Critical points of Green's function and geometric function theory

TL;DR

This paper investigates the critical points of Green's functions in multiply connected domains, exploring their behavior, geometric properties, and connections with differential operators and invariant structures in complex analysis.

Contribution

It introduces a unified approach using affine and projective connections to analyze critical points and their limits, linking Green's function critical points with other domain invariants.

Findings

Critical points relate to zeros of the Bergman kernel

In doubly connected domains, critical points, zeros, and boundary limits are interconnected

The projective properties of associated differential operators are characterized

Abstract

We study questions related to critical points of the Green's function of a bounded multiply connected domain in the complex plane. The motion of critical points, their limiting positions as the pole approaches the boundary and the differential geometry of the level lines of the Green's function are main themes in the paper. A unifying role is played by various affine and projective connections and corresponding M\"obius invariant differential operators. In the doubly connected case the three Eisenstein series $E_2$, $E_4$, $E_6$ are used. A specific result is that a doubly connected domain is the disjoint union of the set of critical points of the Green's function, the set of zeros of the Bergman kernel and the separating boundary limit positions for these. At the end we consider the projective properties of the prepotential associated to a second order differential operator depending canonically on the domain.