On spectral convergence of vector bundles and convergence of principal bundles

TL;DR

This paper investigates how the eigenvalues of the connection Laplacian on vector bundles vary continuously under geometric convergence, introducing a new topology for principal G-bundles with G-connections.

Contribution

It introduces the asymptotically G-equivariant measured Gromov-Hausdorff topology and applies it to analyze spectral convergence of vector and principal bundles.

Findings

Eigenvalues of the connection Laplacian are continuous under the new topology.

Established a framework for the convergence of principal G-bundles with G-connections.

Demonstrated the applicability of the topology to Riemannian manifolds with G-actions.

Abstract

In this article we consider the continuity of the eigenvalues of the connection Laplacian of $G$-connections on vector bundles over Riemannian manifolds. To show it, we introduce the notion of the asymptotically $G$-equivariant measured Gromov-Hausdorff topology on the space of metric measure spaces with isometric $G$-actions, and apply it to the total spaces of principal $G$-bundles equipped with $G$-connections over Riemannian manifolds.