Bounding the invariant spectrum of the Laplacian

Stuart Hall, Newcastle University. Part of the pure mathematics, geometry seminar series.

I will report on an ongoing project with Tommy Murphy to understand spectrum of the Laplacian  of a Riemannian metric invariant under the action of a compact Lie Group G. We are interested in questions such as can you bound the kth –invariant eigenvalue independent of the metric?  Are such bounds sharp and are there smooth metrics that achieve them? The only large family of manifolds that have a substantial set of answers to such questions are toric Kahler manifolds. I’ll try to detail what is known here and how techniques might be generalised to other settings.

Stuart Hall, Newcastle University