Complete two-sided $δ$-stable minimal hypersurfaces in $\mathbf R^{n+1}$

In this paper, we study complete $δ$-stable minimal hypersurfaces in $\mathbf R^{n+1}$. We prove that complete two-sided $δ$-stable minimal hypersurfaces have Euclidean volume growth if $3\leq n\leq 5$ and $δ>δ_0(n)$, where $δ_0(3)=1/3$, $δ_0(4)=1/2$ and $δ_0(5)=21/22$. We also give a sufficient condition such that complete two-sided $δ$-stable minimal hypersurfaces in $\mathbf R^{n+1}$ is the hyperplane. Furthermore, we prove that a complete two-sided $δ$-stable minimal hypersurface is the hyperplane if $3\leq n\leq 5$ and $δ>δ_1(n)$, where $δ_1(3)=3/8$, $δ_1(4)=2/3$ and $δ_1(5)=21/22$.