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We study hypersurfaces in a nearly $\mathrm{G}_2$ manifold. We define variousquantities associated to such a hypersurface using the $\mathrm{G}_2$ structureof the ambient manifold and prove several relationships between them. Inparticular, we give a necessary and sufficient condition for a hypersurfacewith an almost complex structure induced from the $\mathrm{G}_2$ structure ofthe ambient manifold, to be nearly K$\ddot{\text{a}}$hler. Then using thenearly $\mathrm{G}_2$ structure on the round sphere $S^7$, we prove that for acompact minimal hypersurface $M^6$ of constant scalar curvature in $S^7$ withthe 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 $6$, thusgeneralizing an earlier result about nearly K$\ddot{\text{a}}$hler $S^6$.

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