Magnetic ground states and the conformal class of a surface

Abstract

On a closed, orientable Riemannian surface ${\Sigma }_g$ of arbitrary genus $g\ge 1$ and Riemannian metric $h$ we study the magnetic Laplacian with magnetic potential given by a harmonic $1$-form $A$. Its lowest eigenvalue (magnetic ground state energy) is positive, unless $A$ represents an integral cohomology class. We isolate a countable set of ground state energies which we call $\text{ground state spectrum}$ of the metric $h$. The main result of the paper is to show that the ground state spectrum determines the volume and the conformal class of the metric $h$. In particular, hyperbolic metrics are distinguished by their ground state spectrum. We also compute the magnetic spectrum of flat tori and introduce some magnetic spectral invariants of $({\Sigma }_g,h)$ which are conformal by definition and involve the geometry of what we call the Jacobian torus of $({\Sigma }_g,h)$ (in Algebraic Geometry, the Jacobian variety of a Riemann surface).