Spectral properties of magnetic fields on sub-Riemannian contact manifolds

Abstract

Motivated by some recent studies of the magnetic Laplacian on Riemannian manifolds, we focus on the first eigenvalue of the magnetic horizontal Laplacian on contact manifolds. We characterize conditions for positive spectral shift, and provide some sharp upper bounds. In the Riemannian setting, a genus 1 assumption is known to force the underlying metric to be flat when equality holds in the sharp upper bounds. Interestingly, we show that the equivalent topological condition in the three--dimensional contact setting consists of having first Betti number equal to 2. In this case, equality in our upper bounds implies that the structure is that of a Heisenberg left--invariant nilmanifold. We conclude by showing that, in some specific three--dimensional contact settings, the knowledge of the first eigenvalue of the magnetic Laplacian uniquely determines the manifold Chern class, fully determining the topology of the underlying manifold.