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

Abstract

In this paper, we study complete $\hat{I}´$-stable minimal hypersurfaces in ${\mathbf{R}}^{n+1}$. We prove that complete two-sided $\hat{I}´$-stable minimal hypersurfaces have Euclidean volume growth if $3\le n\le 5$ and $\hat{I}´>\hat{I}{´}_0(n)$, where $\hat{I}{´}_0(3)=1/3$, $\hat{I}{´}_0(4)=1/2$ and $\hat{I}{´}_0(5)=21/22$. We also give a sufficient condition such that complete two-sided $\hat{I}´$-stable minimal hypersurfaces in ${\mathbf{R}}^{n+1}$ is the hyperplane. Furthermore, we prove that a complete two-sided $\hat{I}´$-stable minimal hypersurface is the hyperplane if $3\le n\le 5$ and $\hat{I}´>\hat{I}{´}_1(n)$, where $\hat{I}{´}_1(3)=3/8$, $\hat{I}{´}_1(4)=2/3$ and $\hat{I}{´}_1(5)=21/22$.

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