Stable complete constant mean curvature surfaces in 𝑅³ and 𝐻³

Abstract

We construct some 1 1 -parameter families of complete rotation surfaces with constant mean curvature in the hyperbolic 3 3 -space H 3 {H^3} of constant sectional curvature − 1 -1 , and show that some of them are stable for the variational problem of area together with oriented volume, and that a complete connected, oriented surface with constant mean curvature in the Euclidean 3 3 -space R 3 {R^3} which is stable for the variational problem is a plane.