A generalization of a result of Minakshisundaram and Pleijel

TL;DR

The paper extends Minakshisundaram–Pleijel spectral asymptotics by proving that for a smooth measure $ au=\\psi d\\nu$ supported on a compact codimension $k$ submanifold $N\subset M$, the spectral sum \\(\\sum_{\\lambda_j<T} |\\hat{\\tau}(j)|^2\\) grows like \\frac{T^{k/2}\\int_N |\\psi|^2 d\\nu}{(4\\pi)^{k/2}\\Gamma(k/2+1)} as $T\\to\\infty$, with a geometry-based proof that avoids Fourier integral operator machinery. The authors establish the key heat-flow identity \\|f_{t/2}\\|_2^2 \\sim (4\\pi t)^{-k/2} \\|\\psi\\|_2^2$ as $t\to 0$, and derive the required Tauberian bridge between heat flow and spectral sums. The work also develops sharp geometric control of geodesic-sphere second fundamental forms and Hessians of distance functions, leading to a concrete application: an asymptotic Pythagorean identity for Legendre polynomials via a spherical example. Overall, the paper provides a self-contained, geometry-based route to a broad generalization of classical spectral asymptotics with concrete polynomial-orthogonality consequences.

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.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4
• proof
• Lemma 4.1
• proof