Stability of Catenoids and Helicoids in Hyperbolic Space

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In this paper, we study the stability of catenoids and helicoids in the hyperbolic 3-space H³. (1) For a family of spherical minimal catenoids {Cₐ}ₐ>₀ in H³, there exist two constants 0<ac<aₗ such that Cₐ is an unstable minimal surface with index one if a<ac, Cₐ is a globally stable minimal surface if a≥ac, and Cₐ is a least area minimal surface in the sense of Meeks and Yau if a≥aₗ. (2) For a family of minimal helicoids {Ha}a≥0 in H³, there exists a constant ac=(ac) such that Ha is a globally stable minimal surface if 0≤a≤ac, and Ha is an unstable minimal surface with index infinity if a>ac.