A Construction of Constant Mean Curvature Surfaces in ℍ2 × ℝ and the Krust Property

Abstract

Abstract We show the existence of a $2$-parameter family of properly Alexandrov-embedded surfaces with constant mean curvature $0\leq H\leq \frac {1}{2}$ in ${\mathbb {H}^2\times \mathbb {R}}$. They are symmetric with respect to a horizontal slice and $k$ vertical planes disposed symmetrically and extend the so-called minimal saddle towers and $k$-noids. We show that the orientation plays a fundamental role when $H>0$ by analyzing their conjugate minimal surfaces in $\widetilde {\textrm {SL}}_2(\mathbb {R})$ or $\textrm {Nil}_3$. We also discover new complete examples that we call $(H,k)$-nodoids, whose $k$ ends are asymptotic to vertical cylinders over curves of geodesic curvature $2H$ from the convex side, often giving rise to non-embedded examples if $H>0$. In the discussion of embeddedness of the constructed examples, we prove that the Krust property does not hold for any $H>0$, that is, there are minimal graphs over convex domains in $\widetilde {\textrm {SL}}_2(\mathbb {R})$, $\textrm {Nil}_3$ or the Berger spheres, whose conjugate surfaces with constant mean curvature $H$ in $\mathbb {H}^2\times \mathbb {R}$ are not graphs.