On the Dual Geometry of Laplacian Eigenfunctions
This work addresses a foundational problem in pure and applied mathematics, offering a novel approach to reconstruct dual geometry, but it appears incremental as it builds on known concepts without claiming broad new applications.
The paper tackles the problem of understanding the dual geometry of Laplacian eigenfunctions on compact manifolds and graphs by introducing a similarity measure based on local correlations, which recovers classical duality notions and is applicable to various geometries, as demonstrated through numerical examples.
We discuss the geometry of Laplacian eigenfunctions $-Δφ= λφ$ on compact manifolds $(M,g)$ and combinatorial graphs $G=(V,E)$. The 'dual' geometry of Laplacian eigenfunctions is well understood on $\mathbb{T}^d$ (identified with $\mathbb{Z}^d$) and $\mathbb{R}^n$ (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(φ_λ, φ_μ)$ between eigenfunctions $φ_λ$ and $φ_μ$ is given by a global average of local correlations $$ α(φ_λ, φ_μ)^2 = \| φ_λ φ_μ \|_{L^2}^{-2}\int_{M}{ \left( \int_{M}{ p(t,x,y)( φ_λ(y) - φ_λ(x))( φ_μ(y) - φ_μ(x)) dy} \right)^2 dx},$$ 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.