THE TEICHMÜLLER SPACE OF THE IDEAL BOUNDARY

MASAHIKO TANIGUCHI

Abstract. In this paper, we consider ideal boundaries of Riemann surfaces by themselves, and show that the set of natural equivalence classes of mutually quasiconformally related ideal boundaries admits a complex Banach manifold structure.

1. The ideal boundary

For an open Riemann surface R, we can consider various kinds of compactifications of R. In this note we consider the Royden’s one (cf. [1] and [10]).

To define the Royden compactification, first we take the set R(R) of bounded continuous (complex) functions f on R which is differentiable in distribution sense and that the Dirichlet integral

D(f) = ∫_{R} df ∧ ∗\overline{df}

of f is finite. Then

∥f∥ = sup_{R} |f| + \sqrt{D(f)}

is a norm on R(R), and R(R) is a Banach algebra with respect to this norm. We call this algebra the Royden algebra associated with R.

Now there is a compact Hausdorff space R^{∗}, containing R as an open and dense subset, such that every element in R(R) can be extended to a continuous function on R^{∗} (and hence R(R) can be considered as a subset of the set C(R^{∗}) of all continuous functions on R^{∗}) and that R(R) separates points of R^{∗}, i.e. for every pair of points p_{1} and p_{2} of R^{∗} there is a function in R(R) such that f(p_{1}) ≠ f(p_{2}). Then such an R^{∗} is uniquely determined up to homeomorphisms fixing R pointwise, and we call R^{∗} the Royden compactification of R. Also the compact subset dR = R^{∗} − R is called the Royden boundary of R.

Here there are several ways to construct the Royden compactification canonically. One way is to consider the set X of all characters on R(R). Here a multiplicative linear functional χ on R(R) with χ(1) = 1 is called a character. And equipped with the weak^{∗} topology, X is a compact Hausdorff space. Moreover, by considering the point evaluations, we can regard R as an open and dense subset of X and X gives a representative of the Royden compactification of R.

Remark R(R) is dense in C(R^{∗}) with respect to the uniform topology.

Also we recall the following fact.

1991 Mathematics Subject Classification. Primary 30F25, 30F60; Secondary 30C62.

Key words and phrases. Riemann surfaces, ideal boundaries, quasiconformal maps, Teichmüller spaces.

Proposition 1 ([1],[10]). Every quasiconformal homeomorphism F of a Riemann surface R_{1} onto another R_{2} can be extended to a homeomorphism of R_{1}^{∗} onto R_{2}^{∗}.

Now, we can define another smaller compactification by using, instead of R(R), the set KS(R) of continuous functions f which is a constant on every connected component of the complement of some compact set. The Kerékártó-Stoilow compactification R̂ of R is the compact Hausdorff space uniquely determined (up to homeomorphisms fixing R point-wise) by the conditions that R is open and dense in R̂, that every element of KS(R) can be extended to a continuous function on R̂, and that KS(R) separates points of R̂.

Clearly, there is the canonical projection π from R^{∗} onto the Kerékártó-Stoilow compactification R̂ of R such that π is the identical map on R. We call the closed set dR_{p} = π^{−1}(p) a block of dR over p for every point p ∈ R̂ − R. A block dR_{p} is also open if p is isolated in R̂ − R.

When p ∈ R̂ − R corresponds to a puncture of R, we call p a non-essential point of R̂ − R, and the block dR_{p} a non-essential block. Let N be the subset of R̂ − R consisting of all non-essential points, and set

dR^{o} = dR − ⋃_{p∈N} dR_{p}.

Then dR^{o} is compact, and is called the essential part of dR, or the essential boundary of R.

Definition We say that a pair (Y, R) of a compact topological space Y and a Riemann surface R is a primitive ideal boundary if Y is homeomorphic to the essential part dR^{o} of the Royden boundary of R.

By Proposition 1, if another R′ is quasiconformally equivalent to R, the Royden compactification of R′ is homeomorphic to R^{∗}. So we need to restrict such an R as in the above definition, in order to say that two ideal boundaries are the same. And, considering complex structures near the ideal boundary only, we can define a natural class of primitive ideal boundaries as follows.

Definition We say that primitive ideal boundaries (Y_{1}, R_{1}) and (Y_{2}, R_{2}) are conformally equivalent if there is a homeomorphism F of a neighborhood U of Y_{1} in R_{1}^{∗} into R_{2}^{∗} such that F is conformal on U ∩ R_{1} and F (Y_{1}) = Y_{2}. Here we regard that Y_{j} is the essential boundary of R_{j}^{∗}.

We call the conformal equivalence class of a primitive ideal boundary (Y, R) an ideal boundary, which we denote by [Y, R], or simply by a representative Y if R is clear or not important. Also we call such a Riemann surface R a supporting surface of Y.

We say that an ideal boundary Y is of analytically (in)finite type if a supporting surface of Y is of an analytically (in)finite.

Here note that an ideal boundary [Y, R] is determined uniquely by the complex structure of R near Y.

Proposition 2. Suppose that (Y_{1}, R_{1}) is a primitive ideal boundary. Then if (Y_{1}, R_{1}) and (Y_{2}, R_{2}) are conformally equivalent, then we can take the same Riemann surface R as both of R_{j}, and hence (Y_{1}, R) = (Y_{2}, R) in the sense that the identical map of R to itself can be extended to a homeomorphism of Y_{1} to Y_{2}.

Proof. Let F : U → R_{2}^{∗} be as in the definition of the conformal equivalence between (Y_{1}, R_{1}) and (Y_{2}, R_{2}). Here we may assume that the relative boundary ∂U of U ∩R_{1} in R_{1} consists of a finite number of analytic simple closed curves. Then, there is a compact bordered Riemann surface S such that we can take R = U ∪ S as R_{1}. By identifying U and F (U), we can also take R as R_{2} and hence F is the identitical map on U, which implies the assertion.

Next we say that a subsurface S of a Riemann surface R is almost compact bordered if the closure \overline{S} of S in the subsurface \overline{R}^{p} of R̂, obtained from R by filling all points corresponding to punctures, is compact and the relative boundary ∂S of S in R consists of a finite number of analytic simple closed curves in R. Furthermore, if every component of ∂S divides \overline{R}^{p}, then we call an open set

U = R^{∗} − S ∪ ∂S ∪ (∪_{p∈N, p∈\overline{S}} dR_{p})

a canonical neighborhood of the ideal boundary [Y, R], and call (U ∩ R) an end of R or for Y.

Definition We say that a map f of an ideal boundary [Y_{1}, R_{1}] to another [Y_{2}, R_{2}] is a boundary map (considered as a map of Y_{1} to Y_{2}) if there are a canonical neighborhood U of Y_{1} in R_{1}^{∗} and a homeomorphism F : U → R_{2}^{∗} such that F = f on Y_{1}. Such a map F as above is called a supporting map of f.

If a boundary map f of [Y, R] to itself or to another [Y′, R′] is a surjective homeomorphism (as a map of Y_{1} to itself or to Y′), then we call such an f a boundary self-homeomorphism, or boundary homeomorphism, respectively.

Further, we say that f : Y → Y′ is conformal, quasiconformal, and asymptotically conformal if so is a supporting map F on U ∩ R.

Here recall that f is asymptotically conformal if and only if we can find a (1 + ϵ)- quasiconformal supporting map of f for every ϵ > 0. (For the basic facts about asymptotically conformal maps, see for instance, [5].)

2. Boundary self-homeomorphims

Let BH(Y) be the group of all boundary self-homeomorphisms of an ideal boundary [Y, R]. First we recall the following fact.

Proposition 3 ([8], also see [9]). f is an element of BH(Y) if and only if f is a quasiconformal boundary self-homeomorphism.

Proof. Since ”if”-part is clear, we assume that f ∈ BH(Y). Then there are Riemann surface R supporting Y and a homeomorphism F of a canonical neighborhood U of Y in R^{∗} into R^{∗} which supports f. Then by Corollary in [8], there is a quasiconformal homeomorphism of U ∩ R into R having the boundary value f on Y, which implies the assertion.

Also note that a boundary self-homeomorphism of Y need not necessarily the boundary map of a self-homeomorphism of R.

Theorem 4. There are an ideal boundary Y and an f ∈ BH(Y) such that, for every supporting surface R of Y, every quasiconformal self-homeomorphism of R supports neither f nor f^{−1}.

Proof. Set

R_{0} = {|Im z| < 1} − {n | n ∈ ℤ, n ≥ 0},

and let Y be the ideal boundary supported by R_{0}. Let f be the boundary selfhomeomorphism of Y supported by F_{0}(z) = z + 1. We show that these Y and f are desired ones.

For this purpose, suppose that there were a Riemann surface R_{1} supporting Y and a quasiconformal self-homeomorphism F of R_{1} which, considered as a self-map of R_{1}^{∗}, supports f.

Let U be a canonical neighborhood of Y in R_{0}^{∗} such that F_{0}(U) ⊂ R_{0}^{∗}. Take a smaller canonical neighborhood V in U so that V ∩ R_{0} can be considered also as a subsurface of R_{1} and that F_{0}(V) and F (V) are contained in U. F_{0} and F restricted to V ∩ R_{0} can be extended to quasiconformal self-homeomorphisms of {|Im z| < 1}, which in turn can be identified with {|z| < 1} by a Riemann map. Moreover, they can be extended continuously to {|z| ≤ 1}, where the boundary values coincide by the assumption. Hence denoting by the same notations, we conclude that Φ = F^{−1} ∘F_{0} can be extended to {|z| ≤ 1} by the identical boundary values.

Now since Φ belongs to R({|z| < 1}), so is g(z) = Φ(z) − z, which identically vanishes on {|z| = 1}, and hence Φ gives the identical self-map of Y. Here if there were a sequence of punctures p_{n} of V ∩ R_{0} (considered as a subsurface of {|z| < 1}) such that |p_{n}| tend to 1 and g(p_{n}) ≠ 0 for every n, then since Φ(p_{n}) also tend to {|z| = 1}, by taking a subsequence if necessary, we may further assume that

Φ(p_{n}) ∉ {p_{j}}_{j=1}^{∞}.

Hence we can construct a function P ∈ R(R) such that P (p_{n}) = 1 but P (Φ(p_{n})) = 0 for every n, which would imply that Φ is not the identical map of Y. Indeed, take a mutually disjoint, simply connected neighborhood U_{n} of p_{n} so that Φ(p_{n}) ∉ U_{n} for every n, and map U_{n} onto {|z| < 1} by a Riemann map g_{n} so that g_{n}(p_{n}) = 0. Consider

h_{n}(z) = \frac{− log(2|z|)}{n^{3}}

on W_{n} = {e^{−n^{3}}/2 < |z| < 1/2}, and set P_{n} = h_{n} ∘ g_{n} on g^{−1}(W_{n}). Extend P_{n} to a continuous function by setting 0 or 1 in each connected component of R−g_{n}^{−1}(W_{n}), we have a function P_{n} in R(R) such that D(P_{n}) = 2π/n^{3}. And

P = ∑_{n=1}^{∞} P_{n}

is a desired function.

Thus there is a canonical neighborhood V′ of Y (contained in V) such that F_{0}(p) = F (p), for every puncture p in V′. But then the number of punctures of R_{1} outside V′ is smaller than that of punctures of R_{1} outside F (V′), which is a contradiction.

Since the case of F_{0}^{−1} can be treated similarly, we have the assertion.

Next, there are boundary self-homeomorphism f of Y with no fixed points. For instance, rotations gives such examples. On the other hand, the following fact seems to be non-trivial.

Proposition 5. There is an ideal boundary Y such that every element of BH(Y) fixes the same point of Y.

Proof. In general, the harmonic boundary d_{0}R of the Royden boundary is invariant under quasiconformal boundary homeomorphisms ([10] III.7.C Theorem. Also see [10] III.8.C Theorem), and hence by Proposition 3, d_{0}R∩Y is invariant under every f ∈ BH(Y). On the other hand, if a supporting surface R belongs to O_{HD} − O_{G}, a theorem of Royden states that d_{0}R ∩ Y consists of a single point (cf. [10] III.F Theorem), which implies the assertion.

Finally, eventually trivial conformal equivalence is trivial. Here we say that a conformal boundary self-homeomorphism f : Y → Y is eventually trivial if f is supported by a conformal homeomorphism F of a canonical neighborhood U of Y in R^{∗} into R^{∗} such that F on U ∩ R is homotopic to the identical map of U ∩ R in R.

Proposition 6. Suppose that [Y, R] is an ideal boundary of analytically infinite type. Let f_{1}, f_{2} ∈ BH(Y). If f_{1}^{−1} ∘ f_{2} is an eventually trivial conformal boundary self-homeomorphism, then f_{1} = f_{2}.

Proof. By a theorem of Maitani in [6], F as above is the identical map of U, and hence so is f_{1}^{−1} ∘ f_{2}.

3. The Teichmüller space

Similarly as before, for ideal boundaries [Y, R] and [Y′, R′], we say that a quasiconformal boundary homeomorphisms f : Y → Y′ is homotopic to an asymptotically conformal boundary homeomorphism g : Y → Y′ if there are supporting maps F : U → (R′)^{∗} of f and G : U → (R′)^{∗} of g, where U is a canonical neighborhood of Y in R^{∗}, such that F is quasiconformal on U ∩ R, G is asymptotically conformal on U ∩ R, and F on U ∩ R is homotopic to G on U ∩ R in R.

In particular, if [Y, R] = [Y′, R′] and G is the identical map, then again we say that f and F are eventually trivial.

Theorem 7. For every ideal boundary Y, there is a non-identical, eventually trivial and asymptotically conformal, boundary self-homeomorphism of Y.

Proof. Let U be a canonical neighborhood of Y in R^{∗}, where R is a supporting surface of Y. Take a sequence of points p_{n} on U ∩ R escaping from any compact set of R, and a mutually disjoint, simply connected open neighborhood U_{n} of p_{n} for every n. Map each U_{n} onto {|z| < 1} by a Riemann map g_{n} so that g_{n}(p_{n}) = 0.

Set

φ_{n}(z) = \frac{z + (1/n)}{1 + (1/n)\overline{z}}

on {|z| < 1}, then φ_{n} is a (1/n)-quasiconformal self-homeomorphism of {|z| < 1} and φ_{n}(z) = z on {|z| = 1}. Hence we can define a (1/n)-quasiconformal homeomorphism Φ of U into R^{∗} by setting g_{n}^{−1} ∘ φ_{n} ∘ g_{n} on U_{n} for every n, and to be the identical map outside ∪_{n=1}^{∞} U_{n}. Then Φ gives a eventually trivial and asymptotically conformal boundary self-homeomorphism f of Y.

Next similarly as before, set

h_{n}(z) = \frac{− log(n|z|)}{n^{3}}

on W_{n} = {(1/n)e^{−n^{3}} < |z| < (1/n)}. Then we have an element P_{n} of R(R) by setting P_{n} = h_{n} ∘ g_{n} on g_{n}^{−1}(W_{n}) and extending it by a constant 0 or 1 on each component of R − g_{n}^{−1}(W_{n}). Since D(P_{n}) = 2π/n^{3}, P = ∑_{n=1}^{∞} P_{n} also belongs to R(R), and P (p_{n}) = 1 and P (Φ(p_{n})) = 0 for every n. Thus f is not the identical map.

We say that two ideal boundaries Y_{1} and Y_{2} are quasiconformally related if there is a quasiconformal boundary homeomorphism of Y_{1} onto Y_{2}. Then we can define the Teichmüller space of quasiconformally related ideal boundaries.

Definition For a given ideal boundary Y_{0}, consider pairs (Y, f) of an ideal boundary Y and a quasiconformal boundary homeomorphism f : Y_{0} → Y, which is called a marking of Y.

We say that two pairs (Y_{1}, f_{1}) and (Y_{2}, f_{2}) are Teichmüller equivalent if there is asymptotically conformal boundary homeomorphism of Y_{1} to Y_{2} which is homotopic to f_{2} ∘ f_{1}^{−1}.

We call the set of all Teichmüller equivalence classes [Y, f] (or more precisely [[Y, R], f] of marked ideal boundaries (Y, f) the Teichmüller space of the ideal boundary Y_{0}, which is denoted by T (Y_{0}). A point of T (Y_{0}) is called a marked ideal boundary.

Here note that if Y_{0} is an ideal boundary of analytically finite type, then Y_{0} is empty, and hence T (Y_{0}) consists of a single point (, which can be compared with results in [2],[4]). It is remarkable that the Teichmüller space of every ideal boundary admits a natural complex structure.

Theorem 8. Let Y_{0} be an ideal boundary. Then the Teichmüller space T (Y_{0}) of Y_{0} has a canonical complex Banach manifold structure.

Proof. A theorem of Miyaji in [7] implies that the asymptotic Teichmüller spaces AT (R_{0}) of R_{0} are mutually biholomorphic for all supporting surfaces R_{0} of Y_{0}. Indeed, if R_{1} and R_{2} are such surfaces, then there is another supporting surface R_{3} of Y_{0} and analytically finite Riemann surfaces S_{1} and S_{2} such that R_{3} and S_{j} are obtained from R_{j} by applying a conformal 2-surgery along a dividing simple closed curve for each j. And Reducing Theorem in [7] states that the asymptotic Teichmüller space AT (R_{j}) is biholomorphic to the product AT (S_{j}) × AT (R_{3}) for each j. Here since AT (S_{j}) are trivial, we have a canonical biholomorphism between AT (R_{j}). (For the details of the asymptotic Teichmüller theory, see [5],[2], and [3].)

Next fix a supporting surface R_{0} of Y_{0}. Then we can construct a natural bijection from T (Y_{0}) onto AT (R_{0}) as follows. Take any element [Y, f] of T (Y_{0}). Then there is a quasiconformal homeomorphism F of U ∩ R_{0} of Y in R_{0} into R where U is a canonical neighborhood of Y_{0} in R_{0} and R is a supporting surface of Y. Then F can be extended to a quasiconformal map of R_{0} onto another supporting surface R′ of Y (possibly different from R). which gives a point in AT (R_{0}). By the definitions, we can easily see that this map induces a bijection of T (Y_{0}) to AT (R_{0}). Thus we have prove the assertion.

Remark We say that two boundary self-homeomorpihsms f_{1} and f_{2} in BH(Y_{0}) are AC-equivalent if f_{2} ∘ f_{1}^{−1} is homotopic to a asymptotically conformal selfhomeomorphism of Y. The equivalence class of f is called an AC-mapping class, and denoted by [f].

Now every element f of BH(Y_{0}) naturally induces an automoprhism f^{∗} of T (Y_{0}), by setting

f^{∗}([Y, g]) = [(Y, g ∘ f^{−1})].

Then it is clear from the definition that f_{1}^{∗} = f_{2}^{∗} if and only if [f_{1}] = [f_{2}].

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Department of Mathematics, Graduate school of Science, Kyoto University, Kyoto 606, Japan

  E-mail address: tanig@math.kyoto-u.ac.jp