The Energy Functional of $$G_2$$-Structures Compatible with a Background Metric

The space of $$G_2$$ -structures is naturally stratified by those structures compatible with a fixed Riemannian metric. We study the restriction of the total torsion energy functional to these strata. Precisely we show the short-time existence of its negative gradient flow, we characterise the space of its critical points in terms of spinor fields and, finally, we describe the long-time behaviour of the homogeneous negative gradient flow.