Analysis of the harmonic flow of geometric structures

Abstract

We develop an analysis of the flow of harmonic $H$-structures with the strategy introduced by Chen-Struwe for the harmonic map heat equation when the target does not necessarily have negative sectional curvature, so the Eells-Sampson Theorem cannot apply. In particular, this flow method enables us to find theoretical hypotheses under which the existence of a torsion-free $H$-structure is guaranteed and conditions for which the flow must blow up in finite time. This extends results already known for some specific groups like $\mathrm{U}(n)$, $\mathrm{G}_2$ or $\mathrm{Spin}(7)$.