Geometric Reductions of the $G_2$-Hilbert Functional via Circle Actions

Abstract

In this paper, we study critical points and gradient flows of the $G_2$--Hilbert functional on a manifolds with free ${\mathbb{S}}^1$--actions. We analyze ${\mathbb{S}}^1$--invariant $G_2$--structures under the constant fiber-length non-Kähler transverse ansatz, reducing the variational problem to the $6$--dimensional quotient and we also consider a Gibbons--Hawking-type ansatz with varying fiber length and derive the formal negative $L^2$--gradient flow. We conclude that the unnormalized flow admits only trivial stationary configurations: flat connection, scalar-flat base metric, and constant fiber length.