{+} and κ_{2}^{−}. Let K̃ be the union of the pasted segments. Then we obtain a new finite bordered Riemann surface R̃ and the natural conformal embedding ũ : R \ K → R̃. The pair (ũ, R̃) is an element of the family 𝔉(R, K) ⊂ 𝔉(R, E_{ψ}).

Moreover, from the construction we can extend (ũ^{−1})^{∗}ψ naturally to a holomorphic quadratic differential ψ̃ on R̃, and Eψ̃ = R̃ \ ũ(R \ E_{ψ}) is an allowable slit of R̃ with respect to ψ̃.

Lemma 3.1.

heightψ̃^{d}[γ̃] = height_{ψ^{d}}(ι−1 ; ũ[γ̃])

for every [γ̃] ∈ 𝔖[R̃^{d}].

Proof. We say that a simple closed curve β̃ on R̃^{d} is a ψ̃^{d}-polygon, if β̃ is the union of finitely many horizontal arcs and vertical arcs of ψ̃^{d}. Note that for every [γ̃] ∈ 𝔖[R̃^{d}]

heightψ̃^{d}[γ̃] = inf_{β̃} heightψ̃^{d}(β̃),

where the infimum is taken over all ψ̃^{d}-polygons β̃ freely homotopic to γ̃ in R̃^{d}.

We can now add horizontal segments contained in K to the pre-image (ũ^{d})^{−1}(β̃) of such a ψ̃^{d}-polygon β̃ so that the resulting ψ^{d}-polygon β is a closed curve in the class ι−1 ; ũ[γ̃]. Then, by construction, we obtain

height_{ψ^{d}}(β) = heightψ̃^{d}(β̃).

Hence we conclude that

height_{ψ^{d}}(ι−1 ; ũ[γ̃]) ≤ height_{ψ^{d}}(β) = heightψ̃^{d}(β̃)

for every ψ̃^{d}-polygon β̃ freely homotopic to γ̃, which in turn implies that

height_{ψ^{d}}(ι−1 ; ũ[γ̃]) ≤ heightψ̃^{d}[γ̃]

for every [γ̃] ∈ 𝔖[R̃^{d}].

On the other hand, we can similarly see as above that

heightψ̃^{d}(ιũ[γ]) ≤ height_{ψ^{d}}[γ]

for every [γ] ∈ 𝔖[R^{d}]. Thus we have proved the assertion.

By Lemma 3.1, we see that the holomorphic quadratic differential ψ̃^{d} ∈ A(R̃^{d}) \ {0} satisfies the condition (ii) in Theorem 1.4. Moreover, by definition, Eψ̃ is an allowable slit of R̃ with respect to ψ̃, and

(ψ̃ ∘ ũ)(ũ′)^{2} = ψ on R \ E_{ψ}. (∗)

Lemma 3.2. [R̃^{d}, ιũ] ≠ [R^{d}, id] in T (R^{d}).

Proof. Suppose that [R̃^{d}, ιũ] = [R^{d}, id] in T (R^{d}). Then there exists a conformal map f : R^{d} → R̃^{d} with (f)_{∗} = ιũ which is symmetric in the border Γ.

Take any [γ] ∈ 𝔖[R^{d}]. Then Lemma 3.1 gives

heightψ̃^{d}[f(γ)] = height_{ψ^{d}}[γ].

Since height_{ψ̃^{d}∘f(f′)^{2}}[γ] = heightψ̃^{d}[f(γ)], we obtain

height_{ψ̃^{d}∘f(f′)^{2}}[γ] = height_{ψ^{d}}[γ]

for every [γ] ∈ 𝔖[R^{d}]. Hence the heights mapping theorem [2] implies that

(ψ̃^{d} ∘ f)(f′)^{2} = ψ^{d} on R^{d}. (∗∗)

In particular, the map f sends the zeros of ψ^{d} to those of ψ̃^{d} while keeping multiplicities.

Now from the construction, the zero p_{0} of orders m ≥ 3 breaks into two zeros q̃_{1} and q̃_{2} of ψ̃^{d} of orders k − 2 and m − k, respectively, with 2 ≤ k ≤ (m + 2)/2. Also the endpoints p_{1} of κ_{1} and p_{2} of κ_{2} gather into a zero q̃ of ψ̃^{d} on R̃^{d} of order 2.

Set K̃ = R̃ \ ũ(R \ K). Then the zeros q̃, q̃_{1} and q̃_{2} of ψ̃ on K̃ have orders less than m. Hence we see that

f(p_{0}) ∈ R̃ \ K̃.

Since the conformal embedding ũ maps R \ K onto R̃ \ K̃, (ũ^{d})^{−1} ∘ f(p_{0}) is well defined and (ũ^{d})^{−1} ∘ f(p_{0}) ∉ K. In particular,

(ũ^{d})^{−1} ∘ f(p_{0}) ≠ p_{0}.

Next assume that, for a positive integer n,

((ũ^{d})^{−1} ∘ f)^{n}(p_{0}) ≠ ((ũ^{d})^{−1} ∘ f)^{k}(p_{0})

for every k with 0 ≤ k ≤ n − 1. Then, f ∘ ((ũ^{d})^{−1} ∘ f)^{n}(p_{0}) ∉ K̃, for f ∘ ((ũ^{d})^{−1} ∘ f)^{n}(p_{0}) is a zero of ψ̃ of order m. Hence, similarly as above, ((ũ^{d})^{−1} ∘ f)^{n+1}(p_{0}) ∉ K. In particular,

((ũ^{d})^{−1} ∘ f)^{n+1}(p_{0}) ≠ p_{0}.

Also by the assumption,

((ũ^{d})^{−1} ∘ f)^{n+1}(p_{0}) ≠ ((ũ^{d})^{−1} ∘ f)^{k}(p_{0})

for every k with 1 ≤ k ≤ n.

Thus by induction, we conclude that, for every positive integer n,

((ũ^{d})^{−1} ∘ f)^{n}(p_{0}) ≠ ((ũ^{d})^{−1} ∘ f)^{k}(p_{0})

for every k with 0 ≤ k ≤ n−1. Therefore ψ has infinitely many zeros, which is impossible. So we have shown that

[R̃^{d}, ιũ] ≠ [R^{d}, id]

in T (R^{d}).

Theorem 1.3 follows immediately from Lemma 3.2.

References

1. R. Fehlmann and F. P. Gardiner, Extremal problem for quadratic differentials, Michigan Math. J. 42 (1995), 573-591.

2. F. P. Gardiner, Teichmüller theory and quadratic differentials, Wiley Interscience, 1987.

3. F. P. Gardiner and N. Lakic, Quasiconformal Teichmüller theory, Amer. Math. Soc., 2000.

4. R. Sasai, Generalized obstacle problem, preprint.

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

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