# Regularity estimates for the gradient flow of a spinorial energy functional

## TL;DR

This paper derives regularity estimates for the spinor flow, showing that unbounded second derivatives of the spinor field are the main obstacle to long-term existence, extending previous blow-up criteria to higher dimensions.

## Contribution

It generalizes blow-up criteria for the spinor flow from surfaces to higher dimensions and provides a lower bound on existence time based on initial data.

## Key findings

- Unbounded second derivatives cause flow blow-up.
- Flow metric is equivalent to a modified Ricci flow after diffeomorphism.
- Provides a lower bound for the flow's existence time.

## Abstract

In this note, we establish certain regularity estimates for the spinor flow introduced and initially studied in \cite{AWW2016}. Consequently, we obtain that the norm of the second order covariant derivative of the spinor field becoming unbounded is the only obstruction for long-time existence of the spinor flow. This generalizes the blow up criteria obtained in \cite{Sc2018} 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.