On Holomorphic Contractibility of Teichmüller Spaces

Abstract

The problem of the holomorphic contractibility of Teichmüller spaces $\mathbf{T}(0,n)$ of the punctured spheres ($n>4$) arose in the 1970s in connection with solving algebraic equations in Banach algebras. Recently it was solved by the author. In the present paper, we give a refined proof of the holomorphic contractibility for all spaces $\mathbf{T}(0,n), n>4$ and provide two independent proofs of holomorphic contractibility for low-dimensional Teichmüller spaces, which has intrinsic interest.

1. Preamble

1.1. Holomorphic Contractibility

A complex Banach manifold X is contractible to its point $x_0$ if there exists a continuous map $F:X\times [0,1]\to X$ with $F(x,0)=x$ and $F(x,1)=x_0$ for all $x\in X$. If map F can be chosen so that for every $t\in [0,1]$ the map $F_t:x\mapsto F(x,t)$ of X is holomorphic to itself and $F_t\left(x_0\right)=x_0$, then X is called holomorphically contractible to $x_0$.

The problem of holomorphic contractibility of Teichmüller spaces $\mathbf{T}(0,n)$ of the punctures spheres ($n>4$) arose in the 1970s in connection with solving algebraic equations in Banach algebras. It was caused by the fact that in the space ${\mathbb{C}}^m, m>1$, there are domains (even bounded) that are only topologically but not holomorphically contractible (see [1,2,3,4,5]).

The simplest example of holomorphically contractible domains in complex Banach spaces is given by starlike domains. However, all Teichmüller spaces of sufficiently great dimensions are not starlike (see [6,7]).

Earle [8] established the holomorphic contractibility of two modified Teichmüller spaces related to asymptotically conformal maps.

Recently, this problem was solved positively in [9]. It was established that all spaces $\mathbf{T}(0,n), n>4$, are holomorphically contractible.

Theorem 1.

Any space $\mathbf{T}(0,n)$ with $n>4$ is holomorphically contractible.

The proof of Lemma 3 in that paper contains a wrong assertion (which is not used here) that the map $s_m$, including the space $T\left({\mathsf{\Gamma }}_0\right)$ into $T\left({\mathsf{\Gamma }}_0^m\right)$, is a section of the forgetful map ${\chi }_m: T\left(X_{a^0}^m\right)\to T\left(X_{a^0}\right)$. Such sections do not exist if $n>6$.

In fact, $s_m$ as an open holomorphic map (of a domain onto a manifold) was only used in the proof of Lemma 3 (and of Theorem 1), and the openness is preserved for the limit map $s={lim}_{m\to \infty }{s}_m$, which determines an $(n-3)$-dimensional complex submanifold $s\left(T\left(X_{a^0}\right)\right)$ in the universal Teichmüller space $\mathbf{T}$.

In the present paper, we give a refined proof of holomorphic contractibility for all spaces $\mathbf{T}(0,n), n>4$ and provide two independent proofs of holomorphic contractibility for low-dimensional Teichmüller spaces (of dimensions two and three). The second proof has its own interest in view of the importance of the problem. Its underlying idea is different; the arguments do not extend to higher dimensions.

1.2. Teichmüller Spaces of Low Dimenshions

There are two Teichmüller spaces of dimension two: the space $\mathbf{T}(0,5)$ of the spheres with five punctures and the space $\mathbf{T}(1,2)$ of twice-punctured tori; these spaces are biholomorphically equivalent. Such spheres and tori are uniformized by the corresponding Fuchsian groups $\mathsf{\Gamma }$ and ${\mathsf{\Gamma }}^{\prime }$ so that $\mathsf{\Gamma }$ is a subgroup of index two in ${\mathsf{\Gamma }}^{\prime }$; letting $\mathbf{T}(0,5)=\mathbf{T}\left(\mathsf{\Gamma }\right), \mathbf{T}(1,2)=\mathbf{T}\left({\mathsf{\Gamma }}^{\prime }\right)$, we have $\mathbf{T}\left({\mathsf{\Gamma }}^{\prime }\right)=\mathbf{T}\left(\mathsf{\Gamma }\right)$.

In a similar way, the Teichmüller spaces $\mathbf{T}(0,6)$ of spheres with six punctures and $\mathbf{T}(2,0)$ of closed Riemann surfaces of genus 2 also are biholomorphically equivalent, and in terms of the corresponding Fuchsian groups $\mathsf{\Gamma }$ and ${\mathsf{\Gamma }}^{\prime }$ we have the same relationship $\mathbf{T}\left({\mathsf{\Gamma }}^{\prime }\right)=\mathbf{T}\left(\mathsf{\Gamma }\right)$. We state:

Theorem 2.

The spaces $\mathbf{T}(0,5), \mathbf{T}(1,2), \mathbf{T}(0,6), \mathbf{T}(2,0)$ are holomorphically contractible.

The Teichmüller space $\mathbf{T}(1,3)$ of tori with three punctures also has three dimensions; it will not be involved here.

2. Underlying Results

2.1. Teichmüller Spaces of Punctured Spheres

Consider the ordered n-tuples of points

$$\mathbf{a}=(0,1,a_1,\cdots ,a_{n-3},\infty ), n>4,$$

(1)

with distinct $a_j\in \widehat{\mathbb{C}}\backslash \{1,-1,i\}$ and the corresponding punctured spheres

$$X_{\mathbf{a}}=\widehat{\mathbb{C}}\backslash \{0,1,a_1\cdots ,a_{n-3},\infty \}, \widehat{\mathbb{C}}=\mathbb{C}\cup \{\infty \},$$

regarded as Riemann surfaces of genus zero. Fix a collection ${\mathbf{a}}^0=(0,1,a_1^0,\cdots ,a_{n-3}^0,1,\infty )$ defining the base point $X_{{\mathbf{a}}^0}$ of Teichmüller space $\mathbf{T}(0,n)=\mathbf{T}\left(X_{{\mathbf{a}}^0}\right)$ of n-punctured spheres. Its points are the equivalence classes $\left[\mu \right]$ of Beltrami coefficients from the ball

$$Belt{\left(\mathbb{C}\right)}_1=\{\mu \in L_{\infty }{\left(\mathbb{C}\right): \parallel \mu \parallel }_{\infty }<1\},$$

under the relationship that ${\mu }_1\sim {\mu }_2$ if the corresponding quasiconformal homeomorphisms $w^{{\mu }_1},w^{{\mu }_2}: X_{{\mathbf{a}}^0}\to X_{\mathbf{a}}$ (the solutions of the Beltrami equation $\overline{\partial }w=\mu \partial w$ with $\mu ={\mu }_1,{\mu }_2$) are homotopic on $X_{{\mathbf{a}}^0}$ (and hence coincide at the points $0,1,a_1^0,\cdots ,a_{n-3}^0,\infty$). This models $\mathbf{T}(0,n)$ as the quotient space

$$\mathbf{T}(0,n)=Belt{\left(\mathbb{C}\right)}_1/\sim$$

with a complex Banach structure of dimension $n-3$ inherited from the ball $Belt{\left(\mathbb{C}\right)}_1$.

Another canonical model of $\mathbf{T}(0,n)=\mathbf{T}\left(X_{{\mathbf{a}}^0}\right)$ is obtained using the uniformization of Riemann surfaces and the holomorphic Bers embedding of Teichmüller spaces. Consider the upper and lower half-planes

$$U=\{z=x+iy\in \mathbb{C}: y>0\}, U^{\ast }=\{z=x+iy\in \mathbb{C}: y<0\}$$

The holomorphic universal covering map $h: U\to X_{{\mathbf{a}}^0}$ provides a torsion-free Fuchsian group ${\mathsf{\Gamma }}_0$ of the first kind acting discontinuously on $U\cup U^{\ast }$, and the surface $X_{{\mathbf{a}}^0}$ is represented (up to conformal equivalence) as the quotient space $U/{\mathsf{\Gamma }}_0$. The functions $\mu \in L_{\infty }\left(X_{{\mathbf{a}}^0}\right)=L_{\infty }\left(\mathbb{C}\right)$ are lifted to U as the Beltrami $(-1,1)$-measurable forms $\tilde{\mu }d\overline{z}/dz$ on U with respect to ${\mathsf{\Gamma }}_0$, satisfying $(\tilde{\mu }\circ \gamma )\overline{{\gamma }^{\prime }}/{\gamma }^{\prime }=\tilde{\mu }, \gamma \in {\mathsf{\Gamma }}_0$ and forming the corresponding Banach space $L_{\infty }(U,{\mathsf{\Gamma }}_0)$. We extend these $\tilde{\mu }$ by zero to $U^{\ast }$ and consider the unit ball $Belt{(U,{\mathsf{\Gamma }}_0)}_1$ of this space $L_{\infty }(U,{\mathsf{\Gamma }}_0)$. The corresponding quasiconformal maps $w^{\widehat{\mu }}$ are conformal on the half-plane $U^{\ast }$, and their Schwarzian derivatives.

$$S_w\left(z\right)={\left(\frac{w^{‴}\left(z\right)}{w^{\prime }\left(z\right)}\right)}^{\prime }-\frac{1}{2}{\left(\frac{w^{″}\left(z\right)}{w^{\prime }\left(z\right)}\right)}^2, w=w^{\widehat{\mu }},$$

fill a bounded domain in the complex $(n-3)$-dimensional space $\mathbf{B}\left({\mathsf{\Gamma }}_0\right)=\mathbf{B}(U^{\ast },{\mathsf{\Gamma }}_0)$ of hyperbolically bounded holomorphic ${\mathsf{\Gamma }}_0$-automorphic forms of degree $-4$ on $U^{\ast }$ (i.e., satisfy $(\phi \circ \gamma ){\left({\gamma }^{\prime }\right)}^2=\phi , \gamma \in {\mathsf{\Gamma }}_0$), with norm

$$\parallel \phi \parallel =\underset{U^{\ast }}{sup}4y^2\left|\phi \left(z\right)\right|.$$

This domain models the Teichmüller space $\mathbf{T}\left({\mathsf{\Gamma }}_0\right)$ of the group ${\mathsf{\Gamma }}_0$. It is canonically isomorphic to the space $\mathbf{T}\left(X_{{\mathbf{a}}^0}\right)$ (and to a bounded domain in the complex Euclidean space ${\mathbb{C}}^{n-3}$).

The indicated map $\widehat{\mu }\to S_{w^{\widehat{\mu }}}$ determines a holomorphic map ${\varphi }_{\mathbf{T}}:Belt{(U,{\mathsf{\Gamma }}_0)}_1\to \mathbf{B}\left({\mathsf{\Gamma }}_0\right)$; it has only local holomorphic sections.

Note also that $\mathbf{T}\left({\mathsf{\Gamma }}_0\right)=\mathbf{T}\cap \mathbf{B}\left({\mathsf{\Gamma }}_0\right)$, where $\mathbf{T}$ is the universal Teichmüller space (modeled as a domain of the Schwarzian derivatives of all univalent functions on $U^{\ast }$ admitting quasiconformal extension to U).

2.2. Connection with Interpolation of Univalent Functions

The collections (1) fills a domain ${\mathcal{D}}_n$ in ${\mathbb{C}}^{n-3}$ obtained by deleting from this space the hyperplanes $\{z=(z_1,\cdots ,z_{n-3}): z_j=z_l, j\ne l\}$, and with $z_1=0,z_2=1$. This domain represents the Torelli space of the spheres $X_{\mathbf{a}}$ and is covered by $\mathbf{T}(0,n)$, which is given by the following lemma (cf., e.g., [10]; [11], Section 2.8).

Lemma 1.

The holomorphic universal covering space of ${\mathcal{D}}_n$ is the Teichmüller space $\mathbf{T}(0,n)$. This means that for each punctured sphere $X_{\mathbf{a}}$, there is a holomorphic universal covering

$${\pi }_{\mathbf{a}}:\mathbf{T}(0,n)=\mathbf{T}\left(X_{\mathbf{a}}\right)\to {\mathcal{D}}_n.$$

The covering map ${\pi }_{\mathbf{a}}$ is well defined by

$${\pi }_{\mathbf{a}}\circ {\varphi }_{\mathbf{a}}\left(\mu \right)=(0,1,w^{\mu }\left(a_1\right),\cdots ,w^{\mu }\left(a_{n-3}\right),\infty ),$$

where ${\varphi }_{\mathbf{a}}$ denotes the canonical projection of the ball $Belt{\left(\mathbb{C}\right)}_1$ onto the space $\mathbf{T}\left(X_{\mathbf{a}}\right)$.

Lemma 1 also yields that the truncated collections ${\mathbf{a}}_{\ast }=(a_1,\cdots ,a_{n-3})$ provide the local complex coordinates on the space $\mathbf{T}(0,n)$ and define its complex structure.

These coordinates are simply connected with the Bers local complex coordinates on $\mathbf{T}(0,n)$ (related to basis of the tangent spaces to $\mathbf{T}(0,n)$ at its points, see [12]) via standard variation of quasiconformal maps of $X_{\mathbf{a}}=U/{\mathsf{\Gamma }}_{\mathbf{a}}$

$$\begin{array}{cc}\hfill w^{\mu }\left(z\right)& =z-\frac{z(z-1)}{\pi }\underset{\mathbb{C}}{\int \int }\frac{\mu \left(\zeta \right)}{\zeta (\zeta -1)(\zeta -z)}d\xi d\eta +{O(\parallel \mu \parallel }_{\infty }^2)\hfill \\ & =z-\frac{z(z-1)}{\pi }\sum _{\gamma \in {\mathsf{\Gamma }}_{\mathbf{a}}} \underset{U/{\mathsf{\Gamma }}_{\mathbf{a}}}{\int \int }\frac{\mu \left(\gamma \zeta \right)|{\gamma }^{\prime }{\left(\zeta \right)|}^2}{\gamma \zeta (\gamma \zeta -1)(\gamma \zeta -z)}d\xi d\eta +{O(\parallel \mu \parallel }_{\infty }^2).\hfill \end{array}$$

with a uniform estimate of the ratio ${O(\parallel \mu \parallel }_{\infty }^2{)/\parallel \mu \parallel }_{\infty }^2$ on compacts in $\mathbb{C}$ (see, e.g., [13]).

It turns out that one can obtain the whole space $\mathbf{T}\left(X_{{\mathbf{a}}^0}\right)$ using only the similar equivalence classes $\left[\mu \right]$ of the Beltrami coefficients from the ball $\mu \in Belt{\left(U\right)}_1$ (vanishing on $U^{\ast })$, requiring that the corresponding quasiconformal homeomorphisms $w^{\mu }$ are homotopic on the punctured sphere $X_{{\mathbf{a}}^0}$. Surjectivity of this holomorphic quotient map

$$\chi :Belt{\left(U\right)}_1\to \mathbf{T}(0,n),$$

is a consequence of the following interpolation result from [14].

Lemma 2.

Given two cyclically ordered collections of points $(z_1,\cdots ,z_m)$ and $({\zeta }_1,\cdots ,{\zeta }_m)$ on the unit circle $S^1=\left\{\right|z|=1\}$, there exists a holomorphic univalent function f in the closure of the unit disk $\mathbb{D}=\left\{\right|z|<1\}$ such that $\left|f\right(z\left)\right|<1$ for $z\in \overline{\mathbb{D}}$ distinct from $z_1,\cdots ,z_m$, and $f\left(z_k\right)={\zeta }_k$ for all $k=1,\cdots ,m$. Moreover, there exist univalent polynomials f with such an interpolation property.

It follows that the function f given by Lemma 2 is actually holomorphic and univalent (hence, maps conformally) in a broader disk ${\mathbb{D}}_r, r>1$.

First of all, $f^{\prime }\left(z\right)\ne 0$ on the unit circle $S^1$. Indeed, if $f^{\prime }\left(z_0\right)=0$ at some point $z_0\in S^1$, then in its neighborhood $f\left(z\right)=c_s{(z-z_0)}^s+O({(z-z_0)}^{s+1})=c_s{\zeta }^s$, where $c_s\ne 0$ for some $s>1$, which contradicts the injectivity of $f\left(z\right)$ on $S^1$. Therefore, f is univalent in some disk ${\mathbb{D}}_r=\left\{\right|z|<r\}, r>1$.

Assuming, on the contrary, that f is not globally univalent in any admissible disk ${\mathbb{D}}_r$ with $r>1$, one obtains the distinct sequences $\left\{z_n^{\prime }\right\}, \left\{z_n^{″}\right\}\subset {\mathbb{D}}_r$ with $f\left(z_n^{\prime }\right)=f\left(z_n^{″}\right)$ for any n, whose limit points $z_0^{\prime }, z_0^{″}$ lie on $S^1$. Then, in the limit, we have $f\left(z_0^{\prime }\right)=f\left(z_0^{″}\right)$, which in the case $z_0^{\prime }\ne z_0^{″}$, contradicts the univalence of f on $S^1$ given by Lemma 2, and in the case $z_0^{\prime }=z_0^{″}=z_0$, contradicts the local univalence of f in a neighborhood of $z_0$.

The interpolating function f given by Lemma 2 is extended quasiconformally to the whole sphere $\widehat{\mathbb{C}}$ across any circle $\left\{\right|z|=r\}$ with $r>1$ indicated above. Hence, given a cyclically ordered collection $(z_1,\cdots ,z_m)$ of points on $S^1$, then for any ordered collection $({\zeta }_1,\cdots ,{\zeta }_m)$ in $\widehat{\mathbb{C}}$, there exists a quasiconformal homeomorphism $\widehat{f}$ of the sphere $\widehat{\mathbb{C}}$ carrying the points $z_j$ to ${\zeta }_j, j=1,\cdots ,m$, and such that its restriction to the closed disk $\overline{\mathbb{D}}$ is biholomorphic on $\overline{\mathbb{D}}$.

Taking the quasicircles L passing through the points ${\zeta }_1,\cdots ,{\zeta }_m$ and such that each ${\zeta }_j$ belongs to an analytic subarc of L, one obtains quasiconformal extensions of the interpolating function f, which are biholomorphic on the union of $\overline{\mathbb{D}}$ and some neighborhoods of the initial points $z_1,\cdots ,z_m\in S^1$. Now Lemma 1 provides quasiconformal extensions of f lying in prescribed homotopy classes of homeomorphisms $X_{\mathbf{z}}\to X_{\mathbf{w}}$.

2.3. The Bers fiber space

Pick a space $\mathbf{T}(0,n)=\mathbf{T}\left(X_{{\mathbf{a}}^0}\right)$ with $n\ge 5$ and let

$$X_{{\mathbf{a}}^0}^{\prime }=X_{{\mathbf{a}}^0}\backslash \left\{a_{n-3}^0\right\}=U/{\mathsf{\Gamma }}_0^{\prime }.$$

Due to the Bers isomorphism theorem, the space $T\left(X_{{\mathbf{a}}^0}^{\prime }\right)$ is biholomorphically isomorphic to the Bers fiber space

$$\mathcal{F}(0,n):=\mathcal{F}(T\left(X_{{\mathbf{a}}^0}\right)=\{({\varphi }_{\mathbf{T}}\left(\mu \right),z)\in \mathbf{T}\left(X_{{\mathbf{a}}^0}\right)\times \mathbb{C}: \mu \in Belt{(U,{\mathsf{\Gamma }}_0^{\prime })}_1, z\in w^{\mu }\left(\mathbb{D}\right)\}$$

over$\mathbf{T}\left(X_{{\mathbf{a}}^0}\right)$with holomorphic projection$\pi (\phi ,z)=\phi (\phi \in T\left(X_{{\mathbf{a}}^0}\right)$ (see [15]).

This fiber space is a bounded hyperbolic domain in $\mathbf{B}\left({\mathsf{\Gamma }}_0\right)\times \mathbb{C}$ and represents the collection of domains $D_{\mu }=w^{\mu }\left(U\right)$ (the universal covers of the surfaces $X_{{\mathbf{a}}^0}$) as a holomorphic family over the space $\mathbf{T}(0,n-1)=T\left(X_{{\mathbf{a}}^0}\right)$.

The indicated isomorphism between $\mathbf{T}(0,n+1)$ and $\mathcal{F}(0,n)$ is induced by the inclusion map $j: {\mathbb{D}}_{\ast }\hookrightarrow \mathbb{D}$ forgetting the puncture at $a_n^0$, via

$$\mu \mapsto (S_{w^{{\mu }_1}},w^{{\mu }_1}\left({\widehat{a}}_{n-3}^0\right)) \mathrm{with} {\mu }_1=j_{\ast }\mu :=(\mu \circ \hat{j}{}_0)\overline{\widehat{j^{\prime }}}/\widehat{j^{\prime }},$$

(2)

where $\hat{j}$ is the lift of j to U and ${\widehat{a}}_{n-3}^0$ is one of the points from the fiber over $a_n^0$ under the quotient map $U\to U/{\mathsf{\Gamma }}_0$.

Note also that the holomorphic universal covering maps $h: U^{\ast }\to U^{\ast }/{\mathsf{\Gamma }}_0$ and $h^{\prime }: U^{\ast }\to U^{\ast }/{\mathsf{\Gamma }}_0^{\prime }$ (and similarly, the corresponding covering maps in U) are related by

$$j\circ h^{\prime }=h\circ \hat{j},$$

where $\hat{j}$ is the lift of j. This induces a surjective homomorphism of the covering groups $\theta :{\mathsf{\Gamma }}_0\to {\mathsf{\Gamma }}_0^{\prime }$ by

$$\hat{j}\circ \gamma =\theta \left(\gamma \right)\circ \gamma , \gamma \in {\mathsf{\Gamma }}_0^{\prime }$$

and the norm preserving isomorphism ${\hat{j}}_{\ast }: \mathbf{B}\left({\mathsf{\Gamma }}_0\right)\to \mathbf{B}\left({\mathsf{\Gamma }}_0^{\prime }\right)$ by

$${\hat{j}}_{\ast }\phi =(\phi \circ \hat{j}){\left(\widehat{j^{\prime }}\right)}^2,$$

(3)

which projects to the surfaces $X_{{\mathbf{a}}^0}$ and $X_{{\mathbf{a}}^0}^{\prime }$ as the inclusion of the space $Q\left(X_{{\mathbf{a}}^0}\right)$ of holomorphic quadratic differentials corresponding to $\mathbf{B}\left({\mathsf{\Gamma }}_0\right)$ in the space $Q\left(X_{{\mathbf{a}}^0}^{\prime }\right)$ (cf. [16]).

The Bers theorem is valid for the Teichmüller space $\mathbf{T}(X_0\backslash \left\{x_0\right\})$ of any punctured hyperbolic Riemann surface $X_0\backslash \left\{x_0\right\}$ and implies that $\mathbf{T}(X_0\backslash \left\{x_0\right\})$ is biholomorphically isomorphic to the Bers fiber space $\mathcal{F}\left(\mathbf{T}\left(X_0\right)\right)$ over $\mathbf{T}\left(X_0\right)$.

2.4. Holomorphic Curves and Holomorphic Sections

The group ${\mathsf{\Gamma }}_0$ uniformizing the surface $X_{{\mathbf{a}}^0}$ acts discontinuously on the fiber space $\mathcal{F}\left({\mathsf{\Gamma }}_0\right)$ as a group of biholomorphic maps by

$$\gamma ({\varphi }_{\mathbf{T}}\left(\mu \right),z)=({\varphi }_{\mathbf{T}}\left(\mu \right),{\gamma }^{\mu }z),$$

(4)

where $\mu \in Belt(U,{\mathsf{\Gamma }}_0), z\in w^{\mu }\left(U\right)), \gamma \in {\mathsf{\Gamma }}_0$, and

$${\gamma }^{\mu }\circ w^{\mu }=w^{\mu }\circ \gamma$$

(see [15]). The quotient space

$$\mathcal{V}(0,n):=\mathcal{V}\left({\mathsf{\Gamma }}_0\right)=\mathbf{T}(0,n+1)/{\mathsf{\Gamma }}_0$$

is called the n-punctured Teichmüller curve and depends only on the analytic type of the ${\mathsf{\Gamma }}_0$ group. The projection $\pi : \mathcal{F}(0,n)\to \mathbf{T}(0,n)$ induces a holomorphic projection

$${\pi }_1: \mathcal{V}(0,n)\to \mathbf{T}(0,n).$$

(5)

This curve is also a complex manifold with fibers ${\pi }^{-1}\left(x\right)=X_{\mathbf{a}}$.

Due to the deep Hubbard–Earle–Kra theorem [16,17], the projections $\mathcal{V}(0,n)\to \mathbf{T}(0,n)$ and (4) have no holomorphic sections for any $n\ge 7$ (more generally, for all spaces $\mathbf{T}\left(\mathsf{\Gamma }\right)$ corresponding to groups $\mathsf{\Gamma }$ without elliptic elements, excluding the spaces $\mathbf{T}(1,2)\simeq \mathbf{T}(0,5)$ and $\mathbf{T}(2,0)\simeq \mathbf{T}(0,6)$). Such sections exist for $\mathsf{\Gamma }$ groups containing elliptic elements.

In the exceptional cases of $\mathbf{T}(1,2)$ and $\mathbf{T}(2,0)$, there is a group ${\mathsf{\Gamma }}^{\prime }$ that contains $\mathsf{\Gamma }$ as a subgroup of index two. Then, $\mathbf{T}\left({\mathsf{\Gamma }}^{\prime }\right)=\mathbf{T}\left(\mathsf{\Gamma }\right), \mathcal{F}\left({\mathsf{\Gamma }}^{\prime }\right)=\mathcal{F}\left(\mathsf{\Gamma }\right)$, and the elliptic elements $\gamma \in {\mathsf{\Gamma }}^{\prime }$ produce the indicated holomorphic sections s as the maps

$${\varphi }_{\mathbf{T}}\left(\mu \right)\mapsto ({\varphi }_{\mathbf{T}}\left(\mu \right),w^{\mu }\left(z_0\right)),$$

(6)

where $z_0$ is a fixed point of $\gamma$ in the half-plane U. These sections are called the Weierstrass sections (in view of their connection with the Weierstrass points of the hyperelliptic surface $U/\mathsf{\Gamma }$). We describe these sections following [16].

We also consider the punctured fiber space ${\mathcal{F}}_0\left(\mathsf{\Gamma }\right)$ to be the largest open dense subset of $\mathcal{F}\left(\mathsf{\Gamma }\right)$ on which $\mathsf{\Gamma }$ acts freely and let

$${\mathcal{V}}^{\prime }\left(\mathsf{\Gamma }\right)={\mathcal{F}}_0\left(\mathsf{\Gamma }\right)/\mathsf{\Gamma }.$$

For $\mathsf{\Gamma }$ with no elliptic elements, the universal covering space for ${\mathcal{V}}^{\prime }(g,n)={\mathcal{V}}^{\prime }\left(\mathsf{\Gamma }\right)$ is $\mathbf{T}(g,n+1)$.

If $\mathsf{\Gamma }$ contains elliptic elements $\gamma$, then any holomorphic section $\mathbf{T}\left(\mathsf{\Gamma }\right)\to \mathcal{F}\left(\mathsf{\Gamma }\right)$ is determined by map (6) so that $w^{\mu }\left(z_0\right)$ is exactly one fixed point of corresponding map (4) in the fiber $w^{\mu }\left(U\right)$. These holomorphic sections take their values in the set $\mathcal{V}\left(\mathsf{\Gamma }\right)\backslash {\mathcal{V}}^{\prime }\left(\mathsf{\Gamma }\right)$ and do not provide, in general, sections of projection (5).

In the case of spaces $\mathbf{T}(1,2)$ and $\mathbf{T}(2,0)$, either of the corresponding curves $\mathcal{V}(1,2)$ or $\mathcal{V}(2,0)$ has a unique biholomorphic self-map $\gamma$ of order two, which maps each fiber into itself. The fixed-point locus of $\gamma$ is a finite set of connected closed complex submanifolds of ${\mathcal{V}}^{\prime }(g,n)$, and the restriction of map (5) to one of these submanifolds is a holomorphic map onto $\mathbf{T}(0,n)$; its inverse is a Weierstrass section. The restriction of $\gamma$ to each fiber is a conformal involution of the corresponding hyperelliptic Riemann surface interchanging its sheets, and the fixed points of $\gamma$ are the Weierstrass points on this surface.

In dimension one, there are three biholomorphically isomorphic Teichmüller spaces $\mathbf{T}(1,0), \mathbf{T}(1,1)$ and $\mathbf{T}(0,4)$ (see, e.g., [15,18]). We shall use the last two spaces. Their fiber space $\mathcal{F}(0,4)$ is isomorphic to $\mathbf{T}(0,5)$.

As a special case of the Hubbard–Earle–Kra theorem [16,17], we have:

Lemma 3.

(a) The curve $\mathcal{V}(0,4)$ has, for any of its points x, a unique holomorphic section s with $s\left({\pi }_1\left(x\right)\right)=x$.

(b) If $dim\mathcal{V}{(g,n)}^{\prime }>1$, only curves $\mathcal{V}{(1,2)}^{\prime }$ and $\mathcal{V}{(2,0)}^{\prime }$ have holomorphic sections, which are their Weierstrass sections.

In particular, curve $\mathcal{V}(2,0)$ has six disjoint holomorphic sections corresponding to the Weierstrass points of hyperelliptic surfaces of genus two.

3. Holomorphic Maps of $\mathbf{T}(\mathbf{0},\mathbf{n})$ into Universal Teichmüller Space and Holomorphic Contractibility

3.1. Equivalence Relations

The universal Teichmüller space $\mathbf{T}=Teich\left(U\right)$ is the space of quasisymmetric homeomorphisms of the unit circle factorized by Möbius maps; all Teichmüller spaces have their isometric copies in $\mathbf{T}$.

The canonical complex Banach structure on $\mathbf{T}$ is defined by the factorization of the ball of the Beltrami coefficients

$$Belt{\left(U\right)}_1=\{\mu \in L_{\infty }\left(\mathbb{C}\right): \mu |U^{\ast }={0, \parallel \mu \parallel }_{\infty }<1\}$$

(i.e., supported in the upper-half plane), letting ${\mu }_1,{\mu }_2\in Belt{\left(U\right)}_1$ be equivalent if the corresponding quasiconformal maps $w^{{\mu }_1},w^{{\mu }_2}$ coincide on $\overline{\mathbb{R}}=\mathbb{R}\cup \{\infty \}=\partial U^{\ast }$ (hence, on $\overline{U^{\ast }}$). Such $\mu$ and the corresponding maps $w^{\mu }$ are called $\mathbf{T}$-equivalent. The equivalence classes ${\left[w^{\mu }\right]}_{\mathbf{T}}$ are in one-to-one correspondence with the Schwarzian derivatives $S_w$ in $U^{\ast }$, which fill a bounded domain in the space $\mathbf{B}=\mathbf{B}\left(U^{\ast }\right)$ (see Section 2.1).

The map ${\varphi }_{\mathbf{T}}: \mu \to S_{w^{\mu }}$ is holomorphic and descends to a biholomorphic map of the space $\mathbf{T}$ onto this domain, which we will identify with $\mathbf{T}$. As was mentioned above, it contains the Teichmüller spaces of all hyperbolic Riemann surfaces and of Fuchsian groups as complex submanifolds.

On this ball, we also define another equivalence relationship, letting $\mu , \nu \in Belt{\left(U\right)}_1$ be equivalent if $w^{\mu }(a_j^0)=w^{\nu }(a_j^0)$ for all j and the homeomorphisms $w^{\mu }, w^{\nu }$ are homotopic on the punctured sphere $X_{{\mathbf{a}}^0}$. Let us call such $\mu$ and $\nu$strongly n-equivalent.

Lemma 4.

If the coefficients $\mu ,\nu \in Belt{\left(U\right)}_1$ are $\mathbf{T}$-equivalent, then they are also strongly n-equivalent.

The proof of this lemma is given in [19].

In view of Lemmas 1 and 4, the above factorizations of the ball $Belt{\left(U\right)}_1$ generate (by descending to the equivalence classes) a holomorphic map $\chi$ of the underlying space $\mathbf{T}$ into $\mathbf{T}(0,n)=\mathbf{T}\left(X_{{\mathbf{a}}^{\mathbf{0}}}\right)$.

This map is a split immersion (has local holomorphic sections), which is a consequence, for example, of the following existence theorem from [13], which we present here as

Lemma 5.

Let D be a finitely connected domain on the Riemann sphere $\widehat{\mathbb{C}}$. Assume that there are a set E of positive, two-dimensional Lebesgue measures and a finite number of points $z_1,\dots ,z_m$ distinguished in D. Let ${\alpha }_1,\dots ,{\alpha }_m$ be non-negative integers assigned to $z_1,\dots ,z_m$, respectively, so that ${\alpha }_j=0$ if $z_j\in E$.

Then, for a sufficiently small ${\epsilon }_0>0$ and $\epsilon \in (0,{\epsilon }_0)$, and for any given collection of numbers $w_{sj},s=0,1,\dots ,{\alpha }_j, j=1,2,\dots ,m$, which satisfy the conditions $w_{0j}\in D$,

$$|w_{0j}-z_j|\le \epsilon , |w_{1j}-1|\le \epsilon , |w_{sj}|\le \epsilon (s=0,1,\cdots a_j, j=1,\dots ,m),$$

there exists a quasiconformal automorphism h of domain D, which is conformal on $D\backslash E$ and satisfies

$$h^{\left(s\right)}\left(z_j\right)=w_{sj} \mathrm{for}\mathrm{all} s=0,1,\dots ,{\alpha }_j, j=1,\dots ,m.$$

Moreover, the Beltrami coefficient ${\mu }_h$ of h on E satisfies $\parallel {\mu }_h{\parallel }_{\infty }\le M\epsilon$. The constants ${\epsilon }_0$ and M depends only upon the sets $D,E$ and the vectors $(z_1,\dots ,z_m)$ and $({\alpha }_1,\dots ,{\alpha }_m)$.

3.2. Surjectivity

In fact, we have more, given by the following theorem.

Theorem 3.

Map χ is surjective and generates an open holomorphic map s of the space $\mathbf{T}(0,n)=\mathbf{T}\left(X_{{\mathbf{a}}^{\mathbf{0}}}\right)$ into the universal Teichmüller space $\mathbf{T}$, embedding $\mathbf{T}(0,n)$ into $\mathbf{T}$ as a $(n-3)$-dimensional submanifold.

In particular, this theorem corrects the assertion of Lemma 3 in [9] (mentioned in the preamble).

Proof of Theorem 3.

The surjectivity of $\chi$ is a consequence of Lemma 2. To construct s, we take a dense subset

$$e=\{x_1, x_2, \cdots \}\subset X_{{\mathbf{a}}^0}\cap \mathbb{R}$$

accumulating to all points of $\mathbb{R}$ and considering the punctured spheres $X_{{\mathbf{a}}^0}^m=X_{{\mathbf{a}}^0}\backslash \{x_1,\cdots ,x_m\}$ with $m>1$. The equivalence relations on $Belt{\left(\mathbb{C}\right)}_1$ for $X_{{\mathbf{a}}^0}^m$ and $X_{{\mathbf{a}}^0}$ generate the corresponding holomorphic map ${\chi }_m:\mathbf{T}\left(X_{{\mathbf{a}}^0}^m\right)\to \mathbf{T}\left(X_{{\mathbf{a}}^0}\right)$. □

Uniformizing the surfaces $X_{{\mathbf{a}}^0}$ and $X_{{\mathbf{a}}^0}^m$ by the corresponding torsion-free Fuchsian groups ${\mathsf{\Gamma }}_0$ and ${\mathsf{\Gamma }}_0^m$ of the first kind acting discontinuously on $U\cup U^{\ast }$ and applying the construction from Section 2.3 to $U^{\ast }/{\mathsf{\Gamma }}_0$ and $U^{\ast }/{\mathsf{\Gamma }}_0^m$ (forgetting the additional punctures), one obtains, similar to (3), the norm-preserving isomorphism ${\hat{j}}_{m,\ast }: \mathbf{B}\left({\mathsf{\Gamma }}_0\right)\to \mathbf{B}\left({\mathsf{\Gamma }}_0^m\right)$ by

$${\hat{j}}_{m,\ast }\phi =(\phi \circ \hat{j}){\left(\widehat{j^{\prime }}\right)}^2,$$

which projects to the surfaces $X_{{\mathbf{a}}^0}$ and $X_{{\mathbf{a}}^0}^m$ as the inclusion of the space $Q\left(X_{{\mathbf{a}}^0}\right)$ of quadratic differentials corresponding to $\mathbf{B}\left({\mathsf{\Gamma }}_0\right)$ into the space $Q\left(X_{{\mathbf{a}}^0}^m\right)$, and (since projection ${\eta }_m$ has local holomorphic sections) geometrically, this relation yields a holomorphic embedding of the space $\mathbf{T}\left({\mathsf{\Gamma }}_0\right)$ into $\mathbf{T}\left({\mathsf{\Gamma }}_0^m\right)$ as a $(n-3)$-dimensional submanifold. We denote this embedding by $s_m$.

To investigate the limit function for $m\to \infty$, we compose the maps $s_m$ with the canonical biholomorphic isomorphisms

$${\eta }_m: \mathbf{T}\left(X_{{\mathbf{a}}^0}^m\right)\to \mathbf{T}\left({\mathsf{\Gamma }}_0^m\right)=\mathbf{T}\cap \mathbf{B}\left({\mathsf{\Gamma }}_0^m\right) (m=1,2,\cdots ).$$

Then the elements of $\mathbf{T}\left({\mathsf{\Gamma }}_0^m\right)$ are given by

$${\widehat{s}}_m\left(X_{\mathbf{a}}\right)={\eta }_m\circ s_m\left(X_{\mathbf{a}}\right),$$

and this is a collection of the Schwarzians $S_{f^m}\left(z\right)$ corresponding to the points $X_{\mathbf{a}}$ of $\mathbf{T}\left(X_{{\mathbf{a}}^0}\right)$. Therefore, for any surface $X_{\mathbf{a}}$, we have

$${\widehat{s}}_m\left(X_{\mathbf{a}}\right)=S_{f^m}\left(z\right).$$

(7)

Each ${\mathsf{\Gamma }}_0^m$ is the covering group of the universal cover $h_m: U^{\ast }\to X_{{\mathbf{a}}_0^m}$, which can be normalized (conjugating appropriately ${\mathsf{\Gamma }}_0^m$) by $h_m(-i)=-i, h_m^{\prime }(-i)>0$. Take the fundamental polygon $P_m$ obtained as the union of the circular m-gon in $U^{\ast }$ centered at $z=-i$ with zero angles at the vertices and its reflection with respect to one of the boundary arcs. These polygons increasingly exhaust the half-plane $U^{\ast }$; hence, by the Carathéodory kernel theorem, the maps $h_m$ converge to the identity map locally uniformly in $U^{\ast }$.

Since the set of punctures e is dense on $\mathbb{R}$, it completely determines the equivalence classes $\left[w^{\mu }\right]$ and $S_{w^{\mu }}$ as the points of the universal space $\mathbf{T}$. The limit function $\widehat{s}={lim}_{m\to \infty }{\widehat{s}}_m$ maps $\mathbf{T}\left(X_{{\mathbf{a}}^0}\right)=\mathbf{T}(0,n)$ into the space $\mathbf{T}$ and also is canonically defined by the marked spheres $X_{\mathbf{a}}$.

Similar to (7), the function $\widehat{s}$ is represented as the Schwarzian of some univalent function $f^n$ on $U^{\ast }$ with a quasiconformal extension to $\widehat{\mathbb{C}}$ determined by $X_{\mathbf{a}}$. Then, by the well-known property of elements in the functional spaces with sup norms, $\widehat{s}$ is also holomorphic in the $\mathbf{B}$-norm on $\mathbf{T}$.

Lemma 5 yields that $\widehat{s}$ is a locally open map from $\mathbf{T}\left(X_{{\mathbf{a}}^0}\right)$ to $\mathbf{T}$. Therefore, the image $\widehat{s}\left(\mathbf{T}\left(X_{{\mathbf{a}}^0}\right)\right)$ is an $(n-3)$-dimensional complex submanifold in $\mathbf{T}$, biholomorphically equivalent to $\mathbf{T}\left({\mathsf{\Gamma }}_0\right)$. The proof of Theorem 2 is completed.

The holomorphy property indicated above is based on the following lemma of Earle [20].

Lemma 6.

Let $E,T$ be open subsets of complex Banach spaces $X,Y$ and $B\left(E\right)$ be the Banach space of holomorphic functions on E with sup norm. If $\phi (x,t)$ is a bounded map $E\times T\to B\left(E\right)$ such that $t\mapsto \phi (x,t)$ is holomorphic for each $x\in E$, then map φ is holomorphic.

The holomorphy of $\phi (x,t)$ in t for fixed x implies the existence of complex directional derivatives

$${\phi }_t^{\prime }(x,t)=\underset{\zeta \to 0}{lim}\frac{\phi (x,t+\zeta v)-\phi (x,t)}{\zeta }=\frac{1}{2\pi i}\underset{\left|\xi \right|=1}{\int }\frac{\phi (x,t+\xi v)}{{\xi }^2}d\xi ,$$

while the boundedness of $\phi$ in the sup norm provides the uniform estimate

$$\parallel \phi (x,t+c\zeta v)-\phi (x,t)-{\phi }_t^{\prime }{(x,t)cv\parallel }_{B\left(E\right)}\le M{\left|c\right|}^2,$$

for sufficiently small $\left|c\right|$ and ${\parallel v\parallel }_Y$.

3.3. Explicit Construction of Holomorphic Homotopy

Now we may construct the desired holomorphic homotopy of $\mathbf{T}(0,n)=\mathbf{T}\left(X_{{\mathbf{a}}^0}\right)$ into its base point and establish the general result:

Pick a collection ${\mathbf{a}}^0=(0,1,a_1^0,\cdots ,a_{n-3}^0,\infty )$ and the marked surface $X_{{\mathbf{a}}^0}$ as indicated above, and consider its Teichmüller spaces $\mathbf{T}\left(X_{{\mathbf{a}}^0}\right)$ and $\mathbf{T}\left({\mathsf{\Gamma }}_0\right)$.

Using the canonical embedding of $\mathbf{T}(0,n)$ in $\mathbf{T}$ via $\mathbf{T}\left({\mathsf{\Gamma }}_0\right)$, we define on the space $\mathbf{T}\left({\mathsf{\Gamma }}_0\right)$ a holomorphic homotopy applying the maps

$$W^{\mu }={\sigma }^{-1}\circ w^{\mu }\circ \sigma , \mu \in Belt{\left(U\right)}_1; \sigma \left(\zeta \right)=i(1+\zeta )/(1-\zeta ), \zeta \in \mathbb{D},$$

and $w_t^{\mu }\left(z\right):=w^{\mu }(z,t)=\sigma \circ W_t^{\mu }\circ {\sigma }^{-1}\left(z\right)$; then,

$$S_{w^{\mu }}(·,t)=t^2{S}_{w^{\mu }}(·)=t^{-2}(S_{W^{\mu }}\circ {\sigma }^{-1}){\left({\sigma }^{\prime }\right)}^{-2}.$$

(8)

This point-wise equality determines a bounded holomorphic map by Lemma 6 $\eta (\phi ,t)=S_{w_t^{\mu }}: \mathbf{T}\times \mathbb{D}\to \mathbf{T}$ with $\eta (\mathbf{0},t)=\mathbf{0}, \eta (\phi ,0)=\mathbf{0}, \eta (\phi ,1)=\phi$; its boundedness follows from the estimate

$$S_{W_t^{\mu }}{\left(\zeta \right)<6\left|t\right|}^2/{\left(\right|\zeta |}^2{-1)}^2, \zeta \in U^{\ast }.$$

We apply homotopy (8) to $\phi =S_{w^{\mu }}\in \mathbf{T}\left({\mathsf{\Gamma }}_0\right)$. Since it is not compatible with the group ${\mathsf{\Gamma }}_0$, there are images ${\phi }_t:=\eta (\phi ,t)=S_{w_t^{\mu }}$ that are located in $\mathbf{T}$ outside of $\mathbf{T}\left({\mathsf{\Gamma }}_0\right)$. Map $\chi \circ \eta (\phi ,t)$ carries these images to the points of the space $\mathbf{T}(0,n)=\mathbf{T}\left(X_{{\mathbf{a}}^0}\right)$. We compose this map with the holomorphic map s given by Lemma 3 and with a biholomorphism $\xi : s\left(\mathbf{T}\left(X_{{\mathbf{a}}^0}\right)\right)\to \mathbf{T}\left({\mathsf{\Gamma }}_0\right)$, getting the function

$$\mathsf{\Theta }(\phi ,t)=\xi \circ s\circ \chi \circ \eta (\phi ,t)$$

(9)

which maps holomorphically $\mathbf{T}\left({\mathsf{\Gamma }}_0\right)\times \mathbb{D}$ into $\mathbf{T}\left({\mathsf{\Gamma }}_0\right)$ with $\mathsf{\Theta }(\phi ,0)=\mathbf{0}$.

A crucial step in constructing is to establish that function (9) extends holomorphically to the limit points $(\phi ,1)$ representing the initial Schwarzians $S_{w^{\mu }}$. This property does not extend (in the $\mathbf{B}$-norm) to all points of $\mathbf{T}$.

To prove the limit holomorphy, fix a point ${\phi }_0\in \mathbf{T}\left({\mathsf{\Gamma }}_0\right)$ and consider, in its small neighborhood $V_0$, the local coordinates ${\mathbf{a}}_{\ast }=(a_1,\cdots ,a_{n-3})$ introduced above.

Both maps $\eta$ and $\mathsf{\Theta }$ are holomorphic in the points $({\phi }_0,t)$ of this neighborhood for all t with $\left|t\right|<1$. On the other hand, coordinates ${\mathbf{a}}_{\ast }$ are determined by the corresponding quasiconformal maps $w_t^{\mu }$ and, together with these maps, are uniformly continuous in t in the closed disk $\left\{\right|t|\le 1\}$. This follows from the uniform boundedness of dilatations given by the estimate

$$k\left(w_t^{\mu }\right)=\parallel {\mu }_t{\parallel }_{\infty }\le {\left|t\right|\parallel \mu \parallel }_{\infty }<1$$

(10)

(which holds for generic holomorphic motions) and from non-increasing the Kobayashi metric $d_X(·,·)$ under holomorphic maps. Since this metric on Teichmüller spaces equals their intrinsic Teichmüller metric ${\tau }_{\mathbf{T}\left({\mathsf{\Gamma }}_0\right)}$, one gets from (10),

$${\tau }_{\mathbf{T}\left({\mathsf{\Gamma }}_0\right)}(\mathbf{0},\mathsf{\Theta }(\phi ,t))=d_{\mathbf{T}\left({\mathsf{\Gamma }}_0\right)}(\mathbf{0},\mathsf{\Theta }(\phi ,t))\le {tanh}^{-1}{\left(\right|t|\parallel \mu \parallel }_{\infty }).$$

Hence, function $\mathsf{\Theta }(\phi ,t)$ determines a normal family on $V_0\cap \mathbf{T}\left({\mathsf{\Gamma }}_0\right)$.

Applying the classical Weierstrass theorem about the locally uniform convergent sequences of holomorphic functions in finite-dimensional domains, one derives that the limit function

$$\mathsf{\Theta }(\phi ,1)=\underset{t\to 1}{lim}\mathsf{\Theta }(\phi ,t)$$

is also holomorphic on $V_0\cap \mathbf{T}\left({\mathsf{\Gamma }}_0\right)$ and then on $\mathbf{T}\left({\mathsf{\Gamma }}_0\right)$, which completes the construction of the desired holomorphic homotopy on $\mathbf{T}(0,n)$.

4. Second Proof of Holomorphic Contractibility for Low-Dimensional Teichmüller Spaces

The previous section implies the proof of holomorphic contractibility for all spaces $\mathbf{T}(0,n)$ with $n\ge 5$, which also yields, in particular, Theorem 2. In this section, we provide another proof of this important theorem; it relies on the intrinsic features of the two and three-dimensional Teichmüller spaces mentioned in Section 2.4.

(a)

Case $n=5$ (dimension two). It is enough to establish holomorphic contractibility of the space $\mathbf{T}(0,5)\simeq \mathbf{F}(0,4)$ for the spheres

$$X_{\mathbf{a}}=\widehat{\mathbb{C}}\backslash \{0,1,a_1,a_2,\infty \}.$$

The fibers of $\mathbf{T}(0,5)$ are the spheres with quadruples of punctures $\{0,1,a_1,\infty \}$.

We start with the construction of the needed holomorphic homotopy of the space $\mathbf{T}(0,5)$ to its base point $X_{{\mathbf{a}}^0}$ and first apply the assertion $\left(a\right)$ of Lemma 3 of holomorphic sections over $\mathbf{T}(0,4)$. It implies that for any point

$$x=(S_{w^{{\mu }_1}},w^{{\mu }_1}({\widehat{a}}_{n-3}^0))\in \mathbf{T}(0,5)$$

a unique holomorphic section $s:\mathbf{T}(0,5)\to \mathbf{T}(0,4)$ with $s\left({\pi }_1\left(x\right)\right)=x$. This section has a common point with each fiber ${\pi }^{-1}\left(x\right)$ over $\mathbf{T}(0,4)$.

Since $\mathbf{T}(0,4)$ is (up to a biholomorphic equivalence) a simply connected bounded Jordan domain $D\subset \mathbb{C}$ containing the origin, there is a holomorphic isotopy $h(\zeta ,t):D\times [0,1]\to D$ with $h(\zeta ,0)=\zeta , h(z,1)=0$. Using this isotopy, we define a homotopy $h_1(\phi ,t)$ on $\mathbf{T}(0,5)$, which carries each point $x=(S_{w^{\mu }},w^{\mu }({\widehat{a}}_2^0))\in \mathbf{T}(0,5)$ to its image on the section s passing from x; that is,

$$h_1(\phi ,w^{\mu }({\widehat{a}}_2^0))=(h(\phi ),{\tilde{a}}_2), \phi =S_{w^{\mu }}, \mu \in Belt{(\mathbb{C})}_1,$$

(11)

where ${\tilde{a}}_2$ is the common point of the fiber $h\left(\phi \right)$ and the selected section s. The holomorphy of this homotopy in variables $x=(S_{w^{\mu }},w^{\mu }({\widehat{a}}_2^0))$ for any fixed $t\in [0,1]$ follows from Lemmas 1, 2, and the Bers isomorphism theorem. The limit map

$$h_1^{\ast }\left(x\right)=\underset{t\to 1}{lim}{h}_1(x,t),$$

carries each fiber $w^{\mu }\left(U\right)$ to the initial half-plane U.

There is a canonical holomorphic isotopy

$$h_2(\zeta ,t): U\times [0,1]\to U$$

(12)

of U into its point corresponding to the origin of $\mathbf{T}(0,5)$. Now make $\mathbf{h}(x,t)$ equal to $h_1(x,2t)$ for $t\le 1/2$ and equal to $h_2(x,2t-1)$ for $x=\zeta \in U$ and $1/2\le t\le 1$.

This function is holomorphic at $x\in \mathbf{T}(0,5)$ for any fixed $t\in [0,1]$ and hence provides the desired holomorphic homotopy of the space $\mathbf{T}(0,5)$ into its base point.

(b)

Case $n=6$ (dimension three). This case is more complicated.

We prescribe to each ordered sextuple $\mathbf{a}=\{0,1,a_1,a_2,a_3,\infty \}$ of distinct points the corresponding punctured sphere

$$X_{\mathbf{a}}=\widehat{\mathbb{C}}\backslash \{0,1,a_1,a_2,a_3,\infty \}$$

(13)

and the two-sheeted closed hyperelliptic surface ${\widehat{X}}_{\mathbf{a}}$ of genus two with the branch points $0,1,a_1,a_2,a_3,\infty$. The corresponding Teichmüller spaces $\mathbf{T}(0,6)$ and $\mathbf{T}(2,0)$ coincide up to a natural biholomorphic isomorphism. Note also that the collection $\mathbf{a}=\{0,1,a_1,a_2,a_3,\infty \}$ provides the local complex coordinates on spaces $\mathbf{T}(0,6)$ and $\mathbf{T}(2,0)$.

In view of the symmetry of hyperelliptic surfaces, it suffices to deal with the Beltrami differentials $\mu d\overline{z}/dz$ on ${\widehat{X}}_{\mathbf{a}}$, which are compatible with a conformal involution $J_{\mathbf{a}}$ of ${\widehat{X}}_{\mathbf{a}}$, hence, satisfying $\mu \left(J_{\mathbf{a}}z\right)=\mu \left(z\right)J_{\mathbf{a}}^{\prime }/\overline{J_{\mathbf{a}}^{\prime }}$. In other words, these $\mu$ are obtained by lifting to ${\widehat{X}}_{\mathbf{a}}$ of the Beltrami coefficients on $X_{\mathbf{a}}$. This extends Lemma 2 and its consequences on holomorphy in the neighborhoods of the boundary interpolation points to the corresponding two-sheeted disks on hyperelliptic surfaces.

We fix a base point of $\mathbf{T}(2,0)$, determining a Fuchsian group $\mathsf{\Gamma }$ for which $\mathbf{T}\left(\mathsf{\Gamma }\right)=\mathbf{T}(2,0)$. The corresponding Teichmüller curve $\mathcal{V}(2,0)$ is a 4-dimensional, complex analytic manifold with projection ${\pi }_1: \mathcal{V}(2,0)\to \mathbf{T}(2,0)$ onto $\mathbf{T}(2,0)$ such that for every $\phi \in \mathbf{T}(2,0)$ the fiber ${\pi }_1^{-1}\left(\phi \right)$ is a hyperelliptic surface, determined by $\phi$ (see Section 2.4).

Due to assertion $\left(b\right)$ of Lemma 3, this curve has, for any point

$${\widehat{X}}_{\mathbf{a}}=(S_{w^{{\mu }_1}},w^{{\mu }_1}({\widehat{a}}_{n-3}^0))\in \mathbf{T}(2,0))$$

six distinct holomorphic sections ${\widehat{s}}_1,\cdots ,{\widehat{s}}_6$, corresponding to the Weierstrass points of the surface $X_{\mathbf{a}}$, with ${\widehat{s}}_j\left({\pi }_1\left(X_{\mathbf{a}}\right)\right)=X_{\mathbf{a}}$, and either from these sections has one common point with every fiber over $\mathbf{T}(2,0)$. We lift these sections to the Bers fiber space $\mathcal{F}\left(\mathsf{\Gamma }\right)$ covering $\mathcal{V}(2,0)$.

As mentioned in Section 2.4, these sections are generated by the space $\mathcal{F}\left({\mathsf{\Gamma }}^{\prime }\right)=\mathcal{F}\left(\mathsf{\Gamma }\right)$ corresponding to the extension ${\mathsf{\Gamma }}^{\prime }$ of $\mathsf{\Gamma }$, for which $\mathsf{\Gamma }$ is a subgroup of index two. Every section ${\widehat{s}}_j$ acts on $\mathbf{T}\left({\mathsf{\Gamma }}^{\prime }\right)$ via (6), where $z_0$ is now the corresponding Weierstrass point of hyperelliptic surface ${\widehat{X}}_{\mathbf{a}}$, and ${\widehat{s}}_j$ is compatible with action (2) of the Bers isomorphism.

Thus each ${\widehat{s}}_j$ descends to a holomorphic map $s_j: \mathbf{T}(0,6)\to \mathcal{V}(0,6)$ of the underlying space $\mathbf{T}(0,6)$ for the punctured spheres (10). We choose one from these maps and denote it by s.

The features of sections ${\widehat{s}}_j$ provide that the descended map s also determines, for each point $z_0\in X_{\mathbf{a}}$, its unique image on every fiber $w^{\mu }\left(X_{\mathbf{a}}\right)$ with $\mu \in Belt{\left(X_{\mathbf{a}}\right)}_1$, and this image is the point $w^{\mu }\left(z_0\right)$.

The next preliminary construction consists of embedding space $\mathbf{T}(0,5)$ into $\mathbf{T}(0,6)$, using the forgetting map (3). Its image $j_{\ast }\mathbf{T}(0,5)$ is a connected submanifold in $\mathbf{T}(0,6)$, and the corresponding fibers of the curve $\mathcal{V}(0,6)$ over the points $j_{\ast }\phi \in j_{\ast }\mathbf{T}(0,5)$ are the surfaces $w^{j_{\ast }\mu }\left(X_{\mathbf{a}}\right)$ with $j_{\ast }\mu \left(z\right)=\mu \left(\hat{j}\left(z\right)\right)\widehat{j^{\prime }}\left(z\right)/\overline{\widehat{j^{\prime }}\left(z\right)}$. The covering domains $w^{j_{\ast }\mu }\left(U\right)$ over these surfaces fill a submanifold $\tilde{\mathbf{T}}(0,7)\subset \mathbf{T}(0,7)$, which is biholomorphically equivalent to the space $\mathbf{T}(0,6)$.

Using the biholomorphic equivalence of space $\mathbf{T}(0,5)$ to its image $j_{\ast }\mathbf{T}(0,5)$ in $\mathbf{T}(0,6)$, we carry over to $j_{\ast }\mathbf{T}(0,5)$ the result of the previous step $\left(a\right)$ on the holomorphic contractibility of $\mathbf{T}(0,5)$, which provides a holomorphic homotopy

$$h(j_{\ast }\phi ,t): j_{\ast }\mathbf{T}(0,5)\times [0,1]\to j_{\ast }\mathbf{T}(0,5) \mathrm{with} h(j_{\ast }\phi ,0)=j_{\ast }\phi , h(j_{\ast }\phi ,1)=\mathbf{0}$$

(14)

(here, $\mathbf{0}$ stands for the origin of $j_{\ast }\mathbf{T}(0,5)$).

Now we may construct the desired holomorphic homotopy of $\mathbf{T}(0,6)$, contracting this space to its origin.

First, regarding $\mathbf{T}(0,6)$ as the Bers fiber space $\mathcal{F}(0,5)$ over $\mathbf{T}(0,5)$ (whose fibers are the covers of surfaces $X_{{\mathbf{a}}^{\prime }}$ with collections of five punctures ${\mathbf{a}}^{\prime }=(0,1,a_1,a_2,\infty )$), we apply homotopy (11) and define, for any pair $x=(j_{\ast }\phi ,z)$ with $\phi \in \mathbf{T}(0,5)$ and $z\in X_{\mathbf{a}}$, the map

$${\tilde{h}}_1((j_{\ast }\phi ,z),t)=(h\left(j_{\ast }\phi \right),t),w_t^{j_{\ast }\mu }\left(z\right)), \phi \in \mathbf{T}(0,5),$$

(15)

noting that the image point $w_t^{j_{\ast }\mu }\left(z\right)$ is uniquely determined on surface $w^{h\left(j_{\ast }\mu \right)}\left(X_{\mathbf{a}}\right)$ by map s, as indicated above.

The pairs $(j_{\ast }\phi ,z)$ are located in the space $\mathcal{F}(0,6)$ and fill a three-dimensional submanifold $\tilde{\mathbf{T}}(0,6)$ biholomorphically equivalent to $\mathbf{T}(0,6)$.

Homotopy (15) is well defined on $\tilde{\mathbf{T}}(0,6)\times [0,1]$ and contracts the set $\tilde{\mathbf{T}}(0,6)$ into fiber $\tilde{U}$ over the base point. It is holomorphic with respect to the space variable $x=(j_{\ast }\phi ,z)$ for any fixed $t\in [0,1]$ and continuous in both variables.

In view of the biholomorphic equivalence of $\tilde{\mathbf{T}}(0,6)$ to $\mathbf{T}(0,6)$, (15) generates a holomorphic homotopy $h_1(x,t)$ of the space $\mathbf{T}(0,6)$ onto the initial fiber (half-plane) U over the origin of $\mathbf{T}(0,5)$.

It remains to combine this homotopy $h_1$ with the additional homotopy (12) of U into its point corresponding to the origin of $\mathbf{T}(0,6)$. This provides the desired homotopy h and completes the proof of Theorem 1.