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

Timothy Buttsworth; Stepan Hudecek; Artem Pulemotov

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

Differential Geometry 2026-06-28 v1

Authors: Timothy Buttsworth , Stepan Hudecek , Artem Pulemotov

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.

Keywords

G2-structure Hodge Laplacian SU(3)-structure Gauss formula Poisson equation cohomogeneity one

Cite

@article{arxiv.2606.29659,
  title  = {Integral Gauss formula and the Poisson equation for the $G_2$-Laplacian},
  author = {Timothy Buttsworth and Stepan Hudecek and Artem Pulemotov},
  journal= {arXiv preprint arXiv:2606.29659},
  year   = {2026}
}

Comments

28 pages