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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 beRiemaniann and we prove that the eigenfunctions of the associated LaplaceBeltrami operator (with no boundary conditions) are the orthogonal polynomialswith respect to the pluripotential equilibrium measure $\mu_E$ of $E.$ Weconjecture that this may hold in a wider generality.  The considered differential operators have been already introduced in theframework of orthogonal polynomials and studied in connection with certainsymmetry groups. In this work instead we highlight the relationships betweenorthogonal polynomials with respect to $\mu_E$ and the Riemaniann structurenaturally arising from Pluripotential Theory

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