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arXiv · 2606.29659

Integral Gauss formula and the Poisson equation for the $G_2$-Laplacian

Timothy Buttsworth·Stepan Hudecek·Artem Pulemotov
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Abstract

We produce a formula, analogous to the Gauss-Codazzi equation, which relates the geometry of a $G_2$-structure and its Hodge Laplacian to the geometry of the induced $SU(3)$-structure on an embedded hypersurface. As an application, we obtain necessary conditions for the solvability of the Poisson equation for (not necessarily closed) $G_2$-structures in a neighbourhood of this hypersurface. Next, we prove that our conditions are sufficient in the cohomogeneity one setting, assuming the symmetry group is compact and simple.

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Timothy Buttsworth, Stepan Hudecek, Artem Pulemotov. 2026-06-28. Integral Gauss formula and the Poisson equation for the $G_2$-Laplacian. https://arxiv.org/abs/2606.29659

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