Average number of zeros and mixed symplectic volume of Finsler sets

Abstract

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.

Crofton formulas in pseudo-Riemannian space forms

“…), where we write Λ 1 1 for oriented null lines. Let Σ M F be its suspension with poles at two copies of L 0 with opposite orientations.…”

“…Proof. For ζ > 1 2 , Q ζ is positive definite, and so e * m ζ k = m ζ k by the uniqueness of probability measure on the Grassmannian invariant under the positive definite orthogonal group, as e * : M ∞ (Gr p+q+l+r−k (R p+l,q+r )) → M ∞ (Gr p+q−k (R p,q )) is essentially the pushforward operation under intersection with R p,q . The statement then follows by analytic extension in ζ, combined with Proposition 5.10.…”

“…Among the many applications in integral geometry of such formulas, we mention the construction of a basis of the space of unitarily invariant valuations on C n by Alesker [3], the interpretation of Alesker's product of smooth and even valuations in terms of Crofton measures [11], the Holmes-Thompson intrinsic volumes of projective metrics [10]. Applications outside integral geometry include isoperimetric inequalities in Riemannian geometry [20], symplectic geometry [31], systolic geometry [37], algebraic geometry [1] and more. Crofton formulas are employed also outside of pure mathematics, in domains such as microscopy and stereology, see [27].…”

“…On spheres and hyperbolic spaces, a natural class of subsets are the totally geodesic submanifolds of a fixed dimension, endowed with the invariant measure. On the 2-dimensional unit sphere, we have a formula similar to (1), with affine lines replaced by equators. This formula is the main ingredient in the proof of the Fáry-Milnor theorem that the total curvature of a knot in R 3 is bigger than 4π if the knot is non-trivial.…”

How many roots of a system of random trigonometric polynomials are real?

“…mixed volume of some ellipsoids ell(Λ i ), depending on the supports Λ i . This theorem is a specialization of [AK,Theorem 1] to the case X = T n and V i = V (Λ i ); see also [ZK]. The second theorem states that for almost all tuples of n real Laurent polynomials with supports Λ 1 , .…”