Recent zbMATH articles in MSC 30F45 https://www.zbmath.org/atom/cc/30F45 2021-04-16T16:22:00+00:00 Werkzeug A generalized Hurwitz metric. https://www.zbmath.org/1456.30077 2021-04-16T16:22:00+00:00 "Arstu" https://www.zbmath.org/authors/?q=ai:arstu. "Sahoo, Swadesh Kumar" https://www.zbmath.org/authors/?q=ai:sahoo.swadesh-kumar Summary: In 2016, the Hurwitz metric was introduced by D. Minda in arbitrary proper subdomains of the complex plane and he proved that this metric coincides with the Poincaré's hyperbolic metric when the domains are simply connected. In this paper, we provide an alternate definition of the Hurwitz metric through which we could define a generalized Hurwitz metric in arbitrary subdomains of the complex plane. This paper mainly highlights various important properties of the Hurwitz metric and the generalized metric including the situations where they coincide with each other. On the Hardy number of a domain in terms of harmonic measure and hyperbolic distance. https://www.zbmath.org/1456.30024 2021-04-16T16:22:00+00:00 "Karafyllia, Christina" https://www.zbmath.org/authors/?q=ai:karafyllia.christina The author establishes the Hardy number of a domain in terms of harmonic measure and hyperbolic distance. For a domain $$D\subset\mathbb C$$ and $$z\in D$$ and a Borel subset $$E$$ of the closure $$\overline D$$ of $$D$$, let $$\omega_D(z,E)$$ be the harmonic measure at $$z$$ of $$\overline E$$ with respect to the component of $$D\setminus\overline E$$ containing $$z$$, and let $$d_D(z,w)$$ be the hyperbolic distance between $$z$$ and $$w$$ in $$E$$ for a simply connected domain $$D\ne\mathbb C$$. Denote by $$\psi$$ a conformal map on the unit disk $$\mathbb D=\{z\in\mathbb C:|z|<1\}$$, $$\psi(0)=0$$, and $$F_{\alpha}=\{z\in\mathbb D:|\psi(z)|=\alpha\}$$, $$\alpha>0$$. The Hardy number of $$\psi$$ is given by $$h(\psi)=\sup\{p>0: \psi\in H^p(\mathbb D)\}$$, where $$H^p(\mathbb D)$$ is the Hardy space on $$\mathbb D$$. The following theorem expresses $$h(\psi)$$ in terms of the hyperbolic distance. Theorem 1.1. If $$h(\psi)$$ denotes the Hardy number of $$\psi$$, then $h(\psi)=\liminf_{\alpha\to+\infty}\frac{d_{\mathbb D}(0,F_{\alpha})}{\log\alpha}.$ Denote $L=\lim_{\alpha\to+\infty}(\log\omega_{\mathbb D}(0,F_{\alpha})^{-1}/\log\alpha),\;\;\;\mu=\lim_{\alpha\to+\infty}(d_{\mathbb D}(0,F_{\alpha})/\log\alpha).$ Theorem 1.3. If $$\mu$$ exists, then $$L$$ exists and $$L=\mu$$. Let $$N(\alpha)\in\mathbb N\cup\{\infty\}$$ denote the number of components of $$F_{\alpha}$$, $$\alpha>0$$, and let $$F_{\alpha}^i$$ denote each of these components, $$i=1,\dots,N(\alpha)$$. Denote by $$F_{\alpha}^*$$ a component of $$F_{\alpha}$$ such that $$\omega_{\mathbb D}(0,F_{\alpha}^*)=\max_i\{\omega_{\mathbb D}(0,F_{\alpha}^i)\}$$. Theorem 1.5. If $$L$$ exists, then $$\mu$$ exists if and only if $\limsup_{\alpha\to+\infty}\frac{\log_{\mathbb D}\omega(0,F_{\alpha}^*)^{-1}}{\log\alpha}=L.$ If $$\mu$$ exists, then $$\mu=L$$. Reviewer: Dmitri V. Prokhorov (Saratov) Polyakov-Alvarez type comparison formulas for determinants of Laplacians on Riemann surfaces with conical singularities. https://www.zbmath.org/1456.58024 2021-04-16T16:22:00+00:00 "Kalvin, Victor" https://www.zbmath.org/authors/?q=ai:kalvin.victor Summary: We present and prove Polyakov-Alvarez type comparison formulas for the determinants of Friederichs extensions of Laplacians corresponding to conformally equivalent metrics on a compact Riemann surface with conical singularities. In particular, we find how the determinants depend on the orders of conical singularities. We also illustrate these general results with several examples: based on our Polyakov-Alvarez type formulas we recover known and obtain new explicit formulas for determinants of Laplacians on singular surfaces with and without boundary. In one of the examples we show that on the metrics of constant curvature on a sphere with two conical singularities and fixed area $$4 \pi$$ the determinant of Friederichs Laplacian is unbounded from above and attains its local maximum on the metric of standard round sphere. In another example we deduce the famous Aurell-Salomon formula for the determinant of Friederichs Laplacian on polyhedra with spherical topology, thus providing the formula with mathematically rigorous proof.