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

2017/03/24 by Federico Piazzon, Piazzon, Federico
Physics and Astronomy · #Advanced Differential Geometry Research #Complex Variables (math.CV) #Cosmology and Gravitation Theories #FOS: Mathematics #Quantum chaos and dynamical systems #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.1703.08392

openalex publication_date 2017/03/24 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

Abstract

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

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