Average number of zeros and mixed symplectic volume of Finsler sets - Akhiezer, Dmitri; Kazarnovskii, Boris (Geometric and functional analysis, 2018)
Let X be an n -dimensional manifold and V 1 , . . . , V n ⊂ C ∞ ( X , R ) finite-dimensional vector spaces with Euclidean metric. We assign to each V i a Finsler ellipsoid, i.e., a family of ellipsoids in the fibers of the cotangent bundle to X . We prove that the average number of isolated common zeros of f 1 ∈ V 1 , . . . , f n ∈ V n is equal to the mixed symplectic volume of these Finsler ellipsoids. If X is a homogeneous space of a compact Lie group and all vector spaces V i together with their Euclidean metrics are invariant, then the average numbers of zeros satisfy the inequalities, similar to Hodge inequalities for intersection numbers of divisors on a projective variety. This is applied to the eigenspaces of Laplace operator of an invariant Riemannian metric. The proofs are based on a construction of the ring of normal densities on X , an analogue of the ring of differential forms. In particular, this construction is used to carry over the Crofton formula to the product of spheres.