Explicit descriptions of spectral properties of Laplacians on spheres $${\mathbb {S}}^{N}\,(N\ge 1)$$: a review

Abstract

In their remarkable paper, Minakshisundaram and Pleijel established by using the parametrix for the heat equation the asymptotic expansion of the heat kernel on compact Riemannian manifolds. The result has since been extensively used in the spectral analysis of the Laplace-Beltrami operator, and in particular, in proving Weyl’s law for the asymptotic distribution of eigenvalues and various direct and inverse problems in spectral geometry. However, the question of describing the explicit values of the corresponding heat trace coefficients associated with an arbitrary compact Riemannian manifold has remained an interesting task. In this paper, we review results on Minakshisundaram-Pleijel coefficients associated with the Laplacian on spheres $${\mathbb {S}}^{N}$$ ( $$N\ge 1$$ ) and other associated spectral invariants, namely, the Minakshisundaram-Pleijel zeta functions & their residues, and the zeta-regularised determinants of the Laplacian on spheres. The results reviewed deal mainly with closed-form formulae for the afore-mentioned spectral invariants and the explicit values of the first few of these spectral invariants are given.