Nonlinear Dirac equations on Riemann surfaces

TL;DR

This paper develops analytical techniques for nonlinear Dirac equations on Riemann surfaces, including regularity, singularity removal, and energy identities, with applications to geometric surface representations.

Contribution

It introduces new analytical methods for nonlinear Dirac equations, addressing regularity and singularity issues in geometric contexts.

Findings

Established small energy regularity results.

Proved removable singularity theorems.

Derived energy identities for solutions.

Abstract

We develop analytical methods for nonlinear Dirac equations. Examples of such equations include Dirac-harmonic maps with curvature term and the equations describing the generalized Weierstrass representation of surfaces in three-manifolds. We provide the key analytical steps, i.e., small energy regularity and removable singularity theorems and energy identities for solutions.