The spinorial energy functional on surfaces

This is a companion paper to arXiv:1207.3529 where we introduced the spinorial energy functional and studied its main properties in dimensions equal or greater than three. In this article we focus on the surface case. A salient feature here is the scale invariance of the functional which leads to a plenitude of critical points. Moreover, via the spinorial Weierstra{\ss} representation it relates to the Willmore energy of periodic immersions of surfaces into $\mathbb{R}^3$.