Magnetic ground states and the conformal class of a surface

TL;DR

This work analyzes the magnetic Laplacian on closed orientable surfaces and shows that the ground state spectrum, obtained from energies corresponding to harmonic potentials, encodes both the surface volume and its conformal class. By establishing sharp asymptotics for small magnetic potentials and leveraging Torelli’s theorem, the authors prove that the magnetic ground state spectrum determines the Jacobian (and hence the conformal class) of the surface; in genus ≥2 this yields isometry for hyperbolic metrics within a conformal class, while in genus 1 the flat metric is uniquely determined by the spectrum. They introduce conformal spectral invariants built from the Jacobian torus and compute the magnetic spectrum for flat tori, revealing sharp extremal results (e.g., hexagonal lattice maximizing certain invariants). The paper thus provides a novel link between magnetic spectral data and conformal geometry, offering new tools (Jacobian-based invariants) for distinguishing surfaces beyond the classical Laplace spectrum. It also extends the analysis to higher-dimensional flat tori and lays out a detailed framework for recovering the conformal structure from a countable ground state sequence.

Abstract

On a closed, orientable Riemannian surface $Σ_g$ of arbitrary genus $g\geq 1$ 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 $\textit{ground state spectrum}$ 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 $(Σ_g,h)$ which are conformal by definition and involve the geometry of what we call the Jacobian torus of $(Σ_g,h)$ (in Algebraic Geometry, the Jacobian variety of a Riemann surface).

Figures (6)

• Figure 1: The spectrum of a flat torus for a potential $A=2\pi(\alpha\,dx_1+\beta\,dx_2)$ is given by the squared distances of $P_A=(\alpha,\beta)$ from the points of $L^{\star}$ (i.e., the squared lengths of the segments in the picture), up to a factor $4\pi^2$. The first eigenvalue corresponds to the shortest distance, i.e., to the length of the bold segment in the picture.
• Figure 2: When $P_A\in L^{\star}$ we get the usual Laplace-Beltrami spectrum.
• Figure 3: Moduli space of flat tori.
• Figure 4: The point in the acute triangle $\tau$ at maximum distance from the lattice $L^{\star}$ is the circumcenter of the triangle.
• Figure 5:

Theorems & Definitions (48)

• Lemma 1
• Theorem 2
• Theorem 3
• Theorem 4
• Corollary 5
• Theorem 6
• Remark 7
• Corollary 8
• Theorem 9
• Theorem 10