Rotational cmc surfaces in terms of Jacobi elliptic functions

Denis Polly

doi:10.1515/advgeom-2025-0032

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Rotational cmc surfaces in terms of Jacobi elliptic functions

Denis Polly

Published/Copyright: October 28, 2025

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From the journal Advances in Geometry Volume 25 Issue 4

Abstract

We give a classification of rotational cmc surfaces in non-Euclidean space forms in terms of explicit parametrizations using Jacobi elliptic functions. Our method hinges on a Lie sphere geometric description of rotational linear Weingarten surfaces and can thus be applied to a more general class of surfaces. As another application of this framework, we give explicit parametrizations of a class of rotational constant harmonic mean curvature surfaces in hyperbolic space. In doing so, we close the last gaps in the classification of all channel linear Weingarten surfaces in space forms, started in [24].

Keywords: Constant mean curvature surface; Weingarten surface; space form

MSC 2010: 53A10 (primary); 53C42; 53A35; 53A40; 33E05 (secondary)

1 Introduction

The study of surfaces of constant mean curvature (cmc surfaces for short) has been a subject of interest in surface theory since the 19^th century. It was Delaunay [14] who discovered that the profile curves of cmc surfaces of revolution, today called Delaunay surfaces, can be constructed by rolling conic sections along the axis of revolution and tracing their focal points. This alludes to a connection between the profile curves of Delaunay surfaces and conics, also on the level that Jacobi elliptic differential equations can be used to describe them. This connection extends to the wider class of linear Weingarten surfaces (lW) of revolution, that is, surfaces with an affine relationship between their Gauss and mean curvature; see [23].

Surfaces invariant under rotations with constant mean curvature have been of interest in many contexts, extending to space forms of constant curvature κ ≠ 0. While rotational surfaces in spherical space forms (κ > 0) are invariant under a 1-parameter family of Euclidean rotations in ℝ⁴, the situation is much richer in cases in hyperbolic space forms (κ < 0), where rotations come in three different flavors, see [17].

The situation is even richer in hyperbolic space forms, given that the family of surfaces of constant mean curvature H splits into three distinct classes depending on the sign of H² + κ: surfaces with H² + κ ≥ 0 have been subject to many publications, being closely related to cmc (minimal) surfaces in ℝ³ and thus allowing for investigation via representations like the DPW method [19] or the Bryant representation [8]. There is not as much literature in the H² + κ < 0 case, however some publications investigate general cmc surfaces, hence apply also to this case; see [1; 12; 19] and [28].

The study of rotational cmc surfaces has come back into focus recently, providing multiple classification theorems in hyperbolic and spherical space forms from differing points of view. This includes description of the profile curves of these surfaces as energy minimizers in [3], as level sets of an auxiliary potential in [4] and their explicit parametrization in terms of one of their principal curvatures in [20].

Some of the references just mentioned consider the wider class of rotational lW surfaces in space forms. This class was considered in [24], where the authors proved that every lW surface enveloping a 1-parameter family of spheres is rotational in its space form. Furthermore, the authors shed light on the connection between rotational lW surfaces and Jacobi elliptic functions by presenting explicit parametrizations of a wide class of rotational lW surfaces in terms of these functions, namely those that are parallel to surfaces of constant Gauss curvature (cGc), see Figure 1 a, c and e. However, the specific case of surfaces parallel to cmc H surfaces in hyperbolic space forms with H² + κ < 0 was unresolved: these surfaces are not related to a constant Gauss curvature surface via parallel transformations, hence no parametrizations for them can be obtained from results stated in [24].

Figure 1

Fruits in the linear Weingarten: a) cocoa pod surface (constant Gauss curvature in ℍ³, hyperbolic rotation) b) saturn peach surface (cmc in 𝕊³) c) star fruit surface (constant Gauss curvature in ℍ³, hyperbolic rotation) d) cantaloupe surface (cmc in ℍ³, hyperbolic rotation) e) peach front (intrinsically flat in ℍ³, parabolic rotation). Plotted and rendered using Rhino and mathematica.

The main motivator of this paper is to close this gap in the classification of channel lW surfaces in space forms via Jacobi elliptic functions. We will develop a new approach to derive differential equations governing the profile curves of rotational lW surfaces in non-Euclidean space forms. This new approach stems from a characterization of lW surfaces via enveloped isothermic sphere congruences with additional properties; see [10]. This description belongs to the realm of Lie sphere geometry and will prove particularly useful once applied to the specific case of cmc surfaces. Further, the following advantages of our setup shall be emphasized:

since spherical and hyperbolic space form geometries appear as subgeometries of Lie sphere geometry, we can carry out our analysis for both ambient spaces simultaneously, breaking symmetry only at the very end to obtain space form specific parametrizations.
surfaces of constant harmonic mean curvature can be investigated with similar ease. This is useful, because another class of surfaces, not covered in the classification results of [24], are surfaces with constant harmonic mean curvature greater than or equal to 1.

The basic idea to derive parametrizations of rotational cmc surfaces in space forms was laid out in the authors PhD thesis [31]. Our goal is to bring this idea to fruition.

This paper is organized as follows: in Section 2 we set the stage of Lie sphere geometry and its space form subgeometries. We will discuss briefly how lW and rotational surfaces can be investigated using this setup but refer the details-seeking reader to publications that discuss this setup in more detail, such as [11; 13; 24]. This section will also introduce a characterization of lW surfaces in terms of isothermic sphere congruences that take values in certain families of spheres [10] and are, in some sense, Christoffel dual (Lemma 2.6). In Section 3 we will further investigate these isothermic sphere congruences in the case of rotational lW surfaces in non-Euclidean space forms, leading to differential equations governing their profile curves in Propositions 3.2 and 3.5. Section 4 will see us applying these differential equations to the case of rotational cmc surfaces. This is particularly convenient as it will turn out that one of the isothermic sphere congruences in this case is the surface itself. Thus, we arrive at explicit parametrisations of rotational cmc surfaces in non-Euclidean space forms. Our results split along the lines of ambient space form, type of rotation and the sign of H² + κ (Theorems 4.5 to 4.7, 4.13 and 4.14).

Finally, in Section 5, we finish the classification of channel lW surfaces in terms of Jacobi elliptic functions, started in [24]. We show how the formulas derived in Section 3 can be applied to the case of surfaces with constant harmonic mean curvature, which will allow us to close the gap in the classification of rotational lW surfaces in hyperbolic space forms (Theorems 5.3 and 5.6).

2 Preliminaries

In this section, we introduce the light cone model of Lie sphere geometry and explain how to investigate linear Weingarten surfaces, channel surfaces and rotationally invariant surfaces in this set-up. Our description of linear Weingarten surfaces in Lie sphere geometry was originally given in [10].

We are interested in rotational cmc surfaces. However, we will discuss the more general class of channel linear Weingarten surfaces in preparation of the special cases dealt with in the following sections. A more detailed account of these preparations can be found in [24], where a Lie sphere geometric investigation of channel linear Weingarten surfaces was laid out. In this note we state the necessary results and refer to [10] and [24] for proofs.

2.1 Lie sphere geometry

Let ℝ^4,2 be the 6-dimensional real vector space with non-degenerate inner product (⋅, ⋅) of signature (++++−−). A vector 𝔳 ∈ ℝ^4,2 is called timelike, spacelike or lightlike if (𝔳, 𝔳) is negative, positive or vanishes respectively. We denote by L^4,1 the set of all lightlike vectors (the light cone).

We call a basis of ℝ^4,2 orthonormal if the inner product takes the form

( v , w ) = − v 0 w 0 + v 1 w 1 + v 2 w 2 + v 3 w 3 + v 4 w 4 − v 5 w 5 ,

and pseudo-orthonormal, if the inner product takes the form

( v , w ) = − v 0 w 5 − v 5 w 0 + v 1 w 1 + v 2 w 2 + v 3 w 3 + v 4 w 4 .

We define the set of (L-)spheres in a space form as the union of the sets of points, oriented hyperspheres and oriented hyperplanes (i.e. complete, totally geodesic hypersurfaces). In the light cone model of Lie sphere geometry, originally introduced by Lie in [27] (see [13] for a modern account), L-spheres of 3-dimensional space forms are represented by points in the projective light cone

L = P L 4 , 1 = 〈 v 〉 ⊂ R 4 , 2 : v ≠ 0 is lightlike

which is called the Lie quadric. Two L-spheres s₁, s₂ ∈ ℒ are in oriented contact if and only if there are homogeneous coordaintes 𝔰_i ∈ s_i —which we also call lifts of s_i —such that

s 1 , s 2 = 0 .

Note that this equation is independent of the choice of lifts. We call maps s : Σ → ℒ into the Lie quadric sphere congruences.

The transformation group of Lie sphere geometry is derived from the projective orthogonal group of ℝ^4,2: this group O(4, 2)/{± Id} acts on the Lie quadric in the way linear transformations typically act on projective spaces (see [13, Corollary 3.3]). It maps oriented spheres to oriented spheres and preserves oriented contact. Further, it preserves linear sphere complexes: each vector 𝔭 ∈ ℝ^4,2 defines a family of spheres

{ s ∈ L : ( p , s ) = 0 ∀ s ∈ s }

called the (linear) sphere complex defined by 𝔭. We also call 𝔭 itself a (linear) sphere complex. Note that two parallel vectors define the same sphere complex. We call a sphere complex timelike, lightlike and spacelike if the spanning vector 𝔭 has the respective causal character.

We may break symmetry and recover models for the space form geometries: let 𝔭 be a unit timelike sphere complex and 𝔮 a perpendicular sphere complex. The affine sub-quadric

ℜ p , q = v ∈ L 4 , 1 : ( v , p ) = 0 , ( v , q ) = − 1

has constant sectional curvature κ = −(𝔮, 𝔮), see [9, Section 2]. Since we define L-spheres in the sphere complex 𝔭 to be points in the space form, we call 𝔭 a point sphere complex. Similarly, 𝔮 defines the curvature of the space form, hence we call it the space form vector. For a point sphere s we call 𝔰 ∈ ℜ_𝔭,𝔮 with 〈𝔰〉 = s a space form lift. Together with the subgroup Iso_𝔭,𝔮 of Lie sphere transformations fixing 𝔭 and 𝔮, ℜ_𝔭,𝔮 is a model for a space form geometry, see [9, Section 2] or [13, Chapter 2]. The set of oriented hyperplanes in ℜ_𝔭,𝔮 is represented by points in the affine sub-quadric

T p , q = v ∈ L 4 , 1 : ( v , p ) = − 1 , ( v , q ) = 0 .

Sphere congruences taking values in the sphere complex 𝔮 are thus called plane congruences.

The study of hypersurfaces in space forms is carried out in Lie sphere geometry via the utilization of Legendre lifts: let 𝔣 : Σ → ℜ_𝔭,𝔮 be a parametrization of a hypersurface in the space form ℜ_𝔭,𝔮 and 𝔫 : Σ → 𝔗_𝔭,𝔮 its tangent plane congruence. Then 𝔣 and 𝔫 satisfy

(1) ( f , n ) = 0 , and ( d f , n ) = 0 .

This implies that the (projective) line congruence Λ = 〈𝔣, 𝔫〉 is a Legendre immersion; see [13, Section 4.2]. We call (1) the contact conditions. Additionally we assume the immersion condition

∀ p ∈ Σ ∀ s ∈ Γ Λ : d p s ( X ) ∈ Λ ( p ) ⇒ X = 0 .

We say that Λ envelopes a sphere congruence s = 〈𝔰〉 if 𝔰(p) ∈ Λ(p) for all p ∈ Σ. We denote the set of sphere congruences enveloped by Λ by ΓΛ (similarily, we denote the set of maps into L^4,1 such that 𝔰(p) ∈ s(p) for a given sphere congruence as Γs)^[1].

Lie sphere transformations naturally act on Legendre lifts. Together with the following lemma, this implies that Lie sphere transformations act on surfaces in space forms in a meaningful way.

Lemma 2.1

([13, Section 2.5]). Given a point sphere complex 𝔭 and a Legendre immersion Λ, there is precisely one point sphere congruence f enveloped by Λ.

Similarly, any Legendre immersion Λ generically envelopes precisely one (tangent) plane congruence ^[2]for any space form vector 𝔮. Additionally, Λ envelopes curvature sphere congruences, see [13, Section 4.4]: there are coordinates (u, v) and sphere congruences s₁, s₂ ∈ ΓΛ such that

s 1 u , s 2 v ∈ Γ Λ

for any lifts si∈Γsi,i=1,2. The coordinates (u, v) are then curvature line coordinates for any space form lift of any point sphere envelope. Away from umbilics we have Λ = s₁ ⊕ s₂.

2.2 Channel linear Weingarten surfaces

This subsection is a brief summary of the Lie sphere geometric treatment of channel linear Weingarten surfaces given in [24]. For the purposes of this paper, we present Theorem 2.2 and Theorem 2.5 and point to [24] and references therein for details. We conclude with a short proof of a lemma that will be instrumental in the following investigations.

Given a point sphere complex 𝔭 and a space form vector 𝔮, the curvature of an L-sphere s in ℜ_𝔭,𝔮 can be computed via

κ s = ( s , q ) ( s , p ) ,

which is lift-independent; see [10]. We can apply this to recover the principal curvatures κ₁, κ₂ of a surface 𝔣 with values in ℜ_𝔭,𝔮 from the curvature sphere congruences s₁, s₂ of its Legendre lift Λ. We define the mean curvature H and the (extrinsic) Gauss curvature K of 𝔣 via

H = κ 1 + κ 2 2 , K = κ 1 κ 2 .

A surface parametrized by 𝔣 is linear Weingarten (lW) if there exists a (non-trivial) triple of constants a, b, c ∈ ℝ such that

(2) a K + 2 b H + c = 0.

The discriminant D of an lW surface is defined as D = b² − ac and we call the surface tubular if D = 0. The class of lW surfaces contains important sub-classes: for a = 0 we obtain constant mean curvature (cmc) surfaces, for b = 0 constant Gauss curvature (cGc) surfaces. For c = 0 we learn that the harmonic mean curvature H := K/H is constant; we call such surfaces constant harmonic mean curvature (chc) surfaces.

As shown in [10], linear Weingarten surfaces in space forms can be characterized in Lie sphere geometry as a subclass of the class of Ω-surfaces. We will state this result and point to the original source for a proof.

We call a sphere congruence s isothermic if it possesses a Moutard lift, that is,

∃ s ∈ Γ s : s u v ∥ s

for suitable coordinates (u, v) which are then curvature line parameters. Following Demoulin [16; 15], we call a Legendre immersion an Ω-surface if it envelopes a (possibly complex conjugated) pair of isothermic sphere congruences that separate the curvature sphere congruences harmonically. We also call a surface 𝔣 in a space form Ω if its Legendre lift is an Ω-surface.

Theorem 2.2

([10]). The Legendre lifts of non-tubular linear Weingarten surfaces are those Ω-surfaces Λ = 〈𝔰⁺, 𝔰⁻〉 whose isothermic sphere congruences s^± take values in fixed linear sphere complexes 𝔭^±. The plane 〈𝔭⁺, 𝔭⁻〉 is spanned by the point sphere complex 𝔭 and the space form vector 𝔮.

Remark 2.3

The isothermic sphere congruences s^± of a linear Weingarten surface are complex conjugate if and only if D < 0.

We therefore generally call an Ω-surface Λ a linear Weingarten surface if its isothermic sphere congruences take values in fixed linear sphere complexes 𝔭^±. To emphasize this, we denote the lW surface by (Λ, 𝔭^±).

Example 2.4

Let 𝔣 : Σ → ℜ_𝔭,𝔮 be a parametrization of a cmc H surface. According to [9, Section 4.2], the sphere complexes 𝔭^± are given by

p + = p , p − = q − H p .

Since the point sphere envelope of a Legendre immersion is unique, this implies that the surface itself is one of the isothermic sphere congruences spanning (Λ, 𝔭^±), s⁺ = 〈𝔣〉. The other one is given by

s − = n + H f .

This is the sphere congruence enveloped by Λ that has in each point the same mean curvature as the surface. According to [5, § 67] this is the central sphere congruence of the surface (or its conformal Gauss map, as introduced in [7]). Further, 𝔰^±, with 𝔰⁺ = 𝔣 are Christoffel dual lifts of s^± (see Lemma 2.6 below).

Somewhat dual to this is the description of chc surfaces: let 𝔣 : Σ → ℜ_𝔭,𝔮 be a parametrization of a chc H surface. Then we have

p + = q , p − = p − 1 H ‾ q ⇒ s + = n , s − = f + 1 H ‾ n .

This means that one isothermic sphere congruence is given by the tangent plane congruence and the other is the sphere congruence that has the same harmonic mean curvature as the surface (this is called the middle sphere congruence in [5, § 67] or the Laguerre Gauss map in [29]).

Next, consider a channel lW surface, that is, the surface envelopes a 1-parameter family of spheres. The enveloped sphere curve s is then automatically a curvature sphere congruence (as it has a lift that only depends on one curvature line parameter). The simplest example is given by rotational surfaces which we describe in the next subsection.

The following theorem has been shown in [24], which we point to for a proof.

Theorem 2.5

([24, Theorem 4.7]). Every non-tubular channel linear Weingarten surface in a space form ℜ_𝔭,𝔮 is a rotational surface.

By this theorem, all channel cmc surfaces are rotational in their space form. Further, because D = b² > 0 for cmc surfaces, they envelope a real pair of isothermic sphere congruences. Thus, it will be our goal to describe rotational Ω surfaces (Λ, 𝔭^±) with 𝔭^± a real pair.

We close this section with the following lemma, introducing a property of the isothermic sphere congruences enveloped by an Ω-surface that is important for our later investigations.

Lemma 2.6

([30, Subsection 4.2.2]). Let Λ be an Ω-surface spanned by two isothermic sphere congruences s^± and let (u, v) be curvature line coordinates. Then there are lifts 𝔰⁺ ∈ Γs⁺ and 𝔰⁻ ∈ Γs⁻ such that

(3) s u − = C U s u + s u + 2 a n d s v − = − C V s v + s v + 2 ,

where U, V are functions of (only) u and v respectively. These lifts are unique up to constant rescaling. We call them the Christoffel dual lifts.

Proof. We only sketch the proof of this lemma, for details see [31, Lemma 3.6] ^[3]. The existence of Christoffel dual lifts follows from the following characterization of Ω surfaces: there exist lifts 𝔰_i ∈ s_i of their curvature sphere congruences such that

s 1 u = ϕ u s 2 , s 2 v = ε 2 ϕ v s 1

where ε ∈ {i, 1}, see [31, Proposition 2.4]. With these special lifts it is straightforward to prove that the lifts

c ± = e ∓ ε ϕ ‾ s 1 ± ε s 2 ,

are Christoffel dual lifts of s^±.

Now, assume that there is another pair of Christoffel dual lifts, given by α^±𝔠^±. Checking (3) then proves that α^± have to be constants. □

Remark 2.7

The functions U, V have the property that

c u + , c u + U = c v + , c v + V .

The existence of functions with these properties follows from the existence of a Moutard lift; see [31, Section 2.3].

2.3 Rotational surfaces

The goal of this subsection is to give a quick overview over the unifying description of rotational surfaces in space forms within the realm of Lie sphere geometry. For brevity we refer the interested reader to [24, Section 3].

As we discussed, the subgroup Iso_𝔭,𝔮 of Lie sphere transformations that fix a point sphere complex 𝔭 and a space form vector 𝔮 models the isometry group of the space form ℜ_𝔭,𝔮. Let Π be a 2-plane with Π ⊥ 〈𝔭, 𝔮〉; we call a 1-parameter subgroup ρ of Iso_𝔭,𝔮 that acts as the identity on Π^⊥ a subgroup of rotations. We call ρ

elliptic if Π has signature (++)
parabolic if Π has signature (+0)
hyperbolic if Π has signature (+−).

As 𝔭 is timelike, these are the only signatures that can occur; note that not all types of rotations exist in 𝕊³ and ℝ³. We will use a basis {𝔳, 𝔢} of Π that is orthogonal with 𝔢 being unit spacelike and (𝔳, 𝔳) encoding the signature.

Definition 2.8

A sphere congruence s : Σ² → ℒ is called rotational if it can be parametrized as ^[4]

s ( θ , t ) := ρ θ c ( t ) ,

where ρ is a 1-parameter subgroup of rotations in a space form ℜ_𝔭,𝔮 with respect to Π ⊥ 〈𝔭, 𝔮〉 and c := s(0, ⋅) : I → 〈𝔢〉^⊥ ∩ ℒ. We call c its planar profile curve. A Legendre immersion Λ is called rotational with respect to ρ if it is spanned by two rotational sphere congruences.

As a special case, a surface 𝔣 : Σ² → ℜ_𝔭,𝔮 is called rotational if its lift f = 〈𝔣〉 is rotational. Note that not every sphere congruence enveloped by a rotational Legendre immersion is rotational. However, in the case of Legendre lifts, some prominent envelopes are.

Lemma 2.9

The Legendre lift Λ of a rotational surface f and its curvature sphere congruences are rotational. If f is an Ω-surface, then the enveloped isothermic sphere congruences s^± are rotational as well.

Proof. Let 𝔠 ∈ Γc be the space form lift of the profile curve of f. There is a unique curve 𝔫₀ : Σ² → 〈𝔢〉^⊥ ∩ 𝔗_𝔭,𝔮 satisfying

c , n 0 = c ˙ , n 0 = 0 ,

where c˙ is the derivative of 𝔠 with respect to the parameter t. It is then straightforward to show that 𝔫(θ, ⋅) = ρ_θ𝔫₀ is the tangent plane congruence of 𝔣, hence two rotational sphere congruences span Λ = 〈𝔣, 𝔫〉. Further, the principal curvatures of 𝔣 only depend on t, hence the curvature spheres s_i(θ, ⋅) = ρ_θ 〈𝔫₀ + κ_i𝔠〉 are rotational.

Finally, as mentioned in the proof of Lemma 2.6, an Ω-surface can be characterized by the existence of lifts 𝔰_i ∈ s_i of the curvature sphere congruences such that

s 1 θ = ϕ θ s 2 , s 2 t = ε 2 ϕ t s 1 ,

where ε ∈ {1, i}. Then 𝔰^± = 𝔰₁ ± ε_𝔰2, the isothermic sphere congruences, are also rotational, because ϕ_θ = 0. □

Remark 2.10

More generally, the curvature spheres and isothermic sphere congruences of a rotational Ω-surface are rotational. The above proof, written in terms of any rotational sphere congruences spanning Λ, can be refined to show this.

3 Rotational LW surfaces

The goal of this section is the derivation of the differential equations describing rotational lW surfaces in non-flat space forms. We will restrict our attention to lW surfaces with real sphere complexes, where we can use Lemma 2.6. Our formalism will include ambient spaces of arbitrary (non-vanishing) constant sectional curvature and (in the case of hyperbolic space forms) all types of rotation. The case of parabolic rotations in ℍ³(κ) will be dealt with separately, because the setup for this case involves lightlike vectors and thus degenerates to some extent. It turns out that the Christoffel dual lifts of rotational lW surfaces are determined by Jacobi elliptic differential equations. The main results of this section are Propositions 3.2 and 3.5 where the saught after equations are given. These results will prove particularly useful in the case of cmc surfaces, investigated in Section 4.

Let (Λ, 𝔭^±) be a rotational lW surface with real sphere complexes 𝔭^±. We will describe explicit parametrizations of the isothermic sphere congruences s^± in terms of coordinate functions that satisfy a system of equations posed in Propositions 3.2 and 3.5.

Let us start with the non-isotropic cases. Assume that the 1-parameter subgroup of rotations ρ¹ that fixes Λ is non-parabolic. Then, we have the orthogonal splitting of the ambient space ℝ^4,2 = Π ⊕_⊥ Π₁ ⊕_⊥ Π₂, where Π = 〈𝔭⁺, 𝔭⁻〉, Π₁ is the rotation plane of ρ₁ and Π₂ is the orthogonal complement of Π ⊕ Π₁. For i = 1, 2, we may choose an orthonormal basis {𝔳_i, 𝔢_i} for Π_i such that 𝔢_i is unit spacelike and 𝔳_i carries the signature of Π_i. This results in the basis {𝔭⁺, 𝔭⁻, 𝔳₁, 𝔢₁, 𝔳₂, 𝔢₂} of ℝ^4,2 satisfying the multiplication table

(4) p + p − v 1 e 1 v 2 e 2 p + κ + ε p − ε κ − v 1 κ 1 0 e 1 0 1 v 2 κ 2 0 e 2 0 1

Given this basis, a the parametrization of a lift of a (ρ¹-)rotational sphere congruence is given by

s ( θ , t ) = ρ θ 1 ( a ( t ) , b ( t ) , r ( t ) , 0 , h ( t ) , k ( t ) ) ⊤ ,

with suitable functions a, b, r, h, k parametrizing a lift of the profile curve 𝔠 of s, see Subsection 2.3. We call r, h, k the coordinate functions of the surface (or of the planar profile curve). Further we define the constant Δ = κ⁺κ⁻ − ε².

Let s^± be the isothermic sphere congruences enveloped by a rotational linear Weingarten surface (Λ, 𝔭^±). According to Lemma 2.9, s^± and their Christoffel dual lifts are rotational as well. We fix Christoffel dual lifts 𝔰^± ∈ Γs^± such that ^[5]

s ± , p ± = 0 , and s ± , p ∓ = Δ .

This yields the following parametrizations for 𝔰^±:

(5) s + ( θ , t ) = ρ θ 1 − ε , κ + , r + ( t ) , 0 , h + ( t ) , k + ( t ) ⊤ s − ( θ , t ) = ρ θ 1 κ − , − ε , r − ( t ) , 0 , h − ( t ) , k − ( t ) ⊤

for suitable functions r^±, h^± and k^±. We will focus on deriving an equation for r⁺ and omit the decorations for the coordinate functions of 𝔰⁺ from now on.

We define polar coordinates: for suitable functions ψ, d (of t) we have

h v 2 + k e 2 = d ρ ψ 2 v 2 ,

where ρ² is a 1-parameter subgroup of rotations in Π₂.

Remark 3.1

Note that this form assumes that the Π₂-part of 𝔰⁺ has the same causal character as 𝔳². We will see in Section 4 that this is the case for cmc surfaces. We call polar coordinates of this form causal polar coordinates, and those of the form hv2+ke2=dρψ2e2 spacelike polar coordinates.

Thus, the parametrization of 𝔰⁺ in (5) takes the form

(6) s + ( θ , t ) = ρ θ 1 ρ ψ ( t ) 2 − ε , κ + , r ( t ) , 0 , d ( t ) , 0 ⊤ .

At this point, note that we fixed the parametrization up to a choice of speed

(7) v 2 = κ 1 r ′ 2 + κ 2 h ′ 2 + k ′ 2

of the profile curve.

With this notation, we pose the following proposition.

Proposition 3.2

The sphere congruence 𝔰⁺ can be parametrized such that its coordinate functions in (6) satisfy the following system of equations:

(8) r ′ 2 = κ − Δ 2 r 2 − C 1 r 2 − C 2

(9) ψ ′ = κ 1 κ 2 Δ ε r 2 + κ + C κ + Δ + κ 1 r 2

(10) d 2 = − κ + Δ + κ 1 r 2 κ 2 ,

where C₁, C₂ are the (real) roots of the polynomial

(11) p C ( x ) = κ 1 κ − x 2 + Δ 2 + 2 κ 1 ε C x + κ 1 κ + C 2

Remark 3.3

Note that sgn Δ is negative if Π is Lorentzian. Thus, κ1κ2Δ is always positive and (9) allows for real solutions. Further, the specific form of (8) assumes κ⁻ ≠ 0. This case shall be considered separately in Example 3.6.

Proof. First, we note that the second equation in (3) yields ^[6]

s θ − = C s θ + s θ + 2 ⇒ r − = C r .

This further implies the contact condition s−,sθ+=0. Before we introduce polar coordinates, we use that the other contact conditions

s − , s + = s − , s t + = 0

become two equations for the two functions h⁻, k⁻, which we therefore express in terms of r, h and k:

h − k − = 1 κ 2 h k ′ − k h ′ k ′ − k − κ 2 h ′ κ 2 h − κ 1 C + Δ ε − κ 1 C r ′ r ,

where we assume hk′ − kh′ ≠ 0. A lengthy (but uneventful) computation now yields, because 𝔰⁻ is lightlike,

0 = κ − Δ + κ 1 C 2 r 2 + 1 κ 2 h ′ k − k ′ h 2 v 2 κ 1 C − Δ ε 2 − r ′ 2 κ 1 ε 2 Δ 2 − κ + Δ κ 1 C r ′ r 2 .

At this point, we introduce polar coordinates for 𝔰⁺ to express h and k and arrive at

(12) 0 = κ 1 Δ 3 r ′ 2 + v 2 κ 1 C − ε Δ 2 + κ 2 d 2 κ − Δ v 2 + κ 1 C 2 v 2 r 2 .

Note that, because 𝔰⁺ is lightlike,

κ 2 d 2 = − κ + Δ − κ 1 r 2 ,

hence we already obtained (10) and (12) only depends on r and v. We fix the speed of the profile curve via v² = r² (thus introducing isothermic coordinates for 𝔰⁺) and finally obtain

(13) r ′ 2 = 1 κ 1 Δ 2 κ 1 κ − r 4 + Δ 2 + 2 κ 1 ε C r 2 + κ 1 κ + C 2 = 1 κ 1 Δ 2 p C r 2 .

Using the roots C_1,2 of the polynomial p_C we arrive at (8).

Finally, note that in polar coordinates,

r 2 = v 2 = − κ 1 κ + Δ r ′ 2 + κ 2 ψ ′ d 2 2 κ 2 d 2 ,

hence (13) and (10) imply (9). □

Remarks 3.4

(Solution of the equations). (a) Equation (8) (as (17) later) is solved by Jacobi elliptic functions. However, the solutions are only real if p_C has real roots. Thus, we first need to investigate the roots of p_C whose signs will determine the form of the solution to (8).

(b) Note that ψ′ is only determined up to sign. This stems from the fact that ψ ↦ −ψ amounts to a rigid motion of the surface given in polar coordinates.

Next, we consider parabolic rotational surfaces in hyperbolic space. In this case Π₁ has signature (+0) and is spanned by a pair of orthogonal vectors 𝔢₁, 𝔳 where 𝔢₁ is unit spacelike and 𝔳 is lightlike. Chose a lightlike vector 𝔬, perpendicular to Π, such that (𝔳, 𝔬) = −1 and a spacelike unit vector 𝔢₂ to complete a pseudo-orthonormal basis {𝔭, 𝔮, 𝔳, 𝔢₁, 𝔬, 𝔢₂} of ℝ^4,2. The multiplication table then reads as in (4) except

v o v 0 − 1 o − 1 0

The action of the parabolic rotation ρⁱ (i = 1, 2) is then given by

ρ θ i : v , e i , o ↦ v i , e i + θ v , o + θ e i + 1 2 θ 2 o .

This implies the following parametrizations for Christoffel dual lifts 𝔰^± of isothermic sphere congruences enveloped by a parabolic rotational linear Weingarten surface (Λ, 𝔭^±):

(14) s + ( θ , t ) = ρ θ 1 − ε , κ + , h ( t ) , 0 , r ( t ) , k ( t ) ⊤

(15) s − ( θ , t ) = ρ θ 1 κ − , − ε , h − ( t ) , 0 , r − ( t ) , k − ( t ) ⊤ .

These lifts are again uniquely determined by requiring that (𝔰^±, 𝔭^∓) = Δ, however via reparametrization we can still choose the speed

v 2 = k ′ 2 − 2 r ′ h ′ ,

of the profile curve of 𝔰⁺.

Polar coordinates, in this case, are obtained via rotations ρ² in the plane 〈𝔳, 𝔢₂〉 to write

k ( t ) e 2 + h ( t ) o = ρ ψ ( t ) 2 o + d ( t ) v .

This yields the polar coordinate parametrization

(16) s + ( θ , t ) = ρ θ 1 ρ ψ ( t ) 2 − ε , κ + , d ( t ) , 0 , r ( t ) , 0 ,

for the Christoffel dual lift 𝔰⁺ ∈ Γs⁺.

Proposition 3.5

The sphere congruence 𝔰⁺ can be parametrized such that its coordinate functions in (16) satisfy the following system of equations:

(17) r ′ 2 = κ − Δ 2 r 2 − c 1 r 2 − c 2 ,

(18) ψ ′ = − 1 Δ ε r 2 + κ + C r 2

(19) d = κ + Δ 2 r ,

where C₁, C₂ are the (real) roots of the polynomial

(20) p C ( x ) = κ − x 2 + 2 C ε x + κ + C 2 .

The proof of this proposition follows the same strategy as the one for Proposition 3.2.

Example 3.6

The form we chose for (8) and (17) assumes that the polynomial p_C is indeed quadratic, hence κ⁻ ≠ 0. We will consider the case of 𝔭⁻ being lightlike here: this corresponds to the class of either minimal surfaces in ℝ³ or Bryant-type surfaces in ℍ³; see [21]. In the case of Bryant type surfaces, the first part of the proof of Proposition 3.2 applies, but (13) becomes

r ′ 2 = 1 κ 1 + 2 ε C Δ 2 r 2 + κ + Δ 2 C 2 .

The solution to this depends on the signs of the constants κ+and A:=1κ1+2εCΔ2:

κ + < 0 , A < 0 : no solution κ + < 0 , A > 0 : r ( t ) = − κ + C 2 A Δ 2 cosh ( A t ) κ + > 0 , A < 0 : r ( t ) = κ + C 2 − A Δ 2 sin ( − A t ) κ + > 0 , A > 0 : r ( t ) = κ + C 2 A Δ 2 sinh ( A t ) .

For parabolic rotations, the solution space is the same, but A:=2εCΔ2. The function ψ is found via simple integration of these trigonometric/hyperbolic functions.

4 Rotational cmc surfaces

In this section, we apply the results of Section 3 to the specific case of rotational cmc surfaces. We will split our investigation into two parts, considering surfaces with H² + κ > 0 (which we call Delaunay-type surfaces) and surfaces with H² + κ < 0 (called sub-horospherical surfaces) separately. This separation is motivated by the fact that Delaunay-type surfaces are closely related to cmc surfaces in ℝ³ (via the Lawson correspondence) while sub-horospherical surfaces are not. Further, we will see that the form of (8) degenerates for H² + κ = 0 (see Example 4.2), hence it is natural to consider it separately for the given cases. Further, this section will include application of the explicit parametrizations we obtain to recover computational proofs of some well-known facts about rotational cmc surfaces (see Theorem 4.9).

From now on, we assume that for cmc surfaces H ≥ 0. Given any cmc surface, this can be achieved via a change of orientation of the Gauss map.

4.1 Delaunay-type surfaces

In this subsection, we apply the equations derived in Subsection 3 to the case of rotational Delaunay-type surfaces in 𝕊³ and ℍ³. Delaunay-type surfaces, that is, rotational cmc surfaces with H² + κ ≥ 0, have a rich history and have been described from multiple viewpoints, see for instance [8; 18]. We will add to this history by parametrizing Delaunay-type surfaces in the 3-sphere and hyperbolic 3-space in terms of Jacobi elliptic functions. This does provide formulas that are easy to use in most tools used to visualize and investigate surfaces.

Furthermore, these surfaces are parallel to cGc surfaces, hence one can obtain parametrizations in terms of Jacobi elliptic functions via parallel transformation from the parametrizations given for rotational cGc surfaces in [24]. However, we shall use the equations derived in Subsection 3, because this approach is

more direct,
particularly useful in the cmc case and
also applicable for surfaces with H² + κ ≤ 0 (which only exist in ℍ³).

The last point is our motivation to go through the computations in detail such that the next subsection can be brief.

As we have mentioned in Example 2.4, the sphere complexes of a cmc surface (Λ, 𝔭^±) in a space form ℜ_𝔭,𝔮 take the form

(21) p + = p , p − = q − H p

and Christoffel dual lifts of the isothermic sphere congruences s^± are given as combinations of the space form lifts 𝔣, 𝔫 of the point sphere envelope and the tangent plane congruence via

(22) s + = f , s − = n + H f ,

up to constant rescalings.

Remark 4.1

The fact that s⁺ represents the surface itself implies that we have to use causal polar coordinates (see Remark 3.1): whenever Π₂ has signature (++) this is obvious. For signature (+−) (which only occurs in hyperbolic space forms), the Π₂-part of 𝔰⁺ has to be timelike, or 𝔰⁺ would be spacelike.

To apply Proposition 3.2 (or Proposition 3.5 in the parabolic rotation case), we note that

(23) κ + = − 1 , κ − = − H 2 + κ , ε = H ⇒ Δ = κ .

Since (𝔰⁺, 𝔭⁻) = κ, the Christoffel dual lift described in Proposition 3.2 coincides with the space form lift 𝔣 for κ ∈ {±1} (up to sign). This explains why the equations derived in Subsection 3 are particularly useful in the case of cmc surfaces.

Example 4.2

The surfaces considered in Example 3.6 contain the class of cmc surfaces in hyperbolic space with κ⁻ = H² + κ = 0. These surfaces, dubbed horospherical in [22], are thus parametrized by the coordinate functions

r ( t ) = c 2 A cosh ( A t ) , ψ ( t ) = − κ 1 κ 2 1 κ 1 t + κ 1 c 2 − 1 A + κ 1 C 2 artanh tanh ( A t ) 1 + κ 1 C 2 A

where A=1κ1+2C needs to be positive, hence C>−12κ1(A=2C and C>0 in the parabolic case). This gives (upon integration of (18) in the parabolic case) a complete classification of rotational horospherical surfaces including, for instance, catenoid cousins; compare [8; 32].

We start with the case of non-parabolic surfaces and investigate the polynomial (11): for H² + κ > 0,

p C ( x ) = − κ 1 H 2 + κ x 2 + κ 2 + 2 κ 1 H C x − κ 1 C 2

has real roots C₁, C₂ if and only if

D ( C ) := κ 2 + 2 κ 1 H C 2 − 4 κ 1 2 C 2 H 2 + κ > 0 .

This poses a condition on the constant C, which we shall investigate case by case. However, in all cases we have

C 1 C 2 = C 2 H 2 + κ ≥ 0 ,

with C = 0 yielding an equal sign. This implies that C₁ and C₂ have the same sign. However, under the assumption C₁, C₂ < 0, (8) becomes

r ′ 2 = − H 2 + κ κ 2 r 2 + − C 1 r 2 + − C 2 ,

which allows for no real solutions. Thus, we further restrict C to values where C₁, C₂ ≥ 0. From now on, we name the roots such that C₁ > C₂ ≥ 0.

Note that

D ( x ) = 0 ⇔ x ± = κ 2 κ 1 H ± H 2 + κ ,

which determines the conditions on C:

> 0, κ₁ > 0 In this case, C ∈ [x⁻, x⁺] guarantees real roots of p_C(x). Note that C₁ > 0 at C = 0 ∈ [x⁻, x⁺], hence for all feasable values C₁ > C₂ ≥ 0 and at the bounds C₁ = C₂.
< 0, κ₁ > 0 In this case, C ∈ (−∞, x⁻]∪[x⁺, ∞)guarantees real roots. However, 0 ∈ [x⁺, ∞), which we know to be a valid value. Indeed, for C ∈ (−∞, x⁻] we have C₁, C₂ < 0. We conclude that we restrict C to C ∈ [x⁺, ∞) and have C₁ > C₂ ≥ 0 and C₁ = C₂ at C = x⁺.
< 0, κ₁ < 0 In the case of hyperbolic rotations in ℍ³, finally, C ∈ (−∞, x⁺] ∪ [x⁻, ∞). In this case, however, C = 0 is not a feasible value. Thus, we conclude C ∈ [x⁻, ∞), which yields C₁ > C₂ ≥ 0 and C₁ = C₂ at C = x⁻.

With these preparations, we prove in the following proposition that the solution r of (8) can be expressed as a Jacobi elliptic function and the solution ψ of (9) can be expressed as an elliptic integral: Π_k(p; s) denotes the incomplete elliptic integral of the third kind with modulus p and parameter k as defined in [26, Section 17.2], that is,

Π p ( k ; s ) = ∫ 0 s d u 1 − k sn p 2 ( u ) .

(Defining the incomplete integral of the third kind by Π(k;s,p)=∫0s11−ksin2(u)du1−p2sin2(u), as is often done, we obtain the relationship Π_p(k; s) = Π(k; am_p(s), p).)

Proposition 4.3

Assume that H² + κ > 0 and let C be chosen such that the polynomial p_C has two real roots C₁, C₂ satisfying C₁ > C₂ ≥ 0. Then the solutions of (8) and (9) are given by

(24) r ( t ) = C 1 dn p ( Ξ t ) w i t h p 2 = C 1 − C 2 C 1 ∈ [ 0 , 1 ] , ψ ( t ) = κ 1 κ 2 κ 1 κ 1 H t − κ 1 C − κ H κ 1 C 1 − κ 1 Ξ Π p κ 1 C 1 − C 2 κ 1 C 1 − κ ; Ξ t ,

where Ξ = H 2 + κ κ 2 C 1 .

Proof. Since C₁ > 0 we define a (real-valued) function y satisfying r=C1y. We then have

r ′ 2 = − H 2 + κ κ 2 r 2 − C 1 r 2 − C 2 ⟺ y ′ 2 = H 2 + κ κ 2 C 1 1 − y 2 − C 2 C 1 + y 2 ,

the real solution of which is

y ( t ) = dn p H 2 + κ κ 2 C 1 t with p 2 = C 1 − C 2 C 1 .

Note that p ∈ [0, 1] because C₁ > C₂. This proves the claimed form of r.

For ψ, we put the expression for r into (9). Using the Jacobi identity dnp2+p2snp2=1, this yields

ψ ′ ( t ) = κ 1 κ 2 κ H C 1 dn p 2 H 2 + κ κ 2 C 1 t − C − κ + κ 1 C 1 dn p 2 H 2 + κ κ 2 C 1 t = κ 1 κ 2 κ 1 κ 1 H − κ 1 C − κ H κ 1 C 1 − κ 1 1 − κ 1 C 1 − C 2 κ 1 C 1 − κ sn p 2 H 2 + κ κ 2 C 1 t

By integration we obtain the desired solution ψ. □

Remark 4.4

The given form for ψ is invalid if κ₁C₁−κ = 0. However, this happens if and only if κ₁C−κH = 0 in which case ψ(t) = Ht as one quickly deduces from (9) (up to sign ambiguity). We will not emphasize this special case from now on.

We now apply this result to give explicit parametrizations of Delaunay-type surfaces in 𝕊³ and ℍ³ in the following two theorems. These directly follow from Proposition 4.3 under the right choice of constants (which we will highlight).

We regard 𝕊³ as the unit sphere in ℝ⁴, which is equipped with an orthonormal basis. This corresponds to κ = κ₁ = κ₂ = 1 in Proposition 4.3.

Theorem 4.5

(Classification of Delaunay-type surfaces in 𝕊³). Every rotational constant mean curvature H surface in 𝕊³ ⊂ ℝ⁴ is given by

( θ , t ) ↦ r ( t ) cos θ , r ( t ) sin θ , 1 − r 2 ( t ) cos ψ ( t ) , 1 − r 2 ( t ) sin ψ ( t )

where

r ( t ) = C 1 dn p ( Ξ t ) w i t h p 2 = C 1 − C 2 C 1 ∈ [ 0 , 1 ] , ψ ( t ) = H t − C − H C 1 − 1 1 Ξ Π p C 1 − C 2 C 1 − 1 ; Ξ t ,

where Ξ = H 2 + 1 C 1 a n d C 1 > C 2 are the non-negative roots of

p C ( x ) = H 2 + 1 x 2 − ( 1 + 2 H C ) x + C 2 .

For the following theorem, we view ℍ³ in the Minkowski model, that is, as a subset in ℝ^3,1, which is equipped with an orthonormal basis (𝔢₀, 𝔢₁, 𝔢₂, 𝔢₃), where 𝔢₀ is unit timelike. This amounts to κ = −1, κ₁ = −κ₂ and κ₁ = 1 for elliptic and κ₁ = −1 for hyperbolic rotations.

Theorem 4.6

(Classification of non-parabolic Delaunay-type surfaces in ℍ³). Every non-parabolic rotational constant mean curvature surface with H ² > 1 in ℍ³ ⊂ ℝ^3,1 is given by one of the following parametrizations:

elliptic rotations: κ₁ = 1

( θ , t ) ↦ 1 + r 2 ( t ) cosh ψ ( t ) , 1 + r 2 ( t ) sinh ψ ( t ) , r ( t ) cos θ , r ( t ) sin θ ,

hyperbolic rotations: κ₁ = −1

( θ , t ) ↦ r ( t ) cosh θ , r ( t ) sinh θ , r 2 ( t ) − 1 cos ψ ( t ) , r 2 ( t ) − 1 sin ψ ( t )

where

r ( t ) = C 1 dn p ( Ξ t ) w i t h p 2 = C 1 − C 2 C 1 ∈ [ 0 , 1 ] , ψ ( t ) = H t − C + κ 1 H C 1 + κ 1 1 Ξ Π p C 1 − C 2 C 1 + κ 1 ; Ξ t ,

where Ξ = H 2 − 1 C 1 a n d C 1 > C 2 are the non-negative roots of

p C ( x ) = H 2 − 1 x 2 − κ 1 + 2 H C x + C 2 .

In order to finish the classification, we consider parabolic Delaunay-type surfaces in ℍ³. Again, we regard ℍ³ as subset of ℝ^3,1, however we equip ℝ^3,1 with a pseudo-orthonormal basis (𝔳, 𝔢₁, 𝔬, 𝔢₂). This corresponds to κ = −1 in Proposition 3.5. Solving the differential equations therein leads to Theorem 4.7. First, however, we discuss the polynomial p_C for this case.

The roots of p_C take the particularly simple form

C ± = C H ± 1

For negative C, these are both negative, hence no real solution of (17) exists. For C = 0, p_C degenerates and (17) does not have real solutions. For positive C, we have C⁻ > C⁺ > 0, and the solutions of (17) and (18) are now obtained in a way similar to the proof of Proposition 4.3. Thus, we arrive at the following theorem.

Theorem 4.7

(Classification of parabolic Delaunay-type surfaces in ℍ³). Every parabolic rotational constant mean curvature surface with H² > 1 in ℍ³ ⊂ ℝ^3,1 is given by

( θ , t ) ↦ 1 + r 2 ψ 2 + θ 2 2 r , r ( t ) θ , r ( t ) , r ( t ) ψ ( t )

and the functions

r ( t ) = C − dn p ( Ξ t ) w i t h p 2 = 2 H + 1 ∈ ( 0 , 1 ) ψ ( t ) = H t − C C − 1 Ξ Π p p 2 ; Ξ t ,

where Ξ = C ( H + 1 ) a n d C ± = 1 H ± 1 .

Remark 4.8

As in the Euclidean case, Delaunay-type surfaces in 𝕊³ and ℍ³ come in two flavors. Namely, unduloids and the nodoids can be characterized by whether ψ is monotone or oscillating. This bifurcation appears in the parametrizations of Theorems 4.5 and 4.6 in the following way: unless ψ′ is constant, we have Aψ′ = Hr²−C, where A is a positive function. This implies that the sign of ψ′ is constant if and only of CH is not in the image [C₂, C₁] of r². We can plug in specific values for C (namely 0, H and 2H) to obtain situations where ψ′ has constant sign (unduloid case) or oscillates (nodoid case). In the parabolic rotational case, all surfaces are nodoids.

As an application of our parametrizations, we give a simple construction of embedded cmc tori in 𝕊³. As was proved in [2], such surfaces are always rotational and can thus be constructed using Theorem 4.5. For given mean curvature value H the question becomes which values for the constant C yield closed profile curves. This is answered in the following theorem.

Theorem 4.9

Given a Delaunay-type surface in 𝕊³, parametrized as in Theorem 4.5, the profile is periodic if there exist n ∈ ℕ and C ∈ ℝ feasible such that

(25) n Ξ F p H − C − H C 1 − 1 Π p C 1 − C 2 C 1 − 1 = π ,

where FpandΠpC1−C2C1−1 denote the complete elliptic integrals of the first and third kind respectively. Additionally, if C < 0, the profile curve has no self-intersections and the resulting surface is an embedded cmc torus of revolution.

Remark 4.10

A similar theorem can be stated for hyperbolic rotational Delaunay-type surfaces in ℍ³, where we can use a formula similar to (25) to find complete and embedded hyperbolic rotational cmc surfaces.

Proof. The radius function given in Theorem 4.5 is periodic with period α=2FpΞ. Using this and

Π p k ; x + n F p = Π p ( k ; x ) + n Π p ( k ) ,

we see that (25) is equivalent to ψ(s+α) = ψ(s)+2π. Because cos ψ and sin ψ are thus α-periodic, so is the profile curve of the surface. The fact that it has no self-intersections for C < 0 follows from Remark 4.8. □

Example 4.11

Figure 2 depicts two cmc tori with H = 2 in the sphere after stereographic projection into ℝ³. These correspond to the cases n = 5, 6 (and suitable values for C) in (25).

Figure 2

Two cmc H = 2 tori in 𝕊³ after stereographic projection (n = 5, 6). Plotted and rendered using Rhino and mathematica.

4.2 Sub-horospherical surfaces in ℍ3

In this subsection, we apply Proposition 3.2 (and Proposition 3.5) to the case of surfaces with H² + κ < 0 in ℍ³, which we call sub-horospherical. For sub-horospherical surfaces in hyperbolic space forms, less literature is available than for their Delaunay-type counterparts. This stems from the fact that they are less similar to the well-studied class of cmc surfaces in ℝ³, i.e., there is no Lawson correspondence, and the DPW method [18] in its original form could not be used to construct surfaces with H ² + κ < 0. However, the special case of rotational surfaces was considered in some publications catering to rotational surfaces of constant curvature (with any value H). For instance [6] uses a new loop group method (see [19]) to construct examples of rotational cmc surfaces and [20; 28] give explicit parametrizations of their profile curves.

In our setup, the significant difference in the solutions to (8) arises in the form of the roots of the polynomial p_C. We analyze these differences to obtain explicit parametrization of rotational sub-horospherical surfaces in terms of Jacobi elliptic functions. These results also apply to the well-studied class of minimal surfaces (H = 0) in hyperbolic space; see for example [6; 25].

The setup is as in Subsection 4.1: we have

κ + = − 1 , κ − = − H 2 + κ , ε = H ⇒ Δ = κ < 0 ,

and (11) takes the form

p C ( x ) = − H 2 + κ κ 1 x 2 + κ 2 + 2 κ 1 H C x − κ 1 C 2 .

The main difference to Subsection 4.1 lies in the following facts: p_C has real roots C⁺, C⁻ for all values of C but they satisfy

C + C − = C 2 H 2 + κ < 0 ,

hence, with the right choice of nomenclature, C⁺ > 0 ≥ C⁻ (C⁻ vanishes for C = 0).

This warrants a different analysis of equation (8), which we give in the proof of the following proposition.

Proposition 4.12

Assume that H² + κ < 0 in a hyperbolic space form and choose C ∈ ℝ. Then the solutions of (8) and (9) are given by

(26) r ( t ) = C + cn p ( Ξ t ) , ψ ( t ) = κ 1 κ 2 κ 1 κ C t + C + H κ − C κ 1 κ 1 C + − κ 1 Ξ Π p κ κ − κ 1 C + ; Ξ t

with p 2 = − C − C + − C − ∈ [ 0 , 1 ] a n d Ξ = − H 2 + κ C + − C − .

Proof. The manipulation of the differential equation (8) follows the treatment described in the proof of Proposition 4.3: define y=rC+, a real function that satisfies

y ′ 2 = H 2 + κ κ 2 C + − C − 1 − y 2 − C − C + − C − + C + C + − C − y 2

due to (8). This implies that

y ( s ) = cn p ˜ i − H 2 + κ κ 2 C + − C − s with p ˜ 2 = C + C + − C − < 0 ,

hence, using Jacobi’s imaginary transformation (see [24, App. A]),

y ( s ) = 1 cn p − H 2 + κ κ 2 C + − C − s , with p 2 = − C − C + − C − .

This proves the correctness of the stated solution r. Clearly, p² is positive but smaller than 1 (recall that C⁻ ≥ 0). We can use this solution in (9) to obtain

ψ ′ ( s ) = κ 1 κ 2 κ H C + n c p 2 − H 2 + κ C + − C − s − C − κ + κ 1 C + n c p 2 − H 2 + κ C + − C − s = κ 1 κ 2 κ 1 κ C + C + H κ − C κ 1 κ 1 C + − κ 1 1 − κ κ − κ 1 C + sn p 2 − H 2 + κ C + − C − s .

Integration leads to the claimed form of ψ. □

As application of Proposition 4.12 we give the classification of non-parabolic sub-horospherical surfaces in ℍ³. We arrive at this by fixing κ = −1 and an orthonormal basis {𝔳₁, 𝔢₁, 𝔳₂, 𝔢₂} of ℝ^3,1 ≅ 〈𝔭^±〉^⊥ such that κ₁ = −κ₂ ∈ {±1}.

Theorem 4.13

(Classification of non-parabolic sub-horospherical surfaces in ℍ³). Every non-parabolic rotational constant mean curvature H surface with H² < 1 in ℍ³ is given by one of the following parametrizations:

elliptic rotations: κ₁ = 1

( θ , t ) ↦ 1 + r 2 ( t ) cosh ψ ( t ) , 1 + r 2 ( t ) sinh ψ ( t ) , r ( t ) cos θ , r ( t ) sin θ ,

hyperbolic rotations: κ₁ = −1

( θ , t ) ↦ r ( t ) cosh θ , r ( t ) sinh θ , r 2 ( t ) − 1 cos ψ ( t ) , r 2 ( t ) − 1 sin ψ ( t )

with

r ( t ) = C + cn p ( Ξ t ) , w i t h p 2 = − C − C + − C − ∈ [ 0 , 1 ] , ψ ( t ) = C t − C + C + κ 1 H C + + κ 1 1 Ξ Π p 1 κ 1 C + + 1 ; Ξ t ,

where Ξ = − H 2 − 1 C + − C − a n d C 1 > 0 ≥ C 2 are the (real) roots of

p C ( x ) = H 2 − 1 x 2 − κ 1 + 2 H C x + C 2 .

We are left with the case of parabolic rotations: as in Subsection 4.1, equip ℝ^3,1 with a pseudo-orthonormal basis (𝔳, 𝔢₁, 𝔬, 𝔢₂). This corresponds to κ = −1 in Proposition 3.5. The roots of the polynomial (20) are

C 1 , 2 = C H ± 1 .

Since H² < 1, for all values of C, one of these roots is positive and one negative (excluding the degenerate case C = 0). Let us denote the positive root by C⁺ and the negative one by C⁻. We then see, as in the proof of Theorem 4.7, that

r ( t ) = C + cn p ( Ξ t )

where Ξ=|2C| and

p 2 = H + 1 2 for C > 0 − H + 1 2 for C < 0.

With this expression for r, we obtain

ψ ′ ( t ) = 1 − 2 d n p 2 ( Ξ t )

from (18) (up to sign). Integration provides the last piece in the proof of the following theorem.

Figure 3

Surfaces with cmc H = 0.3 obtained using the parametrizations of Theorems 4.13 and 4.14 (elliptic, hyperbolic and parabolic rotation from left to right). Choosing the right parameter for the hyperbolic rotational case, such that the profile curve closes, amounts to a problem similar to the one discussed in Theorem 4.9. Plotted and rendered using Rhino and mathematica.

Theorem 4.14

(Classification of parabolic rotational sub-horospherical surfaces in ℍ³). Every parabolic rotational constant mean curvature surface with H² < 1 in ℍ³ ⊂ ℝ^3,1 is given by

( θ , t ) ↦ 1 + r 2 ψ 2 + θ 2 2 r , r ( t ) θ , r ( t ) , r ( t ) ψ ( t )

and the functions

r ( t ) = C + c n p ( Ξ t ) w i t h p 2 = 1 + H 2 for C > 0 1 − H 2 for C < 0 ψ ( t ) = s − 2 Ξ E p ∘ am p ( Ξ t )

where Ξ = | 2 C | .

5 Classification of channel linear Weingarten surfaces in ℍ³

In [24] a complete and transparent classification of channel linear Weingarten (clW) surfaces in space forms in terms of Jacobi elliptic functions was given for

all clW surfaces in 𝕊³, see [24, Theorem 6.2],
“most” clW surfaces in ℍ³, see [24, Theorem 6.10].

The surfaces missing in ℍ³ were those in parallel families that do not contain surfaces of constant Gauss curvature. We will close this gap here. We will state a Bonnet-type theorem that classifies lW surfaces in ℍ³ in terms of constant curvature surfaces in their parallel families (Proposition 5.1) and then collect the needed parametrizations to complete [24, Theorem 6.10]. For simplicity, we restrict our attention to κ = −1, but the results extend to other hyperbolic space forms.

Recall that we call a surface lW, if its Gauss and mean curvature satisfy a (non-trivial) relationship of the form

(27) a K + 2 b H + c = 0 ,

for constants a, b, c ∈ ℝ³. Because the principal curvatures can be expressed in terms of the curvature spheres s_i via κi=si,qsi,p, this can be written as

s 1 , q , s 1 , p W s 2 , q , s 2 , p ⊤ = 0 , where W = a b b c .

As described in [9, Section 4.8], parallel transformations of surfaces in a space form can be viewed as application of rotations P^t in the plane 〈𝔭, 𝔮〉, resulting in a change of the linear Weingarten condition of the form

W ↦ P t W P t ⊤ .

In the specific case of ℍ³, we have

P t = cosh ( t ) sinh ( t ) sinh ( t ) cosh ( t ) .

This can be used to prove the following proposition (for a proof see [9, Section 4.8] or [31, Theorem 3.15]).

Proposition 5.1

Let f : Σ² → ℍ³ be a non-tubular linear Weingarten surface satisfying (27). If D = − det W < 0, then f is parallel to a surface of (negative) constant Gauss curvature. For D > 0 there are three cases:

If a+c2>|b|, then the parallel family of f contains a surface of constant (positive) Gauss curvature.
If a+c2=|b|, then f is of Bryant type and its parallel family either consists of intrinsically flat surfaces (K + 1 = 0) or contains either a surface of constant mean curvature H = 1 or constant harmonic mean curvature H = 1.
If a+c2<|b|, then the parallel family of f contains either a constant mean curvature H < 1 surface or a constant harmonic mean curvature H > 1 surface.

This proposition (together with Theorem 2.5) can be used to classify channel linear Weingarten surfaces with explicit parametrizations up to parallel transformation. Relying on the results of [24] and Section 4, we still have to give explicit parametrizations of constant harmonic mean curvature (chc) |H| ≥ 1 surfaces.

For a chc surface (Λ, 𝔭^±) the sphere complexes 𝔭^± take the form

(28) p + = q , p − = p − 1 H ‾ q ,

and Christoffel dual lifts of the isothermic sphere congruences s^± are given as combinations of the space form lifts 𝔣, 𝔫 of the point sphere envelope and the tangent plane congruence via

s + = n , s − = f + 1 H ‾ n ,

see Example 2.4. Note that s⁺ is the tangent plane congruence of Λ (which corresponds to the hyperbolic Gauss map of the surface, see [13]) and s⁻ is the middle sphere congruence (see [5, § 67]). One could use the equations developed in Subsection 3 to parametrize s⁺ and s⁻ and look for the space form lift. However, noting that 𝔣 and 𝔫 satisfy

( f , n ) = f , n t = f , n θ = 0 ,

we can determine 𝔣 as the re-scaled (ℝ^3,1-)cross product of 𝔫, 𝔫_t and 𝔫_θ:

( u × v × w ) = det − e 0 e 1 e 2 e 3 u 0 u 1 u 2 u 3 v 0 v 1 v 2 v 3 w 0 w 1 w 2 w 3

for an orthonormal basis (𝔢₀, 𝔢₁, 𝔢₂, 𝔢₃) of ℝ^3,1 with timelike 𝔢₀.

From (28) we obtain

κ + = 1 , κ − = 1 H ‾ 2 − 1 , ε = − 1 H ‾ ⇒ Δ = − 1 .

As in the previous section, we equip ℝ^3,1 with an orthonormal basis (𝔢₀, 𝔢₁, 𝔢₂, 𝔢₃) for non-parabolic rotations and with a pseudo-orthonormal basis (𝔳, 𝔢₁, 𝔬, 𝔢₂) for parabolic rotations. We use this in (8) and (9) (and (17), (18)) to obtain the following theorem.

Example 5.2

As in Section 4, we treat the case H = 1 — where the polynomial p_C degenerates —- separately. In this instance, we learn (for non-parabolic rotations)

r ( s ) = C 2 / A sinh ( A s ) ,

where A=1κ1+2C(A=2C). Note that, contrary to the cmc case, negative values of A also yield a real solution.

Theorem 5.3

(Classification of constant harmonic mean curvature H² > 1 surfaces in ℍ³). The Gauss map of every rotational constant harmonic mean curvature surface with H² > 1 in ℍ³ ⊂ ℝ^3,1 can be parametrized by

elliptic rotations: κ₁ = 1

( θ , t ) ↦ r ( t ) 2 − 1 cosh ψ ( t ) , r ( t ) 2 − 1 sinh ψ ( t ) , r ( t ) cos θ , r ( t ) sin θ , w h e n | r | > 1 1 − r ( t ) 2 sinh ψ ( t ) , 1 − r ( t ) 2 cosh ψ ( t ) , r ( t ) cos θ , r ( t ) sin θ , w h e n | r | < 1

hyperbolic rotations: κ₁ = −1

( θ , t ) ↦ r ( t ) cosh θ , r ( t ) sinh θ , r 2 ( t ) + 1 cos ψ ( t ) , r 2 ( t ) + 1 sin ψ ( t )

with

r ( t ) = C + cn p ( Ξ t ) , p 2 = C + C + − C − ψ ( t ) = 1 H ‾ − κ 1 t + H ‾ C − κ 1 κ 1 C + − 1 1 Ξ Π p κ 1 C + κ 1 C + − 1 ; Ξ t ,

where Ξ = 1 − 1 H ¯ 2 C + − C − a n d C + > 0 > C − are the (real) roots of

(29) p C ( x ) = 1 H ‾ 2 − 1 x 2 + κ 1 − 2 C H ‾ x + C 2 ,

parabolic rotations:

( θ , t ) ↦ − 1 + r 2 ψ 2 + θ 2 2 r , r ( t ) θ , r ( t ) , r ( t ) ψ ( t )

with

r ( t ) = C + cn p ( Ξ t ) , p 2 = C + C + − C − ψ ( t ) = − 1 H ˉ t + C C + 1 Ξ Π p ( 1 ; Ξ t ) ,

where Ξ = 2 | C | and

C ± = C 1 H ˉ ± sign C .

Remark 5.4

In the case of elliptic rotations, the polar coordinates used in Π₂ are not necessarily causal (see Remark 3.1). However, this does not effect the solution of (8), only the polar coordinates in Π₂. This explains the different ways we write the parametrization in this instance.

Proof. Using (8) and (9) (or (17) and (18) for the parabolic case), we obtain a parametrization of s⁺, i.e., the tangent plane congruence. The computation is very similar to the cmc H < 1 case, described in Subsection 4.2. We therefore only give a basic outline of this step and refer for details to the proof of Proposition 4.12. Given the specific parametrizations, we assume κ₁ = −κ₂ ∈ {±1} for the non-parabolic case.

First, the polynomial (11) takes the form (29). Its roots, given by

C ± = κ 1 − 2 C H ˉ ∓ 1 − 4 κ 1 C H ˉ + 4 C 2 2 1 − 1 H ˉ 2 .

are real for all values of C and satisfy C⁺ ≥ 0 ≥ C⁻. This holds mutatis mutandis for the polynomial (20) in the parabolic case.

The solution of (8) (or (17)) is given by

r ( t ) = C + cn p ( Ξ t ) , with p 2 = C + C + − C − .

(Note that Ξ² is non-negative because κ⁻ < 0, which is the key difference to the sub-horospherical case investigated in Subsection 4.2.) This solution and (9), (18) yield

n o n − p a r a b o l i c : ψ ′ ( t ) = 1 H ˉ − κ 1 + H ˉ C − κ 1 κ 1 C + − 1 1 1 − κ 1 C + κ 1 C + − 1 sn p 2 ( Ξ t ) , p a r a b o l i c : ψ ′ ( t ) = − 1 H ˉ + C C + 1 1 − sn p 2 ( Ξ t ) .

which integrates to the desired solution. □

Remark 5.5

Due to the duality mentioned in [9, Section 2], the Gauss map 𝔫 of a chc surface 𝔣 is itself a spacelike cmc surface in de Sitter space. An investigation of spacelike lW surfaces in Lorentz space forms shall be carried out in a future project.

Together with [24, Theorem 6.10] and Examples 4.2 and 5.2, the following theorem provides a complete classification of channel linear Weingarten surfaces in hyperbolic space ℍ³.

Theorem 5.6

Every channel linear Weingarten surface in ℍ³ with a+c2<|b| is parallel to a rotational surface determined by one of the parametrizations given in Theorems 4.13, 4.14 and 5.3.

Funding statement: This work was done while the author was a JSPS International Research Fellow (Graduate School of Science, Kobe University) and has been supported by the JSPS Grant-in-Aid for JSPS Fellows 22F22701.

Acknowledgements

The author would like to express his gratitude to Udo Hertrich-Jeromin for many years of mentoring and suggesting interesting avenues of research, like this one. Further, the author thanks Joseph Cho, Yuta Ogata, Mason Pember, and Wayne Rossman for many fruitful discussions about this subject. The author acknowledges TU Wien Bibliothek for financial support through its Open Access Funding Programme.

Communicated by: T. Leistner

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Received: 2024-12-06

Revised: 2025-04-14

Published Online: 2025-10-28

Published in Print: 2025-10-27

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/advgeom-2025-0032

Keywords for this article

Constant mean curvature surface; Weingarten surface; space form

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Rotational cmc surfaces in terms of Jacobi elliptic functions

Article

Abstract

1 Introduction

2 Preliminaries

2.1 Lie sphere geometry

Lemma 2.1

2.2 Channel linear Weingarten surfaces

Theorem 2.2

Remark 2.3

Example 2.4

Theorem 2.5

Lemma 2.6

Remark 2.7

2.3 Rotational surfaces

Definition 2.8

Lemma 2.9

Remark 2.10

3 Rotational LW surfaces

Remark 3.1

Proposition 3.2

Remark 3.3

Remarks 3.4

Proposition 3.5

Example 3.6

4 Rotational cmc surfaces

4.1 Delaunay-type surfaces

Remark 4.1

Example 4.2

Proposition 4.3

Remark 4.4

Theorem 4.5

Theorem 4.6

Theorem 4.7

Remark 4.8

Theorem 4.9

Remark 4.10

Example 4.11

4.2 Sub-horospherical surfaces in ℍ3

Proposition 4.12

Theorem 4.13

Theorem 4.14

5 Classification of channel linear Weingarten surfaces in ℍ3

Proposition 5.1

Example 5.2

Theorem 5.3

Remark 5.4

Remark 5.5

Theorem 5.6

Acknowledgements

References

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Articles in the same Issue

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5 Classification of channel linear Weingarten surfaces in ℍ³