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On the geodesics of the Szegö metric

Anjali Bhatnagar

TL;DR

This paper analyzes the geodesic structure of the Szegö metric on $C^\infty$-smoothly bounded strongly pseudoconvex domains, focusing on closed geodesics and geodesic spirals and drawing connections to the Bergman metric. It establishes the completeness of the Szegö metric and its domination of the Carathéodory metric, then proves that every nontrivial loop class contains a closed geodesic, with the Hadamard-Cartan property ruling out such geodesics only in the unit ball. For infinitely sheeted universal covers, it further shows the existence of geodesic spirals through points not lying on closed geodesics by leveraging compactness arguments tied to a defining function $\rho$ and the Fefferman kernel asymptotics. These results advance understanding of Szegö geometry, provide a parallel to known Bergman dynamics, and outline limitations and open questions in non-simply connected domains.

Abstract

We explore the existence of closed geodesics and geodesic spirals for the Szegö metric in a $C^{\infty}$-smoothly bounded strongly pseudoconvex domain $Ω\subset\mathbb{C}^n$, which is not simply connected for $n \geq 2$.

On the geodesics of the Szegö metric

TL;DR

This paper analyzes the geodesic structure of the Szegö metric on -smoothly bounded strongly pseudoconvex domains, focusing on closed geodesics and geodesic spirals and drawing connections to the Bergman metric. It establishes the completeness of the Szegö metric and its domination of the Carathéodory metric, then proves that every nontrivial loop class contains a closed geodesic, with the Hadamard-Cartan property ruling out such geodesics only in the unit ball. For infinitely sheeted universal covers, it further shows the existence of geodesic spirals through points not lying on closed geodesics by leveraging compactness arguments tied to a defining function and the Fefferman kernel asymptotics. These results advance understanding of Szegö geometry, provide a parallel to known Bergman dynamics, and outline limitations and open questions in non-simply connected domains.

Abstract

We explore the existence of closed geodesics and geodesic spirals for the Szegö metric in a -smoothly bounded strongly pseudoconvex domain , which is not simply connected for .
Paper Structure (3 sections, 10 theorems, 43 equations)

This paper contains 3 sections, 10 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Closed geodesics
  3. Geodesic spirals

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb {C}^{n}$ be a $C^\infty$-smoothly bounded strongly pseudoconvex domain which is not simply connected, equipped with the Szegö metric $ds_{s_\Omega}^2$. We have

Theorems & Definitions (21)

  • Example 1
  • Example 2
  • Definition 1
  • Theorem 1.1
  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • proof : Proof of Theorem \ref{['main']} (a)
  • Remark 1
  • ...and 11 more