A spinorial energy functional: critical points and gradient flow

TL;DR

This paper introduces a spinorial energy functional on the universal bundle of unit spinors, characterizes its critical points as Ricci-flat metrics with parallel spinors, and studies the associated spinor flow including short-time existence and uniqueness.

Contribution

It defines a new energy functional related to spinors, characterizes its critical points, and analyzes the spinor flow dynamics.

Findings

Critical points correspond to Ricci-flat metrics with parallel spinors.

Established short-time existence and uniqueness of the spinor flow.

Provided foundational properties of the spinorial energy functional.

Abstract

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.