Minimal surfaces with symmetries

Franc Forstnerič

Abstract Let $G$ be a finite group acting on a connected open Riemann surface
$X$ by holomorphic automorphisms and acting on a Euclidean space
$\mathbb{R}^{n}$ $(n\geq 3)$ by orthogonal transformations. We identify a
necessary and sufficient condition for the existence of a $G$-equivariant
conformal minimal immersion $F:X\to\mathbb{R}^{n}$. We show in particular that
such a map $F$ always exists if $G$ acts without fixed points on $X$.
Furthermore, every finite group $G$ arises in this way for some open Riemann
surface and $n=2|G|$. We obtain an analogous result for minimal surfaces
having complete ends with finite total Gaussian curvature, and for discrete
groups acting on $X$ properly discontinuously and acting on $\mathbb{R}^{n}$
by rigid transformations.

Keywords: minimal surface, equivariant conformal minimal immersion, Oka
manifold

MSC (2020): Primary 53A10; Secondary 32E30, 32H02, 32Q56

Date: 16 February 2024

## 1\. Introduction

Objects with symmetries are of special interest in any mathematical theory. In
this paper, we study the existence of immersed orientable minimal surfaces in
Euclidean spaces $\mathbb{R}^{n}$ with a given finite or countable group of
symmetries induced by rigid transformations of $\mathbb{R}^{n}$.

An immersed minimal surface in $\mathbb{R}^{n}$ for $n\geq 3$ is the image of
a conformal harmonic immersion $F:X\to\mathbb{R}^{n}$ from an open conformal
surface $X$, which can be taken to be a Riemann surface if it is orientable;
see [47, 5]. We shall call such $F$ a conformal minimal immersion. Euclidean
isometries of $\mathbb{R}^{n}$ form an affine group generated by the
orthogonal group $O(n,\mathbb{R})$ and the additive group $(\mathbb{R}^{n},+)$
acting by translations. By also adding dilations, we obtain the group of rigid
transformations. Postcomposition by a rigid transformation of $\mathbb{R}^{n}$
maps minimal surfaces to minimal surfaces, and rigid transformations are the
largest class of self-maps of $\mathbb{R}^{n}$ with this property. Hence, it
is of interest to find minimal surfaces which are invariant under a given
group of rigid transformations. Symmetries of specific minimal surfaces were
studied by many authors, but we are interested in general existence results.

Let $X$ be a connected open Riemann surface and $G$ be a finite subgroup of
the group $\mathrm{Aut}(X)$ of holomorphic automorphisms of $X$. The
stabiliser $G_{x}=\\{g\in G:gx=x\\}$ of any point $x\in X$ is a cyclic
subgroup of $G$, which is trivial for points in the complement of a closed
discrete subset of $X$ (see [45, Corollary 3.5, p. 93]). Assume that $G$ also
acts on $\mathbb{R}^{n}$ by orthogonal transformations. The following result
and Remark 1.2 provide a necessary and sufficient condition for the existence
of a $G$-equivariant conformal minimal immersion $X\to\mathbb{R}^{n}$.

######  Theorem 1.1.

Let $G$ be a finite group acting effectively on a connected open Riemann
surface $X$ by holomorphic automorphisms, and acting on $\mathbb{R}^{n}$
$(n\geq 3)$ by orthogonal transformations. If for every nontrivial stabiliser
$G_{x}$ $(x\in X)$ there is a 2-plane $\Lambda_{x}\subset\mathbb{R}^{n}$ on
which $G_{x}$ acts effectively by rotations, then there exists a conformal
minimal immersion $F:X\to\mathbb{R}^{n}$ such that

(1.1) $F(gx)=gF(x)\ \ \text{holds for all $x\in X$ and $g\in G$},$

and the image $F(X)$ is not contained in any affine hyperplane of
$\mathbb{R}^{n}$.

A map $F:X\to\mathbb{R}^{n}$ is said to be $G$-equivariant if condition (1.1)
holds, and nondegenerate if $F(X)$ is not contained in any affine hyperplane
of $\mathbb{R}^{n}$. Note that the image $S=F(X)\subset\mathbb{R}^{n}$ of a
$G$-equivariant map is $G$-invariant, i.e., $gS=S$ for all $g\in G$. Since the
fixed-point-set of a nontrivial linear map on $\mathbb{R}^{n}$ is a proper
linear subspace of $\mathbb{R}^{n}$, nondegeneracy of $F$ implies that every
$g\in G$ which acts effectively on $\mathbb{R}^{n}$ also acts effectively on
$F(X)\subset\mathbb{R}^{n}$. Thus, if $H$ is the normal subgroup of $G$
consisting of all elements $g\in G$ which act trivially on $\mathbb{R}^{n}$,
then $G/H$ is a symmetry group of the minimal surface $F(X)$ in Theorem 1.1.
Any additional symmetries can be eliminated by using a general position
argument in the proof of the theorem.

Our proof of Theorem 1.1 gives several additions concerning approximation,
interpolation, and the control of the flux; see Theorem 5.1 and compare with
the results in the non-equivariant case [5, Theorems 3.6.1 and 3.6.2]. In
particular, the map $F$ in Theorem 1.1 can be chosen to be the real part of a
$G$-equivariant null holomorphic immersion $X\to\mathbb{C}^{n}$; see Theorem
5.3.

In Section 6 we construct $G$-equivariant minimal surfaces with complete ends
of finite total curvature on given finitely many orbits of $G$ on the Riemann
surface $X$; see Theorem 6.3 and Corollary 6.5. However, we do not know
whether all ends can be made complete and of finite total curvature; see
Problem 6.2. In particular, the construction of complete $G$-equivariant
minimal surfaces of finite total curvature remains an open problem.

Finally, in Section 7 we show that the analogue of Theorem 1.1 also holds if
$G$ is an infinite discrete group acting on $\mathbb{R}^{n}$ by rigid
transformations, and acting on a Riemann surface $X$ properly discontinuously
by holomorphic automorphisms such that the quotient surface $X/G$ is
noncompact; see Theorem 7.1. This case is only relevant if $X$ has genus at
most one, since every Riemann surface of genus $\geq 2$ has at most finitely
many automorphisms by a theorem of Hurwitz [34] (see also [45, Theorem 3.9]).

Minimal surfaces with symmetries appeared in the very origin of the theory;
indeed, most classical examples have symmetries (the catenoid, the helicoid,
Scherk’s surfaces, Riemann’s minimal examples, Schwarz’s surfaces, etc.). All
mentioned examples have infinite groups of symmetries and are parameterized by
plane domains. See also the discussion in Example 1.8. Finding examples with
given groups of symmetries from Riemann surfaces of genus $\geq 1$ is a more
difficult task due to the problem of controlling the periods of their
Weierstrass data, and only a few examples have been described explicitly. In
this paper we give general existence results for such minimal surfaces. The
techniques developed in the paper also seem promising for constructing minimal
surfaces with given symmetries and satisfying various additional conditions
such as being complete or proper; see Problems 1.12 and 1.13.

######  Remark 1.2.

The conditions on stabilisers in Theorem 1.1 are necessary. Indeed, let $x\in
X$ be a point with a nontrivial stabiliser $G_{x}$ of order $k>1$. There is a
local holomorphic coordinate $z$ on a neighbourhood $U\subset X$ of $x$, with
$z(x)=0$, in which a generator $g$ of the cyclic group $G_{x}$ is the rotation
$gz=\mathrm{e}^{\mathfrak{i}\phi}z$ through the angle $\phi=2\pi/k$ (see
Miranda [45, Corollary 3.5, p. 93]). Assume that $F:X\to\mathbb{R}^{n}$ is a
$G$-equivariant conformal immersion, not necessarily harmonic. Differentiating
the identity (1.1) and taking into account that $g$ acts linearly on
$\mathbb{R}^{n}$ gives

$dF_{x}\circ dg_{x}=g\circ
dF_{x}:T_{x}X\to\Lambda_{x}:=dF_{x}(T_{x}X)\subset\mathbb{R}^{n}.$

Since $dF_{x}:T_{x}X\to\Lambda_{x}$ is a conformal linear isomorphism, we
infer that $\Lambda_{x}$ is a $G_{x}$-invariant $2$-plane in $\mathbb{R}^{n}$
on which $g$ acts as the rotation $R_{\phi}$ through the angle $\phi$, so the
conditions in Theorem 1.1 hold. Conversely, these conditions imply that the
local conformal linear embedding $U\to\Lambda$ is $G_{x}$ equivariant. These
conditions are superfluous for minimal surfaces with branch points.

######  Remark 1.3.

In Theorem 1.1 and its corollaries presented below, the Riemann surface $X$ is
assumed to be connected. However, these results generalize to the case when
for every connected component $X^{\prime}$ of $X$ the stabiliser group
$G_{X^{\prime}}=\\{g\in G:gX^{\prime}=X^{\prime}\\}$ acts effectively on
$X^{\prime}$. This is equivalent to asking that the stabiliser $G_{x}$ of a
generic point $x\in X$ is trivial.

The first immediate reduction is to the case when $G$ acts transitively on the
set of connected components of $X$. (See the argument preceding [38, Theorem
4.1] by Kutzschebauch et al.) Assuming this to be the case, fix a component
$X^{\prime}$ of $X$. Since $G_{x}\subset G_{X^{\prime}}$ holds for every $x\in
X^{\prime}$, Theorem 1.1 provides a $G_{X^{\prime}}$-equivariant conformal
minimal immersion $F^{\prime}:X^{\prime}\to\mathbb{R}^{n}$. If
$X^{\prime\prime}$ is another component of $X$ and $h\in G$ is such that
$h(X^{\prime\prime})=X^{\prime}$, we define $F$ on $X^{\prime\prime}$ by
$F(x)=h^{-1}F^{\prime}(hx)$ for $x\in X^{\prime\prime}$. It is immediate that
the resulting map $F:X\to\mathbb{R}^{n}$ is a $G$-equivariant conformal
minimal immersion.

If on the other hand the stabiliser $G_{X^{\prime}}$ of some component
$X^{\prime}$ of $X$ does not act effectively on $X^{\prime}$, then the
conditions in Theorem 1.1 must be adjusted. We shall not consider this case.

Theorem 1.1 is a special case of Theorem 5.1, which also involves
approximation and interpolation of a given $G$-equivariant minimal immersion
on a suitable $G$-invariant subset of $X$ by global $G$-equivariant conformal
minimal immersions. The proof relies on two main ingredients. The classical
Enneper–Weierstrass representation of minimal surfaces in $\mathbb{R}^{n}$
reduces the problem to constructing holomorphic maps (the so-called
Weierstrass data) from the given Riemann surface $X$ into the punctured null
quadric ${\mathbf{A}}_{*}$ in $\mathbb{C}^{n}$ (see (2.1)) having suitable
integrals (periods) on a system of curves in $X$ whose union contains a basis
of the homology group $H_{1}(X,\mathbb{Z})$ and some other arcs which are used
to guarantee the interpolation conditions. In our case, the Weierstrass data
are $G$-equivariant holomorphic maps from $X$ to the projective
compactification of the null quadric, and we formulate a Weierstrass
representation theorem for $G$-equivariant minimal surfaces; see Theorem 2.4.
The main point is to approximate such maps on certain $G$-invariant Runge
subsets of $X$ by global $G$-equivariant holomorphic maps having suitable
periods. We combine the approach developed in [2] (see also [5, Theorem
3.6.1]) with [38, Theorem 4.1] due to Kutzschebauch et al., which shows how to
reduce Oka-theoretic problems for certain $G$-equivariant holomorphic maps to
the nonequivariant case for sections of an associated holomorphic map having
ramification points; see Section 3. Ultimately, the main complex-analytic tool
that we use is an Oka-theoretic result for sections of ramified holomorphic
maps (see [17, Theorem 2.1] and [18, Theorem 6.14.6]), combined with the
techniques from [2] and [5, Chapter 3] which enable us to control periods of
maps $X\to{\mathbf{A}}_{*}$. The main step is Lemma 4.5, and Theorem 1.1 is
then proved in Section 5. The same method applies if $G$ is an infinite
discrete group acting on $X$ properly discontinuously; see Theorem 7.1.

In the remainder of this introduction, we give several corollaries to Theorem
1.1 and we place our results in the context of what is known. The following
corollary is immediate.

######  Corollary 1.4.

If $G$ is a finite group acting freely (without fixed points) on an open
Riemann surface $X$ by holomorphic automorphisms, then for every action of $G$
by orthogonal maps on $\mathbb{R}^{n}$ $(n\geq 3)$ there exists a
nondegenerate $G$-equivariant conformal minimal immersion
$X\to\mathbb{R}^{n}$, which can be chosen to be the real part of a
$G$-equivariant null holomorphic immersion $X\to\mathbb{C}^{n}$.

If $G$ is a finite group acting on a connected Riemann surface $X$ by
holomorphic automorphisms, then the union $X_{0}$ of fixed point sets of
elements of $G$ is a closed discrete subset of $X$ (see (2.3)), which is
finite if $X$ has finite genus but may be infinite otherwise; see Section 2.
Removing from $X$ any closed $G$-invariant subset $X^{\prime}$ containing
$X_{0}$, the group $G$ acts freely on the open Riemann surface $X\setminus
X^{\prime}$, and hence Corollary 1.4 applies to the pair $(X\setminus
X^{\prime},G)$.

######  Corollary 1.5.

For every connected open Riemann surface $X$ and finite subgroup
$G\subset\mathrm{Aut}(X)$ of order $n\geq 2$ there are an effective action of
$G$ by orthogonal transformations on $\mathbb{R}^{2n}$ and a nondegenerate
$G$-equivariant conformal minimal immersion $X\to\mathbb{R}^{2n}$.

###### Proof.

I wish to thank Urban Jezernik for the following argument. Consider the
regular representation of $G$ on the complex Euclidean space $\mathbb{C}^{n}$
with the basis vectors $e_{g}$ for $g\in G$, where an element $h\in G$ acts by
$he_{g}=e_{hg}$. For a fixed $g\in G$ of order $k>1$ let $\Sigma_{g}$ denote
the $k$-dimensional $\mathbb{C}$-linear subspace of $\mathbb{C}^{n}$ spanned
by the vectors $e_{g^{j}}$ $(j=0,1,\ldots,k-1)$ corresponding to the elements
of the cyclic group $\langle g\rangle$. Clearly, $\Sigma_{g}$ is $g$-invariant
and the eigenvalues of the $\mathbb{C}$-linear isomorphism
$g:\Sigma_{g}\to\Sigma_{g}$ are precisely all the $k$-th roots of $1$. In
particular, there is a vector $0\neq w\in\Sigma_{g}$ with
$gw=\mathrm{e}^{\mathfrak{i}2\pi/k}w$. Identifying $\mathbb{C}^{n}$ with
$\mathbb{R}^{2n}$, the $2$-plane $\Lambda_{g}\subset\mathbb{R}^{2n}$
determined by the complex line $\mathbb{C}w$ is $g$-invariant and $g$ acts on
it as a rotation through the angle $2\pi/k$. Since every stabiliser $G_{x}$ in
Theorem 1.1 is a cyclic subgroup of $G$, the conditions of Theorem 1.1 hold
for this representation of $G$. ∎

Given a smooth surface $X$, an immersion $F:X\to\mathbb{R}^{n}$ induces on $X$
a unique structure of a conformal surface such that $F$ is a conformal
immersion (see [5, Sect. 1.10]). This conformal structure is clearly invariant
under postcomposition of $F$ by rigid motions of $\mathbb{R}^{n}$. In
particular, if $F$ is an embedding and the image surface
$\widetilde{X}=F(X)\subset\mathbb{R}^{n}$ is $G$-invariant for a finite
subgroup $G$ of $O(n,\mathbb{R})$ (i.e., $g\widetilde{X}=\widetilde{X}$ holds
for all $g\in G$), there is a unique action of $G$ on $X$ by conformal
automorphisms such that $F$ is a $G$-equivariant conformal embedding. If in
addition the surface $X$ is oriented and every $g\in G$ preserves the
orientation on $X$, then $G$ acts on $X$ by holomorphic automorphisms. If this
action is effective, Remark 1.2 shows that the conditions on stabilisers hold,
so Theorem 1.1 implies the following corollary.

######  Corollary 1.6.

Assume that $G$ is a finite subgroup of the orthogonal group $O(n,\mathbb{R})$
for some $n\geq 3$ and $X\subset\mathbb{R}^{n}$ is a smoothly embedded,
connected, oriented, noncompact, $G$-invariant surface such that every $g\in
G$ preserves the orientation on $X$, and $g$ induces the identity map on $X$
only if $g=1\in G$. Then, $X$ endowed with the complex structure induced by
the embedding $X\hookrightarrow\mathbb{R}^{n}$ and with the induced action of
$G$ on $X$ by holomorphic automorphisms admits a nondegenerate $G$-equivariant
conformal minimal immersion $F:X\to\mathbb{R}^{n}$.

######  Remark 1.7.

In the context of Corollary 1.6, it is natural to ask whether there is a
regular homotopy of $G$-equivariant (conformal) immersions
$F_{t}:X\to\mathbb{R}^{n}$ $(t\in[0,1])$ connecting the initial embedding
$F_{0}:X\hookrightarrow\mathbb{R}^{n}$ to a conformal minimal immersion
$F_{1}:X\to\mathbb{R}^{n}$. An inspection of our proof of Theorem 1.1 shows
that there is a homotopy of $G$-equivariant maps $f_{t}:X\to Y$ $(t\in[0,1])$
into the projective closure $Y$ (2.8) of the null quadric such that
$f_{0}=2\partial F_{0}/\theta$ (see (2.6) for the definition of the
holomorphic 1-form $\theta$ on $X$), the map $f_{1}$ is holomorphic, and
$f_{1}=2\partial F_{1}/\theta$. However, we do not know whether the maps
$f_{t}$ for $0<t<1$ can be chosen such that they integrate to immersions
$F_{t}:X\to\mathbb{R}^{n}$ with $2\partial F_{t}=f_{t}\theta$.

######  Example 1.8 (Equivariant minimal surfaces of genus zero).

Let $S$ be the unit sphere in $\mathbb{R}^{3}$. The induced Riemann surface
structure on $S$ is that of the Riemann sphere
$\mathbb{C}\cup\\{\infty\\}=\mathbb{CP}^{1}$, the unique complex structure on
$S$ up to biholomorphisms. The special orthogonal group $SO(3,\mathbb{R})$
acts on $S$ by orientation preserving isometries, hence by holomorphic
automorphisms, and it forms a real 3-dimensional subgroup of the holomorphic
automorphism group

$\mathrm{Aut}(S)=\Big{\\{}z\mapsto\frac{az+b}{cz+d}:\ \ a,b,c,d\in\mathbb{C},\
ad-bc=1\Big{\\}}.$

Finite subgroups of $SO(3,\mathbb{R})$ are called spherical von Dyck groups.
Besides the cyclic and the dihedral groups, there are the symmetry groups of
Platonic solids, the so-called crystallographic groups: the alternating group
$A_{4}$ of order $12$ is the group of symmetries of the tetrahedron, the
symmetric group $S_{4}$ of order $24$ is the group of symmetries of the cube
and the octahedron, and the alternating group $A_{5}$ of order $60$ is the
group of symmetries of the icosahedron and the dodecahedron. Corollary 1.6
shows that every spherical von Dyck group of order $m>1$ is a group of
symmetries of a minimal surface in $\mathbb{R}^{3}$ parameterized by a
complement of $m$ points in $\mathbb{CP}^{1}$. However, this case is already
known. After the initial work of Goursat [24], it was shown by Xu [54], using
explicit functions in the Enneper–Weierstrass representation, that any closed
subgroup $G$ of $SO(3,\mathbb{R})$, which is not isomorphic to
$SO(2,\mathbb{R})$ or $SO(3,\mathbb{R})$, is the symmetry group of a complete
immersed minimal surface in $\mathbb{R}^{3}$ of genus zero with finite total
curvature and embedded ends. In the genus zero case, the only period vanishing
conditions are those coming from the ends, which amount to vanishing of the
residues of the Weierstrass data at such points. Examples of (families of)
minimal surfaces in $\mathbb{R}^{3}$ with groups of $SO(3,\mathbb{R})$
symmetries were given by Jorge and Meeks [36], Rossman [50], Small [52], and
others. Choi, Meeks and White proved in [13] that if $X$ is a minimal surface
in $\mathbb{R}^{3}$ with a catenoidal end, then every intrinsic local isometry
of $X$ extends to a rigid motion of $\mathbb{R}^{3}$. As a corollary due to Xu
[54, Corollary 2.2], one sees that if such $X$ has finite total curvature and
embedded ends, at least one of which is catenoidal, then the symmetry group of
$X$ is a closed subgroup of $SO(3)$.

It is natural to ask which finite groups arise in the context of Theorem 1.1
for Riemann surfaces of genus $\mathfrak{g}\geq 1$. The study of finite groups
$G$ acting effectively on a connected Riemann surface $X$ by holomorphic
automorphisms is based on the observation that the orbit space $X/G$ has the
structure of a Riemann surface such that the quotient projection $\pi:X\to
X/G$ is holomorphic, it is ramified precisely at the points $x\in X$ with
nontrivial stabiliser group $G_{x}$, and the ramification index at such a
point equals $k_{x}=|G_{x}|$, the order of the stabiliser. Furthermore,
stabilisers of points in the $G$-orbit of $x$ are conjugate cyclic subgroups
of $G$, so there are $|G|/k_{x}$ of them (see Miranda [45, Proposition 3.3, p.
77]). By the uniformization theorem for Riemann surfaces, we have that
$X=\mathbb{U}/K$ where $\mathbb{U}$ is either the Riemann sphere
$\mathbb{CP}^{1}=\mathbb{C}\cup\\{\infty\\}$, the complex number field
$\mathbb{C}$, or the upper halfplane

(1.2) $\mathbb{H}=\\{z=x+\mathfrak{i}y\in\mathbb{C}:y>0\\},$

and $K$ is a subgroup of $\mathrm{Aut}(\mathbb{U})$ acting properly
discontinuously and without fixed points. If $\mathbb{U}=\mathbb{CP}^{1}$ then
$K$ is the trivial group, and if $\mathbb{U}=\mathbb{C}$ then $K$, if
nontrivial, is a free cyclic group of rank one or two generated by one or two
translations. The case $\mathbb{U}=\mathbb{H}$ is more complicated and will be
discussed in Example 1.9. Any subgroup $G\subset\mathrm{Aut}(X)$ is then
isomorphic to a quotient group $\Gamma/K$, where $\Gamma$ is a subgroup of
$\mathrm{Aut}(\mathbb{U})$ containing $K$ as a normal subgroup. By analysing
these conditions, the Riemann–Hurwitz formula provides limitations on the
number and type of finite or discrete groups acting on a given compact Riemann
surface $X$; see Miranda [45, Chapter III]. These results also apply to open
Riemann surfaces of finite genus. Indeed, by Maskit [42] every open Riemann
surface $X$ of finite genus embeds in a compact Riemann surface $X^{*}$ of the
same genus such that every holomorphic automorphism of $X$ extends to a
holomorphic automorphism of $X^{*}$. Applying this technique, Miranda [45, pp.
80–82] discusses finite subgroups of $\mathrm{Aut}(X)$ for compact Riemann
surfaces. In the simplest case when $X=\mathbb{CP}^{1}$, the quotient
projection $\pi:\mathbb{CP}^{1}\to\mathbb{CP}^{1}$ has either two or three
ramification points. The case of two ramification points corresponds to cyclic
groups of rotations on $\mathbb{C}$. In the case of three ramification points
and considering $\mathbb{CP}^{1}$ as the round sphere in $\mathbb{R}^{3}$, we
have the dihedral group and the crystallographic groups $A_{4},\ S_{4}$, and
$A_{5}$ mentioned in Example 1.8.

######  Example 1.9 (Equivariant minimal surfaces of genus $\geq 2$).

The projective special linear group $PSL(2,\mathbb{R})=SL(2,\mathbb{R})/\\{\pm
I\\}$ of degree two over the real numbers can be realised as the group of
orientation preserving isometries of the hyperbolic plane. The Poincaré
halfplane model is given by the upper halfplane $\mathbb{H}$ (1.2), endowed
with the metric $\frac{dx^{2}+dy^{2}}{y^{2}}$ of constant negative curvature,
on which $PSL(2,\mathbb{R})$ acts by holomorphic automorphisms

(1.3) $\mathbb{H}\ni z\mapsto\frac{az+b}{cz+d}\quad\text{for}\
a,b,c,d\in\mathbb{R},\ ad-bc=1.$

This action realises $PSL(2,\mathbb{R})$ as the holomorphic automorphism group
$\mathrm{Aut}(\mathbb{H})$. (One can also use the Poincaré disc model
$\mathbb{D}=\\{z\in\mathbb{C}:|z|<1\\}$ with the Poincaré metric
$\frac{4|dz|^{2}}{(1-|z|^{2})^{2}}$ and $PSL(2,\mathbb{R})$ acting as the
group $\mathrm{Aut}(\mathbb{D})$.) A subgroup $\Gamma\subset
PSL(2,\mathbb{R})$ is called a Fuchsian group if it acts on $\mathbb{H}$ (by
maps (1.3)) properly discontinuously. General Fuchsian groups were first
studied by Poincaré [48], who was motivated by Fuchs [23].

Every Riemann surface $X$ of genus $\mathfrak{g}\geq 2$ is a quotient
$X=\mathbb{H}/K$, where $K\subset\mathrm{Aut}(\mathbb{H})$ is a Fuchsian group
acting without fixed points. Every group $G\subset\mathrm{Aut}(X)$ is then of
the form $G\cong\Gamma/K$, where $\Gamma\subset\mathrm{Aut}(\mathbb{H})$ is
subgroup containing $K$ as a normal subgroup. If $X$ is compact then $\Gamma$
is of a special form described by Moore in [46, p. 923].

For a compact Riemann surface $X$ of genus $\mathfrak{g}\geq 2$, Hurwitz’s
automorphism theorem [34] (see also [45, Theorem 3.9, p. 96, and Chapter VII])
says that the automorphism group $\mathrm{Aut}(X)$ is finite of order at most
$84(\mathfrak{g}-1)$. In view of the aforementioned theorem by Maskit [42],
the same holds on every noncompact Riemann surface of finite genus
$\mathfrak{g}\geq 2$. The maximal size $84(\mathfrak{g}-1)$ can arise if and
only if $X$ admits a branched cover $X\to\mathbb{CP}^{1}$ with three
ramification points, of indices 2, 3, and 7. A group for which the maximum is
achieved is called a Hurwitz group, and the corresponding Riemann surface is a
Hurwitz surface. Klein’s quartic curve of genus 3 (see [37]) is a Hurwitz
surface of lowest genus. From Klein’s result, Macbeath [40] deduced the
existence of Hurwitz surfaces of infinitely many genuses. The next smallest
genus of a Hurwitz surface is $\mathfrak{g}=7$; see Macbeath [41] for an
explicit description. Most Riemann surfaces of genus $\mathfrak{g}\geq 2$ do
not have any nontrivial holomorphic automorphisms.

Greenberg [26] proved that every countable group $G$ is the automorphism group
of a noncompact Riemann surface, which can be taken to have a finitely
generated fundamental group if $G$ is finite. He also proved [27, Theorem 6’]
that every finite group is the automorphism group of a compact Riemann surface
(see also Jones [35]). Greenberg’s theorem, together with Corollary 1.5,
implies the following result.

######  Corollary 1.10.

For every finite group $G$ of order $n>1$ there exist an open connected
Riemann surface $X$, effective actions of $G$ by holomorphic automorphisms on
$X$ and by orthogonal transformations on $\mathbb{R}^{2n}$, and a
nondegenerate $G$-equivariant conformal minimal immersion
$X\to\mathbb{R}^{2n}$. The surface $X$ can be chosen to be the complement of
$n$ points in a compact Riemann surface.

Since the conditions in Theorem 1.1 pertain to nontrivial isotropy subgroups
of a given automorphism group $G\subset\mathrm{Aut}(X)$, it is of interest to
understand the possible number of fixed points of holomorphic automorphisms of
Riemann surfaces. There is a considerable literature on this subject. Hurwitz
[34] proved that every nontrivial holomorphic automorphism of a compact
Riemann surface of genus $\mathfrak{g}$ has at most $2\mathfrak{g}+2$ fixed
points. In view of the result of Maskit [42], Hurwitz’s theorem also holds on
every open Riemann surface of finite genus. Moore [46] determined the number
of fixed points of each element of a cyclic group of automorphisms of a
compact Riemann surface with genus at least two. It was shown by Minda [44,
Theorem 1] that if $X$ is a Kobayashi hyperbolic Riemann surface and
$\phi:X\to X$ is a holomorphic self-map with at least two fixed points, then
$\phi$ is an automorphism of $X$ of finite order. We refer to the survey in
Miranda [45, Chapter 3] for further information on this topic.

######  Problem 1.11.

Let $X$ be an open Riemann surface of genus $\mathfrak{g}\geq 2$ with a
nontrivial automorphism group $\mathrm{Aut}(X)$. Which subgroups $G$ of
$\mathrm{Aut}(X)$ are symmetry groups of conformal minimal surfaces
$X\to\mathbb{R}^{n}$ for a given $n\geq 3$? (By Corollary 1.5 every such group
arises for $n=2|G|$.)

Of particular interest are minimal surfaces of finite total Gaussian
curvature. We discuss this case in Section 6 and obtain an analogue of Theorem
1.1 for $G$-equivariant minimal surfaces having some ends of finite total
Gaussian curvature; see Problem 6.2 and Theorem 6.3.

One may also ask whether the Calabi–Yau problem for minimal surfaces (see [5,
Chapter 7] for background and a survey on this problem) has an affirmative
answer for minimal surfaces with symmetries. Explicitly, we pose the following
problem.

######  Problem 1.12.

Assume the hypotheses of Theorem 1.1, and let $M$ be a compact, smoothly
bounded, $G$-invariant domain in the Riemann surface $X$ such that no element
of $G$ has any fixed point on $bM$. Does there exist a continuous
$G$-equivariant map $F:M\to\mathbb{R}^{n}$ whose restriction to the interior
$\mathring{M}=M\setminus bM$ is a complete conformal minimal immersion?

An affirmative answer for the trivial group is given by [1, Theorem 1.1] (see
also [5, Theorem 7.4.1]), where in addition the map
$F|_{bM}:bM\to\mathbb{R}^{n}$ is a topological embedding.

Here is another interesting problem.

######  Problem 1.13.

In the context of Theorem 1.1, is there a proper nondegenerate $G$-equivariant
conformal minimal immersion $X\to\mathbb{R}^{n}$?

We expect the answer to be affirmative. Without the equivariance condition,
proper conformal minimal immersions $X\to\mathbb{R}^{n}$ $(n\geq 3)$ from an
arbitrary open Riemann surface $X$ exist in great abundance; see Alarcón and
López [7] for a construction of such surfaces in $\mathbb{R}^{3}$ which
project properly to a plane $\mathbb{R}^{2}\subset\mathbb{R}^{3}$, and [5,
Theorem 3.10.3] for any dimension $n\geq 3$. In [5, Section 10.3] the reader
can also find a survey of the history of this subject.

It seems likely that an analogue of Theorem 1.1 holds for nonorientable
conformal surfaces; however, the nature of the isotropy groups can be more
complicated, and the fixed point set of a Euclidean isometry restricted to the
surface may contain curves. We shall not study this case here. Recall that
every conformal minimal immersion $X\to\mathbb{R}^{n}$ from a nonorientable
open conformal surface $X$ is given by a $\mathfrak{I}$-invariant conformal
minimal immersion $\widetilde{X}\to\mathbb{R}^{n}$ from the orientable
2-sheeted cover $\widetilde{X}\to X$ whose deck transformation is a fixed-
point-free antiholomorphic involution
$\mathfrak{I}:\widetilde{X}\to\widetilde{X}$. For the theory of such surfaces,
see [8, 4].

The problem treated in this paper can be considered for conformal minimal
surfaces in any Riemannian manifold of dimension $\geq 3$ with a nontrivial
group of isometries. Although the connection to complex analysis is lost in
general, there are some other special cases (besides the Euclidean spaces)
which could possibly be approached with these techniques. One of them concerns
superminimal surfaces in self-dual or anti-self-dual Einstein four-manifolds.
This case can be treated using the Bryant correspondence in Penrose twistor
spaces, thereby reducing problems on superminimal surfaces to those concerning
holomorphic Legendrian curves in complex contact three-manifolds. We refer to
the survey of this subject in [19] where the Calabi–Yau problem (see Problem
1.12) was solved affirmatively for superminimal surfaces in such Riemannian
four-manifolds. (The special case concerning the four-sphere with the
spherical metric was obtained beforehand by Alarcón et al. in [3].) See also
[21] for the construction of proper superminimal surfaces in the hyperbolic
four-space.

Important examples of minimal surfaces are holomorphic curves in complex
Euclidean spaces $\mathbb{C}^{n}$ $(n>1)$ and, more generally, in Kähler
manifolds. In principle, equivariant holomorphic maps are easier to construct
than general minimal surfaces since many more operations and techniques are
available. Heinzner proved in [29] that if $G$ is a reductive complex Lie
group acting on a reduced Stein space $X$ by holomorphic automorphisms, then
$X$ is $G$-equivariantly embeddable in a Euclidean space $\mathbb{C}^{n}$ on
which $G$ acts by $\mathbb{C}$-linear automorphisms if and only if the Luna
slice type of $(X,G)$ is finite. Heinzner’s theorem implies in particular that
every open Riemann surface with an action of a finite group $G$ of holomorphic
automorphisms is equivariantly embeddable in some $\mathbb{C}^{n}$ with a
$\mathbb{C}$-linear action of $G$. See also [30, 31]. Further results were
obtained by Heinzner and Huckleberry [32], Fritsch and Heinzner [22], D’Angelo
[14], among many others. In particular, D’Angelo and Xiao [15] studied
equivariant proper rational maps between balls in complex Euclidean spaces.
For results on the $G$-equivariant Oka principle, also used in this paper, see
the survey by Kutzschebauch et al. [39].

## 2\. Preliminaries

Let $X$ be a connected open Riemann surface. An immersion
$F:X\to\mathbb{R}^{n}$ is conformal if and only if its $(1,0)$-differential
$\partial F=(\partial F_{1},\ldots,\partial F_{n})$ satisfies the nullity
condition

$\sum_{i=1}^{n}(\partial F_{i})^{2}=0,$

and it is harmonic if and only if $\partial F$ is a holomorphic $1$-form on
$X$ (see [47] or [5, Sect. 2.3]). A conformal immersion is harmonic if and
only if its image is a minimal surface, i.e., its mean curvature vector field
vanishes identically.

Pick a nowhere vanishing holomorphic $1$-form $\theta$ on $X$; such exists by
Gunning and Narasimhan [28] and can be chosen to be the differential
$\theta=dh$ of a holomorphic immersion $h:X\to\mathbb{C}$. Then, a conformal
minimal immersion $F:X\to\mathbb{R}^{n}$ satisfies $2\partial F=f\theta$ with
$f=2\partial F/\theta:X\to{\mathbf{A}}_{*}={\mathbf{A}}\setminus\\{0\\}$ a
holomorphic map into the punctured null quadric, where the null quadric in
$\mathbb{C}^{n}$ is the affine subvariety

(2.1)
${\mathbf{A}}=\\{z=(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}:z_{1}^{2}+z_{2}^{2}+\cdots+z_{n}^{2}=0\\}.$

These observations lead to the Enneper–Weiestrass formula (see [5, Theorem
2.3.4]), which says that any conformal minimal immersion
$F:X\to\mathbb{R}^{n}$ is of the form

(2.2) $F(x)=F(x_{0})+\int_{x_{0}}^{x}\Re(f\theta),\quad x,x_{0}\in X,$

where $f:X\to{\mathbf{A}}_{*}$ is a holomorphic map such that
$\int_{C}\Re(f\theta)=0$ for any closed path $C$ in $X$ (so the integral in
(2.2) is independent of the path of integration), and $2\partial F=f\theta$.

In the sequel, we shall allow the 1-form $\theta$ on $X$ to have a discrete
zero set and will let $f$ be a meromorphic map such that the vector-valued
1-form $f\theta$ is holomorphic and nowhere vanishing on $X$, i.e., the poles
of $f$ exactly cancel the zeros of $\theta$. (In the proof of Theorem 6.3 we
shall allow $f\theta$ to have poles.)

Let $G$ be a finite group acting faithfully on $X$ by holomorphic
automorphisms. We have already mentioned that the stabiliser $G_{x}$ of any
point $x\in X$ is a cyclic group of some order $k=k(x)\in\mathbb{N}$ which is
generated in a suitable local holomorphic coordinate $z$ on a neighbourhood of
$x\in X$, with $z(x)=0$, by the rotation
$z\mapsto\mathrm{e}^{\mathfrak{i}\phi}z$, where
$\mathrm{e}^{\mathfrak{i}\phi}$ is a primitive $k$-th root of $1$. It follows
that $z^{k}$ is a local holomorphic coordinate on the orbit space $X/G_{x}$,
which is therefore nonsingular. Note that $G_{gx}=gG_{x}g^{-1}$ for all $x\in
X$ and $g\in G$, so the stabilisers of points in a $G$-orbit are pairwise
conjugate subgroups of $G$. We can identify the orbit $Gx=\\{gx:g\in G\\}$
with the set of cosets $\\{gG_{x}:\ g\in G\\}$. For any $g\in
G\setminus\\{1\\}$ the set of fixed points $\mathrm{Fix}(g)=\\{x\in X:gx=x\\}$
is a closed discrete subset of $X$ (see Miranda [45, Proposition 3.2, p. 76]),
which is finite if the surface $X$ has finite genus (see Minda [44]), but it
can be infinite otherwise. Their union

(2.3) $X_{0}=\bigcup_{g\in G\setminus\\{1\\}}\mathrm{Fix}(g)=\\{x\in
X:G_{x}\neq\\{1\\}\\}$

is a closed, discrete, $G$-invariant subset of $X$, and its complement

(2.4) $X_{1}=X\setminus X_{0}=\\{x\in X:gx\neq x\ \text{for all}\ g\in
G\setminus\\{1\\}\\}$

is an open $G$-invariant domain. For every $x\in X$ the orbit $Gx$ has
$|G|/|G_{x}|$ points; this number equals $|G|$ if and only if $x\in X_{1}$.
Since the group $G$ is finite, the orbit space $X/G$ is an open Riemann
surface, the quotient projection $\pi:X\to X/G$ is a holomorphic map which
branches precisely at the points in $X_{0}$, and $\pi:X_{1}\to X_{1}/G$ is a
holomorphic covering projection of degree $|G|$. (See Miranda [45, Theorem
3.4, p. 78] for these facts.) Choose a holomorphic immersion
$\tilde{h}:X/G\to\mathbb{C}$; see [28]. Then, the holomorphic map

(2.5) $h:=\tilde{h}\circ\pi:X\to\mathbb{C}$

is $G$-invariant and it branches precisely at the points of $X_{0}$. Applying
the chain rule to the equation $h\circ g=h$ $(g\in G)$ shows that the
holomorphic $1$-form

(2.6) $\theta=dh=d(\tilde{h}\circ\pi)=\pi^{*}d\tilde{h}$

on $X$ satisfies the following invariance conditions for every $g\in G$:

(2.7) $\theta_{gx}\circ dg_{x}=\theta_{x}\ \ \text{for all $x\in X$,\ \ and
$\theta_{x}=0$ if and only if $x\in X_{0}$}.$

More precisely, $\theta$ has a zero of order $|G_{x}|-1$ at a point $x\in
X_{0}$.

######  Remark 2.1.

Everything said so far in this section also holds if $X$ is disconnected and
for every connected component $X^{\prime}$ of $X$ the stabiliser subgroup
$G_{X^{\prime}}=\\{g\in G:gX^{\prime}=X^{\prime}\\}$ acts effectively on
$X^{\prime}$. (This is equivalent to asking that the stabiliser $G_{x}$ of a
generic point $x\in X$ is trivial.) Without loss of generality, one may always
assume that $G$ acts transitively on the set of connected components of $X$,
so the orbit space $X/G$ is connected. These observations can be used to
justify the generalization of our results mentioned in Remark 1.3.

Suppose now that the group $G$ also acts on $\mathbb{R}^{n}$ $(n\geq 3)$ by
orthogonal maps. Considering $\mathbb{R}^{n}$ as the standard real subspace of
$\mathbb{C}^{n}$, the orthogonal group $O(n,\mathbb{R})$ is a subgroup of the
complex orthogonal group $O(n,\mathbb{C})$, the subgroup of $GL(n,\mathbb{C})$
preserving the $\mathbb{C}$-bilinear form
$(z,w)\mapsto\sum_{i=1}^{n}z_{i}w_{i}$. The punctured null quadric
${\mathbf{A}}_{*}={\mathbf{A}}\setminus\\{0\\}$ (see (2.1)) is smooth and
$O(n,\mathbb{C})$-invariant, hence also $G$-invariant. Consider
$\mathbb{C}^{n}$ as an affine chart in the projective space $\mathbb{CP}^{n}$.
Let $\overline{\mathbf{A}}\subset\mathbb{CP}^{n}$ denote the projective
closure of ${\mathbf{A}}$ and set

(2.8) $\displaystyle Y$ $\displaystyle=$
$\displaystyle\overline{\mathbf{A}}\setminus\\{0\\}={\mathbf{A}}_{*}\cup
Y_{0},$ (2.9) $\displaystyle Y_{0}$ $\displaystyle=$ $\displaystyle
Y\setminus{\mathbf{A}}_{*}=\bigl{\\{}[z_{1}:\cdots:z_{n}]\in\mathbb{CP}^{n-1}:z_{1}^{2}+z_{2}^{2}+\cdots+z_{n}^{2}=0\bigr{\\}}.$

Let $p:\mathbb{C}^{n}\setminus\\{0\\}\to\mathbb{CP}^{n-1}$ denote the
projection $p(z_{1},\ldots,z_{n})=[z_{1}:\cdots:z_{n}]$. The restriction
$p:{\mathbf{A}}_{*}\to Y_{0}$ is a holomorphic fibre bundle with fibre
$\mathbb{C}^{*}=\mathbb{C}\setminus\\{0\\}$, and the natural extension $p:Y\to
Y_{0}$ which equals the identity map on $Y_{0}$ is a holomorphic line bundle.
The action of $O(n,\mathbb{C})$ on $\mathbb{C}^{n}$ extends to an action on
$\mathbb{CP}^{n}$ with the hyperplane at infinity as an invariant complex
submanifold. Hence, the action of $G$ on $\mathbb{C}^{n}$ extends to an action
of $G$ on the manifold $Y$ (2.8) by holomorphic automorphisms.

We denote by $u\,\cdotp v$ the Euclidean inner product on $\mathbb{R}^{n}$ and
by $\|u\|=\sqrt{u\,\cdotp u}$ the Euclidean norm. To any oriented 2-plane
$0\in\Lambda\subset\mathbb{R}^{n}$ we associate a complex line
$L\subset\mathbb{C}^{n}$, contained in the null quadric ${\mathbf{A}}$ (2.1),
by choosing an oriented basis $(u,v)$ of $\Lambda$ such that $\|u\|=\|v\|\neq
0$ and $u\,\cdotp v=0$ (such a pair is called a conformal frame) and setting

(2.10)
$L=L(\Lambda)=\mathbb{C}(u-\mathfrak{i}v)\subset{\mathbf{A}}\subset\mathbb{C}^{n}.$

Clearly, $L$ does not depend on the choice of the oriented conformal frame on
$\Lambda$. A rotation $R_{\phi}$ on $\Lambda$ in the positive direction
corresponds to the multiplication by $\mathrm{e}^{\mathfrak{i}\phi}$ on the
complex line $L(\Lambda)$.

If $F:X\to\mathbb{R}^{n}$ is a conformal immersion then, in any local
holomorphic coordinate $z=x+\mathfrak{i}y$ on $X$, the vectors $\frac{\partial
F}{\partial x}(z)$ and $\frac{\partial F}{\partial y}(z)$ form a conformal
frame and the corresponding complex line $L(\zeta)\subset{\mathbf{A}}$ is
spanned by the vector $\frac{\partial F}{\partial
x}(z)-\mathfrak{i}\frac{\partial F}{\partial y}(z)=2\frac{\partial F}{\partial
z}(z).$

The following proposition summarizes the main properties of immersed
$G$-equivariant conformal minimal surfaces, and it justifies the hypotheses in
Theorem 1.1.

######  Proposition 2.2.

Assume that $X$ is a connected open Riemann surface and $G$ is a finite group
acting effectively on $X$ by holomorphic automorphisms and acting on
$\mathbb{R}^{n}$ $(n\geq 3)$ by orthogonal transformations. Let the sets
$X_{0}\subset X$ and $Y_{0}\subset Y$ be given by (2.3) and (2.9),
respectively. Set $X_{1}=X\setminus X_{0}$, and let $\theta$ be the
holomorphic 1-form on $X$ given by (2.6). If $F:X\to\mathbb{R}^{n}$ is a
$G$-equivariant conformal minimal immersion then

(2.11) $f=2\partial F/\theta:X\to Y$

is a holomorphic $G$-equivariant map satisfying $f^{-1}(Y_{0})=X_{0}$, and the
following assertions hold for every point $x_{0}\in X_{0}$.

  1. (a)

The stabiliser $G_{x_{0}}$ is a cyclic group with a generator $g_{0}$ acting
in a local holomorphic coordinate $z$ on $X$, with $z(x_{0})=0$, by
$g_{0}z=\mathrm{e}^{\mathfrak{i}\phi}z$, where $\phi=2\pi/k$ and
$k=|G_{x_{0}}|$.

  2. (b)

The tangent plane $\Lambda=dF_{x_{0}}(T_{x_{0}}X)\subset\mathbb{R}^{n}$ is
$G_{x_{0}}$-invariant, $g_{0}$ acts on $\Lambda$ by rotation $R_{\phi}$
through the angle $\phi=2\pi/k$ in the positive direction (with respect to the
orientation induced from $T_{x_{0}}X$ by $dF_{x_{0}}$), and $g_{0}$ acts on
the null line $L=L(\Lambda)$ (2.10) as multiplication by
$\mathrm{e}^{\mathfrak{i}\phi}$.

  3. (c)

We have that $g_{0}F(x_{0})=F(x_{0})$, and the vector $F(x_{0})$ is orthogonal
to $\Lambda$.

  4. (d)

We have that $f(x_{0})=p(L)\in
Y_{0}\subset\mathbb{CP}^{n}\setminus\mathbb{C}^{n}$, and $f$ has a pole of
order $|G_{x_{0}}|-1$ at $x_{0}$.

###### Proof.

Recall that $\partial F=(\partial F_{1},\ldots,\partial F_{n})$ is a
holomorphic $1$-form with values in
${\mathbf{A}}_{*}={\mathbf{A}}\setminus\\{0\\}$ (2.1). Since $\theta$ vanishes
precisely on $X_{0}$, the map (2.11) is holomorphic and satisfies

(2.12) $f^{-1}(Y_{0})=\\{x\in X:f(x)\in Y_{0}\\}=X_{0}.$

Differentiation of the $G$-equivariance equation $F\circ g=gF$ (see (1.1)),
taking into account that $g$ acts on $\mathbb{R}^{n}$ as a linear (orthogonal)
transformation, gives

(2.13) $dF_{gx}\circ dg_{x}=g\,dF_{x}\ \ \text{for every $x\in X$ and $g\in
G$}.$

Writing $dF=\partial F+\overline{\partial}F$ we have

$\partial F_{gx}\circ dg_{x}+\overline{\partial}F_{gx}\circ dg_{x}=g\,\partial
F_{x}+g\,\overline{\partial}F_{x}.$

Since $dg_{x}$ is $\mathbb{C}$-linear on $T_{x}X$, the $(1,0)$-part of the
above equation gives

(2.14) $\partial F_{gx}\circ dg_{x}=g\,\partial F_{x}\ \ \text{for every $x\in
X$ and $g\in G$}.$

From this and the condition (2.7) on the 1-form $\theta$ we obtain

(2.15) $f(gx)=\frac{2\partial F_{gx}}{\theta_{gx}}=\frac{2\partial F_{gx}\circ
dg_{x}}{\theta_{gx}\circ dg_{x}}=\frac{g\,2\partial F_{x}}{\theta_{x}}=gf(x),$

which shows that $f:X\to Y$ is $G$-equivariant. (The second equality holds
since the quotient of two $\mathbb{C}$-linear forms on $\mathbb{C}$ is
invariant under precomposition by a linear isomorphism.)

Let $k=|G_{x_{0}}|>1$. By Remark 1.2, a generator $g_{0}$ of $G_{x_{0}}$ acts
in a certain local holomorphic coordinate $z$ on $X$ based at $x_{0}$ by the
rotation through the angle $\phi=2\pi/k$, so (a) holds. Since the map
$dF_{x_{0}}:T_{x_{0}}X\cong\mathbb{R}^{2}\to\Lambda=dF_{x_{0}}(\mathbb{R}^{2})\subset\mathbb{R}^{n}$
is a conformal linear isomorphism, the relation (2.13) at $x=x_{0}$ implies
(b). Since $g_{0}\in\mathrm{Aut}(X)$ fixes $x_{0}$ and $F$ is
$g_{0}$-equivariant, we have $g_{0}F(x_{0})=F(g_{0}x_{0})=F(x_{0})$. As
$g_{0}$ acts on $\mathbb{R}^{n}$ by an orthogonal transformation which
restricts to a nontrivial rotation on $\Lambda$, we infer that $F(x_{0})$ is
either the zero vector or an eigenvector of $g_{0}$ with the eigenvalue $1$
which is orthogonal to $\Lambda$, so (c) holds. Since the holomorphic $1$-form
$f\theta=2\partial F$ has values in ${\mathbf{A}}_{*}$ and
$\\{\theta=0\\}=X_{0}$, $f$ is a meromorphic map to ${\mathbf{A}}_{*}$ with
poles at the points of $X_{0}$, and the order of the pole of $f$ at $x_{0}\in
X_{0}$ equals the order of zero of $\theta$ at $x_{0}$, which is
$|G_{x_{0}}|-1$. This proves (d). ∎

Conversely, given a connected open Riemann surface $X$, a holomorphic 1-form
on $X$ of the form (2.6), and a holomorphic map $f:X\to Y$ such that the
$1$-form $f\theta$ is holomorphic and nowhere vanishing on $X$ and

(2.16) $\Re\int_{C}f\theta=0\ \ \text{holds for every smooth closed curve $C$
in $X$},$

we obtain for any $x_{0}\in X$ and $v\in\mathbb{R}^{n}$ a conformal minimal
immersion $F:X\to\mathbb{R}^{n}$ given by

(2.17) $F(x)=v+\int_{x_{0}}^{x}\Re(f\theta)\ \ \text{for all}\ x\in X.$

Since $f\theta$ is holomorphic, the integral is independent of the path of
integration in view of the period vanishing conditions (2.16). If $x_{0}\in
X_{0}$ then (2.18) implies $gv=v$ for all $g\in G_{x_{0}}$, which is
compatible with Proposition 2.2 (c). If on the other hand $x_{0}\in X\setminus
X_{0}$ then there is no restriction on $v=F(x_{0})\in\mathbb{R}^{n}$. Let us
observe the following.

######  Lemma 2.3.

(Assumptions as above.) The conformal minimal immersion
$F:X\to\mathbb{R}^{n}$, defined by (2.17), is $G$-equivariant if and only if
the map $f:X\to Y$ is $G$-equivariant and

(2.18) $gv=v+\int_{x_{0}}^{gx_{0}}\Re(f\theta)\ \ \text{holds for all $g\in
G$.}$

###### Proof.

Suppose that a map $F:X\to\mathbb{R}^{n}$ of the form (2.17) is
$G$-equivariant. The $G$-equivariance condition at the point $x=x_{0}$ gives

$gv=gF(x_{0})=F(gx_{0})=v+\int_{x_{0}}^{gx_{0}}\Re(f\theta)\quad\text{for
all}\ g\in G,$

so (2.18) holds. By Proposition 2.2 the map $f=2\partial F/\theta:X\to Y$ is
$G$-equivariant as well.

Conversely, assume that $f:X\to Y$ is a $G$-equivariant holomorphic map. Given
a piecewise $\mathscr{C}^{1}$ path $\gamma:[0,1]\to X$, we have in view of
(2.7) for any $g\in G$ that

(2.19)
$\int_{g\gamma}f\theta=\int_{0}^{1}f(g\gamma(t))\,\theta_{g\gamma(t)}(dg_{\gamma(t)}\dot{\gamma}(t))\,dt=\int_{0}^{1}gf(\gamma(t))\,\theta_{\gamma(t)}(\dot{\gamma}(t))\,dt=g\int_{\gamma}f\theta.$

If $f$ also satisfies the period vanishing conditions (2.16) then the integral
of $f\theta$ between a pair of points is independent of the choice of a path.
From (2.17) and (2.18) we obtain

$\displaystyle F(gx)$ $\displaystyle=$ $\displaystyle
v+\int_{x_{0}}^{gx}\Re(f\theta)=v+\int_{x_{0}}^{gx_{0}}\Re(f\theta)+\int_{gx_{0}}^{gx}\Re(f\theta)$
$\displaystyle=$ $\displaystyle gv+g\int_{x_{0}}^{x}\Re(f\theta)=gF(x),$

showing that the map $F$ is $G$-equivariant. ∎

Summarizing, Proposition 2.2 and Lemma 2.3 give the following representation
formula. We explain in Section 7 that the same result holds for infinite
discrete groups $G$ acting on $X$ properly discontinuously provided that $X/G$
is non-compact.

######  Theorem 2.4 (Weierstrass representation of $G$-equivariant minimal
surfaces).

Assume that $X$ and $G$ are as in Theorem 1.1, $\theta$ is given by
(2.5)–(2.6), and $Y$ is given by (2.8). Fix a point $x_{0}\in X$. Then, every
$G$-equivariant conformal minimal immersion $F:X\to\mathbb{R}^{n}$ is of the
form

$F(x)=F(x_{0})+\int_{x_{0}}^{x}\Re(f\theta)\quad\text{for $x\in X$},$

where $f=2\partial F/\theta:X\to Y={\mathbf{A}}_{*}\cup Y_{0}$ is a
$G$-equivariant holomorphic map satisfying $f^{-1}(Y_{0})=X_{0}$ such that
$f\theta$ has no zeros or poles and satisfies the period conditions

$\displaystyle\int_{C}\Re(f\theta)$ $\displaystyle=$ $\displaystyle 0\quad\
\text{for every $[C]\in H_{1}(X,\mathbb{Z})$, and}$ $\displaystyle gF(x_{0})$
$\displaystyle=$ $\displaystyle
F(x_{0})+\int_{x_{0}}^{gx_{0}}\Re(f\theta)\quad\text{for every $g\in G$.}$

## 3\. Constructing holomorphic $G$-equivariant maps $X\to Y$

In this section, we explain the setup and outline the proof of Theorem 1.1;
the details are given in the following two sections. We shall use the notation
from Proposition 2.2. In particular, the holomorphic $1$-form $\theta$ on $X$
is as in (2.6) and the manifolds $Y_{0}\subset Y$ are given by (2.8) and
(2.9).

We begin by defining a $G$-equivariant conformal minimal immersion $F_{0}$
from a neighbourhood of the closed discrete subset $X_{0}$ of $X$ (see (2.3))
to $\mathbb{R}^{n}$. Fix a point $x_{0}\in X_{0}$ and set $k=|G_{x_{0}}|>1$,
where $G_{x_{0}}$ is the (cyclic) stabiliser group of $x_{0}$. By Proposition
2.2 (a), there is a local holomorphic coordinate $z$ on a
$G_{x_{0}}$-invariant disc neighbourhood $\Delta\subset X$ of $x_{0}$, with
$z(x_{0})=0$, such that a generator $g_{0}$ of $G_{x_{0}}$ is the rotation
$g_{0}z=\mathrm{e}^{\mathfrak{i}\phi}z$ with $\phi=2\pi/k$. By the assumption
of Theorem 1.1 there is a $G_{x_{0}}$-invariant plane
$0\in\Lambda\subset\mathbb{R}^{n}$ on which $g_{0}$ acts as the rotation
through the angle $\phi$. Let $L=L(\Lambda)$ (2.10) be the associated complex
line contained in the null quadric ${\mathbf{A}}\subset\mathbb{C}^{n}$ (2.1).
Then, $g_{0}$ acts on $L$ as multiplication by
$\mathrm{e}^{\mathfrak{i}\phi}$. Choose a nonzero vector $y_{0}\in L$ and set

(3.1) $f_{0}(z)=\frac{y_{0}}{z^{k-1}}\ \ \text{for all $z\in\Delta$}.$

Thus, $f_{0}:\Delta\to Y$ is a holomorphic map with the point
$f_{0}(x_{0})=p(y_{0})\in Y_{0}$ at infinity. Since
$\mathrm{e}^{\mathfrak{i}k\phi}=1$, we have that

$f_{0}(g_{0}z)=f_{0}(\mathrm{e}^{\mathfrak{i}\phi}z)=\frac{y_{0}}{\mathrm{e}^{\mathfrak{i}(k-1)\phi}z^{k-1}}=\mathrm{e}^{\mathfrak{i}\phi}\frac{y_{0}}{\mathrm{e}^{\mathfrak{i}k\phi}z^{k-1}}=g_{0}f_{0}(z),$

so $f_{0}$ is $G_{x_{0}}$-equivariant. In the coordinate $z$, we have that

$\theta(z)=d(h(z^{k}))=kh^{\prime}(z^{k})z^{k-1}dz,$

where $h:\Delta\to\mathbb{C}$ is a holomorphic function with nonvanishing
derivative. The 1-form

$(f_{0}\theta)(z)=kh^{\prime}(z^{k})y_{0}dz$

is holomorphic and nonvanishing on $\Delta$. Since $\Delta$ is simply
connected and condition (2.18) in Lemma 2.3 trivially holds on $\Delta$ for
$v=0$ and the group $G_{x_{0}}$, $f_{0}\theta$ integrates to a flat
$G_{x_{0}}$-equivariant conformal minimal immersion
$F_{0}:\Delta\to\Lambda\subset\mathbb{R}^{n}$. (Alternatively, we can observe
that the conformal linear map $F_{0}$ from the $z$-coordinate on $\Delta$ to
the plane $\Lambda$ is $G_{x_{0}}$-equivariant since the generator of
$G_{x_{0}}$ acts as a rotation through the same angle on the domain and
codomain, and take $f_{0}=2\partial F_{0}/\theta$. This may differ from (3.1)
by a multiplicative holomorphic factor.) We extend $f_{0}$ and $F_{0}$ by
$G$-equivariance to the neighbourhood $G\,\cdotp\Delta=\bigcup_{g\in
G}g\Delta\subset X$ of the orbit $Gx_{0}\subset X$. Doing the same at every
point of $X_{0}$ and choosing the neighbourhoods pairwise disjoint yields a
$G$-equivariant holomorphic map $f_{0}:V\to Y$ from a $G$-invariant
neighbourhood $V\subset X$ of $X_{0}$ such that $f_{0}\theta$ is a nowhere
vanishing holomorphic $1$-form on $V$ with values in ${\mathbf{A}}_{*}$, and a
$G$-equivariant conformal minimal immersion $F_{0}:V\to\mathbb{R}^{n}$ with
$2\partial F_{0}=f_{0}\theta$.

To prove Theorem 1.1, we shall find a $G$-equivariant holomorphic map $f:X\to
Y$ which agrees with $f_{0}:V\to Y$ to a given finite order in every point of
$X_{0}$, it satisfies $f(X_{1})\subset{\mathbf{A}}_{*}$ (where
$X_{1}=X\setminus X_{0}$), and conditions (2.16) and (2.18) hold. By Lemma 2.3
(or Theorem 2.4), the map $F:X\to\mathbb{R}^{n}$ given by (2.17) is then a
$G$-equivariant conformal minimal immersion.

Let us first explain how to find a $G$-equivariant holomorphic map $f:X\to Y$
which agrees with $f_{0}$ to a given finite order on $X_{0}$ and satisfies
$f(X_{1})\subset{\mathbf{A}}_{*}$, ignoring the period vanishing conditions
(2.16) and (2.18) for the moment. This is a special case of [38, Theorem 4.1]
due to Kutzschebauch, Lárusson, and Schwarz. For our purposes, some additional
explanations are necessary. Consider the action of $G$ on the product manifold
$X\times Y$ by

(3.2) $g(x,y)=(gx,gy)\ \ \ \text{for}\ \ x\in X,\ y\in Y,\ g\in G.$

Since the projection $X\times Y\to X$ is $G$-equivariant, it induces a
projection

(3.3) $\rho:Z=(X\times Y)/G\to X/G$

onto the open Riemann surface $X/G$. Note that $Z$ is a reduced complex space,
the map $\rho$ is holomorphic, it is branched over the closed discrete subset
$X_{0}/G$ of $X/G$, and the restriction

$\rho:Z_{1}=\rho^{-1}(X_{1}/G)\to X_{1}/G$

is a holomorphic $G$-bundle with fibre $Y$. Since the submanifold
$Y_{0}\subset Y$ and its complement $Y\setminus Y_{0}={\mathbf{A}}_{*}$ are
both $G$-invariant, we have that

$Z_{1}=\left[(X_{1}\times{\mathbf{A}}_{*})/G\right]\cup\left[(X_{1}\times
Y_{0})/G\right]$

where the union is disjoint. The open subset

(3.4) $\Omega=(X_{1}\times{\mathbf{A}}_{*})/G\subset Z$

is without singularities, and its complement

$Z^{\prime}=Z\setminus\Omega=\left[(X_{0}\times
Y)/G\right]\cup\left[(X_{1}\times Y_{0})/G\right]$

is a closed complex subvariety of $Z$ containing the branch locus of $\rho$.
The restricted projection

(3.5) $\rho:\Omega\to X_{1}/G$

is a holomorphic $G$-bundle with fibre ${\mathbf{A}}_{*}$, the punctured null
quadric. To describe the structure of this bundle, fix a point $x_{1}\in
X_{1}$ and let $\tilde{x}_{1}=\pi(x_{1})\in X_{1}/G$. A loop $\gamma:[0,1]\to
X_{1}/G$ with $\gamma(0)=\gamma(1)=\tilde{x}_{1}$ lifts with respect to the
covering projection $\pi:X_{1}\to X_{1}/G$ to a unique path $\lambda:[0,1]\to
X_{1}$ with $\lambda(0)=x_{1}$. Since the fibres of $\pi$ are $G$-orbits of
the free action of $G$ on $X_{1}$, its terminal point satisfies
$\lambda(1)=gx_{1}$ for a unique $g=g(\gamma)\in G$, which only depends on the
homotopy class of $\gamma$ in the fundamental group
$\pi_{1}(X_{1}/G,\tilde{x}_{1})$. Conversely, every $g\in G$ equals
$g(\gamma)$ for some loop $\gamma$ in $X_{1}/G$ based at $\tilde{x}_{1}$, and
the identity $1\in G$ corresponds to loops in the image of the injective
homomorphism
$\pi_{*}:\pi_{1}(X_{1},x_{1})\mapsto\pi_{1}(X_{1}/G,\tilde{x}_{1})$ induced by
the quotient projection $\pi:X_{1}\to X_{1}/G$. In fact, the correspondence
$\gamma\mapsto g(\gamma)$ realises an isomorphism
$\pi_{1}(X_{1}/G,\tilde{x}_{1})/\pi_{*}(\pi_{1}(X_{1},x_{1}))\cong G$. The
monodromy homomorphism of the bundle (3.5) along the loop $\gamma$ is then
given by the action of $g=g(\gamma)$ on the fibre ${\mathbf{A}}_{*}$ of $\rho$
over the point $\tilde{x}_{1}$. A point $z_{1}\in\Omega$ with
$\rho(z_{1})=\tilde{x}_{1}$ is represented by a pair $(\tilde{x}_{1},y_{1})$
for some $y_{1}\in{\mathbf{A}}_{*}$, and the monodromy map detemined by
$\gamma$ identifies it with the point $(\tilde{x}_{1},gy_{1})$.

######  Lemma 3.1.

There is a natural bijective correspondence between (continuous or
holomorphic) sections $\tilde{f}:X/G\to Z$ of the map $\rho:Z\to X/G$ (3.3),
satisfying

$\tilde{f}(X_{1}/G)\subset\Omega=(X_{1}\times{\mathbf{A}}_{*})/G\ \
\text{and}\ \ \tilde{f}(X_{0}/G)\subset(X_{0}\times Y_{0})/G,$

and (continuous or holomorphic) $G$-equivariant maps $f:X\to Y$ satisfying

$f(X_{1})\subset{\mathbf{A}}_{*}\ \ \text{and}\ \ f(X_{0})\subset Y_{0}.$

###### Proof.

Let us first explain this correspondence over the domain $X_{1}=X\setminus
X_{0}$. Given a $G$-equivariant map $f:X_{1}\to{\mathbf{A}}_{*}$, we define a
section $\tilde{f}:X_{1}/G\to\Omega$ of the map $\rho$ (3.5) by
$\tilde{f}(\pi(x))=[(x,f(x))]\in\Omega$ for $x\in X_{1}$, where
$[(x,y)]\in\Omega$ denotes the equivalence class of $(x,y)\in
X_{1}\times{\mathbf{A}}_{*}$ under the action (3.2). Given $x^{\prime}\in
X_{1}$ with $\pi(x^{\prime})=\pi(x)$, we have $x^{\prime}=gx$ for a unique
$g\in G$. From $f(gx)=gf(x)$ we obtain

$[(x^{\prime},f(x^{\prime}))]=[(gx,f(gx))]=[(gx,gf(x))]=[g(x,f(x))]=[(x,f(x))],$

so $\tilde{f}$ is well-defined. Conversely, given a section
$\tilde{f}:X_{1}/G\to\Omega$, we have for every $x\in X_{1}$ that
$\tilde{f}(\pi(x))=[(x,y)]\in\Omega$ for a unique $y\in{\mathbf{A}}_{*}$, and
we define $f(x)=y$. If $[(x^{\prime},y^{\prime})]=[(x,y)]\in\Omega$ then
$(x^{\prime},y^{\prime})=(gx,gy)$ for some $g\in G$, which shows that
$f(gx)=f(x^{\prime})=y^{\prime}=gy=gf(x)$, so the map $f$ is $G$-equivariant.
The conclusion of the lemma now follows by continuity and Riemann’s removable
singularities theorem. ∎

Since the holomorphic map $f_{0}:V\to Y$ from a $G$-invariant neighbourhood
$V\subset X$ of $X_{0}$, defined above, is $G$-equivariant and satisfies
$f_{0}(V\setminus X_{0})\subset{\mathbf{A}}_{*}$ and $f_{0}(X_{0})\subset
Y_{0}$, Lemma 3.1 shows that $f_{0}$ determines a holomorphic section
$\tilde{f}_{0}:V/G\to Z$ such that

(3.6) $\tilde{f}_{0}((V\setminus X_{0})/G)\subset\Omega\ \ \text{and}\ \
\tilde{f}_{0}(X_{0}/G)\subset(X_{0}\times Y_{0})/G.$

Conversely, we have the following lemma.

######  Lemma 3.2.

Let $V\subset X$ be an open $G$-invariant neighbourhood of $X_{0}$ whose
connected components are simply connected and each of them contains precisely
one point of $X_{0}$, and let $\Omega$ be given by (3.4). Assume that
$\tilde{f}_{0}:V/G\to Z$ is a holomorphic section of the map $\rho:Z\to X/G$
(3.3) over $V/G$ satisfying (3.6), and let $f_{0}:V\to Y$ be the associated
$G$-equivariant holomorphic map satisfying $f_{0}(V\setminus
X_{0})\subset{\mathbf{A}}_{*}$ and $f_{0}(X_{0})\subset Y_{0}$ (see Lemma
3.1). If the $1$-form $f_{0}\theta$ has no zeros or poles on $V$, then $f_{0}$
determines a $G$-equivariant conformal minimal immersion
$F_{0}:V\to\mathbb{R}^{n}$ with $2\partial F_{0}=f_{0}\theta$.

###### Proof.

Fix a point $x_{0}\in X_{0}$. The connected component $\Delta$ of $V$
containing $x_{0}$ is simply connected, $G_{x_{0}}$-invariant, and the
restricted map $f_{0}:\Delta\to Y$ is $G_{x_{0}}$-equivariant. Choose a vector
$v_{0}\in\mathbb{R}^{n}$ such that $gv_{0}=v_{0}$ for all $g\in G_{x_{0}}$
(this holds e.g. for $v_{0}=0$) and define a conformal minimal immersion
$F_{0}:\Delta\to\mathbb{R}^{n}$ by
$F_{0}(x)=v_{0}+\int_{x_{0}}^{x}\Re(f_{0}\theta)$ for all $x\in\Delta$. The
integral is well-defined since $\Delta$ is simply connected, and $F_{0}$ is
$G_{x_{0}}$-equivariant by Lemma 2.3. (Indeed, condition (2.18) trivially
holds for all $g\in G_{x_{0}}$.) We extend $F_{0}$ by the $G$-equivariance
condition to the domain $G\,\cdotp\Delta=\bigcup_{g\in G}g\Delta\subset X$.
The proof is completed by performing the same construction on the other
$G$-orbits of the closed discrete set $X_{0}\subset X$. ∎

We have now arrived at the main point of our argument. The punctured null
quadric ${\mathbf{A}}_{*}$ is a homogeneous manifold of the complex Lie group
$O(n,\mathbb{C})$ [5, p. 78], hence an Oka manifold by a theorem of Grauert
[25] (see also [18, Proposition 5.6.1]). It is even algebraically elliptic
(see [5, Proposition 1.15.3]). In particular, the map $\rho:\Omega\to X_{1}/G$
in (3.5) is a holomorphic fibre bundle with Oka fibre ${\mathbf{A}}_{*}$. Let
$\tilde{f}_{0}:V/G\to Z$ be a holomorphic section as in Lemma 3.2, satisfying
conditions (3.6). Recall that the branch locus of the holomorphic map
$\rho:Z\to X/G$ projects to $X_{0}/G$. Since the section $\tilde{f}_{0}$ is
holomorphic on a neighbourhood of $X_{0}/G$, the Oka principle for sections of
branched holomorphic maps (see [18, Theorem 6.14.6]) shows that every
continuous section $\tilde{f}^{\prime}_{0}:X/G\to Z$ with
$\tilde{f}(X_{1}/G)\subset\Omega$, which agrees with $\tilde{f}_{0}$ on a
neighbourhood of $X_{0}/G$, is homotopic to a holomorphic section
$\tilde{f}:X/G\to Z$ through a homotopy of sections that are holomorphic on a
smaller neighbourhood of $X_{0}/G$, they agree with $\tilde{f}^{\prime}_{0}$
to any given finite order at every point of $X_{0}$, and they map $X_{1}/G$ to
$\Omega$. (The cited result is an improved version of [17, Theorem 2.1]. The
main addition is that we can control the range of the resulting holomorphic
section $\tilde{f}:X/G\to Z$, ensuring that
$\tilde{f}(X_{1}/G)\subset\Omega$.) The existence of a continuous extension
$\tilde{f}^{\prime}_{0}$ of $\tilde{f}_{0}$ with these properties follows from
the observation that the homotopy type of the open Riemann surface $X/G$ is
that of a bouquet of circles and the fibre ${\mathbf{A}}_{*}$ of the
$G$-bundle (3.5) is connected. (This is a special case of [18, Corollary
5.14.2].)

To prove Theorem 1.1, we must explain how to find a holomorphic section
$\tilde{f}:X/G\to Z$ as above such that the associated $G$-equivariant
holomorphic map $f:X\to Y$, given by Lemma 3.1, integrates to a
$G$-equivariant conformal minimal immersion $F:X\to\mathbb{R}^{n}$ as in
(2.17). This amounts to showing that $f$ can be chosen such that it satisfies
the period conditions (2.16) and (2.18). We shall follow [5, proof of Theorem
3.6.1] with appropriate modifications to ensure $G$-equivariance. The main
lemma is given in the following section (see Lemma 4.5), and Theorem 1.1 is
proved in Section 5 as a special case of Theorem 5.1.

## 4\. The main lemma

The main result of this section is Lemma 4.5, which provides the key step in
the proof of Theorems 1.1 and 5.1. This lemma is a $G$-equivariant analogue of
[5, Proposition 3.3.2].

We begin by adjusting the relevant technical tools from [5, Chapter 3] to
$G$-equivariant minimal surfaces. Recall the following notion; see [5,
Definition 1.12.9].

######  Definition 4.1.

An admissible set in a Riemann surface $X$ is a compact set of the form
$S=K\cup E$, where $K$ is a (possibly empty) finite union of pairwise disjoint
compact domains with piecewise $\mathscr{C}^{1}$ boundaries in $X$ and
$E=S\setminus\mathring{K}$ is a union of finitely many pairwise disjoint,
smoothly embedded Jordan arcs and closed Jordan curves meeting $K$ only at
their endpoints (if at all) and such that their intersections with the
boundary $bK$ of $K$ are transverse.

Note that $\mathring{S}=\mathring{K}$. Since an admissible set $S$ has at most
finitely many holes in $X$, the Bishop–Mergelyan theorem (see Bishop [11] and
[16, Theorem 5]) shows that every function in the algebra
$\mathscr{A}(S)=\mathscr{C}(S)\cap\mathscr{O}(\mathring{S})$ is a uniform
limit of meromorphic functions on $X$ with poles in $X\setminus S$, and of
holomorphic functions on $X$ if $X\setminus S$ has no holes (i.e., $X$ is an
open Riemann surface and $S$ is Runge in $X$). Furthermore, functions of class
$\mathscr{A}^{r}(S)$ for any $r\in\mathbb{N}$ can be approximated in the
$\mathscr{C}^{r}(S)$ topology by meromorphic or holomorphic functions on $X$,
respectively (see [16, Theorem 16]).

We assume in the sequel that $X$ is an open Riemann surface, not necessarily
connected, and $G$ is a finite group acting on $X$ and on $\mathbb{R}^{n}$ as
in Theorem 1.1 such that the action of $G$ is transitive on the set of
connected components of $X$ and the stabiliser of any component $X^{\prime}$
of $X$ acts effectively on $X^{\prime}$ (see Remarks 1.3 and 2.1). These
conditions imply that the set $X_{0}$ (2.3), which is the union of fixed point
sets of all elements of $G$, is a closed, discrete, $G$-invariant subset of
$X$, which is the branch locus of the holomorphic quotient projection

$\pi:X\to\widetilde{X}:=X/G,$

and the orbit space $\widetilde{X}$ is a connected Riemann surface. Set
$\widetilde{X}_{0}=X_{0}/G$. If $S$ is a $G$-invariant admissible subset of
$X$ such that $X_{0}\cap S\subset\mathring{S}$, then $\widetilde{S}=\pi(S)$ is
an admissible subset of $\widetilde{X}$. Conversely, given an admissible
subset $\widetilde{S}\subset\widetilde{X}$ such that
$\widetilde{X}_{0}\cap\widetilde{S}$ is contained in the interior of
${\widetilde{S}}$, its preimage $S=\pi^{-1}(\widetilde{S})$ is a $G$-invariant
admissible subset of $X$ such that $X_{0}\cap S\subset\mathring{S}$.

######  Lemma 4.2.

Let $\pi:X\to\widetilde{X}=X/G$ be as above. A compact set
$\widetilde{S}\subset\widetilde{X}$ is Runge in $\widetilde{X}$ if and only if
its preimage $S=\pi^{-1}(\widetilde{S})\subset X$ is Runge in $X$.

###### Proof.

Recall that a compact set $S$ in an open Riemann surface $X$ is Runge if and
only if it is $\mathscr{O}(X)$-convex. Assume that $\widetilde{S}$ is Runge in
$\widetilde{X}$. Given a point $p\in X\setminus S$, we have that
$\tilde{p}=\pi(p)\in\widetilde{X}\setminus\widetilde{S}$, and hence there is a
holomorphic function $\tilde{h}\in\mathscr{O}(\widetilde{X})$ satisfying
$\tilde{h}(\tilde{p})=1>\sup_{\tilde{x}\in\widetilde{S}}|\tilde{h}(\tilde{x})|$.
The function $h=\tilde{h}\circ\pi\in\mathscr{O}(X)$ then satisfies
$h(p)=1>\sup_{x\in S}|h(x)|$, showing that $S$ is Runge in $X$. Conversely,
assume that $S$ is Runge in $X$. Pick a point
$\tilde{p}\in\widetilde{X}\setminus\widetilde{S}$ and let
$\pi^{-1}(\tilde{p})=\\{p_{1},\ldots,p_{m}\\}\subset X\setminus S$, where the
fibre points $p_{i}$ are listed according to their multiplicities. Since $S$
is Runge in $X$, there exists $h_{0}\in\mathscr{O}(X)$ satisfying $\sup_{x\in
S}|h_{0}(x)|<1$ and $h_{0}(p_{i})=1$ for $i=1,\ldots,m$. The function
$h\in\mathscr{O}(X)$ defined by $h(x)=\frac{1}{m}\sum_{g\in G}h_{0}(gx)$ is
then $G$-invariant and satisfies $h(p_{i})=1>\sup_{x\in S}|h(x)|$ for
$i=1,\ldots,m$. It follows that $h=\tilde{h}\circ\pi$ where
$\tilde{h}\in\mathscr{O}(\widetilde{X})$ satisfies
$\tilde{h}(\tilde{p})=1>\sup_{\tilde{x}\in\widetilde{S}}|\tilde{h}(\tilde{x})|$.
Hence, $\widetilde{S}$ is Runge in $\widetilde{X}$. ∎

The following is a version of [5, Definition 3.1.2], allowing the 1-form
$\theta$ to have zeros.

######  Definition 4.3.

Let $S=K\cup E$ be an admissible set in a Riemann surface $X$ (see Definition
4.1), let $\theta$ be a holomorphic $1$-form on a neighbourhood of $S$ in $X$
without zeros on $bS$, and let $Y$ be the manifold (2.8). A generalized
conformal minimal immersion $S\to\mathbb{R}^{n}$ $(n\geq 3)$ of class
$\mathscr{C}^{r}$ $(r\in\mathbb{N})$ is a pair $(F,f\theta)$, where
$F:S\to\mathbb{R}^{n}$ is a $\mathscr{C}^{r}$ map whose restriction to
$\mathring{S}$ is a conformal minimal immersion and the map
$f\in\mathscr{A}^{r-1}(S,Y)$ satisfies the following conditions:

  1. (a)

$f\theta=2\partial F$ holds on $K$ (in particular, the zeros of $\theta$
cancel the poles of $f$), and

  2. (b)

for any smooth path $\alpha:[0,1]\to X$ parameterizing a connected component
of $E=\overline{S\setminus K}$ we have that
$\Re(\alpha^{*}(f\theta))=\alpha^{*}dF=d(F\circ\alpha)$. (See Remark 4.4.)

Given an admissible set $S\subset X$ and integers $r\geq 1,\ n\geq 3$, we
denote by $\mathrm{GCMI}^{r}(S,\mathbb{R}^{n})$ the space of all generalized
conformal minimal immersions $S\to\mathbb{R}^{n}$ of class $\mathscr{C}^{r}$.
An element $(F,f\theta)\in\mathrm{GCMI}^{r}(S,\mathbb{R}^{n})$ is said to be
nonflat if the image by $F$ of any connected component of $K$ and of $E$ is
not contained in an affine 2-plane in $\mathbb{R}^{n}$. This holds if and only
if the image of any such component by $f$ is not contained in a ray of
${\mathbf{A}}$, compactified with the point at infinity. The identity
principle shows that if $F$ is nonflat on a connected domain $D$ in a Riemann
surface $X$ then its restriction to every arc in $D$ is nonflat; conversely,
if $F$ is nonflat on a nontrivial arc then it is nonflat on every connected
domain containing this arc.

If $G$ is a finite group as in Theorem 1.1 and $S$ is $G$-invariant, then
$(F,f\theta)\in\mathrm{GCMI}^{r}(S,\mathbb{R}^{n})$ is said to be
$G$-equivariant if $F(gx)=gF(x)$ holds for all $x\in S$ and $g\in G$. In this
case, the map $f:S\to Y$ is also $G$-equivariant (see the proof of Proposition
2.2). We denote by

$\mathrm{GCMI}^{r}_{G}(S,\mathbb{R}^{n})$

the space of $G$-equivariant generalized conformal minimal immersions
$S\to\mathbb{R}^{n}$ of class $\mathscr{C}^{r}$.

######  Remark 4.4.

Let $S=K\cup E$ be an admissible set and
$(F,f\theta)\in\mathrm{GCMI}^{r}(S,\mathbb{R}^{n})$. Since $F$ is a conformal
minimal immersion on $\mathring{K}=\mathring{S}$, we have that
$dF=\Re(2\partial F)$ on $K$, and hence condition (b) in Definition 4.3 is
compatible with condition (a) asking that $f\theta=2\partial F$ hold on $K$. A
map $f:S\to Y$ of class $\mathscr{A}^{r-1}(S)$ determines a generalized
conformal minimal immersion
$(F,f\theta)\in\mathrm{GCMI}^{r}(S,\mathbb{R}^{n})$ if and only if
$\Re\int_{\lambda}f\theta=0$ on every closed piecewise $\mathscr{C}^{1}$ path
$\lambda\subset S$, and it suffices to verify this condition on a basis of the
homology group $H_{1}(S,\mathbb{Z})$. (This is a free abelian group of finite
rank. We refer to [5, Lemma 1.12.10] for the construction of a homology basis
with suitable properties that will be used in the sequel.) In particular, if
$F:K\to\mathbb{R}^{n}$ is a conformal minimal immersion satisfying condition
(a) and $E$ is an oriented arc attached with both endpoints $p,q$ to $K$, then
$f|_{E}$ must satisfy the condition $\Re\int_{E}f\theta=F(q)-F(p)$.

The following lemma provides the key ingredient in the proof of Theorem 1.1.

######  Lemma 4.5.

Let $X$ be an open Riemann surface, $G$ be a finite group as in Theorem 1.1
acting transitively on the set of connected components of $X$, $X_{0}$ be the
set (2.3), $\theta$ be a holomorphic 1-form on $X$ as in (2.7) with
$\\{\theta=0\\}=X_{0}$, and let $\pi:X\to X/G=\widetilde{X}$ denote the
quotient projection. Assume that $S$ is $G$-invariant admissible set in $X$
such that $X_{0}\subset\mathring{S}$ and the admissible set
$\widetilde{S}=\pi(S)\subset\widetilde{X}$ is a strong deformation retract of
$\widetilde{X}$. Then, every nonflat $G$-equivariant generalized conformal
minimal immersion
$(F_{0},f_{0}\theta)\in\mathrm{GCMI}^{r}_{G}(S,\mathbb{R}^{n})$
$(r\in\mathbb{N})$ can be approximated in the $\mathscr{C}^{r}$ topology on
$S$ by nonflat $G$-equivariant conformal minimal immersions
$F:X\to\mathbb{R}^{n}$. Furthermore, $F$ can be chosen to agree with $F_{0}$
in any given finite set of points $A\subset K$, and to agree with $F_{0}$ to
any given finite order in the points of $A\cap\mathring{K}$.

###### Proof.

We shall follow the construction in [5, Sections 3.2–3.3], adjusting it to the
$G$-equivariant case by using the approach described in Section 3.

Write $S=K\cup E$ as in Definition 4.1. Since $S$ is $G$-invariant, both $K$
and $E$ are $G$-invariant. The conditions imply that the set $X_{0}$ is finite
and the orbit space $\widetilde{X}=X/G$ is connected. Since the admissible set
$\widetilde{S}=\pi(S)\subset\widetilde{X}$ is a strong deformation retract of
$\widetilde{X}$, it is connected as well. Note that $\widetilde{S}$ contains
the finite set $\widetilde{X}_{0}=X_{0}/G$ in its interior. Since
$\pi:X\setminus X_{0}\to\widetilde{X}\setminus\widetilde{X}_{0}$ is an
unramified covering projection, it follows that $S$ is a strong deformation
retract of $X$. We enlarge the set $A\subset S$ in the lemma so that it
contains $X_{0}$ and the endpoints of all connected components of $E$, and set
$\widetilde{A}=\pi(A)\subset\widetilde{S}$. By [5, Lemma 1.12.10] (or [20,
Lemma 3.1]) and [5, proof of Proposition 3.3.2, p. 142] there is a finite
collection of smooth embedded compact arcs
$\widetilde{\mathscr{C}}=\\{\widetilde{C}_{1},\ldots,\widetilde{C}_{l}\\}$ in
$\widetilde{S}$ (diffeomorphic images of $[0,1]\subset\mathbb{R}$) with the
following properties.

  1. (i)

The intersection of any two distinct arcs in $\widetilde{\mathscr{C}}$ is
either empty or a common endpoint of both arcs.

  2. (ii)

Every point of $\widetilde{A}$ is an endpoint of an arc in
$\widetilde{\mathscr{C}}$.

  3. (iii)

Every point $\tilde{x}_{0}\in\widetilde{X}_{0}$ is an endpoint of a single arc
$\widetilde{C}_{i}\in\widetilde{\mathscr{C}}$, and the other endpoint of
$\widetilde{C}_{i}$ does not belong to $\widetilde{X}_{0}$.

  4. (iv)

The compact set $\widetilde{C}=\bigcup_{i=1}^{l}\widetilde{C}_{i}$ is a strong
deformation retract of $\widetilde{S}$ (and hence of $\widetilde{X}$). In
particular, $\widetilde{C}$ is connected and contains a homology basis of
$\widetilde{S}$ (and hence of $\widetilde{X}$).

Note that (i) and (ii) imply that no point of $\widetilde{A}$ is an interior
point of an arc in $\widetilde{\mathscr{C}}$. We now enlarge the finite sets
$\widetilde{A}$ and $A$ if necessary to also arrange that

  1. (v)

The set of all endpoints of the arcs in $\widetilde{\mathscr{C}}$ equals
$\widetilde{A}$, and $A=\pi^{-1}(\widetilde{A})$.

For each $i=1,\ldots,l$ the preimage $C_{i}=\pi^{-1}(\widetilde{C}_{i})\subset
S$ is the union of arcs $C_{i,1},\ldots,C_{i,m}$, where $m=|G|$ is the degree
of the projection $\pi:X\to\widetilde{X}$. If an arc
$\widetilde{C}_{i}\in\widetilde{\mathscr{C}}$ does not contain any point of
$\widetilde{X}_{0}$ then $\pi$ is a trivial covering projection over
$\widetilde{C}_{i}$, and hence the arcs $C_{i,j}$ for $j=1,\ldots,m$ are
pairwise disjoint. In the opposite case, one of the endpoints $\tilde{x}_{0}$
of $\widetilde{C}_{i}$ belongs to $\widetilde{X}_{0}$, and then several arcs
$C_{i,j}\subset C_{i}$ share an endpoint but are otherwise disjoint. In fact,
if $k=|G_{x_{0}}|>1$ is the order of the stabiliser group of $x_{0}\in X_{0}$
and $\tilde{x}_{0}=\pi(x_{0})$, then $x_{0}$ is the common endpoint of
precisely $k$ arcs $C_{i,j}\subset C_{i}$. The set

(4.1) $C=\pi^{-1}(\widetilde{C})=\bigcup_{i=1}^{l}C_{i}$

is $G$-invariant, Runge in $X$ by Lemma 4.2, and it contains the set $X_{0}$
of branch points of $\pi$. Since $\widetilde{C}$ is a strong deformation
retract of $\widetilde{X}$, $\widetilde{X}_{0}\subset\widetilde{C}$ and $\pi$
is unbranched over $\widetilde{X}\setminus\widetilde{X}_{0}$, it follows that
$C$ is a strong deformation retract of $X$. Our assumptions imply that none of
the arcs $f_{0}(C_{i,j})$ is contained in a compactified ray of the null
quadric ${\mathbf{A}}_{*}$. Hence, [5, Lemma 3.2.1] yields for every
$i=1,\ldots,l$ a $\mathscr{C}^{r-1}$ map $h_{i}:C_{i,1}\times B\to
Y={\mathbf{A}}_{*}\cup Y_{0}$ (see (2.8)), where $B\subset\mathbb{C}^{n}$ is a
ball centred at the origin, with the following properties.

  1. (a)

$h_{i}(x,0)=f_{0}(x)$ for every $x\in C_{i,1}$.

  2. (b)

There is a closed arc $I_{i}$ contained in the relative interior of $C_{i,1}$
such that $h_{i}(x,t)=f_{0}(x)$ for every $x\in C_{i,1}\setminus I_{i}$ and
$t\in B$.

  3. (c)

$h_{i}(x,t)\in{\mathbf{A}}_{*}$ for every $x\in C_{i,1}\setminus X_{0}$ and
$t\in B$.

  4. (d)

The map $h_{i}(x,\cdotp):B\to Y$ is holomorphic for every $x\in C_{i,1}$.

  5. (e)

The following $\mathbb{C}$-linear map is an isomorphism:

(4.2) $\frac{\partial}{\partial
t}\Big{|}_{t=0}\int_{C_{i,1}}h_{i}(\cdotp,t)\theta:\mathbb{C}^{n}\to\mathbb{C}^{n}.$

In the language of [5, Chapter 3], $h_{i}$ is a period dominating spray of
maps $C_{i,1}\to Y$ with the core $f_{0}$. We now extend $h_{i}$ by
$G$-equivariance to a spray $C_{i}\times B\to Y$ over
$C_{i}=\bigcup_{j=1}^{m}C_{i,j}$. Recall that $C$ is given by (4.1). By
condition (b) above, $h_{i}$ extends to a spray $h_{i}:C\times B\to Y$ which
is independent of $t\in B$ on $C\setminus C_{i}$, so it equals $f_{0}$ there.
Let $B^{l}$ denote the Cartesian product of $l$ copies of $B$. Finally, we
combine the sprays $h_{1},\ldots,h_{l}$ into a single spray $h:C\times
B^{l}\to Y$, which is $G$-equivariant with respect to the first variable $x\in
C$, such that, writing the parameter variable as $t=(t^{1},\ldots,t^{l})\in
B^{l}$ with $t^{i}=(t_{i,1},\ldots,t_{i,n})\in B$, we have

(4.3) $h(x,t^{1},\ldots,t^{l})=h_{i}(x,t^{i})\ \ \text{for all $x\in C_{i}$
and $i=1,\ldots,l$}.$

To the collection of arcs $\mathscr{C}=\\{C_{i,j}\\}$ defined above and to any
continuous $G$-equivariant map $f:C\to Y$ such that $f\theta$ assumes values
in the punctured null quadric ${\mathbf{A}}_{*}$ we associate the period
vector
$\mathcal{P}(f)=\big{(}\mathcal{P}_{1}(f),\ldots,\mathcal{P}_{l}(f)\big{)}\in(\mathbb{C}^{n})^{l}$
whose $i$-th component equals

(4.4) $\mathcal{P}_{i}(f)=\int_{C_{i,1}}f\theta\in\mathbb{C}^{n}\ \
\text{for}\ i=1,\ldots,l.$

Our construction clearly implies that

$\frac{\partial}{\partial
t}\Big{|}_{t=0}\mathcal{P}(h(\cdotp,t)):(\mathbb{C}^{n})^{l}\to(\mathbb{C}^{n})^{l}\
\ \text{is an isomorphism}.$

Indeed, the above linear map has a block structure whose $i$-th diagonal
$n\times n$ block equals the map in (4.2) while the off-diagonal blocks
vanish.

In the next step, we approximate $h$ by a spray of maps $H:S\times B^{l}\to Y$
of class $\mathscr{A}^{r-1}(S\times B^{l})$ (where the ball
$B\subset\mathbb{C}^{n}$ is allowed to shrink a little) such that
$H(\cdotp,0)=f_{0}$, $H(\cdotp,t)$ agrees with $f_{0}$ in every point of $A$
and to any given finite order in points of $A\cap\mathring{K}$ (in particular,
in points of the set $X_{0}\subset\mathring{K}$), $H$ is $G$-equivariant in
$x\in S$ for any fixed $t\in B^{l}$, and

(4.5) $\frac{\partial}{\partial
t}\Big{|}_{t=0}\mathcal{P}(H(\cdotp,t)):(\mathbb{C}^{n})^{l}\to(\mathbb{C}^{n})^{l}\
\ \text{is an isomorphism}.$

In other words, $H$ is a $G$-equivariant $\mathcal{P}$-period dominating spray
on $S$ with values in $Y$ and with the core $f_{0}$. To find such $H$, we
proceed as follows.

By the discussion in Section 3, we can view the spray $h$ in (4.3) as a spray
of sections $\tilde{h}:\widetilde{C}\times B^{l}\to Z$ of the map
$\rho:Z=(X\times Y)/G\to X/G=\widetilde{X}$ in (3.3), whose core
$\tilde{h}(\cdotp,0):\widetilde{C}\to Z$ is the section $\tilde{f}_{0}$
restricted to $\widetilde{C}$. Note that $\tilde{h}$ is holomorphic in the
parameter $t\in B^{l}$, and $\tilde{h}(\tilde{x},t)=\tilde{f}_{0}(\tilde{x})$
holds for all $t\in B^{l}$ when $\tilde{x}\in\widetilde{C}$ is sufficiently
near $\widetilde{A}=\pi(A)$.

By Poletsky [49, Theorem 3.1] (see also [16, Theorem 32 and Corollary 7]), the
image of the section $\tilde{f}_{0}:\widetilde{S}\to Z$ has an open Stein
neighbourhood $Z_{0}\subset Z$. (Poletsky’s result is stated for sections of
holomorphic submersions, but in the case at hand, the branch points of
$\rho:Z\to\widetilde{X}$ lie in the interior of $\widetilde{S}$ and a minor
modification of his proof applies. For Stein subvarieties without boundaries,
the existence of open Stein neighbourhoods was proved by Siu [51].) By [17,
Proposition 2.2] the projection $\rho:Z\to X/G=\widetilde{X}$ admits a
holomorphic fibre-spray over the Stein domain $Z_{0}\subset Z$, which is fibre
dominating outside the branch locus of $\rho$, that is, on
$Z_{0}\setminus\rho^{-1}(\widetilde{X}_{0})$, and which is trivial over
$\rho^{-1}(\widetilde{X}_{0})$. By restricting this spray to
$\tilde{f}_{0}(\widetilde{S})\subset Z_{0}$ we obtain a fibre-spray of
sections $s:\widetilde{S}\times W\to Z$, where $0\in W\subset\mathbb{C}^{k}$
is a neighbourhood of the origin in some complex Euclidean space, such that
$s(\cdotp,0)=\tilde{f}_{0}$, $\rho(s(\tilde{x},\zeta))=\tilde{x}$ for all
$\tilde{x}\in\widetilde{S}$ and $\zeta\in W$, and $s$ is fibre dominating over
$\widetilde{S}\setminus\widetilde{X}_{0}$. The fibre domination property of
$s$ and the implicit function theorem imply that we can factor the spray
$\tilde{h}$, constructed above, through the spray $s$. Explicitly, shrinking
the ball $B\subset\mathbb{C}^{n}$ in the domain of $\tilde{h}$ if necessary,
there is a map $\zeta:\widetilde{C}\times B^{l}\to W$ of class
$\mathscr{A}^{r-1}$ such that

$\tilde{h}(\tilde{x},t)=s\bigl{(}\tilde{x},\zeta(\tilde{x},t)\bigl{)}\ \
\text{and}\ \ \zeta(\tilde{x},0)=0\ \ \text{hold for all}\ \
\tilde{x}\in\widetilde{C}\ \ \text{and}\ \ t\in B^{l}.$

Note that $\zeta$ can be chosen such that $\zeta(\tilde{x},t)$ vanishes when
the point $\tilde{x}\in\tilde{S}$ is sufficiently close to a point of
$\widetilde{A}=\pi(A)$. This is because the spray $\tilde{h}$ is supported
(i.e., not identically equal to its core $\tilde{f}_{0}$) on a union of closed
arcs in $\widetilde{C}$ disjoint from $\widetilde{A}$, and over this set we
can find a (necessarily) trivial complex vector subbundle of
$\widetilde{C}\times\mathbb{C}^{k}$ which is mapped by the differential $ds$
isomorphically onto the vertical (with respect to the map
$\rho:Z\to\widetilde{X}$) tangent bundle of $Z$, that is, the kernel of
$d\rho$. Furthermore, by the construction we have that
$\tilde{h}(\tilde{x},t)=\tilde{f}_{0}(\tilde{x})$ for every point
$\tilde{x}\in\tilde{S}$ that is sufficiently close to a point of
$\widetilde{A}$ and for every $t\in B^{l}$.

Consider the Taylor series expansion of $\zeta$ in the $t$-variable:

$\zeta(\tilde{x},t)=\sum t_{i,j}\zeta_{i,j}(\tilde{x})+O(|t|^{2}),$

where the summation is over $i=1,\ldots,l$ and $j=1,\ldots,n$ and the
coefficient functions $\zeta_{i,j}$ are of class
$\mathscr{C}^{r-1}(\widetilde{C})$. Since $\widetilde{C}$ is a piecewise
smooth curve which is Runge in $\widetilde{X}$, Mergelyan’s theorem [16,
Theorem 16] allows us to approximate the functions $\zeta_{i,j}$ in the
$\mathscr{C}^{r-1}(\widetilde{C})$ topology by holomorphic functions
$\tilde{\zeta}_{i,j}$ on $\widetilde{X}$ that vanish to any given finite order
in the points of $\widetilde{A}$. Consider the map
$\widetilde{H}:\widetilde{S}\times B^{l}\to Z$ of class $\mathscr{A}^{r-1}$
defined by

(4.6) $\widetilde{H}(\tilde{x},t)=s\left(\tilde{x},\sum
t_{i,j}\tilde{\zeta}_{i,j}(\tilde{x})\right)\ \ \text{for
$\tilde{x}\in\widetilde{S}$ and $t\in B^{l}$}.$

Note that $\widetilde{H}(\cdotp,0)=\tilde{f}_{0}$, and the partial
differential $\partial_{t}\widetilde{H}|_{t=0}$ is close to
$\partial_{t}\tilde{h}|_{t=0}$ on $\widetilde{C}$. Assuming that the
approximation is close enough, the map $H:S\times B^{l}\to Y$ determined by
$\widetilde{H}$ (see Lemma 3.1) is a $G$-equivariant spray of class
$\mathscr{A}^{r-1}$ with the core $f_{0}$ which is $\mathcal{P}$-period
dominating, i.e., (4.5) holds. Indeed, the period domination condition only
depends on the $t$-derivative of the spray at $t=0$ and is stable under
deformations.

We can now complete the proof of Lemma 4.5. Recall that $X_{1}=X\setminus
X_{0}$, see (2.4). Let $\Omega=(X_{1}\times{\mathbf{A}}_{*})/G\subset Z$ be
the domain in (3.4). Its complement $Z^{\prime}=Z\setminus\Omega$ is a closed
complex subvariety of $Z$ containing the branch locus of $\rho$, and the
restricted projection
$\rho:\Omega\to\widetilde{X}_{1}:=X_{1}/G=\widetilde{X}\setminus\widetilde{X}_{0}$
is a holomorphic $G$-bundle whose fibre ${\mathbf{A}}_{*}$ is an Oka manifold.
Note that the range of the spray $\widetilde{H}$ in (4.6) lies in $\Omega$
except over the points $\tilde{x}\in\widetilde{X}_{0}$, and these points are
contained in the interior of $\widetilde{S}$. Hence, shrinking the ball
$B\subset\mathbb{C}^{n}$ slightly, we can apply the Oka principle for sections
of branched holomorphic maps [17, Theorem 2.1] (see also the improved version
in [18, Theorem 6.14.6] which exactly fits our situation) to approximate
$\widetilde{H}$ on $\widetilde{S}\times B^{l}$ by a holomorphic spray of
sections $\widetilde{\Theta}:\widetilde{X}\times B^{l}\to Z$ which agrees with
$\widetilde{H}$ to a given finite order in the points
$\tilde{x}\in\widetilde{A}$ and maps $\widetilde{X}_{1}\times B^{l}$ to
$\Omega$. (Recall that
$\widetilde{X}_{1}=\widetilde{X}\setminus\widetilde{X}_{0}$. Although [18,
Theorem 6.14.6] is stated for a single section, its proof applies to sprays of
sections. Alternatively, one can treat the parameter variable $t$ as a space
variable, suitably adjusting the spaces under consideration.)

Let $\Theta:X\times B^{l}\to Y$ be the $G$-equivariant holomorphic spray
associated to $\widetilde{\Theta}$ by Lemma 3.1. Assuming that the
approximation of $\widetilde{H}$ by $\widetilde{\Theta}$ is close enough, the
period domination property of $H$ and the implicit function theorem yield a
parameter value $t_{0}\in B^{l}$ close to the origin such that the
$G$-equivariant map $f=\Theta(\cdotp,t_{0}):X\to Y$ approximates $f_{0}$ on
$S$ and satisfies $\mathcal{P}(f)=\mathcal{P}(f_{0})$; see (4.4). From the
last condition and $G$-equivariance it follows that $f$ has the same periods
as $f_{0}$ on the arcs $C_{i,j}\subset C$ for $i=1,\ldots,l$ and
$j=1,\ldots,m$. Furthermore, the construction implies that $f\theta$ is a
holomorphic $1$-form on $X$ with values in ${\mathbf{A}}_{*}$.

Since the $G$-equivariant set $C$ is a strong deformation retract of $X$, it
contains a homology basis of $X$ and also curves which can be used to verify
conditions (2.18) in Lemma 2.3. More precisely, each of these curves can be
chosen to be a union of arcs in the family $\mathscr{C}=\\{C_{i,j}\\}$. Since
the periods of $f\theta$ and $f_{0}\theta$ agree on every arc
$C_{i,j}\in\mathscr{C}$ and $f_{0}$ comes from a generalized $G$-equivariant
conformal minimal immersion $(F_{0},f_{0}\theta)$, it follows that $f\theta$
integrates to a $G$-equivariant conformal minimal immersion
$F:X\to\mathbb{R}^{n}$ satisfying Lemma 4.5. If $X$ is disconnected and the
conditions in Remark 1.3 hold, we apply integration on a connected component
of $X$ and extend it to all of $X$ by $G$-equivariance. ∎

In the proof of Theorem 1.1 we shall also need the following lemma on
approximating flat $G$-equivariant conformal minimal immersions by nonflat
ones.

######  Lemma 4.6.

Let $G=\langle g\rangle$ be a cyclic group of order $k$ whose generator $g$
acts on $\mathbb{C}$ by the rotation $gz=\mathrm{e}^{\mathfrak{i}2\pi/k}z$,
and it acts on $\mathbb{R}^{n}$ $(n\geq 3)$ by an orthogonal transformation.
Let $U\subset\mathbb{C}$ be a disc centred at the origin. Given a flat
$G$-equivariant conformal minimal immersion $F_{0}:U\to\mathbb{R}^{n}$ and an
integer $r\in\mathbb{N}$, there is a nonflat $G$-equivariant conformal minimal
immersion $F:\mathbb{C}\to\mathbb{R}^{n}$ which agrees with $F_{0}$ to order
$r$ in $z=0$ and it approximates $F_{0}$ as closely as desired on a given
compact subset of $U$.

###### Proof.

Assume that $F_{0}$ is defined on an open disc $U\subset\mathbb{C}$ centred at
the origin and containing the closed unit disc
$\overline{\Delta}=\\{z\in\mathbb{C}:|z|\leq 1\\}$. Set $\theta=z^{k-1}dz$ and
write $2\partial F_{0}=h_{0}\theta$, where $h_{0}:U\to Y$ is a $G$-equivariant
holomorphic map such that $h_{0}(U\setminus\\{0\\})\subset\mathbb{C}^{*}y_{0}$
for some $y_{0}\in{\mathbf{A}}_{*}$. Let
$\pi:\mathbb{C}\to\mathbb{C}/G=\mathbb{C}$ denote the quotient projection
$\pi(z)=z^{k}$. Then, $\widetilde{U}=\pi(U)$ is a disc in $\mathbb{C}$
containing $\overline{\Delta}$. By Lemma 3.1, $h_{0}$ determines a holomorphic
section $\tilde{h}_{0}:\widetilde{U}\to Z$ of the holomorphic map
$\rho:Z=(\mathbb{C}\times Y)/G\to\mathbb{C}$ (see (3.3)) over $\widetilde{U}$.
Note that $\rho$ is branched only over the origin. We attach to
$\overline{\Delta}$ the segment
$\widetilde{C}=[1,2^{k}]\subset\mathbb{R}\subset\mathbb{C}$. Note that
$\widetilde{S}=\overline{\Delta}\cup\widetilde{C}$ is an admissible set. The
preimage $C=\pi^{-1}(\widetilde{C})=\bigcup_{i=1}^{k}C_{i}$ is a union of $k$
pairwise disjoint arcs $C_{i}$ attached to $\overline{\Delta}$. One of these
arcs, say $C_{1}$, is the segment $[1,2]\subset\mathbb{R}$ and the other arcs
are obtained by rotating $C_{1}$ through integer multiples of the angle
$2\pi/k$. We define a smooth $G$-equivariant map
$f_{0}:S=\overline{\Delta}\cup C\to Y$ such that $f_{0}=h_{0}$ on
$\overline{\Delta}$, $f_{0}(C)\subset{\mathbf{A}}_{*}$, and
$f_{0}(2)\in{\mathbf{A}}_{*}\setminus\mathbb{C}y_{0}$. Let
$\tilde{f}_{0}:\widetilde{S}=\overline{\Delta}\cup\widetilde{C}\to Z$ denote
the section of $\rho:Z\to\mathbb{C}$ over $\widetilde{S}$ associated to
$f_{0}$ by Lemma 3.1. By [18, Theorem 6.14.6] there is a holomorphic section
$\tilde{f}:\mathbb{C}\to Z$ which approximates $\tilde{f}_{0}$ on
$\widetilde{S}$, it agrees with $\tilde{f}_{0}$ to a given finite order in the
origin $0\in\mathbb{C}$, it agrees with $\tilde{f}_{0}$ in the point $2^{k}$,
and it satisfies
$\tilde{f}(\mathbb{C}^{*})\subset(\mathbb{C}^{*}\times{\mathbf{A}}_{*})/G$.
The section $\tilde{f}$ defines a $G$-equivariant holomorphic map
$f:\mathbb{C}\to Y$ satisfying $f(\mathbb{C}^{*})\subset{\mathbf{A}}_{*}$
which agrees with $h_{0}$ to the given order $r$ in the origin and satisfies
$f(2)=f_{0}(2)\in{\mathbf{A}}\setminus\mathbb{C}y_{0}$. Hence, $f$ is nonflat
on the arc $C_{1}$, and therefore on $\mathbb{C}$. By integrating $f\theta$ we
obtain a nonflat $G$-equivariant conformal minimal immersion
$F:\mathbb{C}\to\mathbb{R}^{n}$ which agrees with $F_{0}$ to order $r$ in
$z=0$ and approximates $F_{0}$ on $\overline{\Delta}$. ∎

## 5\. Proof of Theorem 1.1

In this section, we prove the following result.

######  Theorem 5.1.

Assume that $G$ is a finite group acting effectively on a connected open
Riemann surface $X$ by holomorphic automorphisms and acting on
$\mathbb{R}^{n}$ $(n\geq 3)$ by orthogonal transformations. Let $\pi:X\to X/G$
be the quotient projection and $X_{0}\subset X$ be the set (2.3) of its branch
points. Assume that $S\subset X$ is a $G$-invariant admissible subset (see
Definition 4.1) which is Runge in $X$, $A\subset X\setminus bS$ is a closed
$G$-invariant discrete set containing $X_{0}$, $V\subset X\setminus S$ is an
open $G$-invariant neighbourhood of $A\setminus S$, and $F_{0}:S\cup
V\to\mathbb{R}^{n}$ is such that $F_{0}|_{S}\in\mathrm{GCMI}^{r}_{G}(S)$
$(r\geq 1)$ is a nonflat $G$-equivariant generalized conformal minimal
immersion, and $F_{0}|_{V}$ is a $G$-equivariant conformal minimal immersion.
Then, $F_{0}$ can be approximated as closely as desired in the
$\mathscr{C}^{r}$ topology on $S$ by $G$-equivariant conformal minimal
immersions $F:X\to\mathbb{R}^{n}$ which agree with $F_{0}$ to a given finite
order $k(a)\in\mathbb{N}$ in every point $a\in A$.

By Remarks 1.3 and 2.1, Theorem 5.1 also holds if $X$ is not necessarily
connected and the stabiliser $G_{X^{\prime}}\subset G$ of each connected
component $X^{\prime}$ of $X$ acts effectively on $X^{\prime}$.

We have seen in Section 3 that the hypotheses of Theorem 1.1 imply the
existence of a $G$-equivariant conformal minimal immersion
$F_{0}:V\to\mathbb{R}^{n}$ from a $G$-invariant neighbourhood $V\subset X$ of
the closed discrete set $X_{0}$ (2.3), so Theorem 5.1 implies Theorem 1.1. To
obtain nondegenerate conformal minimal immersions $X\to\mathbb{R}^{n}$ (i.e.,
not lying in any affine hyperplane), it suffices to suitably choose their
values on finitely many $G$-orbits in $X$ and apply the interpolation
statement on the set $A\subset X$ in Theorem 5.1.

###### Proof of Theorem 5.1.

By Lemmas 4.5 and 4.6 we may assume, after shrinking $V$ around the closed
discrete set $A\setminus S$ if necessary, that $F_{0}$ is a nonflat
$G$-equivariant conformal minimal immersion $F_{0}:U\cup V\to\mathbb{R}^{n}$,
where $U\subset X$ is a $G$-invariant open neighbourhood of $S$ such that
$A\cap S=A\cap U$ and $\overline{U}\cap\overline{V}=\varnothing$. Set

$\widetilde{X}=X/G,\quad\widetilde{X}_{0}=\pi(X_{0}),\quad\widetilde{A}=\pi(A),\quad\widetilde{U}=\pi(U),\quad\widetilde{V}=\pi(V).$

Let $\theta$ be a $G$-invariant holomorphic 1-form (2.6) with
$\\{\theta=0\\}=X_{0}$, and let $Y$ be the manifold (2.8). By Proposition 2.2
the map $f_{0}=2\partial F_{0}/\theta:U\cup V\to Y$ is holomorphic and
$G$-equivariant. Let $\tilde{f}_{0}$ be the associated section of the map
$\rho:Z=(X\times Y)/G\to X/G=\widetilde{X}$ (3.3) over
$\widetilde{U}\cup\widetilde{V}$ (see Lemma 3.1). Since $S$ is a $G$-invariant
admissible Runge set in $X$, the image
$\widetilde{S}=\pi(S)\subset\widetilde{X}$ is an admissible set which is Runge
in $\widetilde{X}$ by Lemma 4.2. Hence there is a smooth strongly subharmonic
Morse exhaustion function $\psi:\widetilde{X}\to\mathbb{R}_{+}$ and an
increasing sequence $0<c_{0}<c_{1}<\cdots$ with
$\lim_{i\to\infty}c_{i}=+\infty$ such that, setting $D_{i}=\\{\psi\leq
c_{i}\\}$, we have that $\widetilde{S}\subset\mathring{D}_{0}\subset
D_{0}\subset\widetilde{U}$ and the following conditions hold for every
$i\in\mathbb{Z}_{+}$.

  * •

The number $c_{i}$ is a regular value of $\psi$.

  * •

$\\{\psi=c_{i}\\}\cap\widetilde{A}=\varnothing$.

  * •

The domain $\Gamma_{i+1}=\mathring{D}_{i+1}\setminus D_{i}$ contains at most
one critical point of $\psi$ or at most one point of $\widetilde{A}$, but not
both.

For every $i\in\mathbb{Z}_{+}$ we set

(5.1) $B_{i}=\pi^{-1}(D_{i})=\\{\psi\circ\pi\leq c_{i}\\}\subset X.$

Note that the smoothly bounded compact sets $B_{i}$ are $G$-invariant and they
form a normal exhaustion of $X$.

To prove the theorem, we shall inductively construct a sequence
$(F_{i},f_{i}\theta)\in\mathrm{GCMI}_{G}^{r}(B_{i},\mathbb{R}^{n})$ of
nonflat, $G$-equivariant generalized conformal minimal immersions satisfying
the following two conditions for every $i\in\mathbb{Z}_{+}$:

  1. (a)

$F_{i+1}$ approximates $F_{i}$ in the $\mathscr{C}^{r}$ topology as closely as
desired on $B_{i}$, and

  2. (b)

$F_{i+1}$ agrees with $F_{0}$ to the given order $k(a)$ in every point $a\in
A\cap B_{i+1}$.

Assuming that the approximation conditions are appropriately chosen, the
sequence $F_{i}$ converges to a $G$-equivariant conformal minimal immersion
$F=\lim_{i\to\infty}F_{i}:X\to\mathbb{R}^{n}$ satisfying Theorem 5.1. We refer
to [5, proof of Theorem 3.6.1] for the details.

The initial step is provided by the restriction of $(F_{0},f_{0}\theta)$ to
$B_{0}$. Assuming inductively that we have found
$(F_{i},f_{i}\theta)\in\mathrm{GCMI}_{G}^{r}(B_{i},\mathbb{R}^{n})$ for some
$i\in\mathbb{Z}_{+}$, we shall explain how to find the next map
$(F_{i+1},f_{i+1}\theta)\in\mathrm{GCMI}_{G}^{r}(B_{i+1},\mathbb{R}^{n})$ with
the desired properties. Recall that $\Gamma_{i+1}=\mathring{D}_{i+1}\setminus
D_{i}$ for $i\in\mathbb{Z}_{+}$, so
$D_{i+1}=D_{i}\cup\overline{\Gamma}_{i+1}$. We consider cases.

Case 1: $\Gamma_{i+1}$ does not contain any critical point of $\psi$ or a
point of $\widetilde{A}$. In this case, $D_{i}$ is a strong deformation
retract of $D_{i+1}$, and a generalized conformal minimal immersion
$(F_{i+1},f_{i+1}\theta)\in\mathrm{GCMI}_{G}^{r}(B_{i+1},\mathbb{R}^{n})$ with
the desired properties is furnished by Lemma 4.5.

Case 2: $\Gamma_{i+1}$ contains a critical point $\tilde{x}$ of $\psi$. We can
attach to $D_{i}$ a smooth embedded arc
$\widetilde{E}\subset\mathring{D}_{i+1}\setminus\mathring{D}_{i}$,
intersecting $D_{i}$ only at its endpoints $\tilde{p}$ and $\tilde{q}$, such
that $\widetilde{S}_{i}=D_{i}\cup\widetilde{E}$ is an admissible set (see
Definition 4.1) which is a strong deformation retract of $D_{i+1}$. (One can
distinguish several topologically different subcases as in [5, proof of
Theorem 3.6.1], but this will not affect our discussion.) Since
$\overline{\Gamma}_{i+1}\cap\widetilde{X}_{0}=\varnothing$, the map $\pi$ is
unbranched over $\overline{\Gamma}_{i+1}$ and hence the preimage
$E=\pi^{-1}(\widetilde{E})=\bigcup_{j=1}^{m}E_{j}\subset X$ is a disjoint
union of $m=|G|$ smooth arcs. Let $p_{j},q_{j}\in bB_{i}$ denote the endpoints
of $E_{j}$ with $\pi(p_{j})=\tilde{p}_{j}$ and $\pi(q_{j})=\tilde{q}_{j}$ for
$j=1,\ldots,m$. By [5, Lemma 3.5.4] we can extend the given map
$f_{i}:B_{i}\to Y$ from the induction step to a nonflat $G$-equivariant map
$f^{\prime}_{i}:S_{i}=B_{i}\cup E\to Y$ of class $\mathscr{A}^{r-1}$ such that
$f^{\prime}_{i}(E)\subset{\mathbf{A}}_{*}$ and

(5.2) $\Re\int_{E_{j}}f^{\prime}_{i}\theta=F_{i}(q_{j})-F_{i}(p_{j})\ \
\text{holds for $j=1,\ldots,m$},$

where the arc $E_{j}$ is oriented from $p_{j}$ to $q_{j}$. (It suffices to
ensure the condition (5.2) on the arc $E_{1}$, as it then extends by
$G$-equivariance to the remaining arcs $E_{2},\ldots,E_{m}$. See [5, proof of
Theorem 3.6.1, p. 158] for the details of this argument.) By Remark 4.4, the
map $f^{\prime}_{i}$ determines an extension $F^{\prime}_{i}$ of $F_{i}$ to
the $G$-invariant admissible set $S_{i}=B_{i}\cup
E=\pi^{-1}(\widetilde{S}_{i})$ such that
$(F^{\prime}_{i},f^{\prime}_{i}\theta)\in\mathrm{GCMI}_{G}(S_{i})$. By Lemma
4.5 we can approximate $(F^{\prime}_{i},f^{\prime}_{i}\theta)$ in the
$\mathscr{C}^{r-1}(S_{i})$ topology by
$(F_{i+1},f_{i+1}\theta)\in\mathrm{GCMI}_{G}^{r}(B_{i+1},\mathbb{R}^{n})$
having the desired properties.

Case 3: $\Gamma_{i+1}$ contains a point $\tilde{a}\in\widetilde{A}$. Let
$\widetilde{\Delta}\subset\Gamma_{i+1}\cap\widetilde{V}$ be a small closed
disc around $\tilde{a}$. The initial map $F_{0}$ is then a conformal
$G$-equivariant minimal immersion on
$\Delta=\pi^{-1}(\widetilde{\Delta})\subset V$. Note that
$B_{i}\cap\Delta=\varnothing$. We extend $(F_{i},f_{i}\theta)$ to
$B_{i}\cup\Delta$ by setting $F_{i}=F_{0}$ and $f_{i}=f_{0}$ on $\Delta$.
Choose a point $\tilde{q}\in b\widetilde{\Delta}$ and attach to $D_{i}$ a
smooth embedded arc
$\widetilde{E}\subset\mathring{D}_{i+1}\setminus(\mathring{D}_{i}\cup\mathring{\widetilde{\Delta}})$,
having an endpoint $\tilde{p}\in bD_{i}$ and the other endpoint $\tilde{q}\in
b\widetilde{\Delta}$, such that
$\widetilde{S}_{i}:=D_{i}\cup\widetilde{\Delta}\cup\widetilde{E}$ is an
admissible set in $\widetilde{X}$. Note that $\widetilde{S}_{i}$ is a strong
deformation retract of $D_{i+1}$. Set
$S_{i}=\pi^{-1}(\widetilde{S}_{i})\subset X$. We now proceed as in Case 2,
first extending $(F_{i},f_{i}\theta)$ from $B_{i}\cup\Delta$ to
$(F^{\prime}_{i},f^{\prime}_{i}\theta)\in\mathrm{GCMI}^{r}_{G}(S_{i})$ such
that conditions (5.2) hold and then applying Lemma 4.5 to obtain
$(F_{i+1},f_{i+1}\theta)\in\mathrm{GCMI}_{G}^{r}(B_{i+1},\mathbb{R}^{n})$ with
the desired properties.

This completes the induction step, and hence of the proof of Theorem 5.1. ∎

######  Remark 5.2 (Controlling the flux).

In the proof of Theorem 5.1, one can also control the flux of $G$-equivariant
conformal minimal immersions, provided that the flux homomorphism is
$G$-equivariant. Explicitly, in the proof of Cases 2 and 3 above, we can use
[5, Lemma 3.5.4] to extend the map $f_{i}:B_{i}\to Y$ to a nonflat
$G$-equivariant map $f^{\prime}_{i}:S_{i}=B_{i}\cup E\to Y$ of class
$\mathscr{A}^{r-1}$ such that $f^{\prime}_{i}(E)\subset{\mathbf{A}}_{*}$,
condition (5.2) holds, and in addition the imaginary parts
$\Im\int_{E_{j}}f^{\prime}_{i}\theta$ for $j=1,\ldots,m$ assume any given set
of $G$-equivariant values in $\mathbb{R}^{n}$. In particular, we can obtain
the following analogue of Theorem 1.1 for $G$-equivariant immersed holomorphic
null curves $H=(H_{1},\ldots,H_{n}):X\to\mathbb{C}^{n}$, i.e., $G$-equivariant
holomorphic immersions satisfying $\sum_{i=1}^{n}(dH_{i})^{2}=0$. (See [5,
Theorem 3.6.1] for the basic case when $G$ is the trivial group.)

######  Theorem 5.3 ($G$-equivariant null holomorphic curves).

Let $G$ be a finite group acting effectively on a connected open Riemann
surface $X$ by holomorphic automorphisms, and acting on $\mathbb{C}^{n}$
$(n\geq 3)$ by complex orthogonal transformations in $O(n,\mathbb{C})$. Assume
that for every nontrivial stabiliser $G_{x}$ $(x\in X)$ there is a
$G_{x}$-invariant null complex line
$L_{x}\subset{\mathbf{A}}\subset\mathbb{C}^{n}$ on which a generator of
$G_{x}$ acts by $z\mapsto\mathrm{e}^{\mathfrak{i}\phi}z$ with
$\phi=2\pi/|G_{x}|$. Then there exists a nondegenerate $G$-equivariant
holomorphic null immersion $H:X\to\mathbb{C}^{n}$.

## 6\. Symmetric minimal surfaces with ends of finite total curvature

Let $X$ be an open Riemann surface. An immersed minimal surface
$F:X\to\mathbb{R}^{n}$ has nonpositive Gaussian curvature function
$\mathcal{K}:X\to(-\infty,0]$. The surface is said to have finite total
curvature if

$\int_{X}\mathcal{K}\,d\sigma>-\infty.$

Here, $d\sigma$ denotes the surface area in the induced Riemannian metric
$F^{*}ds^{2}$ on $X$, where $ds^{2}$ is the Euclidean metric on
$\mathbb{R}^{n}$. If in addition $F$ is complete, meaning that the pullback
$F^{*}ds^{2}$ of the Euclidean metric is a complete metric on $X$, then by
Huber [33] the surface $X$ is biholomorphic to $\Sigma\setminus P$ where
$\Sigma$ is a compact Riemann surface and $P$ is a nonempty finite subset of
$\Sigma$. Furthermore, by the Chern–Osserman theorem [12], $\partial F$ is a
meromorphic ${\mathbf{A}}$-valued $1$-form on $X$ with an effective pole in
every point of $P$, and the map $F:X\to\mathbb{R}^{n}$ is proper. Many
classical minimal surfaces are of this kind. Complete minimal surfaces of
finite total curvature have been of major interest in the theory since the
early works of Osserman in the 1960’s; see [12, 36, 47, 10, 55, 5] for
background on this subject.

To motivate our next result (see Theorem 6.3), we first recall the following
Runge-type theorem for complete minimal surfaces with finite total curvature,
due to Alarcón and López [9, Theorem 1.2]. A proof can also be found in [5,
Theorem 4.5.1], and a different proof has been given recently by Alarcón and
Lárusson [6, Corollary 5.3].

######  Theorem 6.1.

Let $\Sigma$ be a compact Riemann surface, $P$ be a nonempty finite subset of
$\Sigma$, $K$ be a smoothly bounded compact Runge domain in the open Riemann
surface $X=\Sigma\setminus P$, and let $A$ and $\Lambda$ be disjoint finite
subsets of $\mathring{K}$. If $F_{0}:K\setminus A\to\mathbb{R}^{n}$ $(n\geq
3)$ is a complete conformal minimal immersion with finite total curvature,
then for any $\epsilon>0$ and integer $k\geq 0$ there is a conformal minimal
immersion $F:X\setminus A\to\mathbb{R}^{n}$ satisfying the following
conditions.

  1. (i)

$F-F_{0}$ extends to a harmonic map on $\mathring{K}$ and satisfies
$|F-F_{0}|<\epsilon$ on $K$.

  2. (ii)

$F-F_{0}$ vanishes at least to order $k$ in every point of $A\cup\Lambda$.

  3. (iii)

$F$ is complete and has finite total curvature.

The conditions in Theorem 6.1 imply that $\partial F_{0}$ has an effective
pole in every point of $A$, and $F$ has a complete end with finite total
curvature in every point of $A\cup P$. Clearly, the result is equivalent to
the special case when $P=\\{p\\}$ is a singleton. We emphasize that one can
not prescribe the pole of $\partial F$ at $p$ in an arbitrary way.

One may wonder whether an analogue of Theorem 6.1 holds in the $G$-equivariant
case. Explicitly, assume that $\Sigma$ is a compact Riemann surface,
$G\subset\mathrm{Aut}(\Sigma)$ is a finite group of automorphisms, and $P$ is
a finite $G$-invariant subset of $\Sigma$. Then, $G$ also acts on the open
Riemann surface $X=\Sigma\setminus P$, and the set $X_{0}=\\{x\in
X:G_{x}\neq\\{1\\}\\}$ (2.3) is finite. Let $K\subset X$ be a smoothly bounded
Runge compact set containing $X_{0}$ in the interior, and let $A$ and
$\Lambda$ be disjoint finite $G$-invariant subsets of $\mathring{K}$. Assume
that $G$ also acts on $\mathbb{R}^{n}$ by orthogonal maps. Under these
assumptions, we pose the following problem.

######  Problem 6.2.

Given a $G$-equivariant complete conformal minimal immersion $F_{0}:K\setminus
A\to\mathbb{R}^{n}$ with finite total curvature, is there a $G$-equivariant
complete conformal minimal immersion $F:X\setminus A\to\mathbb{R}^{n}$ with
finite total curvature satisfying Theorem 6.1?

We do not know the answer to this question, except for the results of Xu [54]
pertaining to planar domains (i.e., minimal surfaces of genus zero, see
Example 1.8). However, the methods used in the proof of Theorem 5.1 give the
following result in this direction.

######  Theorem 6.3.

Assume that $X_{0}\subset X$ and $G$ are as in Theorem 5.1, satisfying the
condition on stabilisers of points $x\in X_{0}$. Let $K$ be a smoothly
bounded, $G$-invariant, Runge compact domain in $X$ such that $X_{0}\cap
bK=\varnothing$, and let $A\subset\mathring{K}\setminus X_{0}$ and
$\Lambda\subset\mathring{K}$ be disjoint $G$-invariant finite sets. Given a
complete $G$-equivariant conformal minimal immersion $F_{0}:K\setminus
A\to\mathbb{R}^{n}$ $(n\geq 3)$ of finite total curvature, there exists for
any $\epsilon>0$ and integer $k\geq 0$ a nondegenerate $G$-equivariant
conformal minimal immersion $F:X\setminus A\to\mathbb{R}^{n}$ satisfying the
following conditions.

  1. (a)

$F-F_{0}$ extends to a harmonic map on $\mathring{K}$ and $|F-F_{0}|<\epsilon$
on $K$.

  2. (b)

$F-F_{0}$ vanishes at least to order $k$ in every point of $A\cup\Lambda$.

  3. (c)

For any compact, smoothly bounded, $G$-invariant set $L$ with $K\subset
L\subset X$, the $G$-equivariant conformal minimal immersion $F:L\setminus
A\to\mathbb{R}^{n}$ is complete and has finite total curvature.

Note that condition (c) follows from (a) and (b). The resulting minimal
surface $F(X\setminus A)\subset\mathbb{R}^{n}$ is $G$-invariant and has a
complete end with finite total curvature in every point of $A$.

###### Proof.

Let $\theta$ be a $G$-invariant holomorphic 1-form (2.6) on $X$ with
$\\{\theta=0\\}=X_{0}$, and let $Y$ be the manifold (2.8). By Proposition 2.2
the map $f_{0}=2\partial F_{0}/\theta:K\to Y$ is holomorphic and
$G$-equivariant, and it maps the finite set $A\cup(K\cap X_{0})$ to $Y_{0}$
(2.9). (That is, the restricted map $f_{0}:K\setminus(A\cup
X_{0})\to{\mathbf{A}}_{*}$ is holomorphic and has an effective pole in every
point of $A\cup(K\cap X_{0})$.) Let $\tilde{f}_{0}$ be the associated
holomorphic section of the map

$\rho:Z=(X\times Y)/G\to X/G$

(see (3.3)) over $K$. An inspection of the proof of Theorem 5.1 shows that it
applies without any changes in this situation, and the resulting
$G$-equivariant conformal minimal immersion $F:X\setminus A\to\mathbb{R}^{n}$
can be chosen such that it satisfies the conclusion of the theorem. Further
details in the non-equivariant situation can be found in [9, proof of Theorem
1.3]. ∎

######  Remark 6.4.

(A) Theorem 6.3 shows that, in the setting of Problem 6.2, we can find a
$G$-equivariant conformal minimal immersion $F:X=\Sigma\setminus
P\to\mathbb{R}^{n}$ with a complete end of finite total curvature in every
point of $P\setminus P_{0}$, where $P_{0}\subset P$ is an orbit of $G$.

(B) An inductive application of Theorem 6.3 gives a Mittag–Leffler type
theorem for $G$-equivariant minimal surfaces having a complete finite total
curvature end in every point of a given closed discrete $G$-invariant subset
$A$ of $X$ with $A\cap X_{0}=\varnothing$, provided that the prescription of
the map in the ends (i.e., in the points of $A$) is $G$-equivariant. The
analogous result without group equivariance is due to Alarcón and López [9,
Theorem 1.3].

We do not know whether Theorem 6.3 still holds if the sets $A$ and $X_{0}$ are
not disjoint, for in this case a problem appears in the proof of Lemma 4.5.

The following immediate corollary to Theorem 6.3 shows that any type of
complete ends with finite total curvature can be realised by a $G$-equivariant
conformal minimal immersion.

######  Corollary 6.5.

Assume that $X_{0}\subset X$ and $G$ are as in Theorem 1.1. Let
$x_{1},\ldots,x_{m}\in X\setminus X_{0}$ be points in distinct $G$-orbits, and
for each $i=1,\ldots,m$ let $F_{i}:U_{i}\setminus\\{x_{i}\\}\to\mathbb{R}^{n}$
be a conformal minimal immersion of finite total curvature defined on a
punctured neighbourhood of $x_{i}$. Set $A=\bigcup_{i=1}^{m}Gx_{i}$. Then
there exists a nondegenerate $G$-equivariant conformal minimal immersion
$F:X\setminus A\to\mathbb{R}^{n}$ which agrees with $F_{i}$ to any given
finite order in $x_{i}$ for $i=1,\ldots,m$.

## 7\. Minimal surfaces with infinite discrete groups of symmetries

Let $X$ be an open Riemann surface, and let $G$ be an infinite discrete group
acting on $X$ by holomorphic automorphisms such that the action is properly
discontinuous, meaning that every pair of points $p,q\in X$ admits open
neighbourhoods $p\in U\subset X$, $q\in V\subset X$ such that the set $\\{g\in
G:gU\cap V\neq\varnothing\\}$ is finite. (Equivalently, for every compact
subset $K\subset X$ the set $\\{g\in G:gK\cap K\neq\varnothing\\}$ is finite.)
Then, $G$ is countable, every orbit $Gx$ $(x\in X)$ is an infinite closed
discrete subset of $X$, the set $X_{0}=\\{x\in X:G_{x}\neq\\{1\\}\\}$ (2.3) is
closed and discrete in $X$, every nontrivial stabiliser $G_{x}$ $(x\in X_{0})$
is a finite cyclic group, the quotient space $X/G$ has the structure of a
Riemann surface, and the quotient projection $\pi:X\to X/G$ is holomorphic.
(See Miranda [45, p. 83] and tom Dieck [53, Chapter 1, Sect. 3] for these
facts.)

Assume that $G$ also acts on $\mathbb{R}^{n}$ by rigid transformations, i.e.,
compositions of orthogonal maps, translations, and dilations. We have the
following generalization of Theorem 1.1.

######  Theorem 7.1.

Let $G$ be a countable discrete group acting properly discontinuously on an
open Riemann surface $X$ by holomorphic automorphisms such that the Riemann
surface $X/G$ is open, and acting on $\mathbb{R}^{n}$ $(n\geq 3)$ by rigid
transformations. If for every point $x\in X$ with nontrivial stabiliser
$G_{x}$ there is a $G_{x}$-invariant affine $2$-plane
$\Lambda\subset\mathbb{R}^{n}$ on which $G_{x}$ acts effectively by rotations,
then there exists a nondegenerate $G$-equivariant conformal minimal immersion
$X\to\mathbb{R}^{n}$.

###### Sketch of proof.

It suffices to inspect the proof of Theorem 5.1, taking into account the
information given above. We point out the relevant facts and the necessary
modifications.

We obtain a $G$-invariant holomorphic $1$-form on $X$ as before by taking
$\theta=d(\tilde{h}\circ\pi)$ (2.6), where $\tilde{h}:X/G\to\mathbb{C}$ is a
holomorphic immersion. Note that the property (2.7) of $\theta$ still holds.

By the hypothesis, every $g\in G$ acts on $\mathbb{R}^{n}$ by a map of the
form

(7.1) $g\mathbf{x}=rO\mathbf{x}+b,\quad\mathbf{x}\in\mathbb{R}^{n}$

for some $r>0$, $O\in O(n,\mathbb{R})$, and $b\in\mathbb{R}^{n}$. Its
differential $dg=rO$ also acts on $\mathbb{C}^{n}$, $\mathbb{CP}^{n}$, the
null quadric ${\mathbf{A}}$, and the manifold $Y={\mathbf{A}}_{*}\cup Y_{0}$
(2.8). The formulas (2.13)–(2.15) must be adjusted by replacing $g$ acting on
the derivative $\partial F$ and on the map $f=2\partial F/\theta:X\to Y$ by
its differential $dg=rO$, so the correct analogue of the equations
(2.14)–(2.15) is

(7.2) $\partial F_{gx}\circ dg_{x}=dg\circ\partial F_{x},\qquad
f(gx)=dg\,\cdotp f(x)=rO\,\cdotp f(x).$

Everything said in the sequel assumes this notion of $G$-equivariance of the
map $f$. Remark 1.2 applies verbatim and shows that the condition on
stabilisers in Theorem 7.1 is necessary. (Note that for $g\in G_{x}$ the
differential $dg$ of its action on $\mathbb{R}^{n}$ (7.1) cannot contain a
dilation by a factor $0<r\neq 1$ since the group $G_{x}$ is finite.)
Proposition 2.2 still holds, except that the second part of condition (c)
(that $F(x_{0})$ is orthogonal to $\Lambda$) need not hold since $g_{0}$ can
be conjugate to a rotation by a translation. Lemma 2.3 and Theorem 2.4 remain
unchanged. The action (3.2) of the group $G$ on $X\times Y$ is now replaced by

$g(x,y)=(gx,dg\,\cdotp y),\quad x\in X,\ y\in Y,\ g\in G.$

This action is properly discontinuous (since it is such on $X$), so the
quotient $Z=(X\times Y)/G$ is a reduced complex space and the projection
$\rho:Z=(X\times Y)/G\to X/G=\widetilde{X}$ (3.3) is a holomorphic map which
is ramified (only) over $\pi(X_{0})=\widetilde{X}_{0}$, and
$\rho:Z\setminus\rho^{-1}(\widetilde{X}_{0})\to\widetilde{X}\setminus\widetilde{X}_{0}$
is a holomorphic $G$-bundle with fibre $Y$ as before. The correspondence
between $G$-equivariant maps $f:X\to Y$ (in the sense of (7.2)) and sections
$\tilde{f}:X\to Z$ of $\rho$, given by Lemma 3.1, remains valid. Lemma 3.2
holds without changes.

In Lemma 4.5, the $G$-invariant admissible set $S$ is no longer compact, but
its projection $\widetilde{S}=\pi(S)\subset\widetilde{X}$ is compact, which is
all that matters in the proof. In fact, we see from (4.2) that the period
domination property of the spray $h_{i}$ on the arcs $C_{i,j}$ in
$C_{i}=\pi^{-1}(\widetilde{C}_{i})=\bigcup_{j}C_{i,j}\subset S$ must only be
arranged on one the arcs, $C_{i,1}$, as it extends by $G$-equivariance to a
spray $C_{i}\times B\to Y$. (Here, $j\in\mathbb{N}$ is the index corresponding
to the elements of $G$.) Lemma 4.6 is of local nature and holds without
changes.

The proof of Theorem 5.1 (see Section 5) is carried out with respect to a
normal exhaustion of the open Riemann surface $\widetilde{X}=X/G$ by an
increasing sequence of compact Runge subsets $D_{i}$. In light of the previous
discussion, it is seen by inspection that all steps hold, so we obtain Theorem
7.1. Note that locally uniform convergence of the resulting sequence of
holomorphic sections $\tilde{f}_{i}:D_{i}\to Z$ implies uniform convergence of
the corresponding sequence $F_{i}:B_{i}=\pi^{-1}(D_{i})\to\mathbb{R}^{n}$ of
$G$-equivariant conformal minimal immersions on compacts in $X$. However, when
the group $G$ is infinite, there are no nonempty compact $G$-invariant sets in
$X$. Hence, an exact analogue of Theorem 5.1 (with uniform approximation on a
$G$-invariant set) is not possible if dilations are involved in the action of
$G$ on $\mathbb{R}^{n}$. ∎

Theorem 7.1 implies the following analogue of Corollary 1.4.

######  Corollary 7.2.

If $G$ is an infinite discrete group acting freely and properly
discontinuously on an open Riemann surface $X$ by holomorphic automorphisms
such that the Riemann surface $X/G$ is open, then for every action of $G$ on
$\mathbb{R}^{n}$ $(n\geq 3)$ by rigid transformations there exists a
nondegenerate $G$-equivariant conformal minimal immersion
$X\to\mathbb{R}^{n}$, which can be chosen to be the real part of a
$G$-equivariant null holomorphic immersion $X\to\mathbb{C}^{n}$.

The analogue of Corollary 1.6 also holds for an infinite discrete group of
rigid transformations on $\mathbb{R}^{n}$ whose induced action on an oriented
embedded surface $X\subset\mathbb{R}^{n}$ is properly discontinuous and the
orbit space $X/G$ is noncompact. The statement of Theorem 6.3 can also be
adjusted to infinite discrete groups acting on $X$ as in Theorem 7.1.

As mentioned in the introduction, every group of automorphisms of a Riemann
surface of genus $\geq 2$ is finite. Hence, Theorem 7.1 and its corollaries
are relevant only when $X$ is an open domain in $\mathbb{C}$ or in a complex
torus. Natural examples of such actions are by groups of translations on
$X=\mathbb{C}$ and on $\mathbb{R}^{n}$, and there are several classical
examples of translation-invariant minimal surfaces (for example, the helicoid,
Scherk’s surfaces, Schwarz’s surfaces, to name a few of the best known ones).
Also, every Riemann’s minimal surface in $\mathbb{R}^{3}$ is translation
invariant and parameterised by a domain in $\mathbb{C}$ on which the group
$\mathbb{Z}$ acts properly discontinuously; see the survey by Meeks and Perez
[43] and [5, Subsec. 2.8.5]. Another natural case is when $X$ is the unit disc
$\mathbb{D}=\\{|z|<1\\}$ (or, equivalently, the upper halfplane $\mathbb{H}$),
since its group of holomorphic automorphisms contains many infinite discrete
subgroups acting properly discontinuously (and even freely).

Acknowledgements. Research is supported by the European Union (ERC Advanced
grant HPDR, 101053085) and by grants P1-0291, J1-3005, and N1-0237 from ARIS,
Republic of Slovenia. I wish to thank Antonio Alarcón for helpful comments,
Urban Jezernik for an idea used in the proof of Corollary 1.5, Frank
Kutzschebauch for helpful discussions concerning the Oka principle for
$G$-equivariant holomorphic maps, Francisco J. López for communicating the
question which is partially answered in the paper, and Tjaša Vrhovnik for a
careful reading of the draft and pointing out several necessary corrections.
Last but not least, I thank the referee for careful reading and inspiring
suggestions.

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Franc Forstnerič

Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19,
SI–1000 Ljubljana, Slovenia

Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI–1000
Ljubljana, Slovenia

e-mail<EMAIL_ADDRESS>