# An estimate for the average number of common zeros of Laplacian eigenfunctions

## TL;DR

This paper provides an upper bound estimate for the average number of common zeros of Laplacian eigenfunctions on homogeneous compact Riemannian manifolds, with an explicit constant and conditions for equality.

## Contribution

It introduces a new estimate for common zeros of eigenfunctions on homogeneous manifolds, utilizing Crofton's formula, and specifies conditions for exactness.

## Key findings

- Upper bound matches Weyl's law expression
- Equality holds when isotropy representation is irreducible
- Explicit constant c(n) is provided

## Abstract

On a compact Riemannian manifold $M$ of dimension $n$, we consider $n$ eigenfunctions of the Laplace operator $\Delta $ with eigenvalue $\lambda$. If $M$ is homogeneous under a compact Lie group preserving the metric then we prove that the average number of common zeros of $n$ eigenfunctions does not exceed $c(n)\lambda^{n/2}{\rm vol}\,M$, the expression known from the celebrated Weyl's law. Moreover, if the isotropy representation is irreducible, then the estimate turns into equality. The constant $c(n)$ is explicitly given. The method of proof is based on the application of Crofton's formula for the sphere.