The spinorial energy functional: solutions of the gradient flow on Berger spheres

TL;DR

This paper investigates the gradient flow of the spinorial energy functional on Berger spheres, demonstrating sphere collapse, stability of certain critical points, and providing examples of non-Killing spinor critical points.

Contribution

It introduces analysis of the flow on Berger spheres, showing collapse phenomena and identifying stable and non-Killing spinor critical points.

Findings

Berger spheres collapse to 2D spheres under the flow

Standard 3-sphere with a Killing spinor is a stable critical point

Existence of non-Killing spinor critical points on the 3-sphere

Abstract

We study the negative gradient flow of the spinorial energy functional (introduced by Ammann, Wei{\ss}, and Witt) on 3-dimensional Berger spheres. For a certain class of spinors we show that the Berger spheres collapse to a 2-dimensional sphere. Moreover, for special cases, we prove that the volume-normalized standard 3-sphere together with a Killing spinor is a stable critical point of the volume-normalized version of the flow. Our results also include an example of a critical point of the volume-normalized flow on the 3-sphere, which is not a Killing spinor.