Spectral estimates on 2-tori

TL;DR

This paper establishes bounds for eigenvalues of the Dirac and Laplace operators on 2-tori, highlighting the influence of spin structures and deriving implications for geometric functionals like the Willmore energy.

Contribution

It provides the first explicit eigenvalue estimates for the Dirac operator on tori that incorporate spin structure information.

Findings

Lower bounds for the first Dirac eigenvalue depending on spin structure

Bounds for Laplace eigenvalues on 2-tori

Comparison of Dirac spectra for different spin structures

Abstract

We prove upper and lower bounds for the eigenvalues of the Dirac operator and the Laplace operator on 2-dimensional tori. In particluar we give a lower bound for the first eigenvalue of the Dirac operator for non-trivial spin structures. It is the only explicit estimate for eigenvalues of the Dirac operator known so far that uses information about the spin structure. As a corollary we obtain lower bounds for the Willmore functional of a torus embedded into S_3. In the final section we compare Dirac spectra for two different spin structures on an arbitrary Riemannian spin manifold.