Polygons and non-starlikeness of Teichmuller spaces

Lemma 1. For any rational function $r_n$ with poles of order two on the boundary circle $\mathbb{S}^1$ of the form $$r_n(z) = \sum_{1}^{n} \frac{c_j}{(z - a_j)^2} + \sum_{1}^{n} \frac{c'_j}{z - a_j}$$ (2) satisfying $\sum_{1}^{n} |c_j| > 0$ , we have the equality $$||r_n||_{\mathbf{B}} = \limsup_{|z| \to 1} (1 - |z|^2)^2 |r_n(z)|.$$ (3) This lemma is a special case of a general deep fact established in [15] for arbitrary quasidisks $D \subset \widehat{\mathbb{C}}$ . It follows from (3) that there is a boundary point $z_0 \in \mathbb{S}^1$ at which the maximal value of $(1-|z|^2)^2|r_n(z)|$ on the closed disk $\overline{\mathbb{D}}$ is attained. Another important fact, which also will be applied in the proof of Theorem 1, is that the Schwarzians of conformal mapping functions of circular polygons are of the form (2). $\mathbf{3^0}$ . We proceed to the proof of Theorem 1 and uniformize the base point $X_0$ of the space $\mathbf{T}(g) = \mathbf{T}(X_0)$ by a Fuchsian group $\Gamma_0$ acting discontinuously on the disks $\mathbb{D}$ , in other wrds, consider this point as the quotient space $X_0 = \mathbb{D}^/\Gamma_0$ . As was mentioned above, there is a surface $X \in \mathbf{T}(X_0)$ corresponding to a point $\varphi \in \mathbf{B}(\Gamma_0)$ with $$\|\varphi\|_{\mathbf{B}(\Gamma_0)} > 2. \tag{4}$$ This surface is represented (uniformized) by a quasifuchsian group $\Gamma_{\mu} = (f^{\mu})^{-1}\Gamma_{0}f^{\mu}$ , where $f^{\mu}(z)$ is a quasiconformal automorphism of $\widehat{\mathbb{C}}$ conformal on $\mathbb{D}^{}$ having the Schwarzian $S_{f^{\mu}}(z) = \varphi(z), \ z \in \mathbb{D}^{}$ (the group $\Gamma_{\mu}$ depends only on $\varphi$ ). We take Ford's fundamental polygon $P(\Gamma_{\mu})$ of this group in domain $D_{\mu}^ = f^{\mu}(\mathbb{D}^)$ . which is the intersection of the exterior domains of all isometric circles of this group: $$P(\Gamma_{\mu}) = \{ z \in D_{\mu}^* : \sup |\gamma'(z)| < 1 \},$$ taking the supremum over all elements of the group $\Gamma_{\mu}$ different from the identity. The sides of this polygon are pairwise $\Gamma_{\mu}$ -equivalent. Its inverse image $(f^{\mu})^{-1}P(\Gamma_{\mu})$ is the fundamental domain of the initial group $\Gamma_0$ in $\mathbb{D}^*$ . Since X is a closed surface, the interior and exterior angles of both polygons $P(\Gamma_{\mu})$ and $(f^{\mu})^{-1}P(\Gamma_{\mu})$ are different from 0, $\pi$ and $2\pi$ . Now, denoting the elements of $\Gamma_{\mu}$ by $\gamma_1 = 1, \gamma_2, \gamma_3, \ldots$ , numerated so that two successive $\gamma_j, \gamma_{j+1}$ determine the adjacent polygons $\gamma_j P(\Gamma_{\mu}), \gamma_{j+1} P(\Gamma_{\mu})$ with a common side consider the increasing connected unions $$\Omega_1 = P(\Gamma_\mu), \quad \Omega_2 = \gamma_1 P(\Gamma_\mu) \bigcup \gamma_2 P(\Gamma_\mu), \quad \Omega_3 = \gamma_1 P(\Gamma_\mu) \bigcup \gamma_2 P(\Gamma_\mu) \bigcup \gamma_3 P(\Gamma_\mu), \dots$$ exhausting the domain $D_{\mu}^*$ . All domains $\Omega_j$ represent the circular polygons with nonzero angles containing inside the infinite point. Consider the corresponding conformal maps $F_j$ of the disk $\mathbb{D}$ onto the polygons $\Omega_j$ and their Schwarzians $S_{F_j}(z)$ . Due to the Ahlfors-Weill theorem [2], all functions $tS_{F_j(z)}$ with $||tS_{F_j(z)}||_{\mathbf{B}} < 2$ are the Schwarzian derivatives of univalent function $W_t(z)$ on $\mathbb{D}$ having quasiconformal extensions to $\mathbb{D}$ with harmonic Beltrami coefficients $$\nu_{F_j}(z) = -\frac{1}{2}(1 - |z|^2)^2 S_{F_j}(1/\overline{z}), \quad |z| < 1.$$ (5) Lemma 1 (actually, the equality (3)) allows one to apply to any $F_j$ rather standard arguments already applied in [14], [16], which provide that the harmonic Beltrami coefficients (4) are extremal in their equivalence classes $[F^{\nu_{S_j}}]$ (i.e., among quasiconformal extensions of $F_j$ onto $\mathbb{D}$ ). Note that the extremality here is doing with respect to the metric $\tilde{\tau}_{\mathbf{T}(G)}$ and means that the extremal rays are geodesic in this metric. Fix a subarc $\gamma \subset \mathbb{S}^1$ and take the conformal map $z = \chi(\zeta)$ of the half-strip $$\Pi_{+} = \{ \zeta = \xi + i\eta : \ \xi > 0, \ 0 < \eta < 1 \}$$ onto $\mathbb{D}^*$ such that $\chi_1^{-1}(z_0) = \infty$ and the pre-images of the endpoints of the arc $\gamma$ are the points 0 and 1. Then the coefficient $\nu_{tS_{F_i}}$ is pulled back to Beltrami coefficient $$t\mu_j(\zeta) := \chi_1^* \nu_{tS_{F_j}}(z) = (\nu_{tS_{F_j}} \circ \chi)(\zeta) \ \overline{\chi'(\zeta)}/\chi'(\zeta)$$ on $\Pi_+$ . We show that either from these coefficients has an essential boundary point at infinity, and a degenerating sequence quadratic differentials at this point; this sequence consists of the functions $$\omega_m(\zeta) = \frac{1}{m} e^{-\zeta/m}, \quad \zeta \in \Pi_+, \quad m = 1, 2, \dots$$ All these functions belong to $A_1^2(\Pi_+)$ and satisfy $\omega_m(\zeta) \to 0$ uniformly on $\Pi_+ \cap \{|\zeta| < M\}$ for any $M < \infty$ ; in addition, $\left| \iint_{\Pi_+} \omega_m(\zeta) d\xi d\eta \right| = 1 - O(1/m)$ . Our goal is to show that for any j, $$\lim_{m \to \infty} \left| \iint_{\Pi_{+}} \mu_{j}(\zeta) \omega_{m}(\zeta) d\xi d\eta \right| = \|\mu_{j}\|_{\infty}. \tag{6}$$ This equality follows from the relations $$\int_{0}^{\infty} \frac{\partial \mu_{j}(\xi + i\eta)}{\partial \xi} e^{-\xi/m} d\xi = \frac{1}{m} \int_{0}^{\infty} \mu_{j}(\xi + i\eta) e^{-\xi/m} d\xi - \mu_{j}(i\eta) = \frac{1}{m} \int_{0}^{\infty} \mu_{j}(\xi + i\eta) e^{-\xi/m} d\xi. \quad (7)$$ The last integral in (7) represents the values of the Laplace transform $$\mathcal{L}\nu = \int_{0}^{\infty} \nu(t)e^{-st}dt$$ of $\nu(\xi + i\eta)$ in the points s = 1/m. Taking into account the Lebesgue theorem on dominated convergence and the properties of the extremal Beltrami coefficient $\mu_j$ (in the metric $\tilde{\tau}_{\mathbf{T}(\Gamma)}$ ), one derives $$\lim_{m \to \infty} \frac{1}{m} \Big| \int_{0}^{\infty} \mu_j(\xi + i\eta) e^{-\xi/m} d\xi \Big| = |\mu_j(\infty) - \nu(i\eta)| = |\mu_j(\infty)|$$ (where $\mu_j(\infty) = \lim_{\xi \to -\infty} \mu_j(\xi + i\eta)$ ). Since $$\lim_{m \to \infty} \left| \iint\limits_{\Pi} \nu_{tS_{F_j}}(\zeta) \omega_m(\zeta) d\xi d\eta \right| = \lim_{m \to \infty} \left| \iint\limits_{D} \mu_j(z) \psi_m(z) dx dy \right| = \|\nu_{tS_{F_j}}\|_{\infty},$$ with $$\psi_m = (\omega_m \circ \chi^{-1})(\chi')^{-2}, \quad m = 1, 2, \dots,$$ (8) we have $$\lim_{m \to \infty} \left| \iint_{D} \mu_{j}(z) \psi_{m}(z) dx dy \right| = \lim_{m \to \infty} \left| \iint_{\Pi_{+}} \nu_{tS_{F_{j}}}(\zeta) \omega_{m}(\zeta) d\xi d\eta \right| = |\mu_{j}(\infty)|, \tag{9}$$ which implies (6). One also obtains that the sequence (8) is degenerated for $\nu_{tS_{F_j}}$ . Consequently, $z_0$ is a substantial point of $\nu_{tS_{F_j}}$ and this Beltrami coefficient is extremal in its equivalence class (hence, this class cannot contain a Teichmüller Beltrami coefficient). The equality (6) implies the extremality of all Beltrami coefficients (4) for the metric $\tilde{\tau}_{\mathbf{T}(G)}$ and also yields that all maps $F_j$ have equal Teichmüller and Grunsky norms. ${\bf 4^0}$ . By the general Carathéodory theorem on convergence of conformal maps, the limit function $$F(z) = \lim_{j \to \infty} F_j(z) \tag{10}$$ maps conformally the disk $\mathbb{D}$ onto the exterior component $D^_{\mu}$ of domain of discontinuity of the group $\Gamma_{\mu}$ . Note that the convergence in (10) is uniform on the disk $\mathbb{D}^*$ (in the spherical metric on $\widehat{\mathbb{C}}$ ). Our goal now is to establish that one of the extremal quoiconformal extensions $\widehat{F}^{\mu}$ of this map inherits the properties of the functions $F_j$ . This is the crucial step in the proof of Theorem 1. It follows from (10) that the Beltrami coefficients (5) are convergent on the disk $\mathbb{D}$ to the harmonic Beltrami coefficient $\nu_{S_F}$ of function F; besides, $$||S_F||_{\mathbf{B}} \le \lim_{j \to \infty} ||S_{F_j}||_{\mathbf{B}} \tag{11}$$ (this inequality also follows from the extremality of $\nu_{tS_{F_j}}$ and general properties of quasiconformal maps). We show that in fact we have in (11) the equality, which yields that for admissible |t|, the harmonic Beltrami coefficient $\nu_{tS_F}$ is extremal (in the distance $\tilde{\tau}_{\mathbf{T}}(\Gamma)$ ) in its equivalence class First observe that normalising the quasiconformal extensions $\widehat{F}(z)$ of univalent functions $F(z) = z + b_0 + b_1 z^{-1} + \dots$ on $\mathbb{D}^*$ to $\widehat{\mathbb{C}}$ additionally by $\widehat{F}(0) = 0$ , one obtains that their Teichmüller norm k(F) is plurisubharmonic on the universal Teichmüller space $\mathbf{T}$ (hence, also on $\mathbf{T}(\Gamma_{\mu})$ with distance $\widetilde{\tau}_{\mathbf{T}}(\Gamma)$ ). Now we pass to the homotopy functions $$F_s(z) = sF(z/s) = z + b_0 s + b_1 s^2 z^{-1} + \dots$$ with $|s| \leq 1$ . Then $k(F_s)$ is circularly symmetric with respect to c, hence continuous on [0,1]. But for any $s \in (0,1)$ , we have $$||S_{F_{i,s}} - S_{F_s}||_{\mathbf{B}} \to 0$$ as $j \to \infty$ , and the properties of extremal quasiconformal maps imply $$\lim_{s \to 1} k(F_{j,s}) = k(F_j) = \|\nu_{S_F}\|_{\infty}$$ (12) (the last equality in (12) is valid only for $||S_F||_{\mathbf{B}} < 2$ ). Together with (4) and (7), this implies the equality in (9). It follows from (12) that all harmonic Beltrami coefficients $\nu_{tS_F}$ with norm less than 1 must be extremal in their classes, and therefore, as $\|\nu_{tS_F}\|_{\mathbf{B}} \to 1$ , the corresponding value $t_0$ must define a boundary point of both spaces $\mathbf{T}(\Gamma_{\mu})$ and $\mathbf{T}$ (i.e., $tS_F \to t_0S_F \in \partial T(\Gamma_{\mu})$ as $t \to t_0$ ). Taking also into account the Abikoff-Bers-Zhuravlev theorem, which states that any the domain $T(\Gamma)$ representing the space $\mathbf{T}(\Gamma)$ has a common boundary with its complementary domain in the space $\mathbf{B}(\Gamma)$ (see [1], [4], [21]), one derives that there are points on the ray $rS_F: r>0$ in $\mathbf{T}(\Gamma_\mu)$ , which must lie in the exterior of $\mathbf{T}(\Gamma_\mu)$ . These points are placed on the interval between $r_0S_F$ and $r_S_F$ , where the second point correspond to the surface $D^_\mu/\Gamma_\mu$ obeying the assumption (4). This completes the proof of the theorem.

Proposition 1. For all Beltrami coefficients $\mu$ compatible with the groups $\Gamma$ uniformizing the closed Riemann surfaces, we have the equality $\varkappa(F^{\mu}) = k(F^{\mu})$ (in the Teichmüller distance on $\mathbf{T}$ ). This result has its intrinsic interest, because the points $S_F \in \mathbf{T}$ with $\varkappa(F) < k(F)$ are dense in the space $\mathbf{T}$ . 3. Generalization of Theorems 1 and 2. The arguments implied in the proof of Theorem 1 provide, in fact, more general result.

Theorem 3. Let $\Gamma$ be a Fuchsian group such that $X_0 = \mathbb{D}/\Gamma$ is a closed Riemann surface of genus $g \geq 2$ , and let G be a subdomain of $\mathbf{T}(\Gamma) \subset \mathbf{B}(\Gamma)$ satisfying: - (a) G contains the origin 0 of $\mathbf{B}(\Gamma)$ and a point $\varphi_0$ with $\|\varphi_0\|_{\mathbf{B}} > 2$ ; - (b) the domain G and its complementary domain $G^*$ in $\mathbf{B}(\Gamma)$ have a common boundary $\partial G$ . Then G is not starlike with respect to the origin. Theorem of such type is also valid for appropriate subdomains of the space $\mathbf{T}(0,6)$ .