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Some remarks on the Carathéodory and Szegö metrics on planar domains

Anjali Bhatnagar, Diganta Borah

Abstract

We study several intrinsic properties of the Carathéodory and Szegö metrics on finitely connected planar domains. Among them are the existence of closed geodesics and geodesic spirals, boundary behaviour of Gaussian curvatures, and $L^2$-cohomology. A formula for the Szegö metric in terms of the Weierstrass $\wp$-function is obtained. Variations of these metrics and their Gaussian curvatures on planar annuli are also studied. Consequently, we obtain optimal universal upper bounds for their Gaussian curvatures and show that no universal lower bounds exist for their Gaussian curvatures. Moreover, it follows that there are domains where the Gaussian curvature of the Szegö metric assumes both negative and positive values. Lastly, it is also observed that there is no universal upper bound for the ratio of the Szegö and Carathéodory metrics.

Some remarks on the Carathéodory and Szegö metrics on planar domains

Abstract

We study several intrinsic properties of the Carathéodory and Szegö metrics on finitely connected planar domains. Among them are the existence of closed geodesics and geodesic spirals, boundary behaviour of Gaussian curvatures, and -cohomology. A formula for the Szegö metric in terms of the Weierstrass -function is obtained. Variations of these metrics and their Gaussian curvatures on planar annuli are also studied. Consequently, we obtain optimal universal upper bounds for their Gaussian curvatures and show that no universal lower bounds exist for their Gaussian curvatures. Moreover, it follows that there are domains where the Gaussian curvature of the Szegö metric assumes both negative and positive values. Lastly, it is also observed that there is no universal upper bound for the ratio of the Szegö and Carathéodory metrics.

Paper Structure

This paper contains 19 sections, 22 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Geodesics
  3. Curvature
  4. $L^2$-cohomology
  5. The Szegö metric on a doubly connected domain
  6. Variations of the Carathéodory and Szegö metrics on planar annuli
  7. Some observations
  8. Localisation of $S_{\Omega}(z)$
  9. Scaling method
  10. Boundary asymptotics
  11. Localisation of $\partial^{k \overline l} S_{\Omega}$ and $\partial^{k \overline l} s_{\Omega}$
  12. Comparison with the classical metrics
  13. Geodesics of the Carathéodory and Szegö metrics
  14. Existence of closed geodesic
  15. Existence of geodesic spirals
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{C}$ be a non-degenerate $n$-connected domain, $n\geq 2$. Equip $\Omega$ with the conformal metric $ds_{m_{\Omega}}=m_{\Omega}(z)\vert dz\vert$, where $m_{\Omega}=c_{\Omega}$ or $s_{\Omega}$. Then

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 23 more