Laplace Beltrami operator in the Baran metric and pluripotential equilibrium measure: the ball, the simplex and the sphere

Abstract

The Baran metric $δ_E$ is a Finsler metric on the interior of $E\subset \R^n$ arising from Pluripotential Theory. We consider the few instances, namely $E$ being the ball, the simplex, or the sphere, where $δ_E$ is known to be Riemaniann and we prove that the eigenfunctions of the associated Laplace Beltrami operator (with no boundary conditions) are the orthogonal polynomials with respect to the pluripotential equilibrium measure $μ_E$ of $E.$ We conjecture that this may hold in a wider generality. The considered differential operators have been already introduced in the framework of orthogonal polynomials and studied in connection with certain symmetry groups. In this work instead we highlight the relationships between orthogonal polynomials with respect to $μ_E$ and the Riemaniann structure naturally arising from Pluripotential Theory