Critical measures for vector energy: Global structure of trajectories of quadratic differentials

Saddle points of a vector logarithmic energy with a vector polynomial external field on the plane constitute the vector-valued critical measures, a notion that finds a natural motivation in several branches of analysis. We study in depth the case of measures μ→=(μ₁,μ₂,μ₃) when the mutual interaction comprises both attracting and repelling forces. For arbitrary vector polynomial external fields we establish general structural results about critical measures, such as their characterization in terms of an algebraic equation solved by an appropriate combination of their Cauchy transforms, and the symmetry properties (or the S-properties) exhibited by such measures. In consequence, we conclude that vector-valued critical measures are supported on a finite number of analytic arcs, that are trajectories of a quadratic differential globally defined on a three-sheeted Riemann surface. The complete description of the so-called critical graph for such a differential is the key to the construction of the critical measures. We illustrate these connections studying in depth a one-parameter family of critical measures under the action of a cubic external field. This choice is motivated by the asymptotic analysis of a family of (non-hermitian) multiple orthogonal polynomials, that is subject of a forthcoming paper. Here we compute explicitly the Riemann surface and the corresponding quadratic differential, and analyze the dynamics of its critical graph as a function of the parameter, giving a detailed description of the occurring phase transitions. When projected back to the complex plane, this construction gives us the complete family of vector-valued critical measures, that in this context turn out to be vector-valued equilibrium measures.