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 $a\ge a_c$, and $\bullet$ ${\mathcal{C}}_a$ is a least area minimal surface in the sense of Meeks and Yau if $a\ge a_l$. (2) For a family of minimal helicoids $\{{\mathcal{H}}_{\bar{a}}{\}}_{\bar{a}\ge 0}$ 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 $0\le \bar{a}\le {\bar{a}}_c$, and $\bullet$ ${\mathcal{H}}_{\bar{a}}$ is an unstable minimal surface with index infinity if $\bar{a}>{\bar{a}}_c$.

This paper replaces both arXiv:1204.4943 and arXiv:1502.04764

Cited by in corpus (1)

• Complex length of short curves and Minimal Fibration in hyperbolic $3$-Manifolds fibering over the circle