Catenaries and minimal surfaces of revolution in hyperbolic space

We introduce the concept of extrinsic catenary in the hyperbolic plane. Working in the hyperboloid model, we define an extrinsic catenary as the shape of a curve hanging under its weight as seen from the ambient space. In other words, an extrinsic catenary is a critical point of the potential functional, where we calculate the potential with the extrinsic distance to a fixed reference plane in the ambient Lorentzian space. We then characterize extrinsic catenaries in terms of their curvature and as a solution to a prescribed curvature problem involving certain vector fields. In addition, we prove that the generating curve of any minimal surface of revolution in the hyperbolic space is an extrinsic catenary with respect to an appropriate reference plane. Finally, we prove that one of the families of extrinsic catenaries admits an intrinsic characterization if we replace the extrinsic distance with the intrinsic length of horocycles orthogonal to a reference geodesic.


Introduction
The catenary is the solution to the problem of minimizing the potential gravitational energy of a chain hanging under its weight when supported only at its ends. Euler proved that by rotating a catenary about the reference line with respect to which the weight is measured, the resulting surface of revolution has zero mean curvature [2,9]. This surface is called a catenoid, the only non-planar minimal surface of revolution in Euclidean space. Recently, the second author extended the notion to the catenary in the sphere and the hyperbolic plane [4]. After we fix a reference geodesic ℓ, the energy functional to be minimized is the potential energy, which is given by integrating the distance of points of the chain to ℓ. A critical point of this functional is called an (intrinsic) catenary.
Embedding the hyperbolic plane (2-sphere) in the hyperbolic space (3sphere, respectively), we may ask, as Euler did in Euclidean space, whether the revolution of a catenary about ℓ is a minimal surface. It turns out that the resulting surface of revolution is not minimal. This problem was circumvented in the spherical case by introducing the notion of an extrinsic catenary. More precisely, instead of measuring the potential using the intrinsic distance, one uses the distance in the ambient Euclidean space to a plane passing by the center of the sphere [4]. This paper aims to provide a similar extension of Euler's result to the hyperbolic space.
Contrarily to the three-dimensional Euclidean space and sphere, in the hyperbolic space, H 3 (r), three types of surfaces of revolution exist. Indeed, considering H 3 (r) as a hypersurface of the 4-dimensional Lorentz-Minkowski space E 4 1 (the hyperboloid model), rotations in H 3 (r) correspond to orthogonal transformations in E 4 1 that leave a 2-dimensional subspace P 2 point-wise fixed. (We refer to P 2 as the axis of revolution.) Thus, depending on the causal character of the axis of revolution P 2 ⊂ E 4 1 , the corresponding surface of revolution is of elliptic, hyperbolic, or parabolic type 1 . Motivated by this classification, instead of measuring the gravitational potential in the hyperbolic plane H 2 (r) from the intrinsic distance to a fixed geodesic of H 2 (r), as in Ref. [4], the potential will be measured using the extrinsic distance. More precisely, the extrinsic catenary problem consists in finding the shape of a curve γ : [a, b] → H 2 (r) ⊂ H 3 (r), which is a critical point of the potential energy where dist E 4 1 (γ(t), P 2 ) is the distance in E 4 1 between P 2 and the point γ(t). Depending on the causal character of P 2 , critical points of the potential energy functional will be called extrinsic catenaries of elliptic, hyperbolic, and parabolic types. Our main theorem will extend Euler's result about the 1 A surface of revolution with axis P 2 is of elliptic type if P 2 is spacelike, hyperbolic type if P 2 is timelike, and parabolic type if P 2 is lightlike. The terminology stands for the fact that the orbits of the revolution are ellipses, hyperbolas, and parabolas, respectively. catenoid to the hyperbolic space. Namely, we have (we prove it as Theorem 9 in Section 5) Main Theorem 1 The generating curves of minimal surfaces of revolution in H 3 (r) of elliptic, hyperbolic, and parabolic types are extrinsic catenaries of spherical, hyperbolic, and parabolic types, respectively.
In addition, we show that one of the families of extrinsic catenaries admits an intrinsic characterization in terms of the so-called horo-catenary. In other words, a horo-catenary is a critical point of the potential energy functional obtained by replacing dist E 4 1 (γ(t), P 2 with the length of a horocycle orthogonal to a fixed reference geodesic. Thus, we obtain the following theorem (we prove it as Theorem 10 in Section 6) Main Theorem 2 Every hyperbolic horo-catenary is an extrinsic catenary in H 2 (r) of the elliptic type. Consequently, the generating curves of minimal surfaces of revolution in H 3 (r) of the elliptic type are horo-catenaries.
Surfaces of revolution in H 3 (r) are well-known objects of investigation. First, Mori found minimal surfaces of revolution of elliptic type [5]. 2 Later, do Carmo and Dajczer obtained all the minimal surfaces of revolution and, among other results, established their relation with helicoids [1]. To our knowledge, this work is the first to characterize minimal surfaces of revolution in hyperbolic space through a variational formulation for their generating curves.
The rest of this paper is divided as follows. In Section 2, we formulate the extrinsic catenary problem in H 2 (r). Theorem 3 characterizes extrinsic catenaries in terms of their curvature function. In Section 3, we characterize extrinsic catenaries by prescribing the curvature in terms of an equation involving Killing vector fields of E 3 1 (Theorem 5). These vector fields indicate the direction of the geodesics in E 3 1 used to measure the weight. To be self-contained, in Section 4, we provide a detailed construction of all surfaces of revolution in hyperbolic space along with the computation of their mean curvature. In Section 5, we prove our main result concerning the characterization of the generating curves of surfaces of revolution in the hyperbolic space (Theorem 9). In Section 6, we prove that horocatenaries are extrinsic catenaries of the elliptic type, thus providing an intrinsic characterization for minimal surfaces of revolution of the elliptic type (Theorem 10). Finally, we present our concluding remarks and formulate some open problems in the last section.

The extrinsic catenary problem
Lorentz-Minkowski space. In the hyperboloid model of the hyperbolic space, we see H n (r) as a hypersurface in E n+1 1 of curvature K = −1/r: We shall denote by ·, · the induced inner product of H n (r). In addition, we embed H 2 (r) into H 3 (r) by the natural inclusion The extrinsic catenary problem is formulated as follows: The extrinsic catenary problem. Given a subspace P 2 of E 4 1 , find the shape of a chain γ : [a, b] → H 2 (r) ⊂ H 3 (r) that optimizes the potential energy where dist(γ(t), P 2 ) is the distance in E 4 1 of the point γ(t) to P 2 . In the definition of extrinsic catenaries, it is implicitly assumed that γ lies in one side of the plane P 2 , which then implies that the integrand in (3) is positive. In contrast to how catenaries are defined in Ref. [4], the weight is now calculated using the extrinsic distance in E 4 1 , rather than the intrinsic distance to a given geodesic in H 2 (r).
In H 3 (r), there are three types of planes P 2 according to their causal character. In general, if P k denotes a k-dimensional subspace of E 4 1 , P k is said to be Lorentzian, Riemannian, or degenerate, if the restriction of the metric ·, · 1 to P k is a Lorentzian, Riemannian, or degenerate metric, respectively. Therefore, we define three types of extrinsic catenaries: γ is an extrinsic catenary of (i) elliptic type, (ii) hyperbolic type, and (iii) parabolic type if γ is a critical point of (3) when P 2 is (i) Lorentzian, (ii) Riemannian, and (iii) degenerate, respectively.
To characterize the extrinsic catenaries, we will obtain the Euler-Lagrange equations of the functional (3). This task can be significantly simplified by choosing a suitable system of coordinates and then applying the standard techniques of the Calculus of Variations.
Without loss of generality, let ℓ be the geodesic in H 2 (r) obtained from the intersection with the plane of equation z = 0: If β v (u) denotes the geodesic with unit velocity X(v) = (0, 0, 1) ∈ T ℓ(v) H 2 (r), X(v), ℓ ′ (v) 1 = 0, then we can parametrize H 2 (r) by This coordinate system is known as the semi-geodesic coordinates [9]. Since the induced metric on H 2 (r) takes the form Now, we compute the distance in E 4 1 from γ(t) to P 2 . Let {e x , e y , e z , e w } be the unit velocity vectors of the canonical Cartesian coordinates (x, y, z, w) Without loss of generality, we can suppose that the subspace P 2 is one of the following: Up to multiplication by a constant, the distance to P 2 can be taken as sinh u r (elliptic), cosh u r cosh v r (hyperbolic), and e −v/r cosh u r (parabolic). Therefore, 1. Extrinsic catenaries of elliptic type are critical points of 2. Extrinsic catenaries of hyperbolic type are critical points of 3. Extrinsic catenaries of parabolic type are critical points of The assumption u > 0 means that γ lies on one side of the plane P 2 .
The functionals W E , W H , and W P are all of the form f (u, v) γ dt. To compute the corresponding Euler-Lagrange equations, we first need the expression of the geodesic curvature κ in H 2 (r) with respect to coordinates (5).

Lemma 2 Consider the functional
where f is some smooth, positive function.
where κ is the geodesic curvature of γ in H 2 (r). If, in addition, γ is regular, then γ is a solution of the Euler-Lagrange equations if, and only if, (14) Proof The Euler-Lagrange equations associated with the Lagrangian L = f γ are These equations are Equivalently, we havė After some manipulations, both equations can be expressed as The identities (13) are now obtained using the above two equations and the expression of κ given in Eq. (11).
Finally, we are in a position to characterize the extrinsic catenaries in the hyperbolic plane.
, v(t)) be a regular curve in H 2 (r). Then, γ is an extrinsic catenary if, and only if, its curvature κ in H 2 (r) satisfies: 2. Hyperbolic type: 3. Parabolic type: Proof Just apply Lemma 2 to the functionals in Eqs. (8), (9), and (10). In other words, choose f in Eq. (14) as sinh u r , cosh u r cosh v r , and e − v r cosh u r , respectively.

Remark 1
In the variational formulation of the extrinsic catenary problem, Section 1, we implicitly assume that the chain's length is prescribed. Thus, there is a constraint b a γ(t) dt = c. Consequently, the quantity dist(γ(t), P 2 ) in (3) should be replaced by dist(γ(t), P 2 ) + λ, where λ is a Lagrange multiplier. In the functional (12), adding a Lagrange multiplier implies that the function f should be replaced by f + λ. The corresponding Euler-Lagrange equations coincide with those of Theorem 3 after a translation in E 4 1 of the plane P 2 .

Characterization of extrinsic catenaries
In this section, we provide a coordinate-free characterization of extrinsic catenaries with the help of Killing vector fields of E 4 1 . The motivation comes from the Euclidean catenary y(x) = 1 a cosh(ax + b) in the (x, y)-plane. This curve is the solution to the hanging chain problem in the Euclidean plane. In this setting, the reference line is the horizontal line ℓ of equation y = 0, and the corresponding Euler-Lagrange equation is The left-hand side is just the curvature κ of the curve y = y(x). The right-hand side is the Euclidean product n · e y , where n = (−y ′ , 1)/ 1 + y ′2 is the unit normal of y = y(x) and −e y = (0, −1) is the direction of gravity in Euclidean plane. Thus, Eq. (19) can be expressed as κ = n · e y dist(γ, ℓ) .
Observe that the direction of gravity −e y also has a geometric interpretation. It gives the direction of the geodesics orthogonal to ℓ used to compute dist(γ, ℓ). We can provide a similar result for extrinsic catenaries in hyperbolic space. However, since the potential energy (3) measures the distance in E 4 1 from the plane P 2 , it is natural to expect that the right-hand side of Eq. (20) will incorporate the extrinsic nature of (3). Such a characterization of hyperbolic extrinsic geodesics will be obtained as a corollary of the more general result where n is the unit normal of γ and ∇f is the gradient vector field of f .
Proof The tangent vector of γ isγ =u∂u +v∂v. Thus, the unit normal n is where {∂u, ∂v} is the coordinate basis determined by the parametrization ψ. The expression of the gradient of f with respect to this basis is It follows from Eqs. (22) and (23) that This equation, together with Eq. (14), finally proves Eq. (21).
Finally, we are in a position to provide the following characterization for extrinsic catenaries.
is an extrinsic catenary if, and only if, its curvature κ in H 2 (r) satisfies where X is the Killing field in E 4 1 of the geodesics in E 4 1 connecting γ to a point of P 2 .
Proof If X is a vector field in E 4 1 , its tangent part X ⊤ in H 2 (r) is given by where ψ is the parametrization (5) of H 2 (r). Consider now the three types of Killing vector fields that determine the distance between a point of H 2 (r) and a subspace P 2 depending on the causal character of P 2 . Denote by {ex, ey, ez} the canonical basis of E 3 1 .
1. If P 2 = [e x , e y ], the Killing vector field is X = e z and, from (25), we have 2. If P 2 = [e y , e z ], the Killing vector field is Y = −e x and, from (25), we have 3. If P 2 = [e y + e x , e z ], the Killing vector field Z = − 1 r √ 2 (e x + e y ) and, from (25), we have Let us compute the gradient ∇f in H 2 (r) of the distances to each of the above planes P 2 .
Finally, applying Theorem 4 gives the desired result. Indeed,

Surfaces of revolution in hyperbolic space
This section describes the three types of surfaces of revolution in H 3 (r). To define them, we first exploit the fact that all isometries of H 3 (r) are induced by orthogonal transformations of E 4 1 . Later, after introducing a convenient coordinate system for each surface type, we compute their mean curvature.
Let P k denote a k-dimensional subspace of E 4 1 and O(P k ) be the set of orthogonal transformations of E 4 1 with positive determinant and that leave P k pointwise fixed. Following do Carmo-Dajczer [1], we have the following classification for surfaces of revolution in H 3 (r). Definition 2 Consider two subspaces P 2 , P 3 ⊂ E 4 1 such that P 2 ⊂ P 3 and P 3 ∩ H 3 (r) = ∅. Let γ : I → H 2 (r) = P 3 ∩ H 3 (r) be a regular curve that does not intersect P 2 . The orbits of γ under the action of O(P 2 ) is called a surface of revolution in H 3 (r) generated by γ and rotated around P 2 . In addition, the surface of revolution is said to be of 1. elliptic type, if P 2 is Lorentzian, 2. hyperbolic type, if P 2 is Riemannian, and 3. parabolic type, if P 2 is degenerate.
Remark 2 In the hyperboloid model, the orbits of the action of a revolution of elliptic, hyperbolic, and parabolic types are ellipses, hyperbolas, and parabolas, respectively. In [1], surfaces of elliptic type are said to be spherical. However, the orbits are elliptic unless P 2 is orthogonal to the x-axis (i.e., the canonical timelike direction). Finally, note that from an intrinsic viewpoint, we can alternatively describe the orbits as circles, hypercycles, and horocycles.
Proposition 6 Let Sγ be a surface of revolution in H 3 (r) ⊂ E 4 1 with generating curve γ(t) = (x(t), y(t), z(t), 0). Let H be the mean curvature of Sγ. We have, 2. If S γ is of hyperbolic type, then

If S γ is of parabolic type, then
Proof First, note we may employ the ternary product × 1 of E 4 1 to define a vector product × in H 3 (r). Indeed, if X, Y ∈ TpH 3 (r), we have where p = (x, y, z, w), X = (x 1 , y 1 , z 1 , w 1 ), and Y = (x 2 , y 2 , z 2 , w 2 ). Here, the vectors of the orthonormal basis {ex, ey, ez, ew} are placed in the last line to guarantee that ex × 1 ey × 1 ez = ew.
Consider a parametrization Φ = Φ(u 1 , u 2 ) of a surface of H 3 (r). Recall that the coefficients (g ij ) of the first fundamental form are g ij = ∂Φ ∂u i , ∂Φ ∂u j . Fix the unit normal The coefficients (h ij ) of the second fundamental form are Then, the mean curvature H of Φ(u 1 , u 2 ) is given by H = g 22 h 11 − 2g 12 h 12 + g 11 h 22 2(g 11 g 22 − g 2 12 ) .
We now compute (32) in each of the three types of surfaces of revolution.
The unit normal ξ of S P is given by Let us compute the second fundamental form's coefficients h ij . The second derivatives of S P are Thus, we have h 12 = 0 and Substitution of h ij and g ij in Eq. (32) gives (31).

Minimal Surfaces of Revolution
This section proves the first of our main results, Theorem 9 below, concerning the characterization of minimal surfaces of revolution in hyperbolic space. To do that, we use Prop. 6 to express the mean curvature in terms of the curvature κ of the generating curve, Prop. 8. Then, we compare the resulting expression with the curvatures of the extrinsic catenaries that appear in Theorem 3. First, we need an expression for the curvature of a generic curve in H 2 (r).

Proposition 8 Let
Sγ be a surface of revolution in H 3 (r) ⊂ E 4 1 with generating curve γ(t) = (x(t), y(t), z(t), 0) ∈ H 2 (r) ⊂ H 3 (r). Then, Sγ is minimal if, and only if, the curvature of its generating curve satisfies 1. if S γ is of elliptic type, then As done for hyperbolic catenaries, we first introduce a coordinate system adapted to the problem. In the hyperboloid model, horocycles correspond to the curves given as the intersection of H 2 (r) with lightlike planes of E 3 1 not passing through the origin [8]. (Thus, horocycles are implicitly written as Euclidean parabolas.) Alternatively, we may describe horocycles as the orbits of the oneparameter subgroup of parabolic rotations [1], i.e., rotations induced in H 2 (r) by the orthogonal transformations of E 3 1 that leave a lightlike plane P 2 pointwise fixed. As a concrete example, consider the plane P 2 = span{e x + e y , e z }, where {e x , e y , e z } is the canonical basis given by the unit velocity vectors associated with the Cartesian coordinates (x, y, z). The orthogonal transformations of E 3 1 that leaves P 2 pointwise fixed correspond to a lightlike rotation L θ with axis (1, 1, 0) and whose matrix is given by [6] In short, the metric of H 2 (r) in the coordinates system φ(u, v) is This coordinate system is similar to the semi-geodesic coordinate system. From now on, we shall refer to φ as the horo-geodesic parametrization. Note that the coordinate curves v → φ(u, v) are geodesics: On the other hand, the coordinate curves u → φ(u, v) are horocycles.
By construction, the coordinate horocycles are orthogonal to the geodesic ℓ(v) = φ(0, v). Thus, the horocycle distance can be computed as Note that d h is nothing but the height of φ with respect to the xy-plane. Therefore, we can provide an intrinsic characterization of extrinsic catenary of spherical type Theorem 10 Every hyperbolic horocatenary in H 2 (r) is an extrinsic catenary of the elliptic type. Consequently, the generating curves of minimal surfaces of revolution in H 3 (r) of the elliptic type are horocatenaries.

Remark 3
The horo-geodesic coordinates are analogous to the semi-geodesic coordinates after we exchange u and v. Thus, the Christoffel symbols are

Concluding remarks
We introduced the concept of extrinsic catenaries in the hyperbolic plane, providing novel insights into the variational formulation of curves in non-Euclidean ambient manifolds. Utilizing the hyperboloid model, we defined extrinsic catenaries as critical points of the gravitational potential functional calculated from the extrinsic distance to a fixed reference plane in the ambient Lorentzian space. We delved into the characterization of extrinsic catenaries in terms of their curvature and as solutions to a prescribed curvature problem involving specific vector fields, and we showed that the generating curve of any minimal surface of revolution in the hyperbolic space is an extrinsic catenary. We note that catenaries of the elliptic type obey a conservation law. Indeed, the Lagrangian L associated with catenaries of the elliptic type, see Eq. (8), does not depend on the v-coordinate, which implies ∂L/∂v is a first integral 3 . Geometrically, this first integral is associated with a Clairaut-like relation, thus providing information about the angle ϑ between an extrinsic catenary of the elliptic type and the v-coordinate curves of the parametrization (5): following Theorem 4.1 of Ref. [7], we can prove that In this context, the v-coordinates curves play the role of parallels if we see the hyperbolic plane as an invariant surface generated by rotations of the hyperbolic type. (We obtain (5) by applying H u/r , Eq. (27), to the geodesic ℓ(v) = r(cosh(v/r), sinh(v/r), 0) followed by the exchange (x, y, z, w) → (x, z, w, y).) Extrinsic catenaries of the hyperbolic and parabolic types have no obvious first integrals. Thus, we may ask whether similar conservation also holds for them or whether the absence of circular coordinates in the Lagrangian is just an artifact of a bad choice of coordinates for the hyperbolic plane.
In many aspects, horocycles behave as extrinsically flat curves in hyperbolic geometry, and replacing geodesics with them often leads to problems with good properties. In this work, we established that extrinsic catenaries of the elliptic type could be intrinsically characterized by replacing the extrinsic distance with the intrinsic length of horocycles orthogonal to a reference geodesic, in analogy with (intrinsic) catenaries [4,7]. The notion of extrinsic catenaries only makes sense when working with the hyperboloid model. Thus, we may also ask whether providing intrinsic characterizations for extrinsic catenaries of the hyperbolic and parabolic types is possible. Consequently, we ask whether it is possible to characterize the generating curve of any hyperbolic minimal surface of revolution without resorting to a specific model for H 3 . A similar question can also be posed concerning minimal surfaces of revolution in S 3 . been partially supported by MINECO/MICINN/FEDER grant no. PID2020-117868GB-I00, and by the "María de Maeztu" Excellence Unit IMAG, reference CEX2020-001105-M, funded by MCINN/AEI/10.13039/501100011033/ CEX2020-001105-M.