Flows of $G_2$-structures, II: Curvature, torsion, symbols, and functionals

Abstract

We continue the investigation of general geometric flows of $G_2$-structures initiated by the third author in "Flows of $G_2$-structures, I." Specifically, we determine the possible geometric flows (up to lower order terms) of $G_2$-structures which are second order quasilinear, by explicitly computing all independent second order differential invariants of $G_2$-structures which are $3$-forms. There are four symmetric $2$-tensors and two vector fields. We do this by deriving explicit computational descriptions of the decompositions of the curvature and the covariant derivative of the torsion into irreducible $G_2$-representations, as well as the decomposition of the $G_2$-Bianchi identity into independent relations. We also show that these six tensors arise as leading order contributions to the Euler-Lagrange equations for the energy functionals of the four independent torsion components, and we establish a $G_2$-analogue of the classical block decomposition of the Riemann curvature operator on oriented $4$-dimensional Riemannian manifolds. Finally, we present a large class of geometric flows of $G_2$-structures which are directly amenable to a deTurck type trick to establish short-time existence and uniqueness, with no initial assumption on the torsion, vastly generalizing an earlier result of Weiss-Witt for the negative gradient flow of the Dirichlet energy. This result is proved through a careful analysis of the principal symbols of the linearizations of these operators, establishing particular linear combinations for which one can prove that the failure of strict parabolicity is due precisely to the diffeomorphism invariance. A detailed introductory section on various foundational results of $G_2$-structure, several of which are not readily available in the literature, should be of wider interest and applicability.