Harmonic flow of $\mathrm{Spin}(7)$structures
Abstract
We formulate and study the isometric flow of $\mathrm{Spin}(7)$structures on compact $8$manifolds, as an instance of the harmonic flow of geometric structures. Starting from a general perspective, we establish Shitype estimates and a correspondence between harmonic solitons and selfsimilar solutions for arbitrary isometric flows of $H$structures. We then specialise to $H=\mathrm{Spin}(7)\subset\mathrm{SO}(8)$, obtaining conditions for longtime existence, via a monotonicity formula along the flow, which actually leads to an $\varepsilon$regularity theorem. Moreover, we prove CheegerGromov and Hamiltontype compactness theorems for the solutions of the harmonic flow, and we characterise Type$\mathrm{I}$ singularities as being modelled on shrinking solitons.We also establish a Bryanttype description of isometric $\mathrm{Spin}(7)$structures, based on squares of spinors, which may be of independent interest.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.06340
 Bibcode:
 2021arXiv210906340D
 Keywords:

 Mathematics  Differential Geometry;
 53C15;
 53C43;
 58J35;
 58J60
 EPrint:
 v2minor changes in the presentation and exposition. 47 pages