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

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.