# Virtio-FPGA: a virtualization solution for SoC-attached FPGAs
††thanks:  This work has received funding from the EU Horizon 2020 Programme
under grant agreement No 957269 (EVEREST) [15].

Anna Panagopoulou  Virtual Open Systems SAS
Grenoble, France
<EMAIL_ADDRESS>Michele Paolino  Virtual Open Systems SAS
Grenoble, France
<EMAIL_ADDRESS>Daniel Raho  Virtual Open Systems SAS
Grenoble, France
<EMAIL_ADDRESS>
###### Abstract

Recently, FPGA accelerators have risen in popularity as they present a
suitable way of satisfying the high-computation and low-power demands of real
time applications. The modern electric transportation systems (such as
aircraft, road vehicles) can greatly profit from embedded FPGAs, which
incorporate both high-performance and flexibility features into a single SoC.
At the same time, the virtualization of FPGA resources aims to reinforce these
systems with strong isolation, consolidation and security. In this paper, we
present a novel virtualization framework aimed for SoC-attached FPGA devices,
in a Linux and QEMU/KVM setup. We use Virtio as a means to enable the
configuration of FPGA resources from guest systems in an efficient way. Also,
we employ the Linux VFIO and Device Tree Overlays technologies in order to
render the FPGA resources dynamically accessible to guest systems. The ability
to dynamically configure and utilize the FPGA resources from a virtualization
environment is described in details. The evaluation procedure of the solution
is presented and the virtualization overhead is benchmarked as minimal (around
10%) when accessing the FPGA devices from guest systems.

###### Index Terms:

FPGAs, virtio, FPGA Manager, VFIO

##  I Introduction

In the recent years, SoC-attached FPGAs has emerged as an architecture of
broad interest in the electric transportation systems[1]. It is a formation
that comes with both performance and flexibility gains, as it combines
general-purpose and highly-specialized computation in a single board [2]. At
the same time, virtualizing the FPGA resources of SoC-attached FPGAs is the
means to build standardized interfaces to access the reconfigurable,
heterogeneous hardware.

The virtualization of FPGA resources is a non-trivial goal, due to the unique
nature of FPGAs; the hardware is modifiable and introduces great challenges in
exposing a unified layer for the virtual resources [3]. The objective of the
proposed solution is to address some of the FPGA virtualization challenges of
ARM SoCs, on the grounds of well-known and accepted APIs and environments. In
this paper, we are going to present Virtio-FPGA; an initial prototype built
for Linux and QEMU/KVM, that aims to provide transparent and dynamic accesses
of the FPGA hardware to guest systems.

The rest of this paper is organized as follows: Section II covers the
background knowledge about the involved technologies. Then, Section III
describes the technical details of the Virtio-FPGA solution, while Section IV
presents the evaluation environment and the obtained results. Section V
outlines related works and Section VI concludes the paper, with discussion on
the future directions.

##  II Background

In this section we will present the basic concepts and ideas of the Linux
technologies involved in the Virtio-FPGA solution. We will first provide
background knowledge on the Virtio standard, then we will present the Linux
FPGA Manager component of the Linux kernel and finally we will provide
background knowledge on the VFIO pass-through for ARM in QEMU and the Device
Tree Overlays technologies.

###  II-A Virtio

Virtio [4] is an I/O virtualization standard, that defines an abstraction
layer to enable communication between guests and host in a paravirtualized
fashion.

The Virtio standard is commonly implemented in the QEMU hypervisor, for
emulating paravirtualized devices. QEMU exposes a virtual device to the guest,
which can be driven by a front-end driver residing in the guest system. At the
other end, the back-end driver is part of QEMU and is responsible for
emulating the device, by interfacing with the host system.

In the base of the Virtio communication, there exists the virtqueue
abstraction. In simple terms, the virtqueues represent memory regions of the
guest system, that are shared with the host. For what concerns the exchange of
data, the guest is responsible to allocate the memory regions as buffers that
will be consumed by the host, and then configure the virtqueues to point to
those buffers. So, the buffers become shared between the two parties and
communication is made possible.

Performance-wise, Virtio is a high-efficient solution to accomplish guest-host
communication, precisely because it is depending on a shared memory model.

###  II-B The Linux FPGA Manager

The Linux FPGA Manager API [5] is a set of kernel functions with the aim of
programming during the runtime an FPGA device with a hardware description
image. The API is implemented as a driver that conceals all the manufacturer-
specific FPGA programming details from the upper software layers. In
particular, the idea is that the manufacturer-related procedures are part of
underlying drivers, that register themselves to the generic FPGA Manager API.
Each of these drivers has to implement a set of operations, so that the
interface with the generic API is properly setup. To mention the most basic of
them, a write_init operation has the role of preparing the FPGA to receive
data, a write operation writes a buffer with data to the FPGA and a
write_complete is triggered after all the image data have been written.

The FPGA Manager supports programming FPGAs using binary data in the form of
either a scatter gather list, a single contiguous buffer, or a firmware file.
All the parsing and interpretation of these vendor exclusive binary data, is
to be performed by the underlying drivers. The relationship between the
generic Linux FPGA Manager core and the underlying drivers is illustrated in
figure 1.

Figure 1: The Linux FPGA Manager as an interface to hide manufacturer-specific
details of the FPGA programming operation.

###  II-C QEMU VFIO pass-through for ARM

VFIO pass-through [6] is a method that allows devices sitting behind an IOMMU
controller to become directly available to guest systems. In a Linux
environment, QEMU is capable of supporting VFIO pass-through for both PCI and
platform (e.g. part of an SoC) devices.

VFIO is a Linux API [7] which permits virtual memory addresses to be used for
managing physical devices. In its base, VFIO interfaces with the IOMMU
subsystem in order to setup specific memory translations, from virtual to
physical memory regions. Consequently, user-space components gain the ability
to program and utilize the exposed device, after proper configuration is done.
Thus, QEMU, being a user-space software layer, becomes capable of driving the
exposed device to make it accessible to guest systems.

To further detail the role of QEMU, it uses the VFIO API in a way that for a
given device, specific host virtual addresses are mapped to device-accessible
physical addresses. These host virtual addresses are basically guest physical
addresses; therefore, the guest drivers can seemingly program the device given
guest physical addresses, although these are transparently mapped to the
actual physical memory regions.

For embedded ARM FPGAs, the solution to achieve direct pass-through of FPGA
devices to guest systems is by utilizing QEMU’s VFIO platform driver. This
QEMU solution can target devices that are bound to Linux VFIO platform driver
[8], and thus, are available for user-space driving. In order for this
exposure to be done safely, the candidate device should be part of the IOMMU
subsystem, namely the SMMU on ARM SoCs. Figure 2 demonstrates the components
of the VFIO pass-through technology, as provided by the QEMU hypervisor, for
an example GPIO device present on the AXI subsystem.

Figure 2: QEMU VFIO-passthrough technology for an example GPIO AXI device.

###  II-D Device Tree overlays

The device tree overlays are blobs that can modify the current, live device
tree of the Linux kernel. The dynamic application of overlays can be enabled
in a Linux kernel through the CONFIG_OF compilation option. Also, a patch to
enable CONFIGFS support for applying the overlays from a user-space context is
required in the kernel.

In general, the re-configuration of FPGAs during runtime, can greatly benefit
from device tree overlays support. Given a hardware configuration image and
also a corresponding device tree blob description, not only the FPGAs can be
re-configured, but also the configured devices can be made visible to the
system on-the-go.

##  III Virtio-FPGA

In this section, we introduce the technical aspects of the Virto-FPGA solution
for SoC-attached FPGAs. Virto-FPGA is an ARM virtualization solution based on
the QEMU/KVM hypervisor, that unveils the configuration and utilization of
FPGA resources to Linux virtual machines. It is an ensemble of software
components which provide a dynamic way for virtual machines to program, then
recognize and interface with the devices that constitute the re-configurable
FPGA hardware.

In the Virtio-FPGA solution, the virtualization challenge is addressed in a
two-fold way: the FPGAs configuration and the FPGA resources utilization from
virtual machines. With this design approach, we aim at reducing the complexity
of the problem by exploiting existing Linux solutions to the greatest extent.

Specifically, the Virtio-FPGA solution is based on the Virtio standard for the
virtual machines to interface with the Linux FPGA Manager component of the
host system. This way, virtual machines become able to dynamically dispatch
configuration requests to FPGA devices. Then, using proper orchestration of
VFIO and Device Tree Overlays technologies, the Virtio-FPGA solution sets-up
an environment for the virtual machines to utilize the FPGA accelerators.

Component-wise, the Virtio-FPGA solution consists of the following elements:

  * •

A QEMU virtual device for FPGA programming

  * •

A guest kernel driver to interface with the QEMU device (Virtio FPGA Manager)

  * •

An extension to QEMU VFIO for ARM during Device Tree setup

  * •

Proper Device Tree Overlay blob files

In our prototype, the solution targets the Xilinx Zynq UltraScale+ MPSoC for
the testing and evaluation purposes. The Virtio-FPGA components are going to
be detailed on the following subsections.

###  III-A FPGAs configuration

In its internals, Virtio-FPGA implements the Virtio specification, to expose a
QEMU virtual device for FPGA programming to the guests. The virtual device has
the role of fulfilling FPGA programming requests that arrive from guest
systems. By interfacing with the virtual device, guests are able to trigger
FPGA programming requests towards the FPGA hardware.

Fundamentally, the virtual device realizes the programming operation by making
use of the Linux FPGA Manager API of the host kernel. The interface between
the virtual device and the Linux FPGA Manager core, resides on the sysfs
firmware entry that is available on the Xilinx Linux FPGA Manager
implementation. Through this sysfs entry, it is possible for user-space
processes to trigger FPGA programming requests by providing the bitstream
configuration file name. Also, the requirement is that the bitstream file
should be present under the /lib/firmware directory, in the host system. The
interface between the QEMU Virtio-FPGA device and the Linux FPGA Manager is
better depicted on figure 3.

Figure 3: The interface between QEMU Virtio-FPGA device and the Linux FPGA
Manager core.

So, given an amount of bitstream files present under the host’s /lib/firmware
directory, guests can choose any of the available configurations to program
FPGAs with. In a possible future Virtio-FPGA extension, the binary data of the
image might be also communicated between the guest and the host system, using
scatter-gather lists.

For what concerns the guest interface, a corresponding Virtio-FPGA driver is
loaded in the guest kernel, responsible to manage the paravirtualized QEMU
device. The main design objectives of the guest kernel driver are:

  * •

To provide transparency, e.g. guests to function as if they were a native
system

  * •

To expose an interface for user-space guest requests

  * •

To communicate with the host introducing a minimum overhead

The Virtio-FPGA solution aims to provide transparency in the guest system. In
fact, the guest FPGA programming interface is built on top of a new, Virtio
FPGA Manager that implements the FPGA Manager set of operations. Kernel
solutions in the guest can take advantage of the Virtio FPGA Manager and
interact with it, to program the FPGAs given a hardware configuration image.

Similarly to the host-side FPGA programming interface, the guest kernel driver
introduces a sysfs entry for triggering the programming requests from a user-
space context. So, as soon as the driver is inserted into the guest system,
the procedure to carry-out an FPGA programming request is as follows:

echo filename >
/sys/class/virtio_fpga/fpga0/firmware

For what concerns the filename, as already mentioned, it should be present
under the host /lib/firmware directory. Then, once the programming operation
terminates, a kernel message is generated to report the status of the FPGA
device (whether the operation was successful or not).

At its base, the Virtio FPGA Manager communicates with the host the necessary
information for configuring the FPGAs. With the Virtio protocol, the guest-
host communication takes place through shared memory regions. In our scenario,
the shared memory regions are two virtqueues; one containing the image
filename residing on the host system, and one ready to accept the status of
the programming operation, once it is over. More in details and with focus on
the introduced overhead for FPGAs configuration, a context-switch to kernel
space and a memory copy are needed to prepare the data for host consumption.
Thus, the configuration overhead is insignificant and for this reason, it is
not being considered during the evaluation procedure.

Figure 4 provides an overview of the components involved, for a guest system
to interact with the QEMU Virtio-FPGA device, in order to trigger FPGA
programming requests.

Figure 4: The internals of the guest interface with the QEMU Virtio-FPGA
device.

###  III-B FPGA resources utilization

As soon as the FPGA is configured, the guest is able to utilize the resources
by making use of QEMU VFIO-passthrough and Device Tree Overlays technologies.

For the Virtio-FPGA solution to function, an extension on QEMU device tree
setup was required to open-up the range of the supported VFIO platform
devices. Specifically, a generic device tree node configuration was introduced
providing basic properties (e.g. interrupts, memory areas) to expose the
device in the guest system. Additionally, the VFIO platform driver in QEMU
does not support hotplugging of devices, so all candidate FPGA devices
introduced through VFIO during QEMU bootup are initially disabled, and remain
at this state until an FPGA configuration request takes place.

For our virtualization purposes, Device Tree Overlays are the key to provide a
dynamic solution to guest systems with no hotplugging support. This technology
is used in Virtio-FPGA in order to dynamically expose FPGA devices to guest
systems after their hardware configuration is carried-out.

More in details, the dynamic exposure of devices to guests happens as follows:
First, the guest Linux kernel is compiled with Device Tree Overlays support
and user-space support for overlays application. Then, QEMU boots, passing-
through some candidate FPGA devices with most basic properties to the guest
system. These FPGA devices are not yet part of the FPGA hardware, but they
might become upon some future configuration. The requirement for this step, is
that the configurations applied should be respected throughout the lifetime of
the host system. To explain more, the basic properties (e.g. physical memory
areas) for the FPGA devices that are not yet physically part of the hardware,
are described early in the host software and become bound to the VFIO and
IOMMU subsystems. This information must be strictly followed by all the
firmware images that might be used for FPGAs configuration, upon a guest
request.

As a next step, the guest can trigger FPGA programming requests to the Virtio-
FPGA and then apply a device tree overlay in the guest system to enable the
corresponding devices. During this procedure, any other device-specific
information can be provided in the overlay (e.g. clocks, compatible strings
etc.).

The use of device tree overlays is necessary in systems where information
about non-present hardware needs to be provided to guests in advance. In the
future, this workaround might be obsolete, in case that hot plug support is
added to platform devices in QEMU.

##  IV Benchmarks

In this section, we present the evaluation procedure and the experimental
results of the Virtio-FPGA prototype. The main objective of the evaluation is
to demonstrate the virtualization overhead of the presented solution. As
already mentioned, the virtualization overhead for FPGAs configuration is
minimal, and thus it is omitted from the evaluations. Thus, we will measure
the virtualization overhead introduced during FPGAs utilization in the scope
of the Virtio-FPGA solution.

To do this, we measured and compared the performance of interfacing with the
FPGA hardware from a guest system and the host system. All the measurements
were performed on a Xilinx Zynq Ultrascale+ MPSOC (ZCU102), which features a
quad-core ARM Cortex-A53, with 4GB DDR4 memory attached to the PS (Processing
System) and 512MB DDR4 memory attached to the PL (Programmable Logic). The
host system is built using the Petalinux tools v2021.2 with Xilinx Linux
version 5.10.0, whereas the guest system is a buildroot-based arm64 QEMU virt
model with Linux version 6.0.

To measure the virtualization overhead of FPGAs utilization we setup a
bitstream file that instructs the FPGAs to feature a CDMA (Central Direct
Memory Access) device, capable of carrying-out DMA transfers between areas of
the on-chip DDR memory. Then, we applied this configuration in the Virtio-FPGA
solution, in a way that the CDMA device of the PL is made available to the
guest system.

Given this setup, we ran a DMA test benchmark from the guest, and obtained
performance results for the DMA transfers. Last, we compared the results with
a scenario where the DMA test benchmark is executed directly in the host
system.

###  IV-A Hardware benchmark application

To further detail the hardware configuration of the FPGAs, we generated the
bitstream file using Vivado 2021.2. In order to prepare the configured CDMA IP
for being passed through with QEMU VFIO, we had to ensure from the design, I/O
Coherency between the PL and the PS. For this reason, we used the HPC0 port of
the AXI, to connect the CDMA IP to the PS. This is because with the HPC0 port,
it is possible for coherent masters in the PL to snoop the CPU caches, given
also that the AxCACHE signals of the port are properly configured. As an
additional requirement, the CCI (Cache Coherent Interconnect) should be
configured to enable snooping, and we addressed this given that we used the
ATF (Arm Trusted Firmware) binary on the host, which configures the CCI by
default. Also, we had to program the LPD_APU register of the LPD_SLCR module,
to enable the broadcasting of CPU caches to the PL masters. We did this, by
setting the register from the boot-up script of the host system. Last but not
least, we setup the AxPROT signals of the HPC0 port, so that accesses to the
CDMA IP from EL1 (e.g. Linux accesses) are allowed. In figure 5 we present the
final design used for the hardware configuration of the FPGAs in this
evaluation.

Figure 5: The design for the evaluation part as generated by Vivado 2021.2.

###  IV-B Software benchmark application

The DMA test benchmark is based on the DMA test client, which is available in
the upstream Linux kernels and can be enabled with CONFIG_DMATEST option
during kernel compilation. For our evaluation purposes, a few tweaks were
applied on the default DMA test client, so that we obtain the performance
numbers. We performed DMA transfers of various memory sizes, ranging from 1
Page (meaning, 4KB data) to 256 Pages transfer size. Also, we carried-out both
interrupt-based and polled-based transfers. As for the evaluation numbers, we
measured:

  1. 1.

The preparation time: Time (in usec) needed to map the memory ranges to the
DMA address space, to be prepared for the DMA transfer.

  2. 2.

The transfer time: Time (in usec) needed from when we program the CDMA device
to start the transfer, till it ends. We evaluated both:

    1. (a)

The interrupt transfer time

    2. (b)

The polled transfer time

For each transfer size, we ran 500 iterations, and extracted the average and
standard deviation values of the preparation and transfer times. For the
preparation times, the results that we obtained are presented in table I.
Also, figure 6 graphically demonstrates the average value for the transfer
time, along with the standard deviation value, in respect to the transfer
size, for the two systems and the two transfer methods.

TABLE I: Preparation time (usec) | Transfer size (4KB Pages)
---|---
| 1 | 16 | 32 | 64 | 128 | 256
Host avg* | 6.14 | 62.5 | 122.3 | 242.37 | 474.334 | 748.6
Host std | 1.79 | 1.53 | 1.54 | 2.42 | 6.11 | 7.74
Guest avg | 4.36 | 42.34 | 82.79 | 162.8 | 292.18 | 411.2
Guest std | 7.65 | 8.84 | 7.98 | 8.56 | 14.59 | 19.66
aAll results are in usec |  |  |

Figure 6: The average transfer time and the standard deviation in respect to
the transfer size, for host and guest setups with interrupt and polled
methods.

###  IV-C Results

From the results, at first we see that the preparation time is always worse in
the native versus the virtualization setup. This behavior can be explained
when looking into how IOMMU translations are generated. In the virtualization
case, the guest system uses a direct model for the DMA mappings, so that it
seemingly programs the DMA using guest physical addresses. All the necessary
IOMMU translations have been already setup for the CDMA IP, thanks to the VFIO
platform driver of QEMU. From the other hand, in the native case the IOMMU
translations are created upon DMA mapping events, and thus no direct model is
being utilized. So, for each DMA mapping event, the native case has to pay the
additional overhead for seting up the translations for the IOMMU subsystem.

With respect to the results obtained for the transfer times, for both guest
and host setups we observe that for the same transfer size, the polled
transfer time is always better than the interrupt transfer time. This is
normal, considering that the transfer sizes are relatively small, and thus it
is not worthy to put the execution thread to sleep and wait for the interrupt
to occur. Related to the standard deviation values, the guest metrics usually
tend to be more unpredictable than the host, as the guest environment has to
deal with its threads being scheduled-out from time to time. In any case, the
highest standard deviation value is noticed for a guest transfer of 1MB data,
with a value of 13.28 usecs.

Considering the virtualization overhead occurring during the transfer, it is
greatly influenced by the transfer size of the DMA operation. In the polled
case, the overhead is decreasing from a value of 152%, for 4KB data, down to a
value of 5%, for 1MB data. In the interrupt case, the higher value for the
overhead is 112% for 4KB data, whereas the lower value is 12.6% for 1MB data.
In all, we conclude that the larger the amount of data to be transferred, the
more the virtualization overhead is hidden. For reasonable transfer sizes
(e.g. 256KB - 1MB) the average overhead for the interrupt case is 15%, whereas
for the polled case is 7%, which is normally lower given the cost of
interrupts virtualization.

##  V Related Work

In the research community, there exists a wide range of academic studies
regarding FPGA resources virtualization. The existing literature in the field
varies, from PCIe and network attached, to SoC-attached FPGAs.

For PCI-e and network-attached FPGAs, there is a plethora of works that aim to
provide FPGAs virtualization. Epigrammatically, the FPGAVirt [9] solution
exploits the Virtio-Vsock layer to enable communication between guest systems
and network-attached FPGAs. Another interesting solution is pvFPGA [10], which
is based on the Xen hypervisor to provide a paravirtualized environment for
PCI-e attached FPGAs. Also, with focus on the scalable management of
homogeneity of the FPGA hardware in the cloud, RC3E [11] presents a hypervisor
that provides an interface for guest systems to interact with virtual FPGA
components.

In the embedded domain, there are few works that provide virtualization of the
FPGA resources with microkernel hypervisors. The work in [12] ports the
CODEZERO hypervisor to a platform with SoC-attached FPGA, involving necessary
extensions to support FPGA hardware tasks management and scheduling. Also, of
much interest is the work in [13], which ports the NOVA hypervisor to an
embedded platform with attached FPGA, in a way that it can delegate the
management of DPR (Dynamic Programmable Regions) resources. In the same
concept, the work in [14] builds a custom hypervisor called Ker-one, which
leverages the FPGA DPR regions as virtual devices to guest systems. Then, the
hypervisor is able to identify and dynamically handle the programming requests
that target the DPR regions. These works deal indeed with many challenges that
FPGAs virtualization present. However, the solutions presented are much
platform-specific, since there is no unified API to carry-out the programming
of FPGA resources for multiple boards. In our solution, the virtualization of
the Linux FPGA Manager creates that unified API which hides the manufacturer
details from the upper software layers.

##  VI Conclusion

In this work we presented Virtio-FPGA: a new virtualization solution for SoC-
attached FPGAs. Its main contribution is that it proves it is feasible to
dynamically re-configure and access FPGA resources, based entirely on Linux
interfaces and APIs. It is a solution that is flexible, in virtue of
interfacing with the generic Linux FPGA Manager API to virtualize the
configuration of FPGAs, and also efficient, as a consequence of utilizing
Virtio and VFIO technologies to eliminate the virtualization overhead. The
virtualization overhead is evaluated with an example CDMA FPGA device, being
no higher than 12% for at least 1MB DMA transfers. Future works include adding
support for partially reconfiguring the FPGA regions, as well as integrating a
monitor tool in the hypervisor to keep track of the hardware tasks.

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# A characterization of rotational minimal surfaces in the de Sitter space

Rafael López  Departamento de Geometría y Topología
Universidad de Granada. 18071 Granada, Spain<EMAIL_ADDRESS>
###### Abstract.

The generating curves of rotational minimal surfaces in the de Sitter space
$\mathbb{S}_{1}^{3}$ are characterized as solutions of a variational problem.
It is proved that these curves are the critical points of the center of mass
among all curves of $\mathbb{S}_{1}^{2}$ with prescribed endpoints and fixed
length. This extends the known properties of the catenary and the catenoid in
the Euclidean setting.

###### Key words and phrases:

de Sitter space, spacelike surface, timelike surface, minimal surface,
catenary

###### 1991 Mathematics Subject Classification:

53C40, 53C42

##  1\. Introduction and motivation

A non-degenerate surface in $3$-dimensional de Sitter space
$\mathbb{S}_{1}^{3}$ is called a minimal surface if the mean curvature $H$
vanishes on the surface. Although the equation $H=0$ is equivalent that the
trace of the second fundamental form vanishes on the surface, there is no a
variational interpretation of these surfaces such as it occurs in Euclidean
space. For spacelike surfaces of $\mathbb{S}_{1}^{3}$, the induced metric is
Riemannian and it makes sense to define the area of a surface. In this case, a
surface of $\mathbb{S}_{1}^{3}$ with zero constant mean curvature is a local
maximizer of the area functional. If the surface is timelike, the induced
metric has index one and there is not a notion of the area of a timelike
surface. Despite these differences with the Euclidean case, and in order to
simplify the statements our results, the terminology of minimal surfaces to
name a non-degenerate surface of $\mathbb{S}_{1}^{3}$ with zero constant mean
curvature is kept in this article.

An interesting class of surfaces of $\mathbb{S}_{1}^{3}$ are the surfaces of
revolution, also called, rotational surfaces. The family of surfaces of
revolution in $\mathbb{S}_{1}^{3}$ is richer than the Euclidean setting
because the rotational axis can be of three different causal character. If, in
addition, we investigate which of these surfaces are minimal, we need to
distinguish if the surface is spacelike and timelike. The property that the
surface is rotational implies that the equation $H=0$ is an ordinary
differential equation given in terms of the generating curve of the surface.
This makes that the classification of rotational minimal surfaces of
$\mathbb{S}_{1}^{3}$ can be obtained: [1, 5, 7, 9]; see also [2, 6, 10].
Recently, rotational surfaces in $\mathbb{S}_{1}^{3}$ with a Weingarten
relation between its curvatures have been investigated because the Weingarten
relation is described again in terms of ordinary differential equations [3,
8].

If the classification of rotational minimal surfaces of $\mathbb{S}_{1}^{3}$
is deduced by solving ordinary differential equations, there is not known a
special property of the generating curves. This contrasts to the Euclidean
situation which it is recalled giving a brief historical account. Euler proved
in 1744 that the catenoid is the only non-planar rotational minimal surface in
$\mathbb{R}^{3}$ being the catenary its generating curve. This curve is also
the solution of the question that asks by the shape of a chain hanging by its
own weight. This problem attracted the interest of many scientistics some
centuries above, beginning with Galileo. The answer, the catenary, was
obtained some decades before to the Euler’s result joining efforts of Hooke,
Leibniz, Huygens and Bernouilli, among others.

Coming back to the de Sitter space, and motivated by the property that the
generating curve of the catenoid is variationally characterized as the shape
of a hanging chain, the purpose of this paper is to answer the following

> Question: Can the generating curves of rotational minimal surfaces of
> $\mathbb{S}_{1}^{3}$ be characterized from a variational viewpoint?

In $\mathbb{R}^{2}$, the problem of a hanging chain is equivalent to find a
curve that minimizes the center of mass where the weight of the curve is
measured with respect to a straight-line of $\mathbb{R}^{2}$. The variational
problem is posed among all curves of $\mathbb{R}^{2}$ with the same endpoints
and the same length. Here the curve is viewed as an ideal incompressible chain
of constant density. In the $2$-dimensional de Sitter space
$\mathbb{S}_{1}^{2}$, the catenary problem asks what is the shape of a hanging
chain in $\mathbb{S}_{1}^{2}$. It is then necessary to precise what it is the
meaning of the ‘weight’ of a curve of $\mathbb{S}_{1}^{2}$ indicating what is
the reference to measure the center of mass. If we are able to solve this
problem, the solution curve will be the analog catenary in
$\mathbb{S}_{1}^{2}$. Now the natural way to proceed is to immerse
$\mathbb{S}_{1}^{2}$ in the $3$-dimensional de Sitter space
$\mathbb{S}_{1}^{3}$, and construct the surface of revolution whose generating
curve is the above catenary. It is natural to ask if the mean curvature of
this surface vanishes constantly on the surface.

A first problem that appears is that we need to distinguish if curves in
$\mathbb{S}_{1}^{2}$ and surfaces in $\mathbb{S}_{1}^{3}$ are spacelike or
timelike. A second problem is how to define the center of mass of a curve in
$\mathbb{S}_{1}^{2}$. Here we need to precise the model of the de Sitter
space. Let $\mathbb{L}^{3}$ be the Lorentz-Minkowski space, that is, the
vector space $\mathbb{R}^{3}$ equipped with the Lorentzian metric
$\langle,\rangle=(dx)^{2}+(dy)^{2}-(dz)^{2}$, where $(x,y,z)$ stand for the
canonical coordinates of $\mathbb{R}^{3}$. The $2$-dimensional de Sitter space
is defined as the set
$\mathbb{S}_{1}^{2}=\\{(x,y,z)\in\mathbb{R}^{3}:x^{2}+y^{2}+z^{2}=1\\}$
endowed with the induced metric from $\mathbb{L}^{3}$. The de Sitter space
$\mathbb{S}_{1}^{2}$ is a Lorentzian surface of constant curvature equal to
$1$.

In order to measure the center of mass of a curve of $\mathbb{S}_{1}^{2}$, we
fix a reference plane $\Pi$ of $\mathbb{L}^{3}$ and the center of mass is
measured using the distance to $\Pi$. Since in $\mathbb{L}^{3}$ there are
three types of planes depending on the causal character, the plane $\Pi$ can
be spacelike, timelike and degenerate. Each one of these planes separates
$\mathbb{S}_{1}^{2}$ in two domains and let us fix one of these domains, which
it is denoted by $(\mathbb{S}_{1}^{2})^{+}$. Let
$\gamma\colon[a,b]\to(\mathbb{S}_{1}^{2})^{+}$, $\gamma=\gamma(t)$, be a non-
degenerate curve. Assuming that the density is constantly $1$, the weight of
$\gamma$ is defined by $\int_{a}^{b}d(t)\,ds,$ where $d(t)$ is the distance
between $\gamma(t)$ and $\Pi$ and $ds$ is the arc-length of $\gamma$. Thus the
center of mass of $\gamma$ is

$\frac{\int_{a}^{b}d(t)\,ds}{\int_{a}^{b}\,ds}=\frac{\int_{a}^{b}d(t)|\gamma^{\prime}(t)|\,dt}{\int_{a}^{b}|\gamma^{\prime}(t)|\,dt},$

where
$|\gamma^{\prime}(t)|=\sqrt{|\langle\gamma^{\prime}(t),\gamma^{\prime}(t)\rangle|}$.
The variational problem posed in $\mathbb{S}_{1}^{2}$ consists to find the
curve that minimizes the center of mass in the class of all curves of
$\mathbb{S}_{1}^{2}$ with prescribed endpoints and fixed length. Introducing a
Lagrange multiplier due to the length constraint, the energy to minimize is

(1)
$\mathcal{E}_{\Pi}[\gamma]=\int_{a}^{b}(d(t)+\lambda)|\gamma^{\prime}(t)|\,dt.$

######  Definition 1.1.

A critical point of $\mathcal{E}_{\Pi}$ is called a catenary with respect to
the plane $\Pi$.

The following step in our investigation consists in constructing a surface of
revolution in $\mathbb{S}_{1}^{3}$ having a catenary as generating curve. For
this, we immersed $\mathbb{S}_{1}^{2}$ into the $3$-dimensional de Sitter
space $\mathbb{S}_{1}^{3}$ as follows. Let $\mathbb{L}^{4}$ be the
$4$-dimensional Lorentz-Minkowski space where the metric is
$(dx_{1})^{2}+(dx_{2})^{2}-(dx_{3})^{2}+(dx_{4})^{2}$ and
$(x_{1},x_{2},x_{3},x_{4})$ are Cartesian coordinates of $\mathbb{R}^{4}$. The
$3$-dimensional de Sitter space $\mathbb{S}_{1}^{3}$ is defined by

$\mathbb{S}_{1}^{3}=\\{p=(x_{1},x_{2},x_{3},x_{4})\in\mathbb{L}^{4}:\langle
p,p\rangle=1\\}$

endowed with the induced metric from $\mathbb{L}^{4}$. The space
$\mathbb{L}^{3}$ is immersed in $\mathbb{L}^{4}$ by $(x,y,z)\mapsto(x,y,z,0)$.
In particular, the $2$-dimensional de Sitter space $\mathbb{S}_{1}^{2}$ is
immersed in $\mathbb{S}_{1}^{3}$ as $\mathbb{S}_{1}^{2}\times\\{0\\}$. The
steps to follow are now clear:

  1. (1)

Let $\gamma$ be a catenary of $\mathbb{S}_{1}^{2}$ with respect to a plane
$\Pi$.

  2. (2)

Let $L$ be the geodesic obtained by intersecting $\mathbb{S}_{1}^{2}$ with
$\Pi$, $L=\mathbb{S}_{1}^{2}\cap\Pi$.

  3. (3)

Let $L\equiv L\times\\{0\\}\subset\mathbb{S}_{1}^{3}$. Consider $\mathcal{G}$
the one-parameter group of rotations of $\mathbb{S}_{1}^{3}$ that leave
pointwise fixed $L$.

  4. (4)

Rotate $\gamma\equiv\gamma\times\\{0\\}$ about $\mathcal{G}$, obtaining a
surface of revolution $S_{\gamma}$ of $\mathbb{S}_{1}^{3}$.

  5. (5)

Compute the mean curvature $H$ of $S_{\gamma}$ and investigate if $H$ is $0$.

As a consequence of all this work, the main result can be stated as follows.

######  Theorem 1.2.

Let $M$ be a non-degenerate surface of revolution in $\mathbb{S}_{1}^{3}$
whose rotation axis is $L=\mathbb{S}_{1}^{2}\cap\Pi$. Then $M$ is a minimal
surface if and only if its generating curve is a catenary with respect to
$\Pi$.

This theorem provides a variational characterization of the generating curves
of rotational minimal surfaces and it shows that our approach to catenaries in
the de Sitter space extends the classical notions of Euclidean catenary and
catenoid.

The proof of Theorem 1.2 is separated in three cases depending in the three
types of surfaces of revolution of $\mathbb{S}_{1}^{3}$. Within a more general
context, these surfaces are described in [4]. To be precise, fix a $2$-plane
$P^{2}\subset\mathbb{L}^{4}$ and consider the one-parameter group
$\mathcal{G}$ of isometries of $\mathbb{L}^{4}$ that leaves $P^{2}$ pointwise
fixed. Let $P^{3}$ be a $3$-dimensional subspace of $\mathbb{L}^{4}$ such that
$P^{2}\subset P^{3}$ and consider $\gamma$ a non-degenerate curve in
$\mathbb{S}_{1}^{3}\cap(P^{3}-P^{2})$. The surface $S_{\gamma}$ obtained by
acting $\mathcal{G}$ on $\gamma$ is called a surface of revolution of
$\mathbb{S}_{1}^{3}$ and $\gamma$ is called its generating curve. This surface
is said to be spherical (resp. hyperbolic, parabolic) if the induced metric on
$P$ from $\mathbb{L}^{4}$ is Riemannian (resp. Lorentzian, degenerate).
Explicit expressions of the groups $\mathcal{G}$ and of the parametrizations
of the rotational surfaces will be described in the following sections:
spherical case (Section 2), hyperbolic case (Section 3) and parabolic case
(Section 4).

Following the above scheme, in each one of the three sections, it is defined
the energy $\mathcal{E}_{\Pi}$ which depends on the distance to the reference
plane $\Pi\subset\mathbb{S}_{1}^{2}$. Catenaries will be obtained as solutions
of the Euler-Lagrange equation of these energies (Theorems 2.2, 3.1 and 4.1).
Next, catenaries will be also characterized as solutions of a prescribing
curvature equation involving the distance to the plane $\Pi$ and the angle
that make the principal normal vector of the curve with the unit vector field
orthogonal to $\Pi$ (Theorems 2.4, 3.2 and 4.2). Finally and answering to the
question proposed in this paper, we prove Theorem 1.2 in each one of these
cases (Theorems 2.6, 3.3 and 4.3). A last Section 5 will investigate the
concept of intrinsic catenary where the center of mass is calculated using the
(intrinsic) distance of $\mathbb{S}_{1}^{2}$.

##  2\. The catenary problem: spherical case

The first case to investigate are the catenaries of $\mathbb{S}_{1}^{2}$ where
the center of mass is measured with respect to a spacelike plane. Without loss
of generality, we can assume that this plane is the plane $\Pi_{xy}$ of
equation $z=0$. The catenary problem consists into find the shape of a curve
contained in
$(\mathbb{S}_{1}^{2})_{xy}^{+}=\\{(x,y,z)\in\mathbb{S}_{1}^{2}:z>0\\}$ whose
shape is obtained as a critical point of the center of mass with respect to
$\Pi_{xy}$. The distance of a point $(x,y,z)\in(\mathbb{S}_{1}^{2})_{xy}^{+}$
to $\Pi$ is $z$.

In order to obtain a manageable expression of this energy, let us parametrize
the de Sitter space $\mathbb{S}_{1}^{2}$ by
$\Psi\colon\mathbb{R}^{2}\to\mathbb{S}_{1}^{2}$, where

$\Psi(u,v)=(\cosh(u)\cos{v},\cosh(u)\sin{v},\sinh(u)).$

Let $\gamma\colon[a,b]\to(\mathbb{S}_{1}^{2})_{xy}^{+}$ be a non-degenerate
curve and let $\gamma(t)=\Psi(u(t),v(t))$. Then
$\langle\gamma^{\prime}(t),\gamma^{\prime}(t)\rangle=v^{\prime
2}\cosh(u)^{2}-u^{\prime 2}$. If $\gamma$ is spacelike (resp. timelike) this
number is positive (resp. negative). Along this paper, we will write
$\epsilon(v^{\prime 2}\cosh(u)^{2}-u^{\prime 2})>0$ where $\epsilon=1$ (resp.
$\epsilon=-1$) if $\gamma$ is spacelike (resp. timelike). Then
$|\gamma^{\prime}(t)|=\sqrt{\epsilon(v^{\prime 2}\cosh(u)^{2}-u^{\prime 2})}$
and
$|\gamma^{\prime}(t)|^{2}=\epsilon\langle\gamma^{\prime}(t),\gamma^{\prime}(t)\rangle$.

The distance of $\gamma(t)$ to $\Pi_{xy}$ is $\sinh u(t)$. According to (1),
the energy of the catenary problem with respect to the plane $\Pi_{xy}$ is

$\mathcal{E}_{xy}[\gamma]=\int_{a}^{b}(\sinh(u)+\lambda)\sqrt{\epsilon(v^{\prime
2}\cosh(u)^{2}-u^{\prime 2})}\,dt,$

where $\lambda$ is a Lagrange multiplier due to the length of the curve being
constant.

We will see that the critical points of $\mathcal{E}_{xy}$ will be expressed
in terms of the curvature $\kappa$ of $\gamma$. Here $\kappa$ is the curvature
of $\gamma$ viewed as a curve in the space $\mathbb{S}_{1}^{2}$, or in other
words, $\kappa$ is the geodesic curvature of $\gamma$. For the rest of
computations of this paper, it is necessary to have an expression of $\kappa$
when $\gamma$ is written as $\gamma(t)=\Psi(u(t),v(t))$.

######  Proposition 2.1.

Let $\gamma(t)=\Psi(u(t),v(t))$ be a non-degenerate curve in
$\mathbb{S}_{1}^{2}$. Then its (geodesic) curvature $\kappa$ is

(2) $\kappa=\epsilon\frac{v^{\prime}(v^{\prime
2}\sinh(u)\cosh(u)^{2}-2u^{\prime
2}\sinh(u))-\cosh(u)(u^{\prime}v^{\prime\prime}-v^{\prime}u^{\prime\prime})}{|\gamma^{\prime}(t)|^{3}}.$

###### Proof.

Let $\xi(p)=-p$, $p\in\mathbb{S}_{1}^{2}$, be the unit normal vector of
$\mathbb{S}_{1}^{2}$. Then

$\kappa=\epsilon\frac{\langle\gamma^{\prime\prime},\gamma^{\prime}\times\xi(\gamma)\rangle}{|\gamma^{\prime}(t)|^{3}}=\epsilon\frac{\mbox{det}(\gamma,\gamma^{\prime},\gamma^{\prime\prime})}{|\gamma^{\prime}(t)|^{3}},$

and (2) follows by a straightforward calculation. ∎

For further purposes, we need the following computations. First,

(3)
$\begin{split}\frac{d}{dt}\left(\frac{u^{\prime}}{|\gamma^{\prime}|}\right)&=\frac{\epsilon
v^{\prime}\cosh(u)}{|\gamma^{\prime}|^{3}}(-u^{\prime
2}v^{\prime}\sinh(u)+\cosh(u)(v^{\prime}u^{\prime\prime}-u^{\prime}v^{\prime\prime}))\\\
&=\frac{\epsilon v^{\prime}\cosh(u)}{|\gamma^{\prime}|^{3}}(u^{\prime
2}v^{\prime}\sinh(u)-v^{\prime
3}\sinh(u)\cosh(u)^{2}+\epsilon\kappa|\gamma^{\prime}|^{3})\\\
&=v^{\prime}\cosh(u)\left(\kappa-\frac{v^{\prime}\sinh(u)}{|\gamma^{\prime}|}\right),\end{split}$

where in the second identity we have replaced the expression
$v^{\prime}u^{\prime\prime}-u^{\prime}v^{\prime\prime}$ in terms of $\kappa$
thanks to (2). A similar computation yields

(4)
$\begin{split}\frac{d}{dt}\left(\frac{v^{\prime}\cosh(u)}{|\gamma^{\prime}|}\right)&=u^{\prime}\left(\kappa-\frac{v^{\prime}\sinh(u)}{|\gamma^{\prime}|}\right)\end{split}$

######  Theorem 2.2.

Let $\gamma(t)=\Psi(u(t),v(t))$ be a non-degenerate curve in
$(\mathbb{S}_{1}^{2})_{xy}^{+}$. Then $\gamma$ is a catenary with respect to
$\Pi_{xy}$ if and only if its curvature $\kappa$ satisfies

(5)
$\kappa=-\frac{v^{\prime}\cosh(u)^{2}}{(\sinh(u)+\lambda)|\gamma^{\prime}|}.$

###### Proof.

The Euler-Lagrange equations of $\mathcal{E}_{xy}$ are calculated using

(6) $\frac{\partial J}{\partial u}-\frac{d}{dt}\left(\frac{\partial
J}{\partial u^{\prime}}\right)=0,\quad\frac{\partial J}{\partial
v}-\frac{d}{dt}\left(\frac{\partial J}{\partial v^{\prime}}\right)=0,$

where $J=J[u,v]$ is the integrand of $\mathcal{E}_{xy}$,

$J[u,v]=(\sinh(u)+\lambda)\sqrt{\epsilon(v^{\prime 2}\cosh(u)^{2}-u^{\prime
2})}.$

Equations (6) are, respectively,

$\frac{v^{\prime
2}\cosh(u)\left(\cosh(u)^{2}+\sinh(u)(\sinh(u)+\lambda)\right)}{|\gamma^{\prime}|}+(\sinh(u)+\lambda)\frac{d}{dt}\left(\frac{u^{\prime}}{|\gamma^{\prime}|}\right)=0.$
$\left(\cosh(u)^{2}+\sinh(u)(\sinh(u)+\lambda)\right)\cosh(u)\frac{u^{\prime}v^{\prime}}{|\gamma^{\prime}|}+\sinh(u)\cosh(u)\frac{d}{dt}\left(\frac{v^{\prime}\cosh(u)}{|\gamma^{\prime}|}\right)=0.$

Using (3) and (4), we obtain

$v^{\prime}\cosh(u)\left(\frac{v^{\prime}\cosh(u)^{2}}{|\gamma^{\prime}|}+\kappa(\sinh(u)+\lambda)\right)=0.$
$u^{\prime}\cosh(u)\left(\frac{v^{\prime}\cosh(u)^{2}}{|\gamma^{\prime}|}+\kappa(\sinh(u)+\lambda)\right)=0.$

This yields the result because $u^{\prime}$ and $v^{\prime}$ cannot
simultaneously vanish. ∎

######  Remark 2.3.

The case $v^{\prime}=0$ is equivalent to $\kappa=0$ and $\gamma$ is a meridian
$\gamma(t)=\Psi(t,v_{0})$ for some constant $v_{0}\in\mathbb{R}$. This
situation also occurs in the Euclidean case: see Remark 2.5 below.

Notice that the Lagrangian $J[u,v]$ does not depend on $v$. Thus the second
equation of (6) gives a first integration of $\gamma$, namely,

$\frac{\partial J}{\partial v}=\frac{\epsilon
v^{\prime}\cosh(u)^{2}(\sinh(u)+\lambda)}{|\gamma^{\prime}|}=c,$

for some non-zero constant $c\in\mathbb{R}$. Without loss of generality, we
can assume that $\gamma(t)=\Psi(u(t),t)$. From the above equation, we deduce

(7)
$u(t)=m\pm\frac{1}{c}\int^{t}\cosh(u)\sqrt{c^{2}-\epsilon\cosh(u)^{2}(\sinh(u)+\lambda)}\,dt,\quad
c,m\in\mathbb{R}.$

We characterize the catenaries with respect to $\Pi_{xy}$ in terms of the
angle that makes the principal normal vector ${\bf n}$ of $\gamma$ with a
vector field of $\mathbb{L}^{3}$. The vector $\mathbf{n}$ is understood as a
unitary tangent of $\mathbb{S}_{1}^{2}$ at $\gamma$ which is orthogonal to
$\gamma^{\prime}$. On the other hand, the vector field of $\mathbb{L}^{3}$ is
the unit vector field orthogonal to the plane $\Pi_{xy}$. Define
$Z=\partial_{z}\in\mathfrak{X}(\mathbb{L}^{3})$.

######  Theorem 2.4.

Let $\gamma(t)=\Psi(u(t),v(t))$ be a non-degenerate curve in
$(\mathbb{S}_{1}^{2})_{xy}^{+}$. Then $\gamma$ is a catenary with respect to
$\Pi_{xy}$ if and only if its curvature $\kappa$ satisfies

(8) $\kappa(t)=\frac{\langle{\bf n}(t),Z\rangle}{d_{xy}(t)+\lambda},$

where $d_{xy}(t)$ is the distance of $\gamma(t)$ to the plane $\Pi_{xy}$.

###### Proof.

Since the unit normal vector $\xi$ to $(\mathbb{S}_{1}^{2})_{xy}^{+}$ along
$\gamma$ is $\xi(\gamma)=-\gamma$, the principal normal vector ${\mathbf{n}}$
is

${\bf
n}=\frac{\xi(\gamma)\times\gamma^{\prime}}{|\gamma^{\prime}|}=-\frac{\gamma\times\gamma^{\prime}}{|\gamma^{\prime}|}.$

Thus

$\langle{\bf
n}(t),Z\rangle=-\frac{\mbox{det}(\gamma,\gamma^{\prime},\partial_{z})}{|\gamma^{\prime}|}=-\frac{v^{\prime}\cosh(u)^{2}}{|\gamma^{\prime}|}.$

Since $d_{xy}=\sinh(u)$, it follows (8) from (5). ∎

######  Remark 2.5.

Equation (8) is analogous to the Euclidean case. In the catenary problem in
$\mathbb{R}^{2}$, the reference line is assumed to be the $x$-axis. The energy
to minimize is $[y]\mapsto\int_{a}^{b}(y+\lambda)\sqrt{1+y^{\prime 2}}\,dx$
for curves $y=y(x)$, $x\in[a,b]$, $y(x)>0$. A critical point $y$ is
characterized by the equation

(9) $\frac{y^{\prime\prime}}{1+y^{\prime 2}}=\frac{1}{y+\lambda},$

whose solution is the catenary

(10) $y(x)=\frac{1}{c}\cosh(cx+a)-\lambda,\quad a,c\in\mathbb{R},c>0.$

The curvature of $y(x)$ is $y^{\prime\prime}/(1+y^{\prime 2})^{3/2}$ and the
principal normal vector $\mathbf{n}$ is $(-y^{\prime},1)/\sqrt{1+y^{\prime
2}}$. If $W(x,y)=\partial_{y}$ is the unit vector field of $\mathbb{R}^{2}$ in
the direction of the gravity, then equation (9) is equivalent to

$\kappa(x)=\frac{\langle\mathbf{n}(x),W\rangle}{d(x)+\lambda},$

where $d(x)=y$ is the distance of $(x,y(x))$ to the $x$-axis. Notice that
vertical straight-lines of $\mathbb{R}^{2}$ are solutions of this equation
because $\kappa=0$ and $\langle{\bf n},W\rangle=0$.

Once the catenaries of $\mathbb{S}_{1}^{2}$ with respect to the plane
$\Pi_{xy}$ have been established, we answer to the Question posed in the
Introduction. We will rotate a catenary curve with respect to the plane
$\Pi_{xy}$ about the geodesic $L_{xy}=\Pi_{xy}\cap\mathbb{S}_{1}^{2}$. For
this, let us immerse $\mathbb{S}_{1}^{2}$ into $\mathbb{S}_{1}^{3}$
considering a Riemannian factor in the fourth coordinate such as it was
explained in Section 1. Consider the geodesic
$L_{xy}\subset\mathbb{S}_{1}^{2}\times\\{0\\}\subset\mathbb{S}_{1}^{3}$, which
will be the rotation axis. The one-parametric group of rotations of
$\mathbb{S}_{1}^{3}$ that leave pointwise the geodesic $L_{xy}$ is
$\mathcal{G}_{xy}=\\{\mathcal{R}_{s}^{xy}:s\in\mathbb{R}\\}$, where

$\mathcal{R}_{s}^{xy}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\
0&0&\cosh(s)&\sinh(s)\\\ 0&0&\sinh(s)&\cosh(s)\end{array}\right).$

Let $\gamma=\gamma(t)$ be a curve in
$(\mathbb{S}_{1}^{2})_{xy}^{+}\subset\mathbb{S}_{1}^{3}$. Let
$S_{\gamma}^{xy}=\\{\mathcal{R}^{xy}_{s}\cdot\gamma(t)\colon
s\in\mathbb{R},t\in[a,b]\\}$ be the surface of revolution obtained by the
orbit of $\gamma$ under the group $\mathcal{G}_{xy}$. Since the plane
$\Pi_{xy}$ containing the rotation axis is spacelike, the surface of
revolution is of spherical type according to the terminology of [4].

######  Theorem 2.6.

The surface of revolution $S_{\gamma}^{xy}$ is minimal if and only if $\gamma$
is a catenary with respect to $\Pi_{xy}$ for the Lagrange multiplier
$\lambda=0$.

###### Proof.

In order to simplify the arguments, we can assume without loss of generality
that $\gamma(t)=\Psi(u(t),t)$. Let us see $\gamma(t)\equiv(\Psi(u(t),t),0)$ as
a curve in $\mathbb{S}_{1}^{3}$. A parametrization of $S_{\gamma}^{xy}$ is

$\mathbf{r}(t,s)=(\cosh(u)\cos(t),\cosh(u)\sin(t),\cosh(s)\sinh(u),\sinh(s)\sinh(u)).$

It is necessary the expression of the mean curvature $H$ of $S_{\gamma}^{xy}$
in terms of the parametrization $\mathbf{r}$. As usually, let $\\{E,F,G\\}$
and $\\{h_{11},h_{12},h_{22}\\}$ be the coefficients of the first and second
fundamental form of $S_{\gamma}^{xy}$ for the parametrization $\mathbf{r}$:

$E=\langle\mathbf{r}_{t},\mathbf{r}_{t}\rangle,\quad
F=\langle\mathbf{r}_{t},\mathbf{r}_{s}\rangle,\quad
G=\langle\mathbf{r}_{s},\mathbf{r}_{s}\rangle,$ $h_{11}=\langle
N,\mathbf{r}_{tt}\rangle,\quad h_{12}=\langle N,\mathbf{r}_{ts}\rangle,\quad
h_{22}=\langle N,\mathbf{r}_{ss}\rangle.$

Then

$H=\frac{\delta}{2}\frac{Eh_{22}-2Fh_{12}+Gh_{11}}{EG-F^{2}},$

where $\delta=1$ if $S_{\gamma}^{xy}$ is spacelike and $\delta=-1$ if
$S_{\gamma}^{xy}$ is timelike. Notice that if $\epsilon=-1$, then necessarily
$\delta=-1$, but if $\epsilon=1$, then $\delta$ may be $1$ or $-1$. In the
proof of this theorem, and in the subsequent sections, the coefficient $F$
vanishes, so

(11) $H=\frac{\delta}{2}\frac{Eh_{22}+Gh_{11}}{EG}.$

In fact, we will investigate when $S_{\gamma}^{xy}$ is minimal, so our
interest focuses under what hypotheses on $\gamma$ we have $Eh_{22}+Gh_{11}$
is $0$ on the surface.

The calculation of $N$ is obtained knowing that $N$ is not only orthogonal to
$\mathbf{r}_{t}$ and $\mathbf{r}_{s}$, but also to $\mathbf{r}$ since $N$ is a
tangent vector of $\mathbb{S}_{1}^{3}$. A straightforward computations leads
to,

$N=\frac{1}{|\gamma^{\prime}|}\left(\begin{array}[]{l}\cos(t)\sinh(u)\cosh(u)-u^{\prime}\sin(t)\\\
u^{\prime}\cos(t)+\sin(t)\sinh(u)\cosh(u)\\\ \cosh(s)\cosh(u)^{2}\\\
\sinh(s)\cosh(u)^{2}\end{array}\right).$

In particular, $\langle N,N\rangle=-\epsilon$. We also have

$E=\cosh(u)^{2}-u^{\prime 2}=\epsilon|\gamma^{\prime}|^{2},\quad
G=\sinh(u)^{2}.$

On the other hand, the coefficients $h_{11}$ and $h_{22}$ of the second
fundamental form are

$\begin{split}h_{11}&=\frac{-\cosh(u)u^{\prime\prime}+2u^{\prime
2}\sinh(u)-\sinh(u)\cosh(u)^{2}}{|\gamma^{\prime}|},\\\
h_{22}&=-\frac{\sinh(u)\cosh(u)^{2}}{|\gamma^{\prime}|}.\end{split}$

Then

$H=-\frac{\delta\sinh(u)}{4|\gamma^{\prime}|EG}\left((1-3\cosh(2u))u^{\prime
2}+\cosh(u)(\cosh(u)+\cosh(3u)+2\sinh(u)u^{\prime\prime})\right).$

In this expression of $H$, the term $u^{\prime\prime}$ is replaced in function
of the curvature $\kappa$. So, from (2) we have

$u^{\prime\prime}=\frac{\epsilon\kappa|\gamma^{\prime}|^{3}-\sinh(u)\cosh(u)^{2}+2u^{\prime
2}\sinh(u)}{\cosh(u)}.$

Then

(12)
$H=-\frac{\delta\epsilon\sinh(u)|\gamma^{\prime}|}{2EG}\left(\cosh(u)^{2}+\kappa\sinh(u)|\gamma^{\prime}|\right).$

Then $H=0$ if and only if $\cosh(u)^{2}+\kappa\sinh(u)|\gamma^{\prime}|$,
proving the result thanks to (5). ∎

######  Remark 2.7.

The fact that $\lambda$ must be $0$ in Theorem 2.6 in order to ensure that
$S_{\gamma}^{xy}$ is minimal is expectable because the same situation occurs
in Euclidean space $\mathbb{R}^{3}$. If we immerse $\mathbb{R}^{2}$ into
$\mathbb{R}^{3}$ by $(x,y)\mapsto(x,0,y)$ and if the catenary $y(x)$ given in
(10) is rotated about the $x$-axis of $\mathbb{R}^{3}$, the mean curvature $H$
of the surface is

$H=\frac{1}{2}\left(\frac{1}{y\sqrt{1+y^{\prime
2}}}-\frac{y^{\prime\prime}}{(1+y^{\prime 2})^{3/2}}\right)=\frac{\lambda
c^{2}}{2\cosh(cx+a)^{2}(\cosh(cx+a)-\lambda c)}.$

Thus $H=0$ if and only if $\lambda=0$.

##  3\. The catenary problem: hyperbolic case

In the case of the catenary problem with respect to a Lorentzian plane, we
have two possible choices of Lorentzian coordinate planes. After a rigid
motion of $\mathbb{S}_{1}^{2}$, we can assume that the plane is the plane
$\Pi_{xz}$ of equation $y=0$. In this section it will be studied the curves of
$\mathbb{S}_{1}^{2}$ that are critical points of the center of mass when it is
measured with respect to the plane $\Pi_{xz}$. The distance of
$(x,y,z)\in\mathbb{S}_{1}^{2}$ to $\Pi_{xz}$ is $|y|$. Let
$(\mathbb{S}_{1}^{2})_{xz}^{+}=\\{(x,y,z)\in\mathbb{S}_{1}^{2}:y>0\\}$. Let
$\gamma=\gamma(t)$, $t\in[a,b]$, be a curve in $(\mathbb{S}_{1}^{2})_{xz}^{+}$
and let $\gamma(t)=\Psi(u(t),v(t))$. The expression of the energy (1) is now

$\mathcal{E}_{xz}[\gamma]=\int_{a}^{b}(\cosh(u)\sin(v)+\lambda)\sqrt{\epsilon(v^{\prime
2}\cosh(u)^{2}-u^{\prime 2})}\,dt.$

######  Theorem 3.1.

Let $\gamma(t)=\Psi(u(t),v(t))$ be a non-degenerate curve in
$(\mathbb{S}_{1}^{2})_{xz}^{+}$. Then $\gamma$ is a catenary with respect to
$\Pi_{xz}$ if and only if its curvature $\kappa$ satisfies

(13)
$\kappa=-\frac{u^{\prime}\cos(v)+v^{\prime}\sinh(u)\cosh(u)\sin(v)}{(\cosh(u)\sin(v)+\lambda)|\gamma^{\prime}|}.$

###### Proof.

We compute the Euler-Lagrange equations (6) for the energy $\mathcal{E}_{xz}$.
The first equation $\frac{\partial J}{\partial
u}-\frac{d}{dt}\left(\frac{\partial J}{\partial u^{\prime}}\right)=0$ writes
as

$\begin{split}&\frac{v^{\prime}\cosh(u)}{|\gamma^{\prime}|}\left(v^{\prime}\cosh(u)\sinh(u)\sin(v)+v^{\prime}\sinh(u)(\cosh(u)\sin(v)+\lambda)+u^{\prime}\cos(v)\right)\\\
&=-(\cosh(u)\sin(v)+\lambda)\frac{d}{dt}\left(\frac{u^{\prime}}{|\gamma^{\prime}|}\right).\end{split}$

Using (3), this equation is simplified into

$\frac{v^{\prime}\sinh(u)\cosh(u)\sin(v)+u^{\prime}\cos(v)}{|\gamma^{\prime}|}=-(\cosh(u)\sin(v)+\lambda)\kappa.$

The second equation of (6) coincides with the above one, obtaining (13). ∎

As in Theorem 2.4, we characterize the catenaries with respect to $\Pi_{xz}$
in terms of the angle that makes the principal normal vector ${\bf n}$ of
$\gamma$ with a vector field of $\mathbb{L}^{3}$. Consider the unit vector
field $Y=\partial_{y}\in\mathfrak{X}(\mathbb{L}^{3})$. Notice that $Y$ is
orthogonal to the plane $\Pi_{xz}$.

######  Theorem 3.2.

Let $\gamma(t)=\Psi(u(t),v(t))$ be a non-degenerate curve in
$(\mathbb{S}_{1}^{2})_{xz}^{+}$. Then $\gamma$ is a catenary with respect to
$\Pi_{xz}$ if and only if its curvature $\kappa$ satisfies

(14) $\kappa(t)=-\frac{\langle{\bf n}(t),Y\rangle}{d_{xz}(t)+\lambda},$

where $d_{xz}(t)$ is the distance of $\gamma(t)$ to the plane $\Pi_{xz}$.

###### Proof.

Following the same arguments as in Theorem 2.4, we have

$\langle{\bf
n}(t),Y\rangle=-\frac{\mbox{det}(\gamma,\gamma^{\prime},\partial_{y})}{|\gamma^{\prime}|}=\frac{v^{\prime}\sinh(u)\cosh(u)\sin(v)+u^{\prime}\cos(v)}{|\gamma^{\prime}|}.$

Since $d_{xz}=\cosh(u)\sin(v)$, it follows the result from (13). ∎

Finally we construct the surfaces of revolution in $\mathbb{S}_{1}^{3}$ of
hyperbolic type using the catenaries with respect to $\Pi_{xz}$. Consider the
geodesic $L_{xz}=\mathbb{S}_{1}^{2}\cap\Pi_{xz}$ viewed as a curve in
$\mathbb{S}_{1}^{3}$. This geodesic $L_{xz}$ will be the rotation axis of a
rotational surface of $\mathbb{S}_{1}^{3}$ of hyperbolic type because the
plane $\Pi_{xz}$ is Lorentzian. The one-parameter family of rotations fixing
$L_{xz}$ is the group $\mathcal{G}_{xz}=\\{R_{s}^{xz}:s\in\mathbb{R}\\}$,
where

$R_{s}^{xz}:=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&\cos s&0&-\sin s\\\
0&0&1&0\\\ 0&\sin s&0&\cos s\end{array}\right).$

If $\gamma=\gamma(t)$ is a curve in $(\mathbb{S}_{1}^{2})_{xz}^{+}$, let
$S_{\gamma}^{xz}$ denote the surface of revolution of $\mathbb{S}_{1}^{3}$
obtained by rotating $\gamma$ with respect to $L_{xz}$,
$S_{\gamma}^{xz}=\\{\mathcal{R}_{s}^{xz}\cdot\gamma(t):s\in\mathbb{R},t\in[a,b]\\}$.

######  Theorem 3.3.

The surface of revolution $S_{\gamma}^{xz}$ is minimal if and only if $\gamma$
is a catenary with respect to $\Pi_{xz}$ for the Lagrange multiplier
$\lambda=0$.

###### Proof.

Without loss of generality, we assume that $\gamma$ is parametrized by
$\gamma(t)=\Psi(u(t),t)$, $t\in[a,b]$. A parametrization of $S_{\gamma}^{xz}$
is

$\mathbf{r}(t,s)=(\cos(t)\cosh(u),\sin(t)\cosh(u)\cos(s),\sinh(u),\sin(s)\sin(t)\cosh(u)).$

The unit normal is

$N=\frac{1}{|\gamma^{\prime}|}\left(\begin{array}[]{c}\cos(t)\sinh(u)\cosh(u)-\sin(t)u^{\prime}\\\
\cos(s)\left(\cos(t)u^{\prime}+\sin(t)\sinh(u)\cosh(u)\right)\\\
\cosh^{2}(u)\\\
\sin(s)\left(\cos(t)u^{\prime}+\sin(t)\sinh(u)\cosh(u)\right)\end{array}\right),$

and $\langle N,N\rangle=-\epsilon$. The coefficients of the first fundamental
form are $F=0$ and

$E=\cosh(u)^{2}-u^{\prime 2}=\epsilon|\gamma^{\prime}|^{2},\quad
G=\sin(t)^{2}\cosh(u)^{2}.$

We also have

$\begin{split}h_{11}&=\frac{2u^{\prime
2}\sinh(u)-\cosh(u)(u^{\prime\prime}+\sinh(u)\cosh(u))}{|\gamma^{\prime}|},\\\
h_{22}&=-\frac{\cosh(t)\sin(t)(u^{\prime}\cos(t)+\sinh(u)\cosh(u)\sin(t))}{|\gamma^{\prime}|}.\end{split}$

Using (11), we have

$\begin{split}H&=\frac{\delta\cosh(u)\sin(t)}{4EG|\gamma^{\prime}|}\\\
&\Big{(}-2\sin(t)\cosh(u)^{2}\left(u^{\prime\prime}+\sinh(2u)\right)+2\cos(t)u^{\prime
3}\\\ &+3\sin(t)u^{\prime
2}\sinh(2u)-2\cos(t)u^{\prime}\cosh^{2}(u)\Big{)}.\end{split}$

Using (2), we replace $u^{\prime\prime}$ in terms of the curvature $\kappa$,
obtaining

$H=-\frac{\delta\epsilon|\gamma^{\prime}|}{2EG}\left(u^{\prime}\cos(t)+\sinh(u)\cosh(u)\sin(t)+\cosh(u)\sin(t)|\gamma^{\prime}|\kappa\right).$

The result is now immediate from (13). ∎

##  4\. The catenary problem: parabolic case

This section is devoted to consider catenaries of $\mathbb{S}_{1}^{2}$ when
the center of mass is calculated with the distance with respect to a
degenerate plane. Without loss of generality, let $\Pi_{y-z}$ be the plane of
equation $y-z=0$. The distance of a point $(x,y,z)$ to $\Pi_{y-z}$ is $|y-z|$
up to the factor $\sqrt{2}$. Let $(\mathbb{S}_{1}^{2})_{y-z}^{+}$ be half-
space $\\{(x,y,z)\in\mathbb{S}_{1}^{2}:y-z>0\\}$. For a curve
$\gamma(t)=\Psi(u(t),v(t))$, $t\in[a,b]$, contained in
$(\mathbb{S}_{1}^{2})_{y-z}^{+}$, define the energy

$\mathcal{E}_{y-z}[\gamma]=\int_{a}^{b}(\cosh(u)\sin(v)-\sinh(u)+\lambda)\sqrt{\epsilon(v^{\prime
2}\cosh(u)^{2}-u^{\prime 2})}\,dt.$

where $\lambda$ is a Lagrange multiplier.

######  Theorem 4.1.

Let $\gamma(t)=\Psi(u(t),v(t))$ be a non-degenerate curve in
$(\mathbb{S}_{1}^{2})_{y-z}^{+}$. Then $\gamma$ is a catenary with respect to
$\Pi_{y-z}$ if and only if its curvature $\kappa$ satisfies

(15)
$\kappa=-\frac{v^{\prime}\cosh(u)(\sinh(u)\sin(v)-\cosh(u))+u^{\prime}\cos(v)}{(\cosh(u)\sin(v)-\sinh(u)+\lambda)|\gamma^{\prime}|}.$

###### Proof.

The computations are similar as in Theorem 2.2. For this, it suffices to
compute $\frac{\partial J}{\partial u}-\frac{d}{dt}\left(\frac{\partial
J}{\partial u^{\prime}}\right)=0$. Then we have

$\begin{split}&\frac{v^{\prime
2}\left(\cosh(u)^{2}(\sinh(u)\sin(v)-\cosh(u))+\sinh(u)\cosh(u)(\cosh(u)\sin(v)-\sinh(u)+\lambda)\right)}{|\gamma^{\prime}|}\\\
&=-\frac{\cosh(u)\cos(v)u^{\prime}v^{\prime}}{|\gamma^{\prime}|}-(\cosh(u)\sin(v)-\sinh(u)+\lambda)\frac{d}{dt}\left(\frac{u^{\prime}}{|\gamma^{\prime}|}\right).\end{split}$

Using (3), and simplifying,

$\begin{split}&\frac{v^{\prime
2}\cosh(u)^{2}(\sinh(u)\sin(v)-\cosh(u))+u^{\prime}v^{\prime}\cosh(u)\cos(v)}{|\gamma^{\prime}|}\\\
&=-(\cosh(u)\sin(v)-\sinh(u)+\lambda)v^{\prime}\cosh(u)\kappa.\end{split}$

∎

We characterize the catenaries of Theorem 4.1 in terms of the angle that makes
the principal normal vector ${\bf n}$ of $\gamma$ with the vector field
$T=\partial_{y}+\partial_{z}\in\mathfrak{X}(\mathbb{L}^{3})$. This vector
field is orthogonal to the plane $\Pi_{y-z}$.

######  Theorem 4.2.

Let $\gamma(t)=\Psi(u(t),v(t))$ be a non-degenerate curve in
$(\mathbb{S}_{1}^{2})_{y-z}^{+}$. Then $\gamma$ is a catenary with respect to
$\Pi_{y-z}$ if and only if its curvature $\kappa$ satisfies

(16) $\kappa(t)=-\frac{\langle{\bf n}(t),T\rangle}{d_{y-z}(t)+\lambda},$

where $d_{y-z}(t)$ is the distance of $\gamma(t)$ to the plane $\Pi_{y-z}$.

###### Proof.

It suffices to compute $\langle{\bf n}(t),T\rangle$,

$\langle{\bf
n}(t),T\rangle=\frac{v^{\prime}\cosh(u)\left(\sinh(u)\sin(v)-\cosh(u)\right)-u^{\prime}\cos(v)}{|\gamma^{\prime}|}$

and use (15). ∎

We now consider rotational surfaces of $\mathbb{S}_{1}^{3}$ about the rotation
axis $L_{y-z}=\Pi_{y-z}\cap\mathbb{S}_{1}^{2}\subset\mathbb{S}_{1}^{3}$. Since
the plane $\Pi_{y-z}$ is degenerate, the surface of revolution is of parabolic
type. The one-parameter group $\mathcal{G}_{y-z}$ of rotations of
$\mathbb{S}_{1}^{3}$ fixing $L_{y-z}$ is the group
$\mathcal{G}_{y-z}=\\{R_{s}^{y-z}:s\in\mathbb{R}\\}$, where

$R_{s}^{y-z}=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&1-\frac{s^{2}}{2}&\frac{s^{2}}{2}&s\\\
0&-\frac{s^{2}}{2}&\frac{s^{2}}{2}+1&s\\\ 0&-s&s&1\\\ \end{array}\right).$

Let $\gamma=\gamma(t)$ be a curve in $\mathbb{S}_{1}^{2}$ and let
$S_{\gamma}^{y-z}=\\{\mathcal{R}_{s}^{y-z}\cdot\gamma(t):s\in\mathbb{R},t\in[a,b]\\}$
the surface of revolution defined by $\gamma$.

######  Theorem 4.3.

The surface of revolution $S_{\gamma}^{y-z}$ is minimal if and only if
$\gamma$ is a catenary with respect to $\Pi_{y-z}$ for the Lagrange multiplier
$\lambda=0$.

###### Proof.

The computations are simplified again if the generating curve is assumed to be
$\gamma(t)=(\Psi(u(t),t),0)$. The parametrization of $S_{\gamma}^{y-z}$ is

$\mathbf{r}(t,s)=\left(\begin{array}[]{l}\cos(t)\cosh(u)\\\
\frac{s^{2}}{2}\sinh(u)-\frac{s^{2}-2}{2}\sin(t)\cosh(u)\\\
\frac{s^{2}+2}{2}\sinh(u)-\frac{s^{2}}{2}\sin(t)\cosh(u)\\\
s(\sinh(u)-\sin(t)\cosh(u))\end{array}\right).$

We compute the mean curvature $H$ of $S_{\gamma}^{y-z}$. The unit normal $N$
of $S_{\gamma}^{y-z}$ is

$N=\frac{1}{|\gamma^{\prime}|}\left(\begin{array}[]{c}\cos(t)\sinh(2u)-2\sin(t)u^{\prime}\\\
-\left(s^{2}-2\right)\cos(t)u^{\prime}+s^{2}\cosh^{2}(u)-\frac{s^{2}-2}{2}\sin(t)\sinh(2u)\\\
-s^{2}\cos(t)u^{\prime}+\left(s^{2}+2\right)\cosh^{2}(u)-\frac{s^{2}}{2}\sin(t))\sinh(2u)\\\
-s\left(2\cos(t)u^{\prime}-2\cosh^{2}(u)+\sin(t)\sinh(2u)\right)\end{array}\right)$

and $\langle N,N\rangle=-\epsilon$. The coefficients of the first fundamental
form are $F=0$ and

$E=\cosh(u)^{2}-u^{\prime 2}=\epsilon|\gamma^{\prime}|^{2},\quad
G=(\sinh(u)-\cosh(u)\sin(t))^{2}.$

The coefficients of the second fundamental form are $h_{12}=0$ and

$\begin{split}h_{11}&=\frac{2\sinh(u)u^{\prime
2}-\sin(u)\cosh(u)^{2}-\cosh(u)u^{\prime\prime}}{|\gamma^{\prime}|},\\\
h_{22}&=\frac{(\cosh(u)\sin(t)-\sinh(u))(\cosh(u)^{2}-\sinh(u)\cosh(u)\sin(t)-u^{\prime}\cos(t))}{|\gamma^{\prime}|}.\end{split}$

As in the above cases, the mean curvature $H$ is calculated using the formula
(11). After substituting $u^{\prime\prime}$ in function of $\kappa$ thanks to
(2), we obtain

$\begin{split}H&=\frac{\delta\epsilon(\cosh(u)\sin(t)-\sinh(u))}{2EG}\\\
&\Big{(}\kappa(\cosh(u)\sin(t)-\sinh(u))|\gamma^{\prime}|+\cosh(u)(\sinh(t)\sin(t)\\\
&-\cosh(u))+u^{\prime}\cos(t)\Big{)}.\end{split}$

Then (15) completes the proof. ∎

InFigure 1 shows pictures of catenaries of $\mathbb{S}_{1}^{2}$ with respect
to the three types of causal character of the plane reference.

Figure 1. Catenaries of $\mathbb{S}_{1}^{2}$: with respect to spacelike plane
(left), timelike plane (middle) and degenerate plane (right).

##  5\. The intrinsic catenary problem

The catenaries studied in the previous sections are critical points of
energies that measure the center of mass with respect to planes of
$\mathbb{L}^{3}$. However, if one wants to pose an ‘intrinsic catenary
problem’ involving only data in $\mathbb{S}_{1}^{2}$, it would be desirable
that the distance were measured in $\mathbb{S}_{1}^{2}$ with respect to a
geodesic of $\mathbb{S}_{1}^{2}$. In this section we will investigate this
problem in the particular case that this geodesic is spacelike. Without loss
of generality, consider the spacelike geodesic
$L=\mathbb{S}_{1}^{2}\cap\Pi_{xy}$, which will serve as reference line to
measure the center of mass of a curve.

The distance of a point $(x,y,z)\in\mathbb{S}_{1}^{2}$ from $L$ is computed as
follows. Let $(x,y,z)=\Psi(u,v)$. Then the geodesic on $\mathbb{S}_{1}^{2}$
passing through to $(x,y,z)$ and orthogonal to $L$ is $\alpha(t)=\Psi(u+t,v)$,
$t\in[-u,0]$. The intrinsic distance of $(x,y,z)$ to $L$ is

$\int_{-u}^{0}|\alpha^{\prime}(t)|\,dt=\int_{0}^{-u}dt=u=\mbox{arcsinh}(z).$

In particular, it is necessary that $z\not=0$. Consider the half-space
$(\mathbb{S}_{1}^{2})_{xy}^{+}=\\{(x,y,z)\in\mathbb{S}_{1}^{2}:z>0\\}$. If
$\gamma$ is a curve in $(\mathbb{S}_{1}^{2})_{xy}^{+}$ parametrized by
$\gamma(t)=\Psi(u(t),v(t))$, the functional energy is defined by

$\mathcal{E}_{in}[\gamma]=\int_{a}^{b}(u+\lambda)\sqrt{\epsilon(v^{\prime
2}\cosh(u)^{2}-u^{\prime 2})}\,dt,$

where $\lambda$ is a Lagrange multiplier that indicates that in the
variational problem it is assumed that all curves have fixed length. A
critical point of $\mathcal{E}_{in}$ is called an intrinsic catenary of
$\mathbb{S}_{1}^{2}$ with respect to the geodesic $L$.

######  Theorem 5.1.

Let $\gamma(t)=\Psi(u(t),v(t))$ be a non-degenerate curve in
$(\mathbb{S}_{1}^{2})_{xy}^{+}$. Then $\gamma$ is an intrinsic catenary with
respect to $L$ if and only if its curvature $\kappa$ satisfies

(17) $\kappa=-\frac{v^{\prime}\cosh(u)}{(u+\lambda)|\gamma^{\prime}|}.$

###### Proof.

We compute the Euler-Lagrange equations of $\mathcal{E}_{in}$ using (6). Both
equations are

$\frac{v^{\prime
2}\cosh(u)(\cosh(u)+(u+\lambda)\sinh(u))}{|\gamma^{\prime}|}=-(u+\lambda)\frac{d}{dt}\left(\frac{u^{\prime}}{|\gamma^{\prime}|}\right),$
$\frac{u^{\prime}v^{\prime}\cosh(u)(\cosh(u)+(u+\lambda)\sinh(u))}{|\gamma^{\prime}|}+(u+\lambda)\cosh(u)\frac{d}{dt}\left(\frac{v^{\prime}\cosh(u)}{|\gamma^{\prime}|}\right)=0.$

Using the expressions (3) and (4), we have

$v^{\prime}\cosh(u)\left((u+\lambda)\kappa+\frac{v^{\prime}\cosh(u)}{|\gamma^{\prime}|}\right)=0,$
$u^{\prime}\cosh(u)\left((u+\lambda)\kappa+\frac{v^{\prime}\cosh(u)}{|\gamma^{\prime}|}\right)=0,$

proving the result. ∎

It is possible to characterize intrinsic catenaries of $\mathbb{S}_{1}^{2}$ as
solutions a prescribing curvature equation involving the principal normal
vector of the curve and vector fields. In contrast to Theorems 2.4, 3.2 and
4.2, it is natural that now the vector field is a vector field on
$\mathbb{S}_{1}^{2}$. Furthermore, this vector field will indicate the
direction of how is measuring the distance to $L$. Since this distance is done
using the geodesics orthogonal to $L$, that is, the meridians of
$\mathbb{S}_{1}^{2}$, the vector field will be tangent to the meridians.
Hence, define the vector field $V\in\mathfrak{X}(\mathbb{S}_{1}^{2})$

$V(\Psi(u,v))=\Psi_{u}=\sinh(u)\cos(v)\partial_{x}+\sinh(u)\sin(v)\partial_{y}+\cosh(u)\partial_{z}.$

######  Theorem 5.2.

Let $\gamma(t)=\Psi(u(t),v(t))$ be a non-degenerate curve in
$(\mathbb{S}_{1}^{2})_{xy}^{+}$. Then $\gamma$ is an intrinsic catenary with
respect to $L$ if and only if its curvature $\kappa$ satisfies

$\kappa(t)=\frac{\langle{\bf n}(t),V\rangle}{d(t)+\lambda},$

where $d(t)$ is the intrinsic distance of $\gamma(t)$ to $L$.

###### Proof.

Using that ${\bf n}=-\gamma\times\gamma^{\prime}/|\gamma^{\prime}|$, from
$\gamma(t)=\Psi(u(t),v(t))$ we have
$\gamma^{\prime}=u^{\prime}\Psi_{u}+v^{\prime}\Psi_{v}$. Thus

$\langle{\bf
n}(t),V\rangle=-\frac{\mbox{det}(\gamma,\gamma^{\prime},V)}{|\gamma^{\prime}|}=-\frac{v^{\prime}}{|\gamma^{\prime}|}\mbox{det}(\Psi,\Psi_{v},\Psi_{u})=-\frac{v^{\prime}\cosh(u)}{|\gamma^{\prime}|}.$

This expression together (17) gives the result. ∎

Finally, notice that the surface of revolution $S_{\gamma}^{xy}$ of spherical
type obtained by rotating an intrinsic catenary $\gamma$ about $L$ is not
minimal. Its mean curvature is given from (12), which together (17) gives for
$\lambda=0$,

$\begin{split}H&=-\frac{\delta}{2|\gamma^{\prime}|\sinh(u)}\left(\cosh(u)^{2}+\kappa\sinh(u)|\gamma^{\prime}|\right)\\\
&=-\frac{\delta\cosh(u)}{2|\gamma^{\prime}|\sinh(u)}\left(\cosh(u)-\frac{v^{\prime}\sinh(u)}{u}\right).\end{split}$

## Acknowledgements

Rafael López is a member of the Institute of Mathematics of the University of
Granada. This work has been partially supported by the Projects I+D+i
PID2020-117868GB-I00, supported by MCIN/ AEI/10.13039/501100011033/,
A-FQM-139-UGR18 and P18-FR-4049.

## Data Availability Statement

My manuscript has no associate data. No funding was received for conducting
this study. The authors have no competing interests to declare that are
relevant to the content of this article.

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