Estimates and monotonicity for a heat flow of isometric $$G_{2}$$-structures

Abstract

Given a $7$-dimensional compact Riemannian manifold $\left( M,g\right) $ that admits $G_{2}$-structure, all the $G_{2}$-structures that are compatible with the metric $g$ are parametrized by unit sections of an octonion bundle over $M$. We define a natural energy functional on unit octonion sections and consider its associated heat flow. The critical points of this functional and flow precisely correspond to $G_{2}$-structures with divergence-free torsion. In this paper, we first derive estimates for derivatives of $V\left( t\right) $ along the flow and prove that the flow exists as long as the torsion remains bounded. We also prove a monotonicity formula and and an $\varepsilon $-regularity result for this flow. Finally, we show that within a metric class of $G_{2}$-structures that contains a torsion-free $G_{2}$-structure, under certain conditions, the flow will converge to a torsion-free $G_{2}$-structure.