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

TL;DR

This paper explores the critical points and gradient flows of the $G_2$-Hilbert functional on manifolds with circle actions, reducing the problem to lower dimensions and analyzing flow behavior.

Contribution

It introduces geometric reductions of the $G_2$-Hilbert functional via circle actions, analyzing invariant structures and their gradient flows under specific ansatzes.

Findings

Flow admits only trivial stationary configurations: flat connection, scalar-flat base metric, constant fiber length.

Reduction to 6-dimensional quotient simplifies the variational problem.

Formal negative $L^2$-gradient flow analysis shows limited non-trivial solutions.

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-K\"ahler 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.