Minimal hypersurfaces in nearly [formula omitted] manifolds

Abstract

We study hypersurfaces in a nearly G2 manifold. We define various quantities associated to such a hypersurface using the G2 structure of the ambient manifold and prove several relationships between them. In particular, we give a necessary and sufficient condition for a hypersurface with an almost complex structure induced from the G2 structure of the ambient manifold, to be nearly Kähler. Then using the nearly G2 structure on the round sphere S7, we prove that for a compact minimal hypersurface M6 of constant scalar curvature in S7 with the shape operator A satisfying |A|2>6, there exists an eigenvalue λ>12 of the Laplace operator on M such that |A|2=λ−6, thus giving the next discrete value of |A|2 greater than 0 and 6, thus generalizing the result of Deshmukh (2010) about nearly Kähler S6.