Laplace Beltrami Operator in the Baran Metric and Pluripotential Equilibrium Measure: The Ball, the Simplex, and the Sphere

The Baran metric δ E is a Finsler metric on the interior of E ⊂ R n arising from pluripotential theory. When E is an Euclidean ball, a simplex, or a sphere, δ E is Riemannian. No further examples of such property are known. We prove that in these three cases, the eigenfunctions of the Laplace Beltra...

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Bibliographic Details
Published in:Computational methods and function theory 2019-12, Vol.19 (4), p.547-582
Main Author: Piazzon, Federico
Format: Article
Language:English
Subjects:
Analysis
Computational Mathematics and Numerical Analysis
Differential equations
Eigenvectors
Functions of a Complex Variable
Mathematics
Mathematics and Statistics
Operators (mathematics)
Polynomials
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Summary:The Baran metric δ E is a Finsler metric on the interior of E ⊂ R n arising from pluripotential theory. When E is an Euclidean ball, a simplex, or a sphere, δ E is Riemannian. No further examples of such property are known. We prove that in these three cases, the eigenfunctions of the Laplace Beltrami operator associated with δ E are the orthogonal polynomials with respect to the pluripotential equilibrium measure μ E of E . We conjecture that this may hold in wider generality. The differential operators that we consider were introduced in the framework of orthogonal polynomials and studied in connection with certain symmetry groups. In this work, we highlight the connections between orthogonal polynomials with respect to μ E and the Riemannian structure naturally arising from pluripotential theory.
ISSN:1617-9447
2195-3724
DOI:10.1007/s40315-019-00286-9