Geodesic
An estimate for the average number of common zeros of Laplacian eigenfunctions
Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 78 (2017) no. 1, pp. 145-154.

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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 the equality. The constant $ c(n)$ is explicitly given. The method of proof is based on the application of Crofton's formula for the sphere. 
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Dmitri Akhiezer; Boris Kazarnovskii. An estimate for the average number of common zeros of Laplacian eigenfunctions. Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 78 (2017) no. 1, pp. 145-154. http://geodesic.mathdoc.fr/item/MMO_2017_78_1_a6/

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