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

## TL;DR

This paper explores the relationship between the Laplace Beltrami operator, orthogonal polynomials, and pluripotential equilibrium measures on specific geometric domains, revealing new connections in pluripotential and differential geometry.

## Contribution

It demonstrates that eigenfunctions of the Laplace Beltrami operator on the ball, simplex, and sphere are orthogonal polynomials with respect to the pluripotential equilibrium measure, linking geometry and polynomial theory.

## Key findings

- Eigenfunctions are orthogonal polynomials w.r.t. equilibrium measure
- The Laplace Beltrami operator is linked to orthogonal polynomials in specific domains
- Conjecture on broader applicability of these relationships

## 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