On the existence of finite critical trajectories in a family of quadratic differentials

TL;DR

This paper investigates the existence of finite critical trajectories in quadratic differentials and provides new proofs related to the support of limit measures for certain orthogonal polynomials with varying parameters.

Contribution

It introduces new results on finite critical trajectories in quadratic differentials and offers novel proofs for the support of limit measures of Laguerre and Jacobi polynomials.

Findings

Finite critical trajectories can connect zeros of quadratic differentials under certain conditions.

New proofs for the support of limit measures of Laguerre and Jacobi polynomials.

Analysis covers both holomorphic and meromorphic quadratic differentials.

Abstract

In this note, we discuss the possible existence of finite critical trajectories connecting two zeros a(t) and b(t) of a family of quadratic differentials satisfying some properties. We treat the cases of holomorphic and meromorphic quadratic differentials, and we give new proofs concerning the supports of limit measures of the root-counting measures of the generalized Laguerre and Jacobi polynomials with varying parameters.