A spinorial energy functional: critical points and gradient flow
  On the universal bundle of unit spinors we study a natural energy functional whose critical points, if dim M \geq 3, are precisely the pairs (g, {\phi}) consisting of a Ricci-flat Riemannian metric g together with a parallel g-spinor {\phi}. We investigate the basic properties of this functional and study its negative gradient flow, the so-called spinor flow. In particular, we prove short-time existence and uniqueness for this flow. 
Bernd Ammann, Hartmut Weiss, and Frederik Witt
math.DG math.AP