Minimal Hypersurfaces in nearly $\mathrm{G}_2$ Manifolds

Abstract

We study hypersurfaces in a nearly $\mathrm{G}_2$ manifold. We define various quantities associated to such a hypersurface using the $\mathrm{G}_2$ 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 $\mathrm{G}_2$ structure of the ambient manifold, to be nearly K$\ddot{\text{a}}$hler. Then using the nearly $\mathrm{G}_2$ structure on the round sphere $S^7$, we prove that for a compact minimal hypersurface $M^6$ of constant scalar curvature in $S^7$ with the shape operator $A$ satisfying $|A|^2>6$, there exists an eigenvalue $\lambda >12$ of the Laplace operator on $M$ such that $|A|^2=\lambda - 6$, thus giving the next discrete value of $|A|^2$ greater than $0$ and $6$, thus generalizing an earlier result about nearly K$\ddot{\text{a}}$hler $S^6$.