Approximations of the connection Laplacian spectra

TL;DR

This paper introduces a convolution-type operator on vector bundles over metric-measure spaces, extending previous work on functions to vector bundles, and demonstrates its spectral approximation properties for Euclidean or Hermitian connections on closed Riemannian manifolds.

Contribution

It extends the convolution Laplacian to vector bundles and proves spectral approximation results for the operator on Riemannian manifolds.

Findings

The spectrum of the operator approximates the connection Laplacian spectrum.

The operator generalizes the graph connection Laplacian.

Spectral convergence is established for Euclidean or Hermitian connections.

Abstract

We consider a convolution-type operator on vector bundles over metric-measure spaces. This operator extends the analogous convolution Laplacian on functions in our earlier work to vector bundles, and is a natural extension of the graph connection Laplacian. We prove that for Euclidean or Hermitian connections on closed Riemannian manifolds, the spectrum of this operator and that of the graph connection Laplacian both approximate the spectrum of the connection Laplacian.