# Lie minimal Weingarten surfaces

Joseph Cho  Institute of Discrete Mathematics and Geometry, TU Wien, Wiedner
Hauptstrasse 8-10/104, 1040 Wien, Austria<EMAIL_ADDRESS>,
Masaya Hara  Department of Mathematics, Graduate School of Science, Kobe
university, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan
<EMAIL_ADDRESS>,  Denis Polly  Institute of Discrete Mathematics and
Geometry, TU Wien, Wiedner Hauptstrasse 8-10/104, 1040 Wien, Austria
<EMAIL_ADDRESS>and  Tomohiro Tada  Department of Mathematics,
Graduate School of Science, Kobe university, 1-1 Rokkodai-cho, Nada-ku, Kobe
657-8501, Japan<EMAIL_ADDRESS>
(Date: v4 )

###### Abstract.

We consider Lie minimal surfaces, the critical points of the simplest Lie
sphere invariant energy, in Riemannian space forms. These surfaces can be
characterized via their Euler-Lagrange equations, which take the form of
differential equations of the principal curvatures. Surfaces with constant
mean curvature that satisfy these equations turn out to be rotational in their
space form. We generalize in flat ambient space: here surfaces where the
principal curvatures satisfy an affine relationship as well as elliptic linear
Weingarten surfaces are rotational as well.

###### Key words and phrases:

Lie minimal surface, minimal surface, constant mean curvature surface, linear
Weingarten surface

###### 2020 Mathematics Subject Classification:

Primary 53A10; Secondary 53A40, 53C42.

##  1\. Introduction

Willmore functional was first considered by Germain [14] as a surface analogue
of the bending energy of curves whose minimizers are the elastic curves of
Bernoulli and Euler. The critical points of the Willmore functional under
compactly supported variations are called _constrained Willmore surfaces_ ,
and they have been widely studied (see, for example, [4, 6]), in part, due to
the now-resolved Willmore conjecture [19, 26]. In particular, the Willmore
functional is conformally invariant, and is in fact the simplest variational
problem that can be considered in conformal geometry, or Möbius geometry [25].
On the other hand, those surfaces not determined by invariants of surfaces in
a geometry are called _deformable surfaces_ , and Cartan has showed that
_isothermic surfaces_ , those surfaces that admit conformal curvature lines
coordinates, are exactly the deformable surfaces in Möbius geometry [8].
Interestingly, constant mean curvature surfaces in space forms are important
examples that are both constrained Willmore and isothermic [25, 24]. Thus,
certain curvature properties, a space form notion, play an important role in
consideration of both minimality and deformability of Möbius geometry.

Similar considerations can be made in other sphere geometries. In the context
of Laguerre geometry, the simplest variational problem revolves around a
functional that is Laguerre invariant called _Weingarten functional_ ; the
critical points of Weingarten functional under compactly supported variations
are called _$\displaystyle L$ -minimal surfaces_ [2] (see also [20]). On the
other hand, the deformable surfaces in Laguerre geometry are exactly
_$\displaystyle L$ -isothermic surfaces_ [1], those surfaces with curvature
line coordinates that are conformal with respect to the third fundamental
form. Again, certain curvature properties play a crucial role in determining
the surfaces with both minimality and deformability, as minimal surfaces in
Euclidean space are both $\displaystyle L$-minimal and $\displaystyle
L$-isothermic [2, 20].

In this paper, we focus on Lie sphere geometry [16], the sphere geometry that
includes both Möbius geometry and Laguerre geometry as its subgeometries, and
consider Lie minimal surfaces that satisfy certain curvature properties in
space forms.

First considered by Blaschke [3], _Lie minimal surfaces_ are surfaces that are
the critical points of the simplest energy $\displaystyle L_{\textrm{Lie}}$,
invariant under Lie sphere transformations, under compactly supported
variations. This class of surfaces has raised interest recently, notably from
two different perspectives: [13] showed that they constitute an integrable
system (an integrable reduction of the Gauss-Codazzi equations of the Lie
sphere frame), while [5] demonstrated that they have harmonic conformal Gauss
maps (a congruence of Lie cyclides).

These two characetrizations are purely Lie sphere geometric in nature.
However, given a surface in a Riemannian space form, the energy $\displaystyle
L_{\textrm{Lie}}$ can be computed using the principal curvatures of the space
form projection [13]. The goal of this paper is gain insights into the
geometry of Lie minimal surfaces whose principal curvatures satisfy additional
properties, namely, relationships of the Weingarten type. We will show that
affine Weingarten and elliptic linear Weingarten surfaces are Lie minimal if
and only if they are rotational in their space forms in the sense of [11].

In Section 2 we will lay the groundwork for our investigations. Starting from
the expression of $\displaystyle L_{\textrm{Lie}}$ via principal curvatures of
the space form projection, we will develop the Euler-Lagrange equations for
Lie minimal surfaces in space forms (Lemma 2.9). These differential equations
for the principal curvatures are then used in Sections 3 and 4 to investigate
Lie minimal surfaces with additional curvature conditions.

As minimal surfaces in Euclidean space are both constrained Willmore and
L-minimal, while cmc surfaces in space forms are constrained Willmore, the
starting point of our investigations are minimal Lie minimal surfaces or, more
generally, Lie minimal surfaces with constant mean curvature
$\displaystyle{H=\tfrac{1}{2}(k_{1}+k_{2})}$. In contrary to the Möbius or
Laguerre geometric cases, we show that these surfaces must be rotational
within their space form in the sense of [11] (Theorem 3.2). Since rotational
cmc surfaces in Riemannian space forms have been completely classified (most
recently in [23]), this provides a complete classification of all Lie minimal
cmc surfaces.

Following this, we consider two more classes of surfaces with constraint
principal curvatures in Section 4: on the one hand, we consider surfaces in
$\displaystyle\mathbb{R}^{3}$ where the principal curvatures satisfy an affine
linear relationship of the form

$\displaystyle xk_{1}+yk_{2}=z.$

This extends the class of cmc surfaces, as these correspond to the special
case $\displaystyle x=y$. We call this class the class of _affine Weingarten
surfaces_. We will demonstrate that Lie minimal affine Weingarten surfaces are
always rotational in $\displaystyle\mathbb{R}^{3}$ (Theorem 4.3). The
classification of rotational affine Weingarten surfaces of, e.g., [18] thus
provides a complete classification of this class as well.

On the other hand, we consider the class of _linear Weingarten_ surfaces,
examples of deformable surfaces in Lie sphere geometry [7]. This class extends
the class of cmc surfaces to those where there is an affine linear
relationship between the mean curvature $\displaystyle H$ and the Gauss
curvature $\displaystyle K=k_{1}k_{2}$, that is

$\displaystyle aK+2bH+c=0.$

According to Bonnet’s theorem (Proposition 4.1), all parallel transformations
of linear Weingarten surfaces are again linear Weingarten. Since the parallel
transformation preserves rotational surfaces and Lie minimality (a Lie sphere
invariant notion), we can prove that all Lie surfaces that are parallel to a
cmc surface are rotational. In the case of Euclidean ambient geometry, this
comprises the class of linear Weingarten surfaces with $\displaystyle
b^{2}-ac>0$, dubbed elliptic linear Weingarten surfaces in [17].

##  2\. Preliminaries

In this section we describe the setup of our investigations. The main goal is
to fix notation we will use for the theory of surfaces in space forms. In
Subsection 2.2 we will define the notion of Lie minimality. The main result of
this section is Lemma 2.9 which we use to prove that channel surfaces are
always Lie minimal (and in later sections).

Let $\displaystyle M_{\kappa}$ be a Riemannian space form with constant
sectional curvature $\displaystyle\kappa$. We think of $\displaystyle
M_{\kappa}$ as either $\displaystyle\mathbb{R}^{3}$ ($\displaystyle\kappa=0$)
or a three dimensional (Riemannian) quadric in $\displaystyle\mathbb{R}^{4}$
($\displaystyle\kappa=1$) or $\displaystyle\mathbb{R}^{3,1}$
($\displaystyle\kappa=-1$). Let $\displaystyle X\mathrel{\mathop{\mathchar
58\relax}}\Sigma\rightarrow M_{\kappa}$ be an umbilic-free immersion with
_curvature line_ coordinates $\displaystyle(u,v)$, that is, the first and
second fundamental form read as

(2.1)
$\mathrm{I}(u,v)=E\operatorname{d\\!}{u}^{2}+G\operatorname{d\\!}{v}^{2}\quad\text{and}\quad\mathrm{II}(u,v)=\mathrm{L}\operatorname{d\\!}{u}^{2}+\mathrm{N}\operatorname{d\\!}{v}^{2}.$

We call the coordinates $\displaystyle(u,v)$ _isothermic_ if they are
additionally conformal, that is, there exists a function $\displaystyle\sigma$
such that

(2.2)
$\mathrm{I}(u,v)=e^{2\sigma}(\operatorname{d\\!}{u}^{2}+\operatorname{d\\!}{v}^{2})\quad\text{and}\quad\mathrm{II}(u,v)=e^{2\sigma}(k_{1}\operatorname{d\\!}{u}^{2}+k_{2}\operatorname{d\\!}{v}^{2}),$

where $\displaystyle k_{1},k_{2}$ denote the principal curvatures. We use
these to define the _mean_ and _(extrinsic) Gauss curvature_ by

$\displaystyle H=\frac{k_{1}+k_{2}}{2},\quad K=k_{1}k_{2},$

respectively. As we are assuming umbilic-free, we have $\displaystyle
k_{1}\neq k_{2}$.

We briefly review the structure equations for surfaces in space forms.

######  Proposition 2.1 ([12, §65]).

Let $\displaystyle X\mathrel{\mathop{\mathchar 58\relax}}\Sigma\rightarrow
M_{\kappa}$ be a surface with curvature line coordinates $\displaystyle(u,v)$
and fundamental forms as in (2.1). Then the Codazzi equations are

(2.3)
$\frac{k_{1,v}}{k_{2}-k_{1}}=\big{(}{\log{\sqrt{E}}}\big{)}_{v}\quad\text{and}\quad\frac{k_{2,u}}{k_{1}-k_{2}}=\big{(}{\log{\sqrt{G}}}\big{)}_{u}.$

If $\displaystyle(u,v)$ are conformal coordinates in addition (so that they
are isothermic coordinates) with fundamental forms as in (2.2), then the Gauss
equation takes the form

(2.4) $\sigma_{uu}+\sigma_{vv}+(\kappa+K)e^{2\sigma}=0.$

while the Codazzi equations simplify to

(2.5)
$\frac{k_{1,v}}{k_{2}-k_{1}}=\big{(}{\log{\sqrt{E}}}\big{)}_{v}=\sigma_{v}\quad\text{and}\quad\frac{k_{2,u}}{k_{1}-k_{2}}=\big{(}{\log{\sqrt{G}}}\big{)}_{u}=\sigma_{u}.$

As rotational surfaces in various space forms are central to our
investigations, let us introduce this notion: consider $\displaystyle
M_{\kappa}$ as a quadric in the appropriate ambient space, equipped with the
proper inner product ($\displaystyle\mathbb{R}^{3}$ can be viewed as
$\displaystyle\\{p\mathrel{\mathop{\mathchar 58\relax}}(v,p)=1\\}$ with
$\displaystyle v\in\mathbb{R}^{4}$ any non-zero vector). Isometries of
$\displaystyle M_{\kappa}$ are then those orthogonal transformations of the
ambient space that fix the space form. We call a surface _rotational_ if it is
invariant under a $\displaystyle 1$-parameter subgroup $\displaystyle\rho$ of
isometries that act as the identity on a $\displaystyle 2$-plane111 Note that
for $\displaystyle\mathbb{R}^{3}$, the plane that is fixed by
$\displaystyle\rho$ must contain $\displaystyle v$. $\displaystyle\Pi$ in
$\displaystyle\mathbb{R}^{4}$. For hyperbolic spaces so that
$\displaystyle\kappa<0$, the definition amounts to three different types of
rotations, depending on the signature of the metric induced on
$\displaystyle\Pi$ (for an extensive treatment of rotations in hyperbolic
space forms see [11]). The next example gives explicit parametrizations of
rotational surfaces along with their principal curvatures:

######  Example 2.2.

For brevity we assume that the induced metric on $\displaystyle\Pi^{\perp}$ is
Riemannian. In this case, the term _surface of revolution_ is used more
frequently. We can parametrize rotational surfaces in $\displaystyle
M_{\kappa}$ via the action of $\displaystyle\rho$ on a (geodesic) planar
profile curve $\displaystyle\gamma=(r,0,h,k)$ ($\displaystyle k\equiv 1$ for
$\displaystyle\kappa=0$). From the parametrization

(2.6) $X(u,v)=\rho(u)(r(v),0,h(v),k(v)),$

and the fact that $\displaystyle\rho$ is a $\displaystyle 1$-parameter
subgroup, one quickly deduces that $\displaystyle\mathrm{I}$ and
$\displaystyle\mathrm{II}$ take the form (2.2) with (we use ′ to denote
$\displaystyle\partial_{v}$)

$\displaystyle
e^{2\sigma}=r^{2},~{}k_{1}=\frac{kh^{\prime}-hk^{\prime}}{r^{2}},~{}k_{2}=\frac{1}{r^{3}}\begin{vmatrix}r&r^{\prime}&r^{\prime\prime}\\\
h&h^{\prime}&h^{\prime\prime}\\\ k&k^{\prime}&\kappa
k^{\prime\prime}\end{vmatrix},$

given a suitable parametrization of $\displaystyle\gamma$. Similar expressions
for the other types of rotations in hyperbolic space can be obtained.

Note that for a rotational surface $\displaystyle k_{1,u}=0$. Thus, rotational
surfaces are an example of channel surfaces as defined next.

######  Definition 2.3.

A surface in a space form $\displaystyle M_{\kappa}$ is a _channel surface_ if
one of its principal curvatures is constant along its principal direction.

######  Remark 2.4.

According to Joachimsthal’s theorem, this definition is equivalent to the more
usual definition of channel surfaces as envelopes of $\displaystyle
1$-parameter families of spheres [3].

###  2.1. Linear Weingarten surfaces

Next, we introduce the class of linear Weingarten surfaces. There are two
different conventions which surfaces to call linear Weingarten: either those
for which Gauss and mean curvature satisfy a linear relationship, or those
where the two principal curvatures do. We call the latter “affine Weingarten”
to distinguish these two surface classes.

######  Definition 2.5.

A surface in a space form $\displaystyle M_{\kappa}$ is called

  * •

_linear Weingarten_ if there exists a non-trivial triple of constants
$\displaystyle a,b,c$ such that

(LW) $aK+2bH+c=0.$

  * •

_affine Weingarten_ if there exists a non-trivial triple of constants
$\displaystyle a,b,c$ such that

(AW) $xk_{1}+yk_{2}+z=0.$

######  Remark 2.6.

The only surfaces that are linear Weingarten and affine Weingarten are
surfaces of constant mean curvature (cmc): set $\displaystyle a=0$ in (LW) and
$\displaystyle x=y$ in (AW).

We call a linear Weingarten surface $\displaystyle X\mathrel{\mathop{\mathchar
58\relax}}\Sigma\to M_{\kappa}$ _tubular_ if $\displaystyle ac-b^{2}=0$. This
is equivalent to one of the principal curvatures of $\displaystyle X$ being
constant. Therefore, tubular linear Weingarten surfaces are also affine
Weingarten ($\displaystyle y=0$) and channel surfaces. We will from now on
assume further that the surfaces we consider are non-tubular.

Regarding channel linear or affine Weingarten surfaces, we have the following
result:

######  Proposition 2.7.

Let a surface be a non-tubular channel surface. If we have either

  * •

it is a linear Weingarten surface in a space form $\displaystyle M_{\kappa}$,
or

  * •

it is an affine Weingarten surface in $\displaystyle\mathbb{R}^{3}$,

then the surface must be rotational.

###### Proof.

The first statement is proven in, for instance, [15].

For the second statement, note that by (AW) $\displaystyle k_{1}=k_{1}(u)$
implies that $\displaystyle k_{2}$ is a function of $\displaystyle u$ alone as
well. The Codazzi equations (2.5) then imply $\displaystyle E=E(u)$ and
$\displaystyle G=U(u)V(v)$ for suitable functions $\displaystyle U,V$ of one
variable. Thus by [9], we have that the surface admits isothermic coordinates.
More precisely, we can set $\displaystyle r=\sqrt{E}$ and define a function
$\displaystyle h$ (of $\displaystyle u$) such that $\displaystyle
h_{u}=Ek_{2}$. The Gauss-Codazzi equations then imply that the fundamental
forms are

$\displaystyle\displaystyle\mathrm{I}(u,v)=r^{2}(\operatorname{d\\!}{u}^{2}+\operatorname{d\\!}{v}^{2})\quad\text{and}\quad\mathrm{II}(u,v)=r^{2}k_{1}\operatorname{d\\!}{u}^{2}+h_{u}\operatorname{d\\!}{v}^{2},$

with $\displaystyle k_{1}=\tfrac{r_{u}^{2}-r_{uu}r}{h_{u}r^{2}}$. A quick
computation shows that the surface of revolution with $\displaystyle r$ and
$\displaystyle h$ chosen such that (2.6) is isothermic ($\displaystyle
r^{2}=r_{u}^{2}+h_{u}^{2}$) has the same fundamental forms, and the claim thus
follows from the fundamental theorem. ∎

###  2.2. Lie minimal surfaces

Next, we define Lie minimal surfaces $\displaystyle
X\mathrel{\mathop{\mathchar 58\relax}}\Sigma\rightarrow M_{\kappa}$.

######  Definition 2.8 ([3, 5, 13, 21]).

The critical points of a functional

(2.7) $\displaystyle\displaystyle L_{Lie}[X]\mathrel{\mathop{\mathchar
58\relax}}=\int_{\Sigma}\frac{k_{1,u}k_{2,v}}{(k_{1}-k_{2})^{2}}\operatorname{d\\!}{u}\wedge\operatorname{d\\!}{v}$

with respect to compactly supported variations are called _Lie minimal
surfaces_.

The functional (2.7) is introduced in [3] as the simplest Lie invariant
integral. Thus, Lie sphere transformations map Lie minimal surfaces to Lie
minimal surfaces.

We now state a characterization of Lie minimal surfaces in terms of a pair of
differential equations that the principal curvatures of $\displaystyle X$ have
to satisfy.

######  Lemma 2.9.

Let $\displaystyle X\mathrel{\mathop{\mathchar 58\relax}}\Sigma\rightarrow
M_{\kappa}$ be a curvature line parametrization of a surface in a space form.
Then $\displaystyle X$ is Lie minimal if and only if the principal curvatures
satisfy

(2.8)
$(k_{2}-k_{1})k_{1,uv}+2k_{1,u}k_{1,v}=0\quad\text{and}\quad(k_{1}-k_{2})k_{2,uv}+2k_{2,u}k_{2,v}=0.$

###### Proof.

Consider a compactly supported variation of the surface $\displaystyle X$,
with variation parameter $\displaystyle\varepsilon$ independent of
$\displaystyle u$ and $\displaystyle v$, that fixes boundaries, and suppose
that it results in, for the principal curvatures
$\displaystyle\hat{k}_{1},\hat{k}_{2}$ at $\displaystyle\varepsilon$,

$\displaystyle\displaystyle k_{1}(u,v)$
$\displaystyle\displaystyle\mapsto\hat{k}_{1}(u,v)=k_{1}(u,v)+\varepsilon
h_{1}(u,v)+\mathcal{O}(\varepsilon^{2})$ $\displaystyle\displaystyle
k_{2}(u,v)$
$\displaystyle\displaystyle\mapsto\hat{k}_{2}(u,v)=k_{2}(u,v)+\varepsilon
h_{2}(u,v)+\mathcal{O}(\varepsilon^{2})$

for suitably chosen functions $\displaystyle
h_{1},h_{2}\mathrel{\mathop{\mathchar 58\relax}}\Sigma\rightarrow\mathbb{R}$
such that, for $\displaystyle i=1,2$, $\displaystyle h_{i}(u,v)=0$ outside of
the compact support of the variation. Setting

$\displaystyle\displaystyle
f=f[k_{1},k_{2},k_{1,u},k_{2,v}]=\frac{k_{1,u}k_{2,v}}{(k_{1}-k_{2})^{2}},$

then

$\left.\frac{\operatorname{d\\!}{L}}{\operatorname{d\\!}{\varepsilon}}\right|_{\varepsilon=0}=\left.\frac{\operatorname{d\\!}{}}{\operatorname{d\\!}{\varepsilon}}\int_{\Sigma}f[\hat{k}_{1}(u,v),\hat{k}_{2}(u,v),\hat{k}_{1,u}(u,v),\hat{k}_{2,v}(u,v)]\operatorname{d\\!}{u}\wedge\operatorname{d\\!}{v}\,\right|_{\varepsilon=0}\\\
\begin{aligned} &=\left.\int_{\Sigma}\left(\frac{\partial f}{\partial
k_{1}}\frac{\partial\hat{k}_{1}}{\partial\varepsilon}+\frac{\partial
f}{\partial
k_{2}}\frac{\partial\hat{k}_{2}}{\partial\varepsilon}+\frac{\partial
f}{\partial
k_{1,u}}\frac{\partial\hat{k}_{1,u}}{\partial\varepsilon}+\frac{\partial
f}{\partial
k_{2,v}}\frac{\partial\hat{k}_{2,v}}{\partial\varepsilon}\right)\operatorname{d\\!}{u}\wedge\operatorname{d\\!}{v}\,\right|_{\varepsilon=0}\\\
&=\int_{\Sigma}\left(\frac{\partial f}{\partial k_{1}}h_{1}+\frac{\partial
f}{\partial k_{2}}h_{2}+\frac{\partial f}{\partial
k_{1,u}}h_{1,u}+\frac{\partial f}{\partial
k_{2,v}}h_{2,v}\right)\operatorname{d\\!}{u}\wedge\operatorname{d\\!}{v}\\\
&=\int_{\Sigma}\left(\left(\frac{\partial f}{\partial
k_{1}}-\frac{\partial}{\partial u}\frac{\partial f}{\partial
k_{1,u}}\right)h_{1}+\left(\frac{\partial f}{\partial
k_{2}}-\frac{\partial}{\partial v}\frac{\partial f}{\partial
k_{2,v}}\right)h_{2}\right)\operatorname{d\\!}{u}\wedge\operatorname{d\\!}{v},\end{aligned}$

using the compact support condition. To find the critical points, we want
$\displaystyle\operatorname{d\\!}{L}/\operatorname{d\\!}{\varepsilon}|_{\varepsilon=0}$
to be zero for arbitrary $\displaystyle h_{1}$ and $\displaystyle h_{2}$, so
we have to show the stationary condition 2 & ∂f∂k1 \- ∂∂u∂f∂k1,u = 0,
∂f∂k2 \- ∂∂v∂f∂k2,v = 0. These equations are called the _Euler-Lagrange
(differential) equations_.

Because

$\displaystyle\displaystyle\frac{\partial f}{\partial k_{1}}$
$\displaystyle\displaystyle=-\frac{2k_{1,u}k_{2,v}}{\left(k_{1}-k_{2}\right)^{3}},$
$\displaystyle\displaystyle\frac{\partial f}{\partial k_{2}}$
$\displaystyle\displaystyle=\frac{2k_{1,u}k_{2,v}}{\left(k_{1}-k_{2}\right)^{3}},$
$\displaystyle\displaystyle\frac{\partial}{\partial u}\frac{\partial
f}{\partial k_{1,u}}$
$\displaystyle\displaystyle=\frac{k_{2,uv}}{\left(k_{1}-k_{2}\right)^{2}}-\frac{2k_{2,v}\left(k_{1,u}-k_{2,u}\right)}{\left(k_{1}-k_{2}\right)^{3}},\quad\text{and}$
$\displaystyle\displaystyle\frac{\partial}{\partial v}\frac{\partial
f}{\partial k_{2,v}}$
$\displaystyle\displaystyle=\frac{k_{1,uv}}{\left(k_{1}-k_{2}\right)^{2}}-\frac{2k_{1,u}\left(k_{1,v}-k_{2,v}\right)}{\left(k_{1}-k_{2}\right)^{3}},$

from (2.2) we have

$\displaystyle\displaystyle
0=-\frac{2k_{1,u}k_{2,v}}{\left(k_{1}-k_{2}\right)^{3}}-\frac{k_{2,uv}}{\left(k_{1}-k_{2}\right)^{2}}+\frac{2k_{2,v}\left(k_{1,u}-k_{2,u}\right)}{\left(k_{1}-k_{2}\right)^{3}}$
(2.9)
$\displaystyle\displaystyle\therefore\quad\frac{(k_{1}-k_{2})k_{2,uv}+2k_{2,u}k_{2,v}}{(k_{1}-k_{2})^{3}}=0.$

Similarly, from (2.2) we have

$\displaystyle\displaystyle
0=\frac{2k_{1,u}k_{2,v}}{\left(k_{1}-k_{2}\right)^{3}}-\frac{k_{1,uv}}{\left(k_{1}-k_{2}\right)^{2}}+\frac{2k_{1,u}\left(k_{1,v}-k_{2,v}\right)}{\left(k_{1}-k_{2}\right)^{3}}$
(2.10)
$\displaystyle\displaystyle\therefore\quad\frac{(k_{2}-k_{1})k_{1,uv}+2k_{1,u}k_{1,v}}{(k_{1}-k_{2})^{3}}=0.$

As we are assuming umbilic-free, the numerators of (2.9) and (2.10) lead to
the desired conclusions (2.8). ∎

######  Corollary 2.10.

Every rotational surface in $\displaystyle M_{\kappa}$ is Lie minimal.

###### Proof.

As we have seen in Example 2.2, $\displaystyle k_{1,u}=k_{2,u}=0$ for
rotational surfaces, hence, (2.8) is satisfied. ∎

##  3\. Lie minimal cmc surfaces

In this section we consider the relationship between Lie minimal and cmc
$\displaystyle H$ (or minimal) surfaces. Recall that every cmc (or minimal)
surface admits isothermic coordinates, which we will denote by
$\displaystyle(u,v)$.

######  Lemma 3.1.

Let $\displaystyle X\mathrel{\mathop{\mathchar 58\relax}}\Sigma\rightarrow
M_{\kappa}$ be Lie minimal with cmc $\displaystyle H$. Then, the principal
curvatures of $\displaystyle X$ satisfy

$\displaystyle k_{1}-k_{2}=\alpha(u)\beta(v),$

where $\displaystyle\alpha$ (resp. $\displaystyle\beta$) is a function only
depending on $\displaystyle u$ (resp. $\displaystyle v$).

###### Proof.

Suppose that $\displaystyle X$ is Lie minimal, and let $\displaystyle
k_{1}-k_{2}>0$ without loss of generality. Since $\displaystyle X$ is Lie
minimal, a simple calculation and Lemma 2.9 show

$\displaystyle\displaystyle(\log(k_{1}-k_{2}))_{uv}=\frac{H_{u}H_{v}}{(k_{1}-k_{2})^{2}},$

which vanishes because $\displaystyle H$ is constant. We conclude that

$\displaystyle\displaystyle\log(k_{1}-k_{2})$
$\displaystyle\displaystyle=\tilde{\alpha}(u)+\tilde{\beta}(v)$
$\displaystyle\displaystyle\therefore\quad k_{1}-k_{2}$
$\displaystyle\displaystyle=\alpha(u)\beta(v),$

where $\displaystyle\alpha$ (resp. $\displaystyle\beta$) is a function only
depending on $\displaystyle u$ (resp. $\displaystyle v$) and $\displaystyle
e^{\tilde{\alpha}(u)}=\alpha(u)$ (resp. $\displaystyle
e^{\tilde{\beta}(v)}=\beta(v)$). ∎

######  Theorem 3.2.

Let $\displaystyle X\mathrel{\mathop{\mathchar 58\relax}}\Sigma\to M_{\kappa}$
be a constant mean curvature $\displaystyle H$ surface. Then, $\displaystyle
X$ is Lie minimal if and only if it is a rotational surface.

###### Proof.

It follows from Corollary 2.10 that every rotational surface is Lie minimal.

Suppose conversely that $\displaystyle X$ has cmc $\displaystyle H$ and is Lie
minimal and is non-tubular. By solving the Codazzi equation (2.5), we find

$\sigma=\frac{1}{2}\left(\log\gamma-\log(k_{1}-k_{2})\right)\quad\therefore\quad
e^{2\sigma}=\frac{\gamma}{k_{1}-k_{2}},$

where $\displaystyle\gamma$ is some positive constant. Because of Lemma 3.1,
we can write this as

$\displaystyle\sigma=\frac{1}{2}\left(\log\gamma-\big{(}\tilde{\alpha}+\tilde{\beta}\big{)}\right)\quad\text{and}\quad\quad
e^{2\sigma}=\frac{\gamma}{\alpha\beta},$

and using the Gauss equation (2.4), we have

$\displaystyle-\frac{1}{2}\left(\tilde{\alpha}^{\prime\prime}+\tilde{\beta}^{\prime\prime}\right)+\left(\kappa+K\right)e^{2\sigma}=0.$

Now we differentiate both sides with respect to $\displaystyle u$ and
$\displaystyle v$ and obtain

$\displaystyle
0=\frac{\alpha^{\prime}\beta^{\prime}}{4\alpha^{2}\beta^{2}}(\kappa+K).$

Thus there are two possibilities. On the one hand, if
$\displaystyle\alpha^{\prime}\beta^{\prime}=0$, then without loss of
generality, $\displaystyle\alpha$ is constant and
$\displaystyle\sigma,k_{1},k_{2}$ only depend on $\displaystyle v$. The
surface is thus a channel linear Weingarten surface and by Proposition 2.7,
rotational.

On the other hand, if $\displaystyle\kappa+K=0$, we have

$\displaystyle(k_{1},k_{2})=\left(H+\sqrt{H^{2}+\kappa},H-\sqrt{H^{2}+\kappa}\right),$

where we assumed $\displaystyle k_{1}-k_{2}>0$ without loss of generality.
Since both principal curvatures are constant, the surface is tubular. ∎

A complete classification of Lie minimal cmc surfaces is thus provided by
explicit parametrizations of all rotational cmc surfaces (see for instance
[23]).

######  Corollary 3.3.

Let $\displaystyle X\mathrel{\mathop{\mathchar
58\relax}}\Sigma\to\mathbb{R}^{3}$ be Lie minimal and minimal and not part of
a plane. Then, $\displaystyle X$ is (part of) the catenoid.

###### Proof.

If $\displaystyle X$ is not restricted to a plane it is rotational according
to Theorem 3.2. Since the catenoid is the only rotational minimal surface in
$\displaystyle\mathbb{R}^{3}$ this finishes the proof. ∎

##  4\. Lie minimal Weingarten surfaces

In this section we study Lie minimal linear and affine Weingarten surfaces in
$\displaystyle\mathbb{R}^{3}$. Many of our results extend to other space
forms, but in order to give a streamlined presentation we restrict ourselves
to $\displaystyle\kappa=0$.

First, we consider the class of linear Weingarten surfaces. For a surface
$\displaystyle X\mathrel{\mathop{\mathchar 58\relax}}\Sigma\to\mathbb{R}^{3}$
satisfying (LW), we define $\displaystyle\Delta=b^{2}-ac$. The surface
$\displaystyle X^{t}=X+tN$, where $\displaystyle N$ denotes the Gauss map of
$\displaystyle X$, is called the _parallel surface at distance $\displaystyle
t$_. It is well-known that obtaining parallel surfaces amounts to applying
certain Lie sphere transformations [10, Sec 4.4], often referred to as
_parallel transformations_.

######  Proposition 4.1 (Bonnet’s theorem [22, Section 3.4]).

Every surface parallel to a linear Weingarten surface is again linear
Weingarten satisfying the linear Weingarten condition

$\displaystyle a^{t}K^{t}+2b^{t}H^{t}+c^{t}=0,$

with

$\displaystyle\Delta^{t}=(b^{t})^{2}-a^{t}c^{t}=\Delta.$

Every linear Weingarten surface with $\displaystyle\Delta>0$ is parallel to
either

  * •

a surface of positive constant Gauss curvature and hence a pair of cmc
surfaces, or

  * •

a minimal surface.

As mentioned in Subsection 2.2, Lie minimality is preserved under Lie sphere
transformations, and hence also preserved under parallel transformations.
These are exactly the transformations that map a surface to one of its
parallel surfaces. Thus, we arrive at the following theorem.

######  Theorem 4.2.

Every non-tubular Lie minimal linear Weingarten surface in
$\displaystyle\mathbb{R}^{3}$ with $\displaystyle\Delta>0$ is a surface of
revolution.

###### Proof.

Assume that $\displaystyle X\mathrel{\mathop{\mathchar
58\relax}}\Sigma\to\mathbb{R}^{3}$ is a Lie minimal linear Weingarten surface
with $\displaystyle\Delta>0$. Then, there is a value $\displaystyle t_{0}$
such that the parallel surface $\displaystyle X^{t_{0}}$ is Lie minimal and
cmc. According to Theorem 3.2, $\displaystyle X^{t_{0}}$ is rotational and
thus has a rotational Gauss map $\displaystyle N$. Therefore, $\displaystyle
X=X^{t_{0}}-t_{0}N$ is rotational as well. ∎

Finally, we consider the case of affine Weingarten surfaces. Recall, that a
surface $\displaystyle X\mathrel{\mathop{\mathchar
58\relax}}\Sigma\to\mathbb{R}^{3}$ is called affine Weingarten if there are
constants $\displaystyle x,y,y$ such that

(4.1) $xk_{1}+yk_{2}+z=0.$

We obtain the following theorem.

######  Theorem 4.3.

Let $\displaystyle X\mathrel{\mathop{\mathchar
58\relax}}\Sigma\rightarrow\mathbb{R}^{3}$ be non-tubular affine Weingarten.
Then it is Lie minimal if and only if it is a surface of revolution.

###### Proof.

Since every surface of revolution is Lie minimal (Corollary 2.10), let us
assume that $\displaystyle X$ is Lie minimal and an affine Weingarten surface.
From Lemma 2.9, we have

$\displaystyle\displaystyle((x+y)k_{1}+z)k_{1,uv}-2xk_{1,u}k_{1,v}$
$\displaystyle\displaystyle=0$
$\displaystyle\displaystyle\textrm{and}\quad((x+y)k_{1}+z)k_{1,uv}-2yk_{1,u}k_{1,v}$
$\displaystyle\displaystyle=0$
$\displaystyle\displaystyle\therefore\quad(x-y)((x+y)k_{1}+z)k_{1,uv}$
$\displaystyle\displaystyle=0.$

There are three cases to consider: for $\displaystyle x-y=0$, the surface is
Lie minimal cmc and thus rotational.

If $\displaystyle k_{1,uv}=0$, we have $\displaystyle k_{1,u}k_{1,v}=0$
because $\displaystyle X$ is Lie minimal and Lemma 2.9. Thus, $\displaystyle
X$ is a channel surface and, according to Proposition 2.7, a surface of
revolution.

Finally, if $\displaystyle(x+y)k_{1}+z=0$ both principal curvatures are
constant. Thus, $\displaystyle X$ is tubular, which goes against our
assumptions. ∎

##  5\. Future work

We have seen that linear Weingarten surfaces in $\displaystyle\mathbb{R}^{3}$
parallel to cmc surfaces are Lie minimal if and only if they are rotational in
Theorem 4.2. This result, of course, extends to the other ambient space forms.
Since linear Weingarten surfaces belong to the realm of Lie sphere geometry,
in the sense that they are those $\displaystyle\Omega$-surfaces that have
constant conserved quantities (see [7]) it seems a worthy goal to employ Lie
sphere geometric methods to investigate Lie minimal linear Weingarten
surfaces. This shall be the subject of a future project.

Acknowledgements. This paper is based on the fourth author’s Master thesis,
the goal of which was to investigate Lie minimal surfaces with additional
curvature properties. The authors would like to express their gratitude to
Professor Wayne Rossman for his support and continued interest in this
project. Further, we thank Professor Udo Hertrich-Jeromin for fruitful
discussions at Kobe University. This work was done while the third author was
a JSPS International Research Fellow (Graduate School of Science, Kobe
University) and has been supported by the JSPS Grant-in-Aid for JSPS Fellows
22F22701.

## References

  * Blaschke [1924a] W. Blaschke.  Über die Geometrie von Laguerre. I. Grundformeln der flächentheorie.  _Abh. Math. Sem. Univ. Hamburg_ , 3(1):176–194, 1924a.  doi:10.1007/BF02954623.
  * Blaschke [1924b] W. Blaschke.  Über die Geometrie von Laguerre. II. Flächentheorie in Ebenenkoordinaten.  _Abh. Math. Sem. Univ. Hamburg_ , 3(1):195–212, 1924b.  doi:10.1007/BF02954624.
  * Blaschke [1929] W. Blaschke.  _Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie III: Differentialgeometrie der Kreise und Kugeln_.  Springer, Berlin, 1929.
  * Bohle et al. [2008] C. Bohle, G. P. Peters, and U. Pinkall.  Constrained Willmore surfaces.  _Calc. Var. Partial Differential Equations_ , 32(2):263–277, 2008.  doi:10.1007/s00526-007-0142-5.
  * Burstall and Hertrich-Jeromin [2002] F. E. Burstall and U. Hertrich-Jeromin.  Harmonic maps in unfashionable geometries.  _Manuscripta Math._ , 108(2):171–189, 2002.  doi:10.1007/s002290200269.
  * Burstall et al. [2002] F. E. Burstall, F. Pedit, and U. Pinkall.  Schwarzian derivatives and flows of surfaces.  In M. A. Guest, R. Miyaoka, and Y. Ohnita, editors, _Differential geometry and integrable systems (Tokyo, 2000)_ , volume 308 of _Contemp. Math._ , pages 39–61. Amer. Math. Soc., Providence, RI, 2002.  doi:10.1090/conm/308/05311.
  * Burstall et al. [2012] F. E. Burstall, U. Hertrich-Jeromin, and W. Rossman.  Lie geometry of linear Weingarten surfaces.  _C. R. Math. Acad. Sci. Paris_ , 350(7-8):413–416, 2012.  doi:10.1016/j.crma.2012.03.018.
  * Cartan [1923] E. Cartan.  Les espaces à connexion conforme.  _Ann. Soc. Pol. Math._ , 2:171–221, 1923.
  * Cayley [1871] A. Cayley.  On the surfaces divisible into squares by their curves of curvature.  _Proc. Lond. Math. Soc._ , 4:120–121, 1871.  doi:10.1112/plms/s1-4.1.120.
  * Cecil [2008] T. E. Cecil.  _Lie sphere geometry_.  Universitext. Springer, New York, second edition edition, 2008.  doi:10.1007/978-0-387-74656-2.
  * do Carmo and Dajczer [1983] M. P. do Carmo and M. Dajczer.  Rotation hypersurfaces in spaces of constant curvature.  _Trans. Amer. Math. Soc._ , 277(2):685–709, 1983\.  doi:10.2307/1999231.
  * Eisenhart [1909] L. P. Eisenhart.  _A treatise on the differential geometry of curves and surfaces_.  Ginn and Company, Boston, 1909.
  * Ferapontov [2000] E. V. Ferapontov.  Lie sphere geometry and integrable systems.  _Tohoku Math. J. (2)_ , 52(2):199–233, 2000.  doi:10.2748/tmj/1178224607.
  * Germain [1821] S. Germain.  _Recherches sur la théorie des surfaces élastiques_.  Courcier, Paris, 1821.
  * Hertrich-Jeromin et al. [2023] U. Hertrich-Jeromin, M. Pember, and D. Polly.  Channel linear Weingarten surfaces in space forms.  _Beitr. Algebra Geom._ , 2023.  ISSN 0138-4821, 2191-0383.  doi:10.1007/s13366-022-00664-w.
  * Lie [1872] S. Lie.  Ueber Complexe, insbesondere Linien- und Kugel-Complexe, mit Anwendung auf die Theorie partieller Differential-Gleichungen.  _Math. Ann._ , 5(1):145–208, 1872.  doi:10.1007/BF01446331.
  * López [2008] R. López.  Rotational linear Weingarten surfaces of hyperbolic type.  _Israel J. Math._ , 167:283–301, 2008.  doi:10.1007/s11856-008-1049-3.
  * López and Pámpano [2020] R. López and Á. Pámpano.  Classification of rotational surfaces in Euclidean space satisfying a linear relation between their principal curvatures.  _Math. Nachr._ , 293(4):735–753, 2020.  doi:10.1002/mana.201800235.
  * Marques and Neves [2014] F. C. Marques and A. Neves.  Min-max theory and the Willmore conjecture.  _Ann. of Math. (2)_ , 179(2):683–782, 2014.  doi:10.4007/annals.2014.179.2.6.
  * Musso and Nicolodi [1996] E. Musso and L. Nicolodi.  A variational problem for surfaces in Laguerre geometry.  _Trans. Amer. Math. Soc._ , 348(11):4321–4337, 1996.  doi:10.1090/S0002-9947-96-01698-4.
  * Musso and Nicolodi [2005] E. Musso and L. Nicolodi.  On the Cauchy problem for the integrable system of Lie minimal surfaces.  _J. Math. Phys._ , 46(11):113509, 15, 2005.  doi:10.1063/1.2116267.
  * Palais and Terng [1988] R. S. Palais and C.-L. Terng.  _Critical point theory and submanifold geometry_ , volume 1353 of _Lecture Notes in Mathematics_.  Springer-Verlag, Berlin, 1988.  doi:10.1007/BFb0087442.
  * Polly [2023] D. Polly.  Rotational cmc surfaces in terms of Jacobi elliptic functions.  Submitted, 2023.  arXiv:2305.15514.
  * Richter [1997] J. Richter.  _Conformal maps of a Riemannian surface into the space of quaternions_.  PhD thesis, Technische Universität Berlin, 1997.
  * Thomsen [1924] G. Thomsen.  Über konforme Geometrie I. Grundlagen der konformen flächentheorie.  _Abh. Math. Sem. Univ. Hamburg_ , 3(1):31–56, 1924.  doi:10.1007/BF02954615.
  * Willmore [1965] T. J. Willmore.  Note on embedded surfaces.  _An. Şti. Univ. “Al. I. Cuza” Iaşi Secţ. I a Mat. (N.S.)_ , 11B:493–496, 1965.