Stability of catenoids and helicoids in hyperbolic space

In this paper, we study the stability of catenoids and helicoids in $3$-dimensional hyperbolic space. We will prove the following results (Theorems 1.4 and 1.5):

1) For a family of spherical minimal catenoids ${\lbrace \mathcal{C}_a \rbrace}_{a \gt 0}$ in hyperbolic $3$-space (see §3 for detailed definitions), there exist a constant $a_l \gt 0$ such that $\mathcal{C}_a$ is a least area minimal surface (see §2.1 for the definition) if $a \geqslant a_l$.

2) For a family of minimal helicoids ${\lbrace \mathcal{H}_{\overline{a}} \rbrace}_{\overline{a} \geqslant 0}$ in hyperbolic $3$-space (see §2.4 for detailed definitions), there exists a constant $\overline{a}_c \gt 0$ such that

• $\mathcal{H}_{\overline{a}}$ is a globally stable minimal surface if $0 \leqslant \overline{a} \leqslant \overline{a}_c$, and

• $\mathcal{H}_{\overline{a}}$ is an unstable minimal surface with Morse index infinity if $\overline{a} \gt \overline{a}_c$.

Keywords

hyperbolic spaces, minimal surfaces, catenoids, helicoids, stability

2010 Mathematics Subject Classification

53A10

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Received 9 February 2017

Accepted 13 October 2017

Published 28 June 2019