Average number of zeros and mixed symplectic volume of Finsler sets

TL;DR

This paper establishes a connection between the average number of common zeros of functions in finite-dimensional spaces on a manifold and the mixed symplectic volume of associated Finsler ellipsoids, with applications to Laplace eigenspaces.

Contribution

It introduces a novel relation between zeros of functions and symplectic geometry, extending Crofton formulas and inequalities to Finsler sets on manifolds.

Findings

Average zeros equal mixed symplectic volume of Finsler ellipsoids

Inequalities similar to Hodge inequalities for invariant cases

Application to Laplace operator eigenspaces

Abstract

Let $X$ be an $n$-dimensional manifold and $V_1, \ldots, V_n \subset C^\infty(X, \mathbb R)$ 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 of $X$. We prove that the average number of isolated common zeros of $f_1 \in V_1, \ldots, f_n \in V_n$ 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 $V_i$ and 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 for a generalization of Crofton formula to the product of spheres.