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Mathematical Research Letters
Volume 28 (2021)
Number 4
Regularity estimates for the gradient flow of a spinorial energy functional
Pages: 1125 – 1173
DOI: https://dx.doi.org/10.4310/MRL.2021.v28.n4.a7
Authors
Abstract
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