Isoperimetric inequalities and sharp upper bounds for Aharonov-Bohm eigenvalues on surfaces

TL;DR

This paper derives isoperimetric inequalities and sharp upper bounds for the first eigenvalue of the magnetic Laplacian on simply connected surfaces, linking geometric properties with spectral bounds.

Contribution

It establishes new inequalities relating eigenvalues to curvature and identifies extremal domains for maximizing the first eigenvalue.

Findings

Maximal first eigenvalue for spherical domains is achieved by geodesic disks with magnetic pole at the center.

On the punctured sphere, the first eigenvalue is maximized when punctures are antipodal.

The bounds depend explicitly on Gaussian curvature and domain geometry.

Abstract

We consider the first eigenvalue of the magnetic Laplacian with zero magnetic field on simply connected compact surfaces and we establish isoperimetric inequalities and upper bounds in terms of a bound on the gaussian curvature. As a corollary, we prove that among all simply connected spherical domains of fixed area, the first eigenvalue is maximal for a geodesic disk with the pole of the magnetic potential at its center; also, for the sphere punctured at two points, the first eigenvalue is maximal when the punctures are antipodal.