A generalization of a result of Minakshisundaram and Pleijel

2023/12/12 by Ankita Sharma, Sharma, Ankita, Mansi Mishra +3
Computer Science · Mathematics · #53B21 #58J35 #Advanced Mathematical Modeling in Engineering #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2312.07078

openalex publication_date 2023/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Minakshisundaram and Pleijel gave an asymptotic formula for the sum of squares of the pointwise values of the eigenfunctions of the Laplace-Beltrami operator on a compact Riemannian manifold, with eigenvalues less than a fixed number. Zelditch later extended this result by replacing the pointwise values with the Fourier coefficients of a smooth measure supported on a compact submanifold. Zelditch's result is very general, and his proof relies on the theory of Fourier integral operators. Here we give a proof based on methods of Riemannian geometry.

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