The Grunsky norm of univalent functions and abelian holomorphic differentials

Lemma 1. The Grunsky norm $\varkappa(f)$ of every function $$f(z) = z + b_0 + b_m z^{-m} + b_{m+1} z^{-(m+1)} + \dots \in \Sigma_Q, \quad m \ge 1$$ satisfies the inequality $$\varkappa(f) \ge \alpha_{\mathbb{D}}(f) := \sup_{\psi \in A_1^2(\mathbb{D}), \|\psi\|_{A_1} = 1} |\langle \mu_0, \psi \rangle_{\mathbb{D}}| = \sup_{\omega \in A_2(\mathbb{D}), \|\omega\|_2 = 1} |\langle \mu_0, \omega^2 \rangle_{\mathbb{D}}|, \tag{4}$$ where $\mu_0$ is an extremal Beltrami coefficient among quasiconformal extensions $f^{\mu}$ of f onto $\mathbb{D}$ . For functions f with equal Grunsky and Teichmüller norms, we have in (4) the equality, and vice versa (see, [8], [12], [17]), but until now there was unknown whether there exist the functions $f \in \Sigma_Q$ with $\varkappa(f) < k(f)$ and $\varkappa(f) = \alpha_{\mathbb{D}}(f)$ . The proof of this lower bound for $\varkappa(f)$ is based on geometric extensions of the Ahlfors Schwarz lemma to metrics of integrable generalized negative curvatures given in [10], [21]. We shall also use the infinitesimal version of the inequality (4). It will be presented later. 1.4. Conformal metrics of negative generalized Gaussian curvature. We shall use some known results on subharmonic conformal metrics $ds = \lambda(t)|dt|$ on the disk $\mathbb{D}$ (with $\lambda(t) \geq 0$ ) of negative generalized Gaussian curvature. Recall that the generalized Gaussian curvature $\kappa_{\lambda}$ of an upper semicontinuous Finsler metric $ds = \lambda |dt|$ in a domain $\Omega \subset \mathbb{C}$ is defined by $$\kappa_{\lambda}(t) = -\frac{\Delta \log \lambda(t)}{\lambda(t)^2},\tag{5}$$ where $\Delta$ is the generalized Laplacian $$\Delta \lambda(t) = 4 \liminf_{r \to 0} \frac{1}{r^2} \left\{ \frac{1}{2\pi} \int_0^{2\pi} \lambda(t + re^{i\theta}) d\theta - \lambda(t) \right\}$$ (provided that $-\infty \leq \lambda(t) < \infty$ ). Similar to $C^2$ functions, for which $\Delta$ coincides with the usual Laplacian, one obtains that $\lambda$ is subharmonic on $\Omega$ if and only if $\Delta\lambda(t) \geq 0$ ; hence, at the points $t_0$ of local maximum of $\lambda$ with $\lambda(t_0) > -\infty$ , we have $\Delta\lambda(t_0) \leq 0$ . The sectional holomorphic curvature of a Finsler metric on a complex Banach manifold X is defined in a similar way as the supremum of the curvatures (5) over appropriate collections of holomorphic maps from the disk into X for a given tangent direction in the image. As is well-known [1], [9], the holomorphic curvature of the Kobayashi-Teichmüller metric $\mathcal{K}_{\mathbf{T}}(x,v)$ of universal Teichmüller space $\mathbf{T}$ equals -4 at all points (x,v) of the tangent bundle $\mathcal{T}(\mathbf{T})$ over $\mathbf{T}$ . Instead, the holomorphic curvature of metric $\lambda_{\varkappa}$ generated on $\mathbb{D}$ by the Grunsky Finsler structure satisfies the inequality $\Delta \log \lambda \geq 4\lambda^2$ , where $\Delta$ is again the generalized Laplacian (see [11]). So, $\lambda_{\varkappa}(t) \leq -4$ on any holomorphic disk in the space $\mathbf{T}$ . This is a special case of metrics whose curvature satisfies the inequality $\Delta \log \lambda \geq K \lambda^2$ with constant K > 0. All such metrics are subharmonic. We shall also apply the upper semicontinuous metrics $\lambda$ satisfying the inequality $$\Delta \log \lambda \ge K\lambda, \quad K = \text{const} > 0$$ (6) (then $u = \log \lambda$ can be negative). For such metrics, we have the following Minda's maximum principle given by

Lemma 2. If a function $u: D \to [-\infty, +\infty)$ is upper semicontinuous in a domain $D \subset \mathbb{C}$ and its generalized Laplacian satisfies the inequality $\Delta u(z) \geq Ku(z)$ with some positive constant K at any point $z \in D$ , where $u(z) > -\infty$ , and if $$\limsup_{z \to \zeta} u(z) \le 0 \quad \text{for all } \zeta \in \partial D,$$ then either u(z) < 0 for all $z \in D$ or $u(z) \equiv 0$ on D. The proof of this lemma related to the Ahlfors-Schwarz lemma is given in [19]; its variations see in [2].

Theorem 1. For any univalent function we have the equality $$\varkappa(f) = \alpha_{\mathbb{D}}(f). \tag{7}$$ It suffices to consider the functions $f \in \Sigma_Q$ , because for functions not admitting quasiconformal extension, both sides of (7) are equal to 1. - 2.2. Applications. We mention two important applications of Theorem 1. - 1. Plurisubharmonicity. Letting $\alpha_{\mathbb{D}}(S_{f^{\mu}}) = \alpha_{\mathbb{D}}(f^{\mu})$ , one can regard this quantity as a function on the universal Teichmüller space T (as well as on the ball Belt( $\mathbb{D}$ )₁. The following corollary to Theorem 1 solves the question posed in [10]. Corollary 1. The function $\alpha_{\mathbb{D}}(S_{f^{\mu}})$ is plurisubharmonic on the space T and on the ball $\operatorname{Belt}(\mathbb{D})_1$ . This follows from the equality (7), because its left-hand side is plurisubharmonic on T. In fact, the indicated plurisubharmonicity is a consequence of the relation (10) and of holomorphy of functions (9) defined below. Another proof is given in [12]. 2. Application to Fredholm eigenvalues of Jordan curves. The Fredholm eigenvalues $\rho_n$ of an oriented smooth closed Jordan curve $L \subset \widehat{\mathbb{C}}$ are the eigenvalues of its double-layer potential, or equivalently, of the integral equation $$u(z) + \frac{\rho}{\pi} \int_{\Gamma} u(\zeta) \frac{\partial}{\partial n_{\zeta}} \log \frac{1}{|\zeta - z|} ds_{\zeta} = h(z),$$ where $n_{\zeta}$ denotes the outer normal and $ds_{\zeta}$ is the length element at $\zeta \in L$ . These values have crucial applications in solving many problems in various fields of mathematics. It is important often to know the least positive eigenvalue $\rho_L = \rho_1$ . This value is naturally connected with conformal and quasiconformal maps related to L and can be defined for any oriented closed Jordan curve L by $$\frac{1}{\rho_L} = \sup \frac{|\mathcal{D}_G(u) - \mathcal{D}_{G}(u)|}{\mathcal{D}_G(u) + \mathcal{D}_{G}(u)},$$ where G and $G$ are, respectively, the interior and exterior of L; $\mathcal{D}$ denotes the Dirichlet integral, and the supremum is taken over all functions u continuous on $\widehat{\mathbb{C}}$ and harmonic on $G \cup G$ . In particular, $\rho_L = \infty$ only for the circle. Note that both sides of the last equality remain invariant under the action of the Moebius group $PSL(2,\widehat{\mathbb{C}})$ . The indicated first eigenvalue is intrinsically connected with the Grunsky coefficients of the exterior conformal mapping function $f_L^: \mathbb{D}^ \to D$ ; this is qualitatively expressed by the remarkable Kühnau-Schiffer theorem, which states that the value $\rho_L$ is reciprocal to the Grunsky norm $\varkappa(f_L^)$ (see [16], [22]). Together with this result, Theorem 1 implies explicitly the first Fredholm eigenvalue $\rho_L$ of every quasiconformal curve $L \subset \widehat{\mathbb{C}}$ . Namely, we have Corollary 2. For any quasicircle $L \subset \widehat{\mathbb{C}}$ , $$\frac{1}{\rho_L} = \sup_{\psi \in A_1^2(D), \|\psi\|_{A_1} = 1} |\langle \mu_0, \psi \rangle_{\mathbb{D}}|, \tag{8}$$ where $\mu_0$ is either of extremal Beltrami coefficients of the appropriately normalized exterior conformal mapping function $f_L$ on which the Teichmüller norm of $f_L$ is attained. Moreover, letting in (8) $\mu_0 = t|\psi|/\psi$ with |t| < 1 and $\psi \in A_1$ , one obtains a dense subset of a possible eigenvalues $\rho_L$ . Conversely, given a quasicircle $L \subset \widehat{\mathbb{C}}$ , take a quasiconformal extension $f^{\mu}$ of the exterior conformal map $f_L$ . Its Beltrami coefficient $\mu$ determines a linear functional $\mu(\psi) = \langle \mu, \psi \rangle_{\mathbb{D}}$ on the linear span $\widetilde{A}$ of the subset $\{\psi \in A_1^2(\mathbb{D}) : \|\psi\|_{A_1} = 1\}$ of the unit sphere in $A_1(\mathbb{D})$ . Its Hahn-Banach extension onto $L_1(\mathbb{D})$ provides an extremal Beltrami coefficient $\mu_0$ in $\mathbb{D}$ for $f_L$ satisfying (8). 3. Teichmüller maps with $\varkappa(f) = k(f)$ . We also mention the following useful property of Teichmüller extremal maps. Corollary 3. If the equivalence class of f is a Strebel point and $\varkappa(f) = k(f)$ , then $\mu_0$ is necessarily of the form $\mu_0(z) = \|\mu_0\|_{\infty} |\psi_0(z)|/\psi_0(z)$ with $\psi_0 \in A_1^2(\mathbb{D})$ , and the relation (8) is completed by $$\varkappa(f^{\mu_0}) = k(f^{\mu_0}) = q_L = \frac{1}{\rho_L} = \sup_{\psi \in A_1^2(D), \|\psi\|_{A_1} = 1} |\langle \mu_0, \psi \rangle_{\mathbb{D}}|,$$ where $L = f^{\mu_0}(\mathbb{S}^1)$ , and $q_L$ denotes its quasireflection coefficient. (equal to the minimum of dilatations $\partial_z W/\partial -\overline{z}W$ of the orientation reversing quasiconformal homeomorphisms of the sphere $\widehat{\mathbb{C}}$ which preserve point-wise the quasicircle $L \subset \widehat{\mathbb{C}}$ and interchange its interior and exterior domains. This was established by the author earlier even for univalent functions in the arbitrary quasidisks (see, e.g., [12] (and in a special case by Kühnau [17]), but it follows from (7) immediately. Recall that quasireflection coefficient of an (oriented) quasicircle $L' \subset \widehat{\mathbb{C}}$ is the minimumal dilatation $\partial_z W/\partial -\overline{z}W$ of the orientation reversing quasiconformal homeomorphisms of the sphere $\widehat{\mathbb{C}}$ , which preserve point-wise the quasicircle $L \subset \widehat{\mathbb{C}}$ and interchange its interior and exterior domains. Theorem 1 also has other important consequences and applications. These results will be presented elsewhere.