Mathematical Research Letters

Volume 28 (2021)

Number 4

Regularity estimates for the gradient flow of a spinorial energy functional

Pages: 1125 – 1173



Fei He (School of Mathematical Sciences, Xiamen University, Xiamen, China)

Changliang Wang (School of Mathematical Sciences, Institute for Advanced Study, Tongji University, Shanghai, China)


In this article, we establish certain regularity estimates for the spinor flow introduced and initially studied in [AWW16]. Consequently, we prove that ${\lvert \nabla^2 \phi \rvert}_{L^\infty (M)} (t)$ goes to $\infty$ as $t$ approaches to the finite singular time. This generalizes the blow up criteria obtained in [Sch18] for surfaces to general dimensions. As another application of the estimates, we also obtain a lower bound for the existence time in terms of the initial data. Our estimates are based on an observation that, up to pulling back by a one-parameter family of diffeomorphisms, the metric part of the spinor flow is equivalent to a modified Ricci flow.

Fen He was partially supported by NSFC11801474, NSFFJ2019J05011, FRFCU11801474, and NSFC11971401.

Received 5 September 2019

Accepted 12 July 2020

Published 22 November 2021