On common zeros of eigenfunctions of the Laplace operator

TL;DR

This paper investigates the average number of common zeros of eigenfunctions of the Laplace operator on certain compact Riemannian manifolds, using geometric methods involving equivariant immersions into spheres.

Contribution

It provides a formula for the average number of common zeros of eigenfunctions on homogeneous manifolds with irreducible isotropy representation.

Findings

Derived the volume of the image of the manifold under an equivariant immersion.

Computed the average number of common zeros for eigenfunctions on specific manifolds.

Connected geometric properties with spectral characteristics of the Laplace operator.

Abstract

We consider the eigenfunctions of the Laplace operator $\Delta $ on a compact Riemannian manifold of dimension $n$. For $M$ homogeneous with irreducible isotropy representation and for a fixed eigenvalue of $\Delta $ we find the average number of common zeros of $n$ eigenfunctions. For this we compute the volume of the image of $M$ under an equivariant immersion into a sphere.