On the Dual Geometry of Laplacian Eigenfunctions

TL;DR

This paper introduces a new measure of similarity between Laplacian eigenfunctions that captures their dual geometry on manifolds and graphs, enabling reconstruction of geometric structures from eigenfunction data.

Contribution

The authors propose a novel global similarity measure for eigenfunctions that generalizes classical duality concepts and applies to various geometries and graphs.

Findings

The similarity measure recovers classical duality notions.

Numerical examples demonstrate applicability to different geometries.

Method effectively reconstructs geometry from eigenfunctions.

Abstract

We discuss the geometry of Laplacian eigenfunctions $-\Delta \phi = \lambda \phi$ 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' $ \alpha(\phi_{\lambda}, \phi_{\mu})$ between eigenfunctions $\phi_{\lambda}$ and $\phi_{\mu}$ is given by a global average of local correlations $$ \alpha(\phi_{\lambda}, \phi_{\mu})^2 = \| \phi_{\lambda} \phi_{\mu} \|_{L^2}^{-2}\int_{M}{ \left( \int_{M}{ p(t,x,y)( \phi_{\lambda}(y) - \phi_{\lambda}(x))( \phi_{\mu}(y) - \phi_{\mu}(x)) dy} \right)^2 dx},$$ where $p(t,x,y)$ is the classical heat kernel and $e^{-t \lambda} + e^{-t \mu} = 1$. 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.