The Baran metric \(\delta _E\) is a Finsler metric on the interior of \(E\subset {\mathbb {R}}^n\) arising from pluripotential theory. When E is an Euclidean ball, a simplex, or a sphere, \(\delta _E\) 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 \(\delta _E\) are the orthogonal polynomials with respect to the pluripotential equilibrium measure \(\mu _E\) 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 \(\mu _E\) and the Riemannian structure naturally arising from pluripotential theory.

中文翻译：

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Baran度量和多能平衡测度中的Laplace Beltrami算子：球，单纯形和球体

Baran度量\（\ delta _E \）是由多能理论产生的\（E \ subset {\ mathbb {R}} ^ n \）内部的Finsler度量。当E为欧几里得球，单形或球体时，\（\ delta _E \）为黎曼。尚无此类属性的其他示例。我们证明在这三种情况下，与\（\ delta _E \）关联的Laplace Beltrami算子的本征函数是关于E的多能平衡测度\（\ mu _E \）的正交多项式。我们推测这可能具有更广泛的普遍性。我们考虑的微分算子是在正交多项式的框架中引入的，并结合某些对称群进行了研究。在这项工作中，我们强调了关于\（\ mu _E \）的正交多项式与自然多能理论自然产生的黎曼结构之间的联系。