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

Abstract

We study the negative gradient flow of the spinorial energy functional (introduced by Ammann, Weiß, 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 volumenormalized 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.

Homogeneous Spinor Flow

“…These are again exactly the equations that Wittmann gets in the proof of his stability result [21] and so any solution has T + = ∞ and converges for t → ∞ and any ǫ > 0 to a Killing spinor on S 3 by [21].…”

“…The spinor flow on Berger spheres has been analyzed by Wittmann in [21] by constructing adapted initial spinor fields and calculating the terms Q 1 and Q 2 explicitly. The system above is exactly the same as Wittmann obtains in [21,Lemma 8] for µ = − a 4 and a ∈ {−2, 2} arbitrary. In particular, we see that for ǫ = 1, the solution is given by x(t) = y(t) = 1 − t 16 .…”