On common zeros of eigenfunctions of the Laplace operator

Abstract

We consider the eigenfunctions of the Laplace operator $\Delta$ on a compact Riemannian manifold M of dimension n. For M homogeneous with irreducible isotropy representation and for a fixed eigenvalue $\lambda$ of $\Delta$ we find the average number of common zeros of n eigenfunctions. It turns out that, up to a constant depending on n, this number equals $\lambda ^{n/2}\mathrm{vol}\,M$, the expression known from the celebrated Weyl’s law. To prove this we compute the volume of the image of M under an equivariant immersion into a sphere.