On spectral convergence of vector bundles and convergence of principal bundles

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.

53C05, 58J50