Stability of Catenoids and Helicoids in Hyperbolic Space

TL;DR

This paper investigates the stability properties of catenoids and helicoids in hyperbolic 3-space, identifying parameter ranges for stability, instability, and least area characteristics, thus advancing understanding of minimal surfaces in hyperbolic geometry.

Contribution

It provides a detailed stability analysis of spherical minimal catenoids and helicoids in hyperbolic space, establishing precise parameter thresholds for different stability regimes.

Findings

Catenoids are unstable with index one for small parameters and stable for larger ones.

Helicoids are stable up to a certain parameter and unstable with infinite index beyond that.

Identifies critical constants separating stability and instability regimes.

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 $\bar{a}>\bar{a}_c$.