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

TL;DR

This paper investigates the properties of complete two-sided $ ext{delta}$-stable minimal hypersurfaces in Euclidean space, establishing conditions for Euclidean volume growth and characterizing when such hypersurfaces are hyperplanes, especially in dimensions 3 to 5.

Contribution

It provides new criteria for Euclidean volume growth and hyperplane characterization of $ ext{delta}$-stable minimal hypersurfaces in specific dimensions.

Findings

Complete two-sided $ ext{delta}$-stable hypersurfaces have Euclidean volume growth for certain $ ext{delta}$ values.

Conditions under which these hypersurfaces are hyperplanes are established.

Explicit $ ext{delta}$ thresholds are identified for dimensions 3 to 5.

Abstract

In this paper, we study complete $\delta$-stable minimal hypersurfaces in $\mathbf R^{n+1}$. We prove that complete two-sided $\delta$-stable minimal hypersurfaces have Euclidean volume growth if $3\leq n\leq 5$ and $\delta>\delta_0(n)$, where $\delta_0(3)=1/3$, $\delta_0(4)=1/2$ and $\delta_0(5)=21/22$. We also give a sufficient condition such that complete two-sided $\delta$-stable minimal hypersurfaces in $\mathbf R^{n+1}$ is the hyperplane. Furthermore, we prove that a complete two-sided $\delta$-stable minimal hypersurface is the hyperplane if $3\leq n\leq 5$ and $\delta>\delta_1(n)$, where $\delta_1(3)=3/8$, $\delta_1(4)=2/3$ and $\delta_1(5)=21/22$.