Stability of catenoids and helicoids in hyperbolic space (Q2000882)

The author studies the stability of catenoids and helicoids in $3$-dimensional hyperbolic space $\mathbb H^3$. Note that spherical minimal catenoids and minimal helicoids in $\mathbb H^3$ are parametrized as a certain natural family $\{\mathcal C_a\}_{a>0}$ and $\{\mathcal H_{\bar{a}}\}_{\bar{a}\geqslant 0}$, respectively. He proves that there exists a constant $a_l>0$ such that $\mathcal C_a$ is a least area minimal surface if $a\geqslant a_l$. He also proves that there exists a constant $\bar{a}_c>0$ such that $\mathcal H_{\bar{a}}$ is a globally stable minimal surface if $0\leqslant\bar{a}\leqslant\bar{a}_c$, and $\mathcal H_{\bar{a}}$ is an unstable minimal surface with Morse index infinity if $\bar{a}>\bar{a}_c$.