On the existence of short trajectories of quadratic differentials related to generalized Jacobi polynomials with non-real varying parameters

The motivation of this paper is to analyze to a large-degree the behavior of the Jacobi (Pnαn,βn) polynomials when the parameters determining the polynomials are complex and depend on the degree n   linearly. We show that the Cauchy transform of the limit (weak) of the root-counting measures of these polynomials satisfies quadratic algebraic equation. We investigate the existence of solutions to these equations as Cauchy transform of compactly supported positive measures. Any connected curve of the support of these measures (if exists) coincides with a horizontal trajectory of some quadratic differential. In this paper, we describe the trajectories of the family of quadratic differentials λ2(z−a)(z−b)(z2−1)2dz2, and we give a necessary and sufficient condition on the complex numbers a,ba,b, and λ for the existence of at least one finite critical trajectory.