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abstract: 'In this paper we prove existence and uniqueness of the Björling problem for the class of immersed surfaces in $\r3$ whose mean curvature is given as an analytic function depending on its Gauss map. As an application, we prove the existence of surfaces with the topology of a Möbius strip for an arbitrary large class of prescribed functions. In particular, we use the Björling problem to construct the first known examples of self-translating solitons of the mean curvature flow with the topology of a Möbius strip in $\mathbb{R}^3$.'
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[**The Björling problem for prescribed mean curvature surfaces in $\r3$ [^1]**]{}\
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Departamento de Geometría y Topología, Universidad de Granada, E-18071 Granada, Spain.\
e-mail: jabueno@ugr.es

Introduction
============

A classical problem in minimal surface theory in $\r3$ is the *Björling problem* [@Bjo]. This problem was posed in 1844 by Björling and asks the following:

> *Given a regular analytic curve $\beta(s)$ in $\r3$ and an analytic distribution of oriented planes $\Pi(s)$ along $\beta(s)$ such that $\beta'(s)\in\Pi(s)$, find *all* minimal surfaces in $\r3$ containing $\beta(s)$ and such that the tangent plane distribution along $\beta(s)$ is given by $\Pi(s)$.*

In 1890 Schwarz [@Sch] solved this problem via an integral representation formula, using holomorphic data. This problem can be applied in other generic situations: for instance, to study surfaces with certain symmetries [@DHKW], and to solve global problems in the minimal surface theory [@AlMi; @GaMi1]; see also [@ACG; @GaMi2; @GaMi3] and references therein for an outline on the development of the *geometric Cauchy problem*. The Björling problem has been also studied when the mean curvature is a non-vanishing constant, see [@BrDo].

Our objective in this paper is to prove existence and uniqueness of the Björling problem for a certain class of *prescribed mean curvature surfaces* immersed in $\R^3$. Specifically, let be $\sig$ an oriented, immersed surface in $\r3$ and $\hi$. We say that $\sig$ has *prescribed mean curvature* $\H$ (in short, $\sig$ is an $\H$-surface) if the mean curvature $H_\sig$ of $\sig$ satisfies $$\label{defHsup}
H_\sig(p)=\H(\eta_p),\hspace{.5cm} \forall p\in\sig,$$ where $\eta:\sig\rightarrow\s2$ is the *Gauss map* of $\sig$.

In general, the study of hypersurfaces in $\rnn$ defined by a prescribed curvature function in terms of the Gauss map goes back, at least, to the famous Christoffel and Minkowski problems for ovaloids, see e.g. [@Chr]. In particular, the existence and uniqueness of ovaloids with prescribed mean curvature was studied among others by Alexandrov and Pogorelov [@Ale; @Pog] but the global geometry of these surfaces remained largely unexplored. In [@GaMi4] the authors studied uniqueness of immersed $\H$-spheres obtaining a Hopf-type theorem for this class of immersed surfaces, and in the making they proved a standing conjecture by Alexandrov. The global properties of $\H$-hypersurfaces immersed in $\rnn$ have been recently developed in [@BGM], where the authors studied several topics such as classification of rotational $\H$-hypersurfaces, a priori height estimates and a structure theorem for properly embedded $\H$-surfaces in $\r3$, and curvature estimates for stable $\H$-surfaces immersed in $\R^3$.

This paper is organized as follows: in **Section \[sec:resolutionbjorling\]** we prove existence and uniqueness of the Björling problem for the class of analytic functions $\H\in C^\omega(\s2)$ by applying Cauchy-Kovalevskaya theorem. This theorem has been used in other works to prove existence and uniqueness of the Björling problem for minimal surfaces in three-dimensional Riemannian and Lorentzian Lie groups via a Weierstrass-type representation formula, see e.g. [@CMO; @MMP; @MeOn].

In **Section \[sec:hmobius\]** we restrict ourselves to the class of analytic functions $\H\in C^\omega(\s2)$ satisfying the symmetry condition $\H(-x)=-\H(x),\ \forall x\in\s2$. This condition on $\H$ ensures us that Equation is independent of the orientation chosen on the $\H$-surface. Bearing this in mind, we use the existence and uniqueness of the Björling problem for adequate Björling data to construct non-orientable $\H$-surfaces with the topology of a Möbius strip. A particular analytic function with this symmetry condition is the one given by $\H(x)=\langle x,e_3\rangle,\ \forall x\in\s2$. The $\H$-surfaces arising for this prescribed function are the *self-translating solitons of the mean curvature flow* in $\r3$, a well studied class of surfaces in the past decades. See e.g. [@CSS; @Hui; @MSHS] and references therein for relevant works regarding this topic. As an application, we construct self-translating solitons of the mean curvature flow in $\r3$ with the topology of a *Möbius strip*. After a detailed search in the literature, we can assert that these *translating Möbius strips* are the first known examples of self-translating solitons with non-orientable topology.

Finally, in **Section \[sec:masejemplos\]** we use the solution of the Björling problem to construct further examples of $\H$-surfaces for analytic prescribed functions. In particular, we construct *helicoidal* $\H$-surfaces, and $\H$-surfaces similar to Enneper’s classical minimal surface.

**Acknowledgment.** The author wants to express his gratitude to his Ph.D. advisor Pablo Mira, for fruitful conversations about this topic.

The Björling problem for $\H$-surfaces {#sec:resolutionbjorling}
======================================

In this paper we will denote by $C^\omega(\s2)$ to the class of *analytic functions* defined on the sphere $\s2$ in $\r3$.

Let be $I\subset\R$ an open interval. A pair of *Björling data* for $\H$-surfaces in $\r3$ is a regular analytic curve $\beta:I\rightarrow\r3$ and an analytic vector field $B:I\rightarrow\r3$ along $\beta(s)$ such that $|\beta'(s)|-|B(s)|=\langle\beta'(s),B(s)\rangle=0,\ \forall s\in I$.

From this definition, we get two obvious consequences: First, given $\H\in C^\omega(\s2)$, a regular analytic curve $\beta(s)$ and an oriented distribution of planes $\Pi(s)$ along $\beta(s)$, we get a pair of Björling data by just defining $B(s)=J\beta'(s)$, where $J$ denotes the $\pi/2$-rotation in the tangent plane $\Pi(s)$. And second, let us denote by $\nabla$ to the Riemannian connection of $\sig$ and suppose that $\beta(s)$ is parametrized by arc-length. Moreover, suppose that $\beta(s)$ is not a straight line. Then there exists $s_0\in\R$ such that $|\beta''(s_0)|\neq 0$. Let $I_0$ be the largest subinterval of $I$ containing $s_0$ such that $|\beta''(s)|\neq 0$ for all $s\in I_0$. If we define $V(s):=\nabla_{\beta'(s)}\beta'(s)$ then $B(s)=V(s)/|V(s)|$ is an analytic unit vector field along $\beta(s)$ such that $\langle\beta'(s),B(s)\rangle=0$. Thus, $B(s)$ determines an oriented distribution of planes $\Pi(s)$ along $\beta(s)$ by defining $\Pi(s)=\beta(s)+B(s)^\bot$. In particular, the Björling problem generalizes the problem of finding a surface which contains a given curve as a geodesic.

Although we do not have a Weierstrass representation for $\H$-surfaces, and thus we cannot solve the Björling problem with an *explicit* integral representation formula just as in the case $\H=0$ (see e.g. [@Mir]), we can prove existence and uniqueness of this problem by applying different methods, as other have done in similar situations; see e.g. [@CMO; @MeOn].

Let $\sig$ be an orientable Riemannian surface and let $\psi:\sig\rightarrow\r3$ be an isometric immersion of $\sig$ in $\r3$. Then, it is well known that the coordinates of $\psi$ satisfy the elliptic PDE $$\label{laplainmersion}
\Delta_\sig\psi=2H_\sig\eta,$$ where $\Delta_\sig$ stands for the Laplace-Beltrami operator on $\sig$, $\eta$ denotes the Gauss map of $\sig$ and $H_\sig$ is the mean curvature of $\sig$ computed with respect to $\eta$.

Recall that $\sig$ inherits a Riemann surface structure induced by its first fundamental form, and thus we can consider a conformal coordinate $z=s+it$ defined in a simply connected domain $\Omega\subset\mathbb{C}$, and we define the usual *Wirtinger operators* $\partial_z=1/2(\partial_s-i\partial_t)$, $\partial_{\overline{z}}=1/2(\partial_s+i\partial_t)$. Then, the induced metric on $\sig$ is expressed as $\langle\cdot,\cdot\rangle=\lambda^2|dz|^2$, where $|dz|^2$ is the flat metric on $\Omega$ and $\lambda^2=\langle\partial_s,\partial_s\rangle=\langle\partial_t,\partial_t\rangle>0$ is the *conformal factor*. For such a conformal coordinate, the operator $\Delta_\sig$ writes as $$\label{laplainmersion2}
\Delta_\sig=\frac{1}{\lambda^2}\Delta_0=\frac{1}{\lambda^2}(\partial_{ss}+\partial_{tt})=\frac{4}{\lambda^2}\partial_{z\overline{z}},$$ where $\Delta_0$ denotes the usual flat Laplacian, and we used the relation of the Laplace-Beltrami operator between two conformal metrics. On the other hand, the Gauss map $\eta$ of $\sig$ has the following expression with respect to $z$: $$\label{normalconforme}
\eta=\frac{2}{i}\frac{\psi_z\wedge\psi_{\overline{z}}}{\sqrt{\langle\psi_z\wedge\psi_{\overline{z}},\psi_z\wedge\psi_{\overline{z}}\rangle}}=\frac{2}{i\lambda^2}\psi_z\wedge\psi_{\overline{z}}.$$

Suppose now that the immersion $\psi:\sig\rightarrow\r3$ defines an $\H$-surface for some $\H\in C^\omega(\s2)$. By Equations and , Equation writes as $$\label{pdeinmersion}
\psi_{z\overline{z}}=-i\H(\eta)\psi_z\wedge\psi_{\overline{z}}.$$ Viewing the immersion in coordinates $\psi=(\psi_1,\psi_2,\psi_3)$, then Equation can be seen as a system of partial differential equations. In this setting, we can prove existence and uniqueness of the Björling problem for the class of $\H$-surfaces, as stated next:

\[thbjorling\] Let be $\H\in C^\omega(\s2)$ and $\beta(s),B(s)$ a pair of Björling data defined on a real interval $I$. Then, there exists an open domain $\Omega\subset\mathbb{C}$ containing $I\times\{0\}$ and a conformal immersion $\psi:\Omega\rightarrow\r3$ that solves the following system $$\label{ecubjorling}
\left\lbrace\begin{array}{l}
\psi_{z\overline{z}}=-i\H(\eta)\psi_z\wedge\psi_{\overline{z}},\\
\psi(s,0)=\beta(s),\\
\psi_t(s,0)=B(s).
\end{array}\right.$$

As a consequence, $\psi$ defines an $\H$-surface $\sig$ that contains the curve $\beta(s)$, and the tangent plane distribution $T_{\beta(s)}\sig$ at each point $\beta(s)\in\sig$ is spanned by the vectors $\beta'(s)$ and $B(s)$.

The system is elliptic and of *Cauchy-Kovalevskaya* type, and thus it has local existence and uniqueness; see [@Pet] for a proof of Cauchy-Kovalevskaya theorem. As the system is elliptic without characteristic points, we have that the existence and uniqueness extends to the whole interval $I$ where $\beta(s)$ and $B(s)$ are defined. Thus, there exist $\delta>0$ and functions $(\psi_1,\psi_2,\psi_3)$ defined in $I\times (-\delta,\delta)\subset\mathbb{C}$ such that $\psi=(\psi_1,\psi_2,\psi_3)$ is a solution of satisfying $\psi(s,0)=\beta(s),\ \psi_t(s,0)=B(s),\ \forall s\in I$.

First, observe that equation $\langle\psi_{z\overline{z}},\psi_z\rangle=0$ holds by substituting $\psi_{z\overline{z}}$ for its expression , and thus the function $\langle\psi_z,\psi_z\rangle$ is holomophic. As $\psi$ satisfies the described initial conditions, the function $\langle\psi_z,\psi_z\rangle$ evaluated at $(s,0)$ is equal to $|\beta'(s)|^2-|B(s)|^2-2i\langle\beta'(s),B(s)\rangle$. This expression vanishes identically, as $\beta'(s)$ and $B(s)$ are orthogonal vector fields with the same length. In this conditions, $\langle\psi_z,\psi_z\rangle$ is a holomorphic function vanishing at the real axis and by the identity principle of holomorphic functions, $\langle\psi_z,\psi_z\rangle$ is identically zero in $I\times(-\delta,\delta)$. We conclude that the map $\psi:I\times(-\delta,\delta)\rightarrow\r3$ is conformal.

By Equation , the regularity of $\beta(s)$, and the orthogonality of $\beta'(s)$ and $B(s)$, it is clear that $\psi$ defines an immersion on an open set $\Omega\subset\mathbb{C}$ containing $I\times\{0\}$. Thus, $\psi$ is a conformal immersion of an $\H$-surface.

This concludes the proof of Theorem \[thbjorling\].

As we pointed out, in general system cannot be explicitly integrable, as happens when $\H=0$. However, we can numerically solve it for producing images of $\H$-surfaces in $\R^3$. Indeed, let us denote by $\psi(s,t)=(\psi_1(s,t),\psi_2(s,t),\psi_3(s,t))$. Given $\H\in C^\omega(\s2)$, $\beta(s),B(s):[s_0,s_1]\rightarrow\r3$ a pair of Björling data and $\delta>0$, the solution of the Björling problem can be plotted using standard software with the command

`ParametricPlot3D Evaluate First\{\psi[s,t]\}/. NDSolve\textcolor{purple}{\{} D[\psi[s,t],s,s]+D[\psi[s,t],t,t]==2\H[\eta[s,t]]D[\psi[s,t],s]\wedge D[\psi[s,t],t], Thread[\psi[s,0]==\beta[s],D[\psi[s,t],t][s,0]==B[s]]\textcolor{purple}{\}}, \{\psi_1,\psi_2,\psi_3\},\{s,s_0,s_1\},\{t,-\delta,\delta\},\{s,s_0,s_1\},\{t,-\delta,\delta\}`.

For example, consider the analytic function $\H(x)=\langle x,e_3\rangle,\ \forall x\in\s2$. The $\H$-surfaces arising for this prescribed function are the *self-translating solitons of the mean curvature flow*. The rotational self-translating solitons of the mean curvature flow are classified as follows: an entire, strictly convex vertical graph that intersects the axis of rotation orthogonally, called the *bowl soliton*; and a one parameter family of properly embedded annuli, with both ends pointing towards the $e_3$ direction, called the *wing-like solitons* or *translating catenoids*. The family of wing-like solitons are parametrized by the *neck size*, i.e. the minimum distance to the axis of rotation, attained at a circumference of radius equal to the *waist*, see [@CSS] for details.

Bearing this in mind, we can recover this family by choosing adequate Björling data. Indeed, consider the one parameter family of Björling data $\beta_\tau(s)=\tau(\cos s,\sin s,0)$ and $B_\tau(s)=(0,0,\tau),\ \forall s\in\R,\ \tau>0$. Then, for each fixed $\tau>0$, the translating soliton $\sig_\tau$ generated by this Björling data corresponds to the wing-like soliton with neck size equal to $\tau$. When $\tau\rightarrow 0$, the sequence $\sig_\tau$ converges smoothly to a double cover of the bowl soliton minus the vertex. See Figure \[fig:hcatbjorling\] for a plot of the wing-like soliton with neck size equal to 1.

![A wing-like soliton for the Björling data $\beta(s)=(\cos s,\sin s,0)$ and $B(s)=(0,0,1),\ s\in\R$.[]{data-label="fig:hcatbjorling"}](hcatbjorling.pdf){width=".75\textwidth"}

$\H$-Möbius strips in $\r3$ {#sec:hmobius}
===========================

The Björling problem adds a large amount of $\H$-surfaces to the ones studied in [@BGM], for any given $\H\in C^\omega(\s2)$. By choosing adequate Björling data, the examples arising may have some prescribed symmetries, as well as some topological properties. In this context, the Björling problem motivates us for the search of *non-orientable* surfaces. However, we must recall that our $\H$-surfaces are supposed to be oriented, in virtue of Definition \[defHsup\], and thus the concept of non-orientable $\H$-surface makes no sense for an arbitrary function $\H$. For instance, if $\H$ is chosen as a positive constant, examples with non-orientable topology do not exist. Thus, some condition on the prescribed function $\H\in C^\omega(\s2)$ must be imposed in order to make Definition \[defHsup\] independent on the orientation chosen on the $\H$-surface. Bearing this in mind, the mildest hypothesis is that $\H$ has to be *antipodally antisymmetric*, that is $\H(-x)=-\H(x)$ for all $x\in\s2$.

Going back to the propose of studying non-orientable $\H$-surfaces, the most recurrent examples are the surfaces with the topology of a *Möbius strip*; we advise [@Mee; @Mir] as two relevant works regarding minimal Möbius strips in $\rn$ given as the solution of the Björling problem. The following result is inspired in the ideas developed in [@Mir].

\[hcintas\] Let $\H\in C^\omega(\s2)$ such that $\H(-x)=-\H(x),\ \forall x\in\s2$, and let $\beta(s),B(s)$ be Björling data such that $\beta(s)$ is $T$-periodic and $B(s)$ is $T$-antiperiodic, i.e. $B(s+T)=-B(s)$, for some $T>0$.

Then, the $\H$-surface generated by Theorem \[thbjorling\] for this Björling data has the topology of a Möbius strip, with fundamental group generated by $\beta(s)$.

Conversely, every $\H$-*Möbius strip* is generated in this way.

We will give a sketch of the proof, since it is an adaptation of the one used to prove Lemma 3 in [@Mir].

First, let $\psi:\M\rightarrow\r3$ be an immersion of an $\H$-Möbius strip, and let $\Gamma(s)$ be a regular, analytic, closed curve in $\M$ that generates its fundamental group. As $\Gamma(s)$ is closed, it is $T$-periodic for some $T>0$. Denote by $\widetilde{\M}$ to its *two sheeted cover*, where we have defined an antiholomorphic involution $I$ without fixed points, and let $\pi:\widetilde{\M}\rightarrow\R^3$ be the canonical projection. In this setting, there exists a regular, analytic, closed curve $\widetilde{\Gamma}(s)$ that generates the fundamental group of $\widetilde{\M}$, defined by $\Gamma(s)=\pi(\widetilde{\Gamma}(s))$, and in particular it is $2T$-periodic.

If we consider $\widetilde{\psi}:\widetilde{\M}\rightarrow\R^3$ given by $\widetilde{\psi}=\psi\circ\pi$, then $\beta(s)=\widetilde{\psi}(\widetilde{\Gamma}(s))$ is a regular, analytic $T$-periodic curve in $\R^3$. Denoting by $\Pi(s)$ the oriented tangent plane distribution of $\widetilde{\psi}$ along $\beta(s)$ we have that $\Pi(s+T)$ and $\Pi(s)$ agree with opposite orientation. Therefore, if $B(s):=J\beta'(s)$ is the $\pi/2$ rotation of $\beta'(s)$ in $\Pi(s)$, it happens that $B(s+T)=-B(s)$. Thus, we have proved that every $\H$-Möbius strip defines a pair of Björling data with the periodicity properties stated.

![A self-translating soliton of the mean curvature flow with the topology of a Möbius stirp, generated by the Björling data $\beta(s)=1/2(\cos 2s,\sin 2s,0),\ B(s)=\cos s(\cos s,\sin s,0)+\sin s(0,0,1),\ \forall s\in\R$.[]{data-label="fig:hmobius"}](hmobius.pdf){width=".8\textwidth"}

Conversely, let $\beta(s)$ and $B(s)$ be a pair of Björling data such that $\beta(s)$ is $T$-periodic and $B(s)$ is $T$-antiperiodic. In particular, they are defined on the entire real line. Let $\psi:\Omega\rightarrow\R$ be the solution of this Björling problem given by Theorem \[thbjorling\]. The $2T$-periodicity of $\beta(s)$ and $B(s)$ along with the uniqueness of the Björling problem, ensures us that $\psi$ is well defined on the quotient $\Omega/(2T\mathbb{Z})$, which is a topological cylinder.

We can suppose that $\Omega$ is symmetric with respect to the conjugation, i.e. $\overline{z}\in\Omega,\ \forall z\in\Omega$, and thus we can define on $\Omega/(2T\mathbb{Z})$ the map $$\begin{array}{cccc}
I&:\Omega/(2T\mathbb{Z})&\longrightarrow&\Omega/(2T\mathbb{Z})\\
& z&\longmapsto& \overline{z}+T.
\end{array}$$

It is also clear that $I$ is an antiholomorphic involution without fixed points, and thus it reverses the orientation of $\Omega/(2T\mathbb{Z})$. Moreover, $I$ defines the following equivalence relation on $\Omega/(2T\mathbb{Z})$: two points $z,w\in\Omega/(2T\mathbb{Z})$ are related if and only if $I(z)=w$. In this situation, the cylinder $\Omega/(2T\mathbb{Z})$ is the orientable two sheeted cover of the space $(\Omega/(2T\mathbb{Z}),I)$, with the canonical projection $\mathfrak{p}:\Omega/(2T\mathbb{Z})\rightarrow(\Omega/(2T\mathbb{Z}),I)$.

Because $\psi$ is an immersion, the unitary vector field $\eta:\Omega\rightarrow\r3$ as defined in is a well defined, unitary, normal vector field for $\psi$. Given $z,w\in\Omega/(2T\Z)$ such that $w=I(z)$, the uniqueness of the Björling problem implies that the unit normals $\eta(z)$ and $\eta(w)$ are opposite. Bearing this in mind, we have: $$\label{pegan}
H_{\psi}(\psi(w))=\H(\eta(w))=\H(\eta(I(z)))=\H(-\eta(z))=-\H(\eta(z))=-H_{\psi}(\psi(z)),$$ and thus the mean curvature at the point $\psi(z)$ has opposite sign to the mean curvature at the point $\psi(I(z))$, for all $z\in\Omega$.

Again, the uniqueness of Theorem \[thbjorling\] allows us to conclude that the quotient map $\widetilde{\psi}(z)=(\psi\circ\mathfrak{p})(z)$ for all $z\in\Omega/(2T\Z)$ is a well defined, conformal immersion of an $\H$-surface in $\r3$ having the topology of a Möbius strip, and in particular is non-orientable. This concludes the proof of Proposition \[hcintas\].

Note that the function $\H(x)=\langle x,e_3\rangle,\ \forall x\in\s2$ lies in the hypothesis of Proposition \[hcintas\]. Thus, we ensure the existence of self-translating solitons of the mean curvature flow with the topology of a Möbius strip, which we will refer to as *translating Möbius strips*, see Figure \[fig:hmobius\]. After a detailed search in the literature, we can assert that this construction gives the first example of a self-translating soliton of the mean curvature flow with non-orientable topology.

In Figure \[fig:noorientable\] we show the construction of a non-orientable translating soliton constructed by half-rotating the vector field $B(s)$ 7 times along the curve $\beta(s)$ before it closes. This surface is homeomorphic to the Möbius strip showed in Figure \[fig:hmobius\]. When the mean curvature vanishes, the minimal surfaces as the one appearing in Figure \[fig:noorientable\] were firstly constructed in [@Mir], see also the independent work of [@MeWe]. These surfaces are commonly known as *bended helicoids*.

![An $\H$-bended helicoid with the topology of a Möbius strip for the analytic function $\H(x)=\langle x,e_3\rangle,\ x\in\s2$.[]{data-label="fig:noorientable"}](noorientable.pdf){width=".65\textwidth"}

Further examples of $\H$-surfaces via the Björling problem {#sec:masejemplos}
==========================================================

In this last Section we show the existence of some examples of $\H$-surfaces immersed in $\r3$, motivated by the analogous examples defined in the minimal surface theory. The main difference with the minimal case $\H=0$ in $\r3$ is that we fail to have a Weierstrass representation, and thus explicit parametrizations of these surfaces are not expected. Even so, we can prove existence and uniqueness by means of Theorem \[thbjorling\] for adequate Björling data with some prescribed symmetries, and obtain $\H$-surfaces that are the analogous to the famous examples in the minimal surface theory.

****

We choose as Björling data the vertical curve $\beta(s)=(0,0,s)$ and a $T$-periodic, analytic, unitary vector field $B(s)$ along $\beta(s)$, for some $T>0$, and let $\sig$ be $\H$-surface given as the solution of the Björling problem for this Björling data.

The unit normal vector field at the $e_3$-axis, namely $\eta(s):=\beta'(s)\wedge B(s)$, is a horizontal vector field satisfying $\eta(s)=\eta(s+T),\ \forall s\in\R$, and the Björling data $\beta(s+T),B(s+T)$ agree with the Björling data $\beta(s)+T e_3,B(s)$. Moreover, as the condition $\eta(s)=\eta(s+T),\ \forall s\in\R$ holds, the points $\beta(s)$ and $\beta(s+T)$ have the same mean curvature.

Bearing this in mind, if we denote by $\Theta(p)=p+T e_3,\ \forall p\in\r3$ , the uniqueness of the Björling problem ensures us that $\sig$ is invariant by the discrete group of translations $T\mathbb{Z}\Theta$ in the $e_3$-direction. Moreover, starting at some $s_0\in\R$, $\sig$ *twists* jointly with $\eta(s)$ around the $e_3$-axis until reaching the instant $\eta(s_0+T)=\eta(s_0)$, generating a simply connected *fundamental part* of $\sig$. Repeating this process, which is just translating this fundamental part of $\sig$ under the action of $\Theta$, we get the whole $\H$-surface $\sig$.

We will refer to these $\H$-surfaces as $\H$-helicoids, since they generalize the usual minimal helicoids in $\r3$. See Figure \[fig:hhelicoide\] for a plot of an $\H$-helicoid for the particular function $\H(x)=\langle x,e_3\rangle$, and the Björling data $\beta(s)=(0,0,s),\ B(s)=(\cos s,\sin s,0)$.

![An $\H$-helicoid for the analytic function $\H(x)=\langle x,e_3\rangle,\ x\in\s2$.[]{data-label="fig:hhelicoide"}](hhelicoide.pdf){width=".25\textwidth"}

****

Finally, we construct $\H$-surfaces based on some curves contained in the well-known Enneper’s surface, one of the most famous examples in the minimal surface theory.

First, consider the curve $$\beta(s)=(0,-s(1-s^2/3)/3,-s^2/3)$$ and the vector field $$B(s)=1/3(1+s^2)(1,0,0).$$ Then, both $\beta(s)$ and $B(s)$ are analytic and satisfy $|\beta'(s)|-|B(s)|=\langle\beta'(s),B(s)\rangle=0,\ \forall s\in\R$, i.e. they can be chosen to be Björling data. If $\H=0$, then the surface given by Theorem \[thbjorling\] is Enneper’s minimal surface. In particular, the curve $\beta(s)$ is obtained as intersecting Enneper’s minimal surface with the plane $\{x=0\}$, which is a plane of reflection symmetry of Enneper’s minimal surface. For the function $\H(x)=\langle x,e_3\rangle,\ \forall x\in\s2$, the $\H$-surface arising is a translating soliton of the mean curvature flow that resembles indeed to Enneper’s minimal surface, see Figure \[fig:henneper\].

![An $\H$-surface constructed over the curve in Enneper’s minimal surface invariant under the reflection with respect to the plane $\{x=0\}$; here $\H(x)=\langle x,e_3\rangle,\ \forall x\in\s2$.[]{data-label="fig:henneper"}](henneper.pdf){width=".6\textwidth"}

We can also choose as Björling data the following $$\begin{array}{c}
\beta(s)=(-\cos s-1/3\cos(3s),\sin s-1/3\sin(3s),\sin(2s)),\\
B(s)=(\cos s+\cos(3s),2\cos(2s)\sin s,-2\sin(2s)).
\end{array}$$ Again, for $\H=0$ the surface given by Theorem \[thbjorling\] is Enneper’s minimal surface, and we will call $\beta(s)$ *Enneper’s core curve*, see [@LoWe]. For the function $\H(x)=\langle x,e_3\rangle,\ \forall x\in\S^2$, the translating soliton arising also resembles to Enneper’s minimal surface, see Figure \[fig:ennepers\], left. This time we cannot guarantee that the *hole* in the middle will eventually close, as we fail to have an explicit parametrization. If we make the vector $B(s)$ twist along the curve $\beta(s)$ and odd number of times, we get another translating soliton with the topology of a Möbius strip; see Figure \[fig:ennepers\], right.

![Two self-translating solitons of the mean curvature flow based on Enneper’s core curve.[]{data-label="fig:ennepers"}](enneperagujero.pdf "fig:"){width=".7\textwidth"} ![Two self-translating solitons of the mean curvature flow based on Enneper’s core curve.[]{data-label="fig:ennepers"}](enneperhelicoide.pdf "fig:"){width=".7\textwidth"}

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The author was partially supported by MICINN-FEDER Grant No. MTM2016-80313-P, Junta de Andalucía Grant No. FQM325 and FPI-MINECO Grant No. BES-2014-067663.

[^1]: Mathematics Subject Classification: 53A10, 53C42.
