The $G_2$ geometry of $3$-Sasaki structures

Abstract

We initiate a systematic study of the deformation theory of the second Einstein metric $g_{1/\sqrt{5}}$ respectively the proper nearly $G_2$ structure ${\phi }_{1/\sqrt{5}}$ of a $3$-Sasaki manifold $(M^7,g)$. We show that infinitesimal Einstein deformations for $g_{1/\sqrt{5}}$ coincide with infinitesimal $G_2$ deformations for ${\phi }_{1/\sqrt{5}}$. The latter are showed to be parametrised by eigenfunctions of the basic Laplacian of $g$, with eigenvalue twice the Einstein constant of the base $4$-dimensional orbifold, via an explicit differential operator. In terms of this parametrisation we determine those infinitesimal $G_2$ deformations which are unobstructed to second order.

Final version.Some proofs were expanded, index of notations added

References in corpus (5)

• Stability and instability of Ricci solitons
• Deformation theory of nearly Kähler manifolds
• Deformation theory of deformed Hermitian Yang-Mills connections and deformed Donaldson-Thomas connections
• Dynamic instability of ${\mathbb{CP}}^N$ under Ricci flow
• Stability of Einstein metrics on fiber bundles