PROJECTIVE STRUCTURES WITH

DISCRETE HOLONOMY REPRESENTATIONS

Hiroshige Shiga and Harumi Tanigawa

Abstract. Let K(X) denote the set of projective structures on a compact Riemann surface X whose holonomy representations are discrete. We will show that each component of the interior of K(X) is holomorphically equivalent to a complex submanifold of the product of Teichmüller spaces and the holonomy representation of every projective structure in the interior of K(X) is a quasifuchsian group.

1. Introduction

Let Σ_{g} be a compact oriented differentiable surface of genus g ≥ 2. A projective structure (or CP^{1}-structure) on Σ_{g} is a maximal system of charts with transition maps in the projective automorphism group of CP^{1}, namely, PSL_{2}(C). Let P_{g} denote the space of all projective structures on Σ_{g} with markings. Since elements of PSL_{2}(C) are holomorphic, a projective structure determines its underlying complex structure. Hence there is a natural map π : P_{g} → T_{g}, where T_{g} denotes the Teichmüller space. For each projective structure, we can take its developing map on the universal and have a holomorphic quadratic differential (or a projective connection) on the Riemann surface (the complex structure under the projective structure) by taking the Schwarzian derivative. It is well-known (cf. section 2 for detail) that this procedure gives a natural identification between π : P_{g} → T_{g} and the bundle of holomorphic quadratic differentials over the Teichmüller space π : Q → T_{g}.

In this paper, we will investigate the set of projective structures on Σ_{g} with discrete holonomy representations with respect to the parametrization by Q. Let K ⊂ Q be the set of all projective structures with discrete holonomy representations. It is well-known that there is an open set QF of Q, which is a neighborhood of the 0-section {0} × T_{g} of Q → T_{g}, consisting of all projective structures with univalent developing maps and quasifuchsian holonomy representations. For any Riemann surface X ∈ T_{g}, the fiber of π|QF : QF → T_{g} over X, which we denote by QF (X), is a domain which is holomorphically equivalent to the Teichmüller space and is called a Bers slice. One of our main result shows that a similar phenomenon occurs for each component of the interior point set int K.

Let Q(X) denote the fiber over X of Q → T_{g}, namely, Q(X) is the space of projective structures on the complex structure X. Let K(X) denote the fiber of π|K : K → X. We will denote by int_{Q(X)} K(X) the interior of K(X) considered

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 1991 Mathematics Subject Classification Primary 32G15: Secondary 30F10.

Research at MSRI is supported by NSF grant #DMS–9022140

in Q(X). We employ this notation to avoid mixing up int_{Q(X)} K(X) with the fiber of int K (interior is considered in Q) over X, although it will turn out these two are the same by our result. As we recalled above, int_{Q(X)} K(X) has a component containing the Bers slice QF (X). (In fact, the Bers slice QF (X) coincides with the component of int_{Q(X)} K(X) containing 0 ([Sh]).) On the other hand, it was shown by Maskit [M] that there are complex structures X on which there exist components of int_{Q(X)} K(X) other than the Bers slice (see also Hejhal [H] and Goldman [G]). Indeed, he showed that given a Fuchsian group isomorphic to π_{1}Σ_{g} there are projective structures outside the Bers slice on some complex structures whose holonomy representation are conjugate to the given Fuchsian group in PSL_{2}(C). However, the Fuchsian holonomy representation hardly gives information about the complex structures under the projective structures when the projective structure is not in the Bers slice, because the developing map is not a covering map ([Kr1]). Here arise natural questions:

(1) For every complex X ∈ T_{g} are there any components of int_{Q(X)} K(X) other than the Bers slice?

(2) What kind of discrete groups appear as holonomy representations in such components?

(3) What kind of analytic properties do such components have?

As for the first question, an affirmative answer is given in [T]. The aim of this paper is to work on the other two questions. We will show that each component of int_{Q(X)} K(X) consists of quasifuchsian groups and is holomorphically equivalent to a submanifold of the product T_{g} × T_{g}.

The authors would like to thank Curt McMullen for stimulating conversations and comments. Considerable part of this work was done at Mathematical Sciences Research Institute. The authors are very grateful for their hospitality.

Remark. All results in this paper holds for bounded projective structures on Riemann surfaces of finite type (i.e. projective structures determined by bounded holomorphic quadratic differentials) by parallel arguments.

2. Notation and basic facts

In this section, we recall some known facts we will use to prove our results. Throughout this paper, Γ denotes a Fuchsian group acting on the upper half

plane H and the lower half plane L such that the quotient X = L/Γ is a compact Riemann surface of genus g ≥ 2.

2.1 Quasiconformal deformations and the Teichmüller space.

For a finitely generated Kleinian group G, we denote by Ω(G) its region of discontinuity and by Λ(G) its limit set.

For a G-invariant open set U ⊂ Ω(G), a measurable (−1, 1)-form μ on the Riemann sphere with the following properties is called a Beltrami differentials for G supported on U:

μ|(Ĉ − U) = 0, ||μ||_{∞} < 1

and

μ ∘ g × \overline{g′}/g′ = μ a.e.

for all g ∈ G. We denote the set of all Beltrami differentials on U by B_{1}(G, U). When U = Ω(G), we abbreviate B_{1}(G, U) by B_{1}(G). The space B_{1}(G) is the unit ball of the complex Banach space of all measurable (−1, 1)-form for G with supremum norm.

By Ahlfors-Bers’ theory, for each Beltrami differential μ on Ĉ with ||μ||_{∞}, there exists a quasiconformal mapping f^{μ} with Beltrami differential μ, that is,

(f^{μ})\overline{z} = μ × (f^{μ})_{z}.

Such a mapping is unique up to post-composition of elements of PSL_{2}(C). In each argument below, we will employ some convenient normalization.

It is easy to see that if μ is in B_{1}(G) then f^{μ} ∘ g ∘ (f^{μ})^{−1} is a Möbius transformation for every g ∈ G and f^{μ}G(f^{μ})^{−1} is a subgroup of PSL_{2}(C) with region of discontinuity f^{μ}(Ω(G)).

Two Beltrami differentials μ_{1} and μ_{2} for G are said to be equivalent if f^{μ_{1}} and f^{μ_{2}} determine an equivalent homomorphism of G, namely, there exists a Möbius transformation A ∈ PSL_{2}(C) such that

f^{μ_{1}} ∘ g ∘ (f^{μ_{1}})^{−1} = A ∘ f^{μ_{2}} ∘ g ∘ (f^{μ_{2}})^{−1} ∘ A^{−1}

for every g ∈ G. A Beltrami differential is called trivial if it is equivalent to 0.

Definition 2.1. The set of all equivalence classes of Beltrami differentials for G is called the quasiconformal deformation space of G and is denoted by QC(G).

For a Fuchsian group Γ acting on the lower half plane L the Teichmüller space T (Γ) of Γ is the quotient space of B_{1}(Γ, L) by the equivalence relation as above.

The space QC(G) has a natural complex structure such that the canonical projection B_{1}(G) → QC(G) is holomorphic. Also, the Teichmüller space T (Γ) has a natural complex structure such that the quotient map B_{1}(Γ, L) → T (Γ) is holomorphic. The Teichmüller space T (Γ) is also regarded as a deformation space of the Riemann surface X = L/Γ, which we denote by T (X). It is well-known that the complex structure of the Teichmüller space of Riemann surfaces of genus g is independent of the choice of the base point X. When we need not mention the base point X we denote the Teichmüller space of Riemann surfaces of genus g by T_{g}.

Here, we recall another definition of T (Γ) as the space of Fuchsian groups. For μ ∈ B_{1}(Γ, L), let f_{μ} denote the quasiconformal homeomorphism of L onto itself with Beltrami differential μ|L fixing 0, 1 and ∞. The mapping f_{μ} is the restriction of a quasiconformal mapping f^{μ′} where μ′ is a Beltrami differential on Ĉ defined by extending μ|L to Ĉ symmetrically beyond R:

μ′(z) = \{ \begin{matrix}μ(z), & if z ∈ L, \\ \overline{μ(\overline{z})}, & if z ∈ H. \end{matrix}

Then the group f_{μ}Γ(f_{μ})^{−1} is a Fuchsian group acting on L. It is easy to see that two Beltrami differentials μ_{1} and μ_{2} in B_{1}(Γ, L) determines the same homomorphism Γ → PSL_{2}(R) if and only if μ′_{1} and μ′_{2} are equivalent in B_{1}(Γ), where μ′_{i} (i = 1, 2) is the symmetric extension of μ_{i} as above. It is also easy to see that μ′_{1} and μ′_{2} are equivalent in B_{1}(Γ) if and only if μ_{1} and μ_{2} are equivalent in B_{1}(Γ).

Definition 2.2 (Teichmüller space as the space of Fuchsian groups). The Teichmüller space T (Γ) is the equivalence classes of the symmetric Beltrami differentials, namely, equivalence classes of Beltrami differentials μ′ ∈ B_{1}(Γ) such that μ′(z) = \overline{μ′(\overline{z})} almost everywhere.

By the above remark, this definition of T (Γ) is the same as that in Definition 2.1, if we regard T (Γ) as a topological space. Actually, these two definition give the same real analytic structures. However, when we regard T (Γ) as the space of Fuchsian groups, it is not considered as a complex manifold, since the space of symmetric Beltrami differentials is not a complex Banach manifold.

See Lehto [L], for example, for more detail on fundamental facts on Teichmüller spaces.

2.2 Projective structures and quadratic differentials. Assume that we are given a projective structure on Σ_{g}. Since the transition mappings are holomorphic, as they are restrictions of elements of PSL_{2}(C), we have a complex structure under the projective structure. Let X denote the Riemann surface and let Γ be a Fuchsian group acting on L such that X = L/Γ. Take a coordinate function of the projective structure. We can take its analytic continuation along any curve on X and have a multivalued holomorphic mapping into the Riemann sphere. This multivalued mapping is lifted to a locally univalent meromorphic function W : L → CP^{1} on the universal covering space L. This mapping is called a developing map of the projective structure. Note that the developing map is determined by the projective structure uniquely up to post compositions of elements of PSL_{2}(C).

When we take the analytic continuation of a local coordinate function along a closed curve and come back to the starting point, the values differ from each other by elements of PSL_{2}(C), since the transition mappings are in PSL_{2}(C). Therefore, we have a homomorphism of π_{1}X to PSL_{2}(C). If we look at this on the universal covering space L, we have a homomorphism χ : Γ → PSL_{2}(C) such that

(2.1)  W ∘ γ(z) = χ(γ) ∘ W (z), z ∈ L

for all γ ∈ Γ. This homomorphism is called the holonomy representation.

Thus the projective structure determines the pair (W, χ) uniquely up to the action of PSL_{2}(C). Conversely, it is easy to see that given a pair (W, χ) of locally univalent meromorphic function W : L → CP^{1} and a homomorphism χ : Γ → PSL_{2}(C), where Γ is a Fuchsian group, satisfying (2.1) we have a projective structure on the Riemann surface L/Γ. Therefore, the space of all projective structures on Σ_{g} is identified with the set of pairs (W, χ) satisfying (2.1) modulo the action of PSL_{2}(C).

Now we recall the parametrization of the space of projective structures by holomorphic quadratic differentials. Given a projective structure on a Riemann surface X = L/Γ, take the Schwarzian derivative of the developing map W and denote it by φ. (Here, Schwarzian derivative of a locally univalent meromorphic function f is defined by (f′′/f′)′ − 1/2(f′′/f′)^{2}.) Then φ is a holomorphic function on L satisfying

(2.2)  φ = φ ∘ γ × γ′^{2}

for all γ ∈ Γ. Such a holomorphic function satisfying (2.2) is called a holomorphic quadratic differential for Γ. A holomorphic quadratic differential for Γ is the lift of a holomorphic quadratic differential on the Riemann surface X = L/Γ. We have seen that given a projective structure on X we have a holomorphic quadratic differential for Γ. Conversely, given a holomorphic quadratic differential φ for Γ, it is well-known (see [H] or [Kr2]) that there is a locally univalent meromorphic function W_{φ} whose Schwarzian derivative is φ. It is easy to see from (2.1) that there is a homomorphism χ_{φ} : Γ → PSL_{2}(C) satisfying (2.1). Thus the space of all projective structure on the surface Σ_{g} is identified with the fiber space Q → T_{g} of holomorphic quadratic differentials on compact Riemann surfaces of genus g.

Notation 2.3. In this paper, we will denote by W_{φ} the developing map and by χ_{φ} the holonomy representation of the projective structure determined by φ. Since they are determined up to elements of PSL_{2}(C), we have to fix normalization in each argument in the following section. The choice is not essential. We shall employ convenient ones each time.

Notation 2.4. Let Rep denote the space of PSL_{2}(C) representations of the fundamental group π_{1}X : Rep = Hom(π_{1}X, PSL_{2}(C))/ PSL_{2}(C). Here, two PSL_{2}(C) representations are equivalent if they differ by conjugation in PSL_{2}(C). We define a mapping h : Q → Rep by sending each φ ∈ Q to the holonomy representation of the projective structure determined by φ, and call it the holonomy map.

We will use the following property of h:

Theorem 2.5 (Hejhal [H, Theorem 1]). The mapping h : Q → Rep is not a covering map. However, it is a local C^{1}-diffeomorphism.

Note that when we regard Q as the space of quadratic differentials for Fuchsian groups on L, we regard T (Γ) as the space of Fuchsian groups, hence not a complex manifold. In particular, the vector bundle Q → T (Γ) is not holomorphic. However, each fiber Q(Γ′) over Γ′ ∈ T (Γ) has a natural complex structure as a finite dimensional complex linear space.

Lemma 2.6. The holonomy map h : Q(Γ) → Rep is holomorphic for each Γ.

Note that the fiber Q(Γ) is identified with the space of projective structure whose underlying complex structure is X = L/Γ.

In the next section, we will discuss the space of projective structures with discrete holonomy representations. We denote by K the subset of Q consisting of all elements φ which determine projective structures with discrete holonomy representations and denote by K(Γ) the fiber over a point Γ. We denote by int_{Q(Γ)} K(Γ) the interior of K(Γ) considered in Q(Γ).

The set of all equivalence classes of B_{1}(Γ, H) is called the Bers slice. From the definition the Bers slice is isomorphic to the Teichmüller space. For each element μ ∈ B_{1}(Γ, H), f^{μ}|L is a univalent meromorphic mapping and f^{μ}Γ(f^{μ})^{−1} is a quasifuchsian group. Therefore, it determines a projective structure on L/Γ. Thus the Bers slice is embedded in Q(Γ). We will also call this image Bers slice.

As we noted in the introduction, for each Γ, int_{Q(Γ)} K(Γ) has a component which contains the Bers slice.

Proposition 2.7 ([Sh]). The component of int_{Q(Γ)} K(Γ) containing 0 coincides with the Bers slice. For any component κ(Γ) of int_{Q(Γ)} K(Γ) which is not the Bers slice, the developing map W_{φ} is surjective onto the Riemann sphere for every φ ∈ κ(Γ). For any two elements φ_{1} and φ_{2} of κ(Γ), χ_{φ_{1}}(Γ) and χ_{φ_{2}}(Γ) are quasi conformally equivalent to each other.

2.3 Holomorphic motions of Kleinian groups. Here we state the theorem of holomorphic motions in a convenient form for later use. (See [EKK] for detail and references for the theory of holomorphic motions.)

Definition 2.8. Let G be a finitely generated discrete group of PSL_{2}(C). A family {θ_{z}}_{z∈Δ} of isomorphisms of G into PSL_{2}(C), parametrized by the unit disc Δ = {z ∈ C; |z| < 1}, is called a holomorphic motion of G or a holomorphic family of isomorphisms of G if it satisfies the following conditions:

(1) θ_{0} is the identity: θ_{0}(g) = g for all g ∈ G.

(2) For every z ∈ Δ, θ_{z}(G) is discrete.

(3) For every g ∈ G, z ↦ θ_{z}(g) is a holomorphic mapping of Δ to PSL_{2}(C).

Theorem 2.9 (Bers [B], Earle-Kra-Krushkal[EKK], Slodokowski [Sl]). Let {θ_{z}}_{z∈Δ} is a holomorphic family of isomorphisms of a discrete group G. Then there is a holomorphic mapping z ↦ μ_{z} from Δ to B_{1}(G) such that θ_{z}(γ) = f^{μ_{z}} ∘ γ ∘ (f^{μ_{z}})^{−1} for all γ ∈ G where f^{μ_{z}} is a quasiconformal mapping with Beltrami differential μ_{z}.

Remark. If we have a holomorphic family of discrete groups, it is actually a holomorphic family of discontinuous groups, unless the family is trivial. In fact, by the above theorem the holomorphic family of isomorphisms are given by a family of quasiconformal deformations. If the discrete group is not discontinuous, there is no non-trivial quasiconformal deformations by the Sullivan’s rigidity theorem ([Su]). Therefore, int_{Q(Γ)} K(Γ) coincides with the interior of set of projective structures with discontinuous holonomy representations.

3. Projective structures with discrete holonomy representations

To show the main result, we will construct a local inverse to the holonomy map h : Q → Rep on each component of int K, which is holomorphic in the direction of fibers of Q → T_{g}. For the construction, we first note the following fact.

Lemma 3.1. Each component κ(Γ) of int_{Q(Γ)} K(Γ) is either consists of totally degenerate groups or consists of quasifuchsian groups which is isomorphic to Γ.

Proof.

First, note that for every φ ∈ κ(Γ) χ_{φ} is an isomorphism which preserves parabolicity (Kra [Kr2] and [Kr3]).

By the result of Maskit [M , Theorem 2 and Theorem 3], χ_{φ}(Γ) is either a quasifuchsian group or a totally degenerate group. By Proposition 2.7, either χ_{φ}(Γ) is a quasifuchsian group for every φ ∈ κ(Γ) or χ_{φ}(Γ) is a totally degenerate group for every φ ∈ κ(Γ).

□

We now define a mapping which plays a key role to analyze the structure of int K or int_{Q(Γ)} K(Γ).

Fix an element φ ∈ κ(Γ) and denote the holonomy representation χ_{φ}(Γ) by G_{φ}, for simplicity. For each Beltrami differential μ ∈ B_{1}(G_{φ}) for G_{φ}, we pull back μ via the developing map W_{φ} to a Beltrami differential μ̂, namely, we set

μ̂ = μ ∘ W_{φ} × \overline{W′_{φ}}/W′_{φ}.

The differential μ̂ is a (−1, 1)-form of the local homeomorphism f^{μ} ∘ W_{φ}, where f^{μ} is a quasiconformal mapping with Beltrami differential μ. It is easy to see that μ̂ is an element of B_{1}(Γ, L), and the assignment Φ̂ : μ → μ̂ is a holomorphic mapping from B_{1}(G_{φ}) to B_{1}(Γ, L).

Proposition 3.2. In the above situation, the mapping Φ̂ : μ → μ̂ descends to a holomorphic mapping Φ : QC(G_{φ}) → T (Γ). (Here, we regard T (Γ) as a complex manifold.)

Proof. We have to show that two equivalent elements μ_{1} and μ_{2} in B_{1}(G_{φ}) are mapped by Φ̂ to equivalent elements μ̂_{1} and μ̂_{2} in B_{1}(Γ, L).

We first show that every trivial differential τ in B_{1}(G_{φ}) is pulled back to a trivial differential τ̂ in B_{1}(Γ, L). Choose three points in Λ(G_{φ}). For each μ ∈ B_{1}(G_{φ}), let f^{μ} denote the quasiconformal mapping with Beltrami differential μ which fixes the three points. If τ is trivial, then f^{τ} fixes every point on Λ(G_{φ}) and maps each component of Ω(G_{φ}) onto itself. Here we note how the locally univalent mapping W_{φ} for φ ∈ int_{Q(Γ)} K(Γ) behaves in L. The preimages (W_{φ})^{−1}(Ω(G_{φ})) and (W_{φ})^{−1}(Λ(G_{φ})) are Γ invariant and mutually disjoint. By Lemma 3.1, the holonomy representation G_{φ} is either a quasifuchsian group or a totally degenerate group. In either case, each component of Ω(G_{φ}) is simply connected. Hence each germ of the local inverse of W_{φ} has an analytic continuation in each component of Ω(G_{φ}). Therefore each component of the preimage (W_{φ})^{−1}(Ω(G_{φ})) is a simply connected domain which is mapped onto a component of Ω(G_{φ}) by W_{φ} injectively. Decompose W_{φ}^{−1}(Ω(G_{φ})) to the disjoint union of connected components and denote it by ∪_{n=1}^{∞}D_{φ,n}. As we have seen above for each n there is a branch W_{φ,n}^{−1} of W_{φ}^{−1} on W_{φ}(D_{φ,n}) such that W_{φ,n}^{−1} ∘ W_{φ} = id. on D_{φ,n}.

We define a mapping f : L → L by

f(z) = \{ \begin{matrix}z & on W_{φ}^{−1}(Λ(G_{φ})) \\ W_{φ,n}^{−1} ∘ f^{τ} ∘ W_{φ}(z) & on D_{φ,n} (n = 1, 2, ...). \end{matrix}

Now we claim that f is a quasiconformal homeomorphism of L. (Intuitively, f is nothing but f^{τ} with the local charts given by W_{φ} rather than the standard coordinate of the Riemann sphere.) We have to care about the behavior of f near W_{φ}^{−1}(Λ(G_{φ})).

Fix a point z_{0} ∈ W_{φ}^{−1}(Λ(G_{φ})) arbitrarily and take a small disc δ around z_{0}. The image W_{φ}(δ) is a neighborhood of W_{φ}(z_{0}) ∈ Λ(G_{φ}). Take a disc δ′ ∋ z_{0} small enough so that f^{τ} (W_{φ}(δ′)) ⊂ W_{φ}(δ). Recall that f^{τ} |W_{φ}(δ) ∩ Λ(G_{φ}) = id. We can take the branch W_{φ,δ′}^{−1} of W_{φ}^{−1} on W_{φ}(δ′) such that W_{φ,δ′}^{−1} ∘ W_{φ} is the identity on δ′. With this branch of W_{φ}^{−1}, we can define a quasiconformal mapping by W_{φ,δ′}^{−1} ∘ f^{τ} ∘ W_{φ} on δ′.

Obviously, W_{φ,δ′}^{−1} ∘ f^{τ} ∘ W_{φ} coincides with f on W_{φ}^{−1}(Λ(G_{φ})). We claim that this mapping coincides with the mapping W_{φ,n}^{−1} ∘ f^{τ} ∘ W_{φ} also on each component of D_{φ,n} ∩ δ′. Then it will follow that f = W_{φ,δ′}^{−1} ∘ f^{τ} ∘ W_{φ} is quasiconformal near z_{0}. (Note that this claim is not quite obvious, because W_{φ}(δ′) ∩ Ω(G_{φ}) may have infinitely many components and f^{τ} might exchange these components.) In order to show the above claim, it suffices to show that f^{τ} maps each component of W_{φ}(δ′)∩Ω(G_{φ}) into the component of W_{φ}(δ)∩Ω(G_{φ}) containing it. To verify this, we take an isotopy {f_{t}}_{0≤t≤1} between f^{τ} and the identity map via quasiconformal mappings fixing every point of Λ(G_{φ}), here, the existence of such an isotopy is due to Earle-McMullen [EM]. Replacing δ′ into a smaller disc, if necessary, we may assume that f_{t}(W_{φ}(δ′)) ⊂ W_{φ}(δ) for all t ∈ [0, 1]. Then for each component u of W_{φ}(δ′) ∩ Ω(G_{φ}) and t, f_{t}(u) is contained in a component of W_{φ}(δ) ∩ Ω(G_{φ}) and the component depends continuously on t. Hence it is independent of t.

Thus we have seen that f is a quasiconformal homeomorphism of L with Beltrami differential τ̂. Since f is the identity map on W_{φ}^{−1}(Λ(G_{φ})) and since this set is Γinvariant, the extension of f to the real axis is the identity. Hence τ̂ is a trivial differential.

Now we proceed to the general case. Assume that μ_{1} and μ_{2} are equivalent in B_{1}(G_{φ}). Then there is a trivial differential τ in B_{1}(G_{φ}) such that f^{μ_{2}} = f^{μ_{1}} ∘ f^{τ}. Now, as we have shown above, τ̂ is trivial in B_{1}(Γ, L) and that W_{φ} ∘ fτ̂ = f^{τ} ∘ W_{φ}, where fτ̂ is the quasiconformal homeomorphism of L with Beltrami differential τ̂ fixing 0, 1 and ∞. Hence we have

f^{μ_{2}} ∘ W_{φ} = f^{μ_{1}} ∘ f^{τ} ∘ W_{φ}
= f^{μ_{1}} ∘ W_{φ} ∘ fτ̂.

Let fμ̂_{i}, i = 1, 2, denote the quasiconformal homeomorphism of L with Beltrami differential μ̂_{i} fixing 0, 1 and ∞. Since the Beltrami differential μ̂_{i} is the Beltrami differential of f^{μ_{i}} ∘ W_{φ} by definition, the above equality implies

fμ̂_{2} = fμ̂_{1} ∘ fτ̂.

Since τ̂ is trivial, μ̂_{1} and μ̂_{2} are equivalent. □

We define another mapping which is essentially obtained in the above argument. Fix a point φ ∈ int_{Q(Γ)} K(Γ). For each μ ∈ B_{1}(G_{φ}), let fμ̂ denote the quasicon-

formal homeomorphism of L with Beltrami differential fixing 0, 1 and ∞. Then by the definition of μ̂, the mapping W^{μ} defined by W^{μ} = f^{μ} ∘ W_{φ} ∘ f −1 ; μ̂ is a locally univalent holomorphic mapping on L to the Riemann sphere. Moreover,

(3.1)  W^{μ} ∘ (fμ̂ ∘ γ ∘ f −1 ; μ̂) = (f^{μ} ∘ χ_{φ}(γ) ∘ (f^{μ})^{−1}) ∘ W^{μ},

for all γ ∈ Γ. Thus W^{μ} is a developing map of a projective structure on L/Γμ̂

where Γμ̂ = fμ̂Γf −1 ; μ̂.

Taking the Schwarzian derivative of W^{μ}, we have a holomorphic quadratic differential ϕ̃(μ) ∈ Q(Γ_{μ}). Thus we have a mapping ϕ̃ : B_{1}(G_{φ}) → Q. The mapping ϕ̃ : B_{1}(G_{φ}) → Q descends to a mapping from QC(G_{φ}) into Q. In fact, if μ_{1} and μ_{2} are equivalent, μ̂_{1} and μ̂_{2} are equivalent by Proposition 3.2. Therefore we have Γμ̂_{1} = Γμ̂_{2} and ϕ̃(μ̂_{1}) and ϕ̃(μ̂_{2}) are in the same fiber Q(Γμ̂_{1}). Furthermore, these two quadratic differentials determine projective structures with the same holonomy representation. Hence by the following Theorem by Poincaré (cf. Kra [Kr1]), these holomorphic differential are the same.

Theorem (Poincaré). Two different projective structures on the same complex structure of a compact surface have different holonomy representations.

We shall denote the mapping QC(G_{φ}) → Q by ϕ. The mapping ϕ : QC(G_{φ}) → Q is a continuous mapping which satisfies ϕ∘h = id., where h is the holonomy map. By Theorem 2.5, ϕ is a C^{1} map. Also, it satisfies Φ = π ∘ ϕ, where π : Q → T (Γ) denotes the projection and Φ : QC(G_{φ}) → T (Γ) is the holomorphic mapping defined in Proposition 3.2.

Proposition 3.3. Let φ be an element of int_{Q(Γ)} K(Γ) and let κ_{φ}(Γ) denote the component of int_{Q(Γ)} K(Γ) containing φ. Then κ_{φ}(Γ) is a subset of a component of ϕ(QC(G_{φ})) ∩ Q(Γ).

Proof.

We shall show every element ψ ∈ κ_{φ}(Γ) is contained in ϕ(QC(G_{φ}))∩Q(Γ). First, we consider the case that ψ is close to φ so that there is a complex analytic disc connecting them, namely, there is a family {φ_{z}}_{z∈Δ} ⊂ κ_{φ}(Γ) depending holomorphically on z such that φ = φ_{0} and φ_{ζ} = ψ for some ζ ∈ Δ (Here, Δ denotes the unit disc of the complex plane). Then the set of holonomy representations χ_{φ_{z}} depends holomorphically on z. Note that χ_{φ_{z}} is isomorphic by Lemma 3.1. Applying Theorem 2.9 to the family χ_{φ_{z}} ∘ χ_{φ_{0}}^{−1}, we have a family of Beltrami differentials {μ_{z}} depending holomorphically on z such that

χ_{φ_{z}}(γ) = f^{μ_{z}} ∘ χ_{φ}(γ) ∘ (f^{μ_{z}})^{−1},

for all γ ∈ Γ and z ∈ Δ. For the Beltrami differential μ̂_{z} = Φ̂(μ_{z}) and the quadratic differential ϕ(μ_{z}) ∈ Q(Γμ̂_{z}) we shall show that ϕ(μ_{z}) = φ_{z} for all z ∈ Δ. The set {z ∈ Δ; ϕ(μ_{z}) = φ_{z}} is clearly a closed set and non-empty. (It contains 0.) On the other hand, since ϕ(μ_{z}) and φ_{z} have the same holonomy representation and the representation map h : Q → Rep is a local homeomorphism (Theorem 2.5), the set {z ∈ Δ; ϕ(μ_{z}) = φ_{z}} is also an open set. Hence ϕ(μ_{z}) = φ_{z} for all z ∈ Δ.

Thus we have shown that all ψ ∈ κ_{φ}(Γ) sufficiently close to φ is contained in Q(Γ) ∩ ϕ(QC(G_{φ})). For general ψ ∈ κ_{φ}(Γ), we take a chain of holomorphic discs connecting ψ and φ and apply the above argument to each disc. □

Now we can prove the main theorem.

Theorem 3.4. Every component of int_{Q(Γ)} K(Γ) consists of quasifuchsian groups. Every such component is holomorphically equivalent to a complex analytic submanifold of T_{g} × T_{g}.

Proof. To prove the first claim, it is sufficient to show that there is no component of int_{Q(Γ)} K(Γ) consisting of totally degenerate groups, by Lemma 3.1.

Assume that there is a component κ(Γ) in int_{Q(Γ)} K(Γ) which consists of totally degenerate groups. We will draw a contradiction. Take an element φ ∈ κ(Γ) and denote the holonomy group by G_{φ} for short. We define a mapping Φ : QC(G_{φ}) → T (Γ) as in Proposition 3.2. Then since κ(Γ) and QC(G_{φ}) are of the same dimension, h(κ(Γ)) is an open set of QC(G_{φ}). It follows that the holomorphic mapping Φ : QC(G_{φ}) → T (Γ) takes the value Γ on the open set h(κ(Γ)) of QC(G_{φ}). Therefore it is a constant mapping: Φ(QC(G_{φ})) ≡ Γ. It follows that ϕ(QC(G_{φ})) ⊂ Q(Γ). As QC(G_{φ}) and Q(Γ) are of the same dimension, ϕ(QC(G_{φ})) is an open subset of Q(Γ). From this fact and Proposition 3.3, we have ϕ(QC(G_{φ})) = κ(Γ).

It follows that there is a point ψ ∈ κ(Γ) such that the holonomy representation χ_{ψ}(Γ) ∈ QC(G_{φ}) has Fuchsian equivalent Γ. Namely, we can choose a point f^{μ}G_{φ}(f^{μ})^{−1} ∈ QC(G_{φ}) such that there is a univalent holomorphic mapping W : L → Ω(f^{μ}G_{φ}(f^{μ})^{−1}) such that W ∘ γ = f^{μ}χ_{φ}(γ)(f^{μ})^{−1} ∘ W for all γ ∈ Γ and ψ = ϕ(μ). Now, the projective structure defined by the univalent map W and the projective structure defined by W_{ψ} have the same holonomy representation and have the same underlying complex structure. By Poincaré’s theorem, W and W_{ψ} coincides up to Möbius transformation. However, this is impossible because W (L) = Ω(f^{μ}G_{φ}(f^{μ})^{−1}) ⫋ Ĉ, while W_{ψ}(L) = Ĉ by Proposition 2.7. (Note that κ(Γ) is not the Bers slice.)

We have seen each component of int_{Q(Γ)} K(Γ) consists of projective structures whose holonomy representations are quasifuchsian. From this fact, we shall show that κ_{φ}(Γ) coincides with the component of ϕ(QC(G_{φ}))∩Q(Γ) containing φ. Recall the proof of Proposition 3.3. For each point φ ∈ int_{Q(Γ)} K(Γ), the component κ_{φ}(Γ) of int_{Q(Γ)} K(Γ) containing φ is an open subset of a component of ϕ(QC(G_{φ}))∩Q(Γ). On the other hand, as G_{φ} is a quasifuchsian group, Q and QC(G_{φ}) is the same dimension. Therefore, h(QC(G_{φ})) is an open subset of Q, hence each component of ϕ(QC(G_{ϕ}))∩Q(Γ) is an open subset of int_{Q(Γ)} K(Γ). Therefore, κ_{φ}(Γ) coincides with the component of ϕ(QC(G_{φ})) ∩ Q(Γ) containing φ.

It follows that κ_{φ}(Γ) is biholomorphically equivalent to the component of Φ^{−1}(Γ) containing G_{φ}, which is a submanifold of QC(G_{φ}) by Lemma 2.6. Now the second claim of the theorem follows from the fact that QC(G_{φ}) is holomorphically equivalent to T_{g} × T_{g}. □

What we have done above is to embed QC(G_{φ}) into Q with the mapping ϕ (in fact, we have seen that ϕ is a local inverse to the holonomy map h : Q → Rep.) and consider the slice along a fiber of Q → T_{g}. It is easy to see that for every Γ′ ∈ T (Γ) each component of the slice over Γ′ is a component of int_{Q(Γ)} K(Γ′)

Corollary 3.5. Each component of int_{Q(Γ)} K(Γ) is complete with respect to the Kobayashi hyperbolic metric and Carathéodory metric. Hence each such component is a domain of holomorphy.

Proof. We showed in the above theorem that each component κ(Γ) of int_{Q(Γ)} K(Γ) is holomorphically equivalent to a submanifold T_{g} × T_{g}. It is well-known that T_{g} is complete with respect to the Carathéodory metric and Kobayashi metric. By a standard argument (see Kobayashi [K]), T_{g} × T_{g} and its submanifolds are complete with respect to these metrics. It is also known that in general a domain in C^{N} complete with respect to Carathéodory distance is a domain of holomorphy. □

Here we give a brief remark on the existence of components of int_{Q(X)} K(X) other than the Bers slice, which is discussed in [T]. In the above discussion, we have shown that any component of int_{Q(Γ)} K(Γ) is a component of int K ∩ Q(Γ). It is shown in [T] that int K has infinitely many component and each component has non-empty intersection with Q(Γ). Therefore we have:

Theorem 3.6. On any complex structure X ∈ T_{g} there are infinitely many components of int_{Q(X)} K(X).

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Shiga: Department of Mathematics, Tokyo Institute of Technology, Tokyo 152 Japan

Tanigawa: Graduate School of Polymathematics, Nagoya University, Nagoya 464-01 Japan