Journal of the Ramanujan Mathematical Society

Volume 41, Issue 2, June 2026 pp. 149–157.

A generalization of a result of Minakshisundaram and Pleijel

Authors: Ankita Sharma, Mansi Mishra and M. K. Vemuri
Author institution:Department of Mathematical Sciences, IIT (BHU), Varanasi 221 005, India.

Summary: 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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