Title: Laplace Beltrami operator in the Baran metric and pluripotential equilibrium measure: the ball, the simplex and the sphere,
Abstract: The Baran metric $\delta_E$ 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 $\delta_E$ 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 $\mu_E$ 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 $\mu_E$ and the Riemaniann structure
naturally arising from Pluripotential Theory