A note on twisted Dirac operators on closed surfaces

TL;DR

This paper establishes an inequality linking nodal sets and eigenvalues of twisted Dirac operators on closed surfaces, providing eigenvalue estimates relevant to Dirac-harmonic maps and Liouville type results.

Contribution

It introduces a new inequality for twisted Dirac operators on surfaces, connecting geometric and spectral properties, with applications to Dirac-harmonic maps.

Findings

Derived an inequality relating nodal sets and eigenvalues

Provided eigenvalue estimates for twisted Dirac operators

Obtained Liouville type results for related geometric equations

Abstract

We derive an inequality that relates nodal set and eigenvalues of a class of twisted Dirac operators on closed surfaces and point out how this inequality naturally arises as an eigenvalue estimate for the $\rm Spin^c$ Dirac operator. This allows us to obtain eigenvalue estimates for the twisted Dirac operator appearing in the context of Dirac-harmonic maps and their extensions, from which we also obtain several Liouville type results.