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

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.

Conjugate Plateau constructions in product spaces

“…The proof of the following two lemmas, sometimes in particular cases, is scattered across the references [105,90,70,66,9,11]. Lemma 3.6 (Horizontal geodesics).…”

“…Note also that Younes and Melo's surfaces can be used as barriers to give ad hoc solutions to more general Jenkins-Serrin problems as in [11,Lem. 3.2].…”

“…For instance, this situation leads to type II self-intersections for certain values a, b < ∞ (i.e., in the case of saddle towers) and H > 0 because the fundamental piece escapes the slab where the boundary is contained (see [11,Prop. 4.6]).…”

“…We have already pointed out that type II self-intersections may occur even if Σ a,b is embedded (see Remark 6.4). Using Proposition 6.3 and the maximum principle with respect to horizontal planes, it is not difficult to see that this behavior is not possible for k = 2, see [11,Prop. 4.6] for the details.…”

“…In the case of H 2 × R and 0 < H ≤ 1 2 , Rodríguez and the first and second authors [11] have produced k-noids and saddle towers that extend Plehnert's, plus another family of H-surfaces, called k-nodoids, with genus 0 and k ≥ 2 ends that approach the asymptotic vertical planes from the convex side (unlike the k-noids, which lie in their concave side).…”