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

TL;DR

This paper investigates hypersurfaces in nearly G2 manifolds, establishing conditions for nearly Kähler structures and analyzing minimal hypersurfaces in the 7-sphere, revealing new spectral properties related to the shape operator.

Contribution

It introduces new relationships between geometric quantities of hypersurfaces in nearly G2 manifolds and characterizes when such hypersurfaces are nearly Kähler, extending spectral results to higher dimensions.

Findings

Necessary and sufficient conditions for nearly Kähler hypersurfaces in nearly G2 manifolds.

Identification of eigenvalues related to the shape operator on minimal hypersurfaces in S^7.

Generalization of spectral properties of minimal hypersurfaces from S^6 to higher dimensions.

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 $S^6$.