Deformations of homogeneous associative submanifolds in nearly parallel \(G_2\)-manifolds (Q1688489)

For any Riemannian manifold \((M,g)\), consider its Riemannian cone \((C(M), \bar{g}) = (\mathbb R_{>0}\times M, dr^2 + r^2g)\). A Riemannian \(7\)-manifold \((M, g)\) is called a nearly parallel \(G_2\)-manifold if the holonomy group of \(\bar{g}\) is contained in \(\mathrm{Spin}(7)\). In other words, it is a spin \(7\)-manifold with a real Killing spinor. There is a special class of calibrated submanifolds called Cayley submanifolds in \(C(M)\). An associative submanifold in \(M\) is a minimal \(3\)-submanifold whose cone is Cayley. The author studies Cayley cone deformations, explicitly when they are homogeneous in the sphere \(S^7\). The homogeneous associative submanifolds in \(S^7\) have been classified by \textit{J.D. Lotay} in [Proc. Lond. Math. Soc. (3) 105, No. 6, 1183--1214 (2012; Zbl 1268.53019)]. There are 8 types of such submanifolds: \(A_1\), \(A_2\) and \(A_3\) not lying in a totally geodesic nearly Kähler \(S^6\), Lagrangian submanifolds \(L_1\), \(L_2\), \(L_3\), and \(L_4\) in \(S^6\), and the totally geodesic \(S^3\). Infinitesimal Lagrangian deformations in \(S^6\) studied previously by \textit{J.D. Lotay} in [Commun. Anal. Geom. 20, No. 4, 803--867 (2012; Zbl 1266.53055)]. In the paper under review, all other infinitesimal deformations are studied. The main results are the following: As an associative submanifold, \(A_1\) is rigid, while \(A_2\) and \(A_3\) are not rigid. The deformation space of \(A_2\) is unobstructed, and all non-trivial associative deformations of \(A_2\) are induced by the \(\mathrm{PGL}(4,\mathbb C)\)-action on \(\mathbb CP^3\) via the Hopf lift (Theorem 1.1). All the associative and non-Lagrangian deformations of the totally geodesic \(S^3\), \(L_1\), \(L_2\), \(L_3\), and \(L_4\) are trivial. In other words, such deformations are induced from \(\mathrm{Spin}(7) \setminus G_2\) (Theorem 1.2).