Stability of Catenoids and Helicoids in Hyperbolic Space

Abstract

In this paper, we study the stability of catenoids and helicoids in the hyperbolic $3$-space $\mathbb{H}^3$. (1) For a family of spherical minimal catenoids $\{\mathcal{C}_a\}_{a>0}$ in $\mathbb{H}^3$, there exist two constants $0<a_c<a_l$ such that $\bullet$ $\mathcal{C}_a$ is an unstable minimal surface with index one if $a<a_c$, $\bullet$ $\mathcal{C}_a$ is a globally stable minimal surface if $a\geq{}a_c$, and $\bullet$ $\mathcal{C}_a$ is a least area minimal surface in the sense of Meeks and Yau if $a\geq{}a_l$. (2) For a family of minimal helicoids $\{\mathcal{H}_{\bar{a}}\}_{\bar{a}\geq{}0}$ in $\mathbb{H}^3$, there exists a constant $\bar{a}_c=\coth(a_c)$ such that $\bullet$ $\mathcal{H}_{\bar{a}}$ is a globally stable minimal surface if $0\leq\bar{a}\leq\bar{a}_c$, and $\bullet$ $\mathcal{H}_{\bar{a}}$ is an unstable minimal surface with index infinity if $\bar{a}>\bar{a}_c$.