On the Dual Geometry of Laplacian Eigenfunctions

We discuss the geometry of Laplacian eigenfunctions -Delta φ = λ φ on compact manifolds (M,g) and combinatorial graphs G=(V,E). The 'dual' geometry of Laplacian eigenfunctions is well understood on mathbbTd (identified with mathbbZd) and ℝn (which is self-dual). The dual geometry is of tremendous role in various fields of pure and applied mathematics. The purpose of our paper is to point out a notion of similarity between eigenfunctions that allows to reconstruct that geometry. Our measure of 'similarity' α(φλ, φμ) between eigenfunctions φλ and φμ is given by a global average of local correlations α(φλ, φμ)2 = | φλ φμ |L2-2intM left( intM p(t,x,y)( φλ(y) - φλ(x))( φμ(y) - φμ(x)) dy right)2 dx, where p(t,x,y) is the classical heat kernel and e-t λ + e-t μ = 1. This notion recovers all classical notions of duality but is equally applicable to other (rough) geometries and graphs; many numerical examples in different continuous and discrete settings illustrate the result.