Quadratic differentials A(z-a)(z-b)dz^2/(z-c)^2 and algebraic Cauchy transform

TL;DR

This paper explores the conditions under which algebraic functions can be represented as Cauchy transforms of signed measures supported on specific geometric structures, analyzing critical trajectories of quadratic differentials.

Contribution

It establishes new criteria for representing algebraic functions as Cauchy transforms supported on semi-analytic curves and isolated points, linked to quadratic differential trajectories.

Findings

Representation of algebraic functions as Cauchy transforms is characterized almost everywhere.

Existence conditions for critical trajectories of quadratic differentials are identified.

Supports of measures include finite semi-analytic curves and isolated points.

Abstract

In this paper, we discuss the representability almost everywhere (a.e.) in the plane of an irreducible algebraic function as the Cauchy transform of a signed measure supported on a finite number of compact semi-analytic curves and a finite number of isolated points. We discuss the existence of critical trajectories of a family of quadratic differentials.