Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A construction of constant mean curvature surfaces in $\mathbb{H}^2\times\mathbb{R}$ and the Krust property

Jesús Castro-Infantes, José M. Manzano, Magdalena Rodríguez

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 a $k$ 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 $\widetilde{\mathrm{SL}}_2(\mathbb{R})$ or $\mathrm{Nil}_3$. 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 $H>0$, i.e., there are minimal graphs over convex domains in $\widetilde{\mathrm{SL}}_2(\mathbb{R})$, $\mathrm{Nil}_3$ or the Berger spheres, whose conjugate surfaces with constant mean curvature $H$ in $\mathbb{H}^2\times\mathbb{R}$ are not graphs.

A construction of constant mean curvature surfaces in $\mathbb{H}^2\times\mathbb{R}$ and the Krust property

Abstract

We show the existence of a -parameter family of properly Alexandrov-embedded surfaces with constant mean curvature in . They are symmetric with respect to a horizontal slice and a vertical planes disposed symmetrically, and extend the so called minimal saddle towers and -noids. We show that the orientation plays a fundamental role when by analyzing their conjugate minimal surfaces in or . We also discover new complete examples that we call -nodoids, whose ends are asymptotic to vertical cylinders over curves of geodesic curvature from the convex side, often giving rise to non-embedded examples if . In the discussion of embeddedness of the constructed examples, we prove that the Krust property does not hold for any , i.e., there are minimal graphs over convex domains in , or the Berger spheres, whose conjugate surfaces with constant mean curvature in are not graphs.

Paper Structure

This paper contains 12 sections, 13 theorems, 28 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conjugation of $H$-surfaces
  4. Minimal graphs
  5. On the solution of the Jenkins-Serrin problem
  6. The case of $\widetilde{\mathrm{SL}}_2(\mathbb{R})$
  7. The case of $\mathrm{Nil}_3$
  8. The conjugate construction
  9. The construction of saddle towers
  10. The construction of $(H,k)$-noids
  11. The construction of $(H,k)$-nodoids
  12. The Krust property

Key Result

Theorem 1.1

For any $H>0$, there are minimal graphs over convex domains in $\mathbb{E}(4H^2-1,H)$ whose conjugate $H$-surfaces in $\mathbb{H}^2\times\mathbb{R}$ are not embedded.

Figures (6)

  • Figure 1: Moduli space of the surfaces $\overline\Sigma_{a,b}^*$ for $H\in[0,\frac{1}{2}]$ and $k\geq 2$ fixed. The black dot represents the limit $\overline\Sigma_{\infty,\infty}^*$ (see Section \ref{['sec:constructions']}) and the dashed lines indicate the minimal $k$-noids in $\mathbb{R}^3$ obtained in the limit after rescaling the metric.
  • Figure 2: Rotation of the normal of the horizontal helicoids $\mathcal{H}_\mu$ in the cases $\mu<\frac{-1}{2}$ (left) and $\mu>\frac{1}{2}$ (right).
  • Figure 3: The domain $\Omega_{a,b}$ with $a,b\in(0,\infty)$ is represented on the left (case $\kappa<0$) and center ($\kappa=0$) for $k=3$. The triangle $T_{a,b}$ is the shaded region and the limit strip $T_{\infty,b}$ appears in lighter color in the central image. On the right, we have sketched $\Sigma_{a,b}$ and the direction of rotation of the normal along $\Gamma_1$ and $\Gamma_2$. We recall that $q_i=F_0(p_i)=(p_i,0)$ for any $i$.
  • Figure 4: Sketch of the domain of a $H$-catenoid (left), a non-embedded $H$-catenodoid (center), and an embedded $H$-catenodoid (right), with $0<H<\frac{1}{2}$.
  • Figure 5: Numerical representation of the domains of $\frac{1}{2}$-catenoids (left) and $\frac{1}{2}$-catenodoids (right), corresponding to $\mathcal{H}_\mu$ with $\mu=-3$ and $\mu=3$, respectively. The darker region is covered twice and illustrates why $\frac{1}{2}$-catenodoids are never embedded.
  • ...and 1 more figures

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3: continuity of the conjugation
  • proof
  • Corollary 2.4: Conjugation of Jenkins-Serrin graphs
  • Remark 3.1
  • Lemma 3.2
  • ...and 20 more