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arXiv · 2605.02074

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

Julieth Saavedra
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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.

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Julieth Saavedra. 2026-05-03. Geometric Reductions of the $G_2$-Hilbert Functional via Circle Actions. https://arxiv.org/abs/2605.02074

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