Minimal cylinders in the three-dimensional Heisenberg group

Abstract

We study minimal cylinders in the three-dimensional Heisenberg group $\textrm{Nil}_3$ using the generalized Weierstrass type representation, the so-called loop group method. We characterize all non-vertical minimal cylinders in terms of pairs of two closed plane curves which have the same signed area. Moreover, as a byproduct of the construction, spacelike CMC cylinders in the three-dimensional Minkowski space can also be obtained.

Appendix A: Basic results

In this appendix we will collect first basic definitions and will then present some results enabling us to use holomorphic potentials in place of meromorphic potentials.

1.1 Notation and definitions

We first define the twisted $\textrm{SL}_2 \mathbb C$ loop group as a space of continuous maps from $\mathbb {S}^1$ to the Lie group $\textrm{SL}_2 \mathbb C$, that is, $\Lambda \textrm{SL}_2 \mathbb C_{\sigma }=\{g: \mathbb {S}^1 \rightarrow \textrm{SL}_2 \mathbb C\;|\; g(-\lambda ) = \sigma g(\lambda ) \}$, where $\sigma ={\text {Ad}}(\sigma _3)$. We restrict our attention to loops in $\Lambda \textrm{SL}_2 \mathbb C_{\sigma }$ such that the associate Fourier series of the loops are absolutely convergent. Such loops determine a Banach algebra, the so-called Wiener algebra, and it induces a topology on $\Lambda \textrm{SL}_2 \mathbb C_{\sigma }$, the so-called Wiener topology. From now on, we consider only $\Lambda \textrm{SL}_2 \mathbb C_{\sigma }$ equipped with the Wiener topology.

Let $D^{\pm }$ denote respective the inside of unit disk and the union of outside of the unit disk and infinity. We define plus and minus loop subgroups of $\Lambda \textrm{SL}_2 \mathbb C_{\sigma }$; $\Lambda ^{\pm } \textrm{SL}_{2} \mathbb C_{\sigma }=\{ g \in \Lambda \textrm{SL}_2 \mathbb C_{\sigma }\;|\; g{ canbeextendedholomorphicallyto}D^{\pm }{} \}$. By $\Lambda _*^+ \textrm{SL}_{2} \mathbb C_{\sigma }$ we denote the subgroup of elements of $\Lambda ^+ \textrm{SL}_{2} \mathbb C_{\sigma }$ which take the value identity at zero. Similarly, by $\Lambda _*^{-} \textrm{SL}_{2} \mathbb C_{\sigma }$ we denote the subgroup of elements of $\Lambda ^{-} \textrm{SL}_{2} \mathbb C_{\sigma }$ which take the value identity at infinity.

We also define the $\textrm{SU}_{1, 1}$-loop group as follows:

$$\begin{aligned} \Lambda \textrm{SU}_{1, 1 \sigma }=\left\{ g \in \Lambda \textrm{SL}_2 \mathbb C_{\sigma }\;|\; \sigma _3 \overline{g(1/\bar{\lambda })}^{t-1} \sigma _3 = g(\lambda )\right\} . \end{aligned}$$

It will be convenient to use $\tau (g)(\lambda ) = \sigma _3 \overline{g(1/\bar{\lambda })}^{t-1} \sigma _3$ for $\lambda \in S^1$. For all geometric quantities, we can assume $\lambda \in \mathbb C^*$.

For potentials $\zeta = \hat{\zeta }dz$ with Lie algebra elements $\hat{\zeta }\in \Lambda \mathfrak {sl}_2 \mathbb C_{\sigma }$ we say that $\zeta$ is an $\mathfrak {su}_{1, 1}$-potential if $\tau (\hat{\zeta })(\lambda ) = - \hat{\zeta }(\lambda )$ holds.

1.2 Holomorphic $C(z,\lambda )$

We should probably be a bit more careful about where from we choose our variables: at one hand it is convenient to choose $\mathbb D$ as complex plane or unit disk, since they are invariant under complex conjugation. But since we consider already very early periodic functions with real period, we actually are interested in strips $\mathbb S$. There are three types of strips: the whole complex plane, the upper half-plane and a strip of finite width. In the first and the last case we can assume w.l.g. that the strip contains the real line “in the middle”. Meaning the real line for the case of $\mathbb S= \mathbb C$, and the real line for a strip of type $-c_0< {\text {Im}}(z) < c_0$. For the third case, the upper half-plane $\mathbb {H} =\{ z \mid {\text {Im}}(z) > 0\}$, it is perhaps best to let $\mathbb {Z}$ translate parallel to the real axis and to choose the base point to be i, instead of moving the upper-half plane down so that the new domain covers the actual real line.

Proposition A1

Let $f: \mathcal {C} \rightarrow \textrm{Nil}_3$ be a minimal (immersed) cylinder in $\textrm{Nil}_3$ with universal cover $\mathbb S$. Then there exists a maximal minimal (immersed) cylinder in $\textrm{Nil}_3$ prolonging $\mathbb S$.

For the notion of “prolongable” see [12, p.207].

Proof

Clearly, if $\mathbb S= \mathbb C$, no extension is possible and the given cylinder already is maximal. If $\mathbb S$ is a finite strip with $\mathbb R$ in the middle, we consider all extensions, realized by finite strips with $\mathbb R$ in the middle. Then by Zorn’s Lemma there exists a maximal object. If $\mathbb S$ is the upper half plane, then any extension has a universal cover which can be realized as a “shifted down” upper half plane. Now a procedure as in the last case yields a maximal element. $\square$

1.3 Holomorphic $\mathfrak {su}_{1, 1}$-potentials

For this we recall that the extended coordinate frame has a meromorphic extension. Let’s have a closer look: The right upper corner of the Maurer-Cartan form of the coordinate frame F is of the form $-\lambda ^{-1}e^{u/2} -\lambda \bar{B} e^{-u/2}$, where u is the metric exponent and B the Hopf differential (see [8]). Since F has a meromorphic extension to $\mathbb D\times \mathbb D$, also the expression above has such an extension. Since F is real-analytic, also $e^{u/2}$ is real analytic. Therefore there exists an open subset ${\mathfrak D}_0$ of $\mathbb D\times \mathbb D$ containing $\mathbb D^\sharp = \{(z, \bar{z}); z \in \mathbb D\}$ to which F extends holomorphically. We thus obtain:

Proposition A2

There exists an open subset $\mathfrak D_0$ of $\mathbb D\times \mathbb D$ containing $\mathbb D^\sharp = \{(z, \bar{z}); z \in \mathbb D\}$ to which F extends holomorphically in (z, w). In particular, also $\zeta$ extends holomorphically in (z, w) to $\mathfrak D_0$.

Remark A3

• (1) If one starts from some potential like $\zeta$, then the maximal minimally immersed cylinder in $\textrm{Nil}_3$ is defined on the largest strip on which the $\textrm{SU}_{1, 1}$-Iwasawa decomposition is real analytic.

• (2) Singularities along the boundary of such a maximal strip indicate several different geometric features of the minimal cylinder.