#  Resonant behavior and stability of a linear three-mirror cavity

Paul Stevens  ID Corresponding author<EMAIL_ADDRESS>Université
Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  Vincent Loriette  ID
Corresponding author<EMAIL_ADDRESS>Université Paris-
Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  Manuel Andia  ID Université
Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  François Glotin  ID
Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  Angélique
Lartaux-Vollard  ID Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay,
France  Nicolas Leroy  ID Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405
Orsay, France  Aymeric van de Walle  ID Université Paris-Saclay, CNRS/IN2P3,
IJCLab, 91405 Orsay, France

###### Abstract

The implementation of Fabry-Perot cavities in gravitational-wave detectors has
been pivotal to improving their sensitivity, allowing the observation of an
increasing number of cosmological events with higher signal-to-noise ratio.
Notably, Fabry-Perot cavities play a key role in the frequency-dependent
squeezing technique, which provides a reduction of quantum noise over the
whole observation frequency spectrum. In this context, linear three-mirror
cavities could be of interest because of the additional control that they can
provide.

In this paper, we develop a complete model to describe the stability behavior
and the properties of transmitted and reflected fields of a linear three-
mirror cavity aiming to be used for design purposes. In particular,
simulations are carried out to show the evolution of the characteristic
“double-peak” as a function of cavity parameters, which is one of the key
features of this system. We show that the double-peak shape is almost freely
adjustable, either in terms of spacing between maxima, their relative height
and their intrinsic width. This is made possible by changing the mirrors’
reflectivity coefficients and their spacing. However, the amount of achievable
realistic configurations is limited by the stability conditions of the cavity.
In particular, if the middle mirror is not close enough to the center of the
cavity, it could be difficult to obtain a stable three-mirror cavity.
Different geometries have been studied to obtain a stable cavity system.

Keywords— Gravitational-wave detectors, quantum noise, frequency-dependent
squeezing, filtering cavity, three-mirror cavity.

##  1 Introduction

Fabry-Perot cavities are among the best-known optical systems. They are used
in many fields, ranging from fundamental physics to engineering. They are
formed by two parallel, partially-transmitting mirrors. The distance between
them, as well as their transmission coefficients, determine the cavity’s
resonance properties. The resulting wavelength selectivity of this system is
an essential feature of Fabry-Perot cavities which can be exploited in a wide
variety of applications such as in lasers to generate a coherent beam or as a
filter to select a specific frequency for telecom networks, or even in
pressure or temperature sensors. They are also crucial in current and future
generations of Gravitational-Wave Detectors (GWD) such as LIGO [1], Virgo [2],
KAGRA [3], Einstein Telescope [4] or Cosmic Explorer [5].

GWD are limited by several sources of noise, of which quantum noise is one of
the most dominant. Despite the efficiency attained by frequency-dependent
squeezing (FDS) [6], the next generation of GWD such as Einstein Telescope –
Low Frequency will need different filtering systems to meet their own quantum
noise targets [4]. For this purpose, different configurations of filter cavity
are currently studied. Among them, linear three-mirror cavities could be
particularly interesting because of the additional control they could offer
for frequency-dependent squeezing. One of the key features of this system is
the doubling of its resonance peak [7, 8, 9, 10]. However, the additional
degrees of freedom of those cavities could make them more difficult to
control.

In this paper, we develop a full model based on electromagnetic field
propagation that we implement in a code to perform a non-exhaustive
exploration of the optical properties and stability behavior of a general
linear three-mirror cavity. Contrary to the study presented in [9], we are not
elaborating here on the fundamental origin of the double peak, but on its
shape variations as a function of the cavity configuration. We will not
compare any further our solutions with the Fabry-Perot model but instead, we
will focus on transmitted and reflected fields to understand how the design of
the cavity impacts the resonance pattern. Some results obtained might not be
relevant to our particular FDS problem but could motivate the use of this
configuration for other goals in different optical systems.

##  2 Three-mirror cavity optical behavior

In the first part of this section, we propose an analytical model based on
field propagation through the three-mirror cavity to explicitly express the
global transmitted and reflected fields on cavity parameters. Then, in the
second part, we will be interested in the evolution of the particular double
peak resonance pattern as a function of those parameters by implementing the
solutions of our model in a numerical simulation.

###  2.1 Three-mirror cavity model

To model the three-mirror cavity as a single optical system, we derive its
overall transmission and reflection amplitude coefficients $r$ and $t$ by
comparing the reflected and transmitted fields to a reference input field. On
the three-mirror cavity scheme presented in Fig. 1, we detail the position of
each field that we use.

Figure 1: Three-mirror cavity scheme, the reflective surface of middle mirror
can either be turn towards input or end mirror.

By following the path of a field entering the cavity and using the convention
for which a reflected field is multiplied by the factor $r_{\text{Surf}}$ of a
reflective surface while a transmitted field is multiplied by an imaginary
transmission factor $it_{\text{Surf}}$ [11], we can construct the equation
system (1). These equations depend on the following configuration parameters:
$r_{1,2,3}$ and $t_{1,2,3}$ the amplitude reflection and transmission
coefficients of input, middle and end mirror (respectively); $k=2\pi/\lambda$
the wave vector with $\lambda$ the wavelength of the input field; $L_{1,2}$
the distance between input and middle mirror or between middle and end mirror
(i.e. first and second sub-cavities). Note that, in this model, we consider
infinitely thin mirrors.

$\displaystyle\begin{cases}\psi_{1}=it_{1}\psi_{\text{In}}+r_{1}\psi_{8}&\psi_{2}=\psi_{1}e^{-ikL_{1}}\\\
\psi_{3}=it_{2}\psi_{2}+r_{2}\psi_{6}&\psi_{4}=\psi_{3}e^{-ikL_{2}}\\\
\psi_{5}=r_{3}\psi_{4}&\psi_{6}=\psi_{5}e^{-ikL_{2}}\\\
\psi_{7}=it_{2}\psi_{6}+r_{2}\psi_{2}&\psi_{8}=\psi_{7}e^{-ikL_{1}}\\\
\psi_{\text{Ref}}=it_{1}\psi_{8}+r_{1}\psi_{\text{In}}&\psi_{\text{Tr}}=it_{3}\psi_{4}\end{cases}$
(1)

From this cavity parameterization, we can derive the expression of reflection
and transmission coefficients of the full three-mirror cavity
$r=\frac{\psi_{\text{Ref}}}{\psi_{\text{In}}}$ and
$t=\frac{\psi_{\text{Tr}}}{\psi_{\text{In}}}$ as a function of configuration
parameters.

The reflection coefficient:

$r=\frac{\psi_{\text{Ref}}}{\psi_{\text{In}}}=\frac{it_{1}\psi_{8}+r_{1}\psi_{\text{In}}}{\psi_{\text{In}}}$
(2)

can be rewritten in terms of $\psi_{2}$ and $\psi_{6}$:

$r=\frac{-t_{1}^{2}e^{-ikL_{1}}(it_{2}\psi_{6}+r_{2}\psi_{2})+r_{1}(\psi_{2}e^{ikL_{1}}-r_{1}e^{-ikL_{1}}(it_{2}\psi_{6}+r_{2}\psi_{2}))}{\psi_{2}e^{ikL_{1}}-r_{1}e^{-ikL_{1}}(it_{2}\psi_{6}+r_{2}\psi_{2}))}$
(3)

Then, expressing $\psi_{2}$ as a function of $\psi_{6}$:

$\psi_{2}=\frac{\psi_{6}}{it_{2}}\left(\frac{e^{2ikL_{2}}}{r_{3}}-r_{2}\right)$
(4)

and, finally, injecting Eq. (4) in Eq. (3), we obtain:

$r=\frac{r_{1}e^{2ik(L_{1}+L_{2})}-r_{1}r_{2}r_{3}e^{2ikL_{1}}-r_{2}(r_{1}^{2}+t_{1}^{2})e^{2ikL_{2}}+r_{3}(r_{1}^{2}+t_{1}^{2})(r_{2}^{2}+t_{2}^{2})}{e^{2ik(L_{1}+L_{2})}-r_{1}r_{2}e^{2ikL_{2}}-r_{2}r_{3}e^{2ikL_{1}}+r_{1}r_{3}(r_{2}^{2}+t_{2}^{2})}$
(5)

In the same way, we can derive the expression of the transmission coefficient:

$t=\frac{-t_{1}t_{2}t_{3}e^{ik(L_{1}+L_{2})}}{e^{2ik(L_{1}+L_{2})}-r_{1}r_{2}e^{2ikL_{2}}-r_{2}r_{3}e^{2ikL_{1}}+r_{1}r_{3}(r_{2}^{2}+t_{2}^{2})}$
(6)

These two equations are consistent with the results presented in [7, 9].

From the above results, it is interesting to note that the global transmission
and reflection coefficients of a three-mirror cavity are composed of both
terms involving each two-mirror combination, comparable to those seen in
Fabry-Perot reflection/transmission coefficients, and terms that involve all 3
mirrors. It follows that, in the general case, a three-mirror cavity cannot
simply be understood as a series of two independent Fabry-Perot cavities nor a
linear combination of intra-cavity fields of 3 cavities formed by each two-
mirror combination, but a more complex system where the overall optical
response depends on all the interdependencies and interactions between fields.
Therefore, however simple the configuration with a third mirror may seem, we
can understand from the above equations that a three-mirror cavity will have a
fundamentally different behavior which cannot be understood as intuitively as
for a Fabry-Perot configuration. However, this increase in complexity makes
the three-mirror cavity system particularly interesting because of the
specific resonance behavior it allows, as we will see in the following part.

###  2.2 Study of the splitting of the resonance

Based on coefficients solutions derived in section 2.1, it is possible to
analyze the optical response of a three-mirror cavity as a function of its
configuration parameters.

For a Fabry-Perot configuration, an amplification of the field inside the
cavity is expected for the resonance condition, i.e. cavity length being equal
to an integer number of input-field half-wavelengths; by extension, scanning
either the cavity length or the input field frequency would periodically lead
to a maximization of transmitted power resulting in a peak distribution whose
shape depends on cavity design and can be described by Airy functions [12].
From a Fabry-Perot configuration, adding a third mirror between the two
initial ones could result, as is represented on Fig. 2, in a resonant pattern
that can no longer be described as a simple peak, as we will discuss in the
last part of this section; there is a splitting of the initial single maxima
[7, 9]. As we will see in this part, the shape of this “double-peak” depends
on the cavity design and can be modulated in strongly different ways by
changing the cavity configuration.

Figure 2: Transmitted field as a function of input-field frequency detuning,
for an initial wavelength of $\lambda=$1064\text{\,}\mathrm{n}\mathrm{m}$$.
The Fabry-Perot cavity has the same overall length as the three-mirror
configuration which is constituted by sub-cavities of length
$L_{1}=L_{2}=$1\text{\,}\mathrm{m}$$. $\Delta\nu=0$ corresponds to either the
resonance of sub-cavities in the three-mirror configuration or the resonance
of Fabry-Perot cavity.

####  2.2.1 Analytical calculation of the spacing between maxima

From Eq. (6), we can derive the analytical relation between three-mirror
cavity parameters and the frequency spacing between each maximum of the double
peak. Squaring that equation and rewriting it in terms of $T_{i}=t_{i}^{2}$
leads to the expression of the global transmission factor:

$T=\frac{T_{1}T_{2}T_{3}}{UU^{*}}$ (7)

where:

$U=e^{2ik(L_{1}+L_{2})}-r_{1}r_{2}e^{2ikL_{2}}-r_{2}r_{3}e^{2ikL_{1}}+r_{1}r_{3}(r_{2}^{2}+t_{2}^{2})$
(8)

and $U^{*}$ its complex conjugate.

Then, using the definition of the wave vector $k$, the product $UU^{*}$ can be
rewritten as a function of the detuning $\Delta\nu$ compared to the input
field frequency:

$\begin{split}UU^{*}&=1+R_{1}R_{2}+R_{2}R_{3}+R_{1}R_{3}(r_{2}^{2}+t_{2}^{2})^{2}\\\
&-2(r_{1}r_{2}+r_{1}r_{2}R_{3}(r_{2}^{2}+t_{2}^{2}))\cos\left(\frac{4\pi}{c}\Delta\nu
L_{1}\right)\\\
&-2(r_{2}r_{3}+R_{1}r_{2}r_{3}(r_{2}^{2}+t_{2}^{2}))\cos\left(\frac{4\pi}{c}\Delta\nu
L_{2}\right)\\\
&+2r_{1}R_{2}r_{3}\cos\left(\frac{4\pi}{c}\Delta\nu(L_{1}-L_{2})\right)\\\
&+2r_{1}r_{3}(r_{2}^{2}+t_{2}^{2})\cos\left(\frac{4\pi}{c}\Delta\nu(L_{1}+L_{2})\right)\end{split}$
(9)

where $R_{i}=r_{i}^{2}$.

To find at which frequency detuning corresponds the maxima as well as the
minima between them, we should solve $dT/d(\Delta\nu)=0$. However, as the
numerator in $T$ is constant, the problem boils down to the search of
solutions of $d(UU^{*})/d(\Delta\nu)=0$, where:

$\begin{split}\frac{d(UU^{*})}{d(\Delta\nu)}&=2(r_{1}r_{2}+r_{1}r_{2}R_{3}(r_{2}^{2}+t_{2}^{2}))\frac{4\pi}{c}L_{1}\sin\left(\frac{4\pi}{c}\Delta\nu
L_{1}\right)\\\
&+2(r_{2}r_{3}+R_{1}r_{2}r_{3}(r_{2}^{2}+t_{2}^{2}))\frac{4\pi}{c}L_{2}\sin\left(\frac{4\pi}{c}\Delta\nu
L_{2}\right)\\\
&-2r_{1}R_{2}r_{3}\frac{4\pi}{c}(L_{1}-L_{2})\sin\left(\frac{4\pi}{c}\Delta\nu(L_{1}-L_{2})\right)\\\
&-2r_{1}r_{3}(r_{2}^{2}+t_{2}^{2})\frac{4\pi}{c}(L_{1}+L_{2})\sin\left(\frac{4\pi}{c}\Delta\nu(L_{1}+L_{2})\right)\end{split}$
(10)

This latter equation involves terms that depend on each part of the cavity. As
a result, in this general case, the spacing of the maxima cannot be linked to
any particular cavity parameter.

From Eq. (10), we could do substantial simplifications by choosing the example
of a symmetrical configuration where each sub-cavity have the same length. It
follows from the condition $L_{1}=L_{2}=L$, that Eq. (10) can be rewritten:

$\begin{split}\frac{d(UU^{*})}{d(\Delta\nu)}&=\frac{8\pi}{c}L\sin\left(\frac{4\pi}{c}\Delta\nu
L\right)\left[r_{1}r_{2}(1+R_{3}(r_{2}^{2}+t_{2}^{2}))\vphantom{\cos\left(\frac{4\pi}{c}\right)}\right.\\\
&\left.+r_{2}r_{3}(1+R_{1}(r_{2}^{2}+t_{2}^{2}))-4r_{1}r_{3}(r_{2}^{2}+t_{2}^{2})\cos\left(\frac{4\pi}{c}\Delta\nu
L\right)\right]\end{split}$ (11)

This new equation has 3 solutions. The more trivial one is $\Delta\nu=0$: this
lets us understand that, for a symmetrical configuration, regardless of the
other cavity parameters, the position of the minima between the two maxima
always remains the same and corresponds to the anti-resonance frequency of a
Fabry-Perot cavity of the same overall length. The second and third solutions
are found for frequencies:

$\Delta\nu_{\pm}=\pm\frac{c}{4\pi
L}\arccos\left[\frac{r_{1}r_{2}(1+R_{3}(r_{2}^{2}+t_{2}^{2}))+r_{2}r_{3}(1+R_{1}(r_{2}^{2}+t_{2}^{2}))}{4r_{1}r_{3}(r_{2}^{2}+t_{2}^{2})}\right]$
(12)

From the latter solutions, which have the same absolute value, we can see that
each maximum will always be equally spaced from $\Delta\nu=0$. As a result,
the resonance patterns achievable for symmetrical configurations are always
symmetrical with maxima spaced by a total frequency range
$\Delta_{\text{Max}}=2\lvert\Delta\nu_{\pm}\rvert$ and centered on zero
detuning.

Finally, by adding the condition of highly reflective mirrors $R_{1}\approx\
R_{2}\approx\ R_{3}\approx\ 1$, Eq. (12) can be simplified even further:

$\Delta\nu_{\pm}\approx\pm\frac{t_{2}c}{4\pi L}$ (13)

This expression, which corresponds to the results presented in [10],
highlights the important role played by the central mirror in highly-
reflective symmetrical cavities.

####  2.2.2 Simulations of resonance behavior as a function of cavity
parameters

To understand in a more general way the variations of the double peak shape as
a function of configuration changes, we realized simulations for which we
computed the three-mirror transmitted field as a function of input field
frequency detuning and cavity-parameter variations. For all of the following
Figures in this section, we started from the same initial meter scale
configuration $L_{1}=L_{2}=$1\text{\,}\mathrm{m}$$, with reflectivities of
mirrors being $R_{1}=R_{2}=R_{3}=0.9$. Note that the frequency detuning value
$\Delta\nu=0$ corresponds to a configuration where each sub-cavity is
microscopically tuned to be resonant for an input field of wavelength
$\lambda=$1064\text{\,}\mathrm{n}\mathrm{m}$$.

The parameters that can be set to design a three-mirror cavity are divided in
two groups: the reflectivities of the mirrors and the lengths of the sub-
cavities. Each category allows for changing in a different way the double-peak
shape. In addition, macroscopic and microscopic variations of sub-cavities’
lengths play different roles. As represented on Fig. 3, from a symmetrical
1m/1m configuration where both maxima are symmetrically distributed around
zero detuning, which corresponds to a horizontal cut at $\Delta L_{1}=0$,
introducing a microscopic asymmetry on the first sub-cavity would result in a
different frequency shift for each maximum. As the asymmetry increases, one
maximum shifts towards $\Delta\nu=0$ while the other shifts away from it. The
limit towards which each peak tends [9] depends on the ratio between the
lengths of the first and second sub-cavities, which is why we can see opposite
behaviors on the upper and lower part of Fig. 3(a). Note that all the
positions of maxima and minima between them can be found by solving Eq. (10),
as shown on Fig. 4.

(a) Transmitted field as a function of input-field detuning and detuning of
the first sub-cavity’s microscopic length. There is two asymptotes, which are
vertical and diagonal, toward which each maxima tend as we increase the
absolute value of $\Delta L_{1}$. The horizontal cuts at $\Delta
L_{1}=$0\text{\,}\mathrm{n}\mathrm{m}$$ and $\Delta
L_{1}=$-50\text{\,}\mathrm{n}\mathrm{m}$$ are represented on plot (3(b))
opposite.

(b) Transmitted field as a function of input-field detuning for two values of
the microscopic length detuning of the first sub-cavity.

Figure 3: Position asymmetry of each double-peak maximum compared to
$\Delta\nu=0$ for different values of the microscopic length detuning of the
first sub-cavity. The reflectivities of mirrors are fixed to
$R_{1}=R_{2}=R_{3}=0.9$, the first sub-cavity’s length is modulated from its
initial value $L_{1}=$1\text{\,}\mathrm{m}$$ while the second sub-cavity’s
length is kept fixed at $L_{2}=$1\text{\,}\mathrm{m}$$. Figure 4: Position of
the extrema of the reflection or transmission factors as a function of $L_{1}$
variation in nanometers. The first sub-cavity’s length is modulated from its
initial value $L_{1}=$1\text{\,}\mathrm{m}$$ while the second sub-cavity’s
length is kept fixed at $L_{2}=$1\text{\,}\mathrm{m}$$.

In addition to the position asymmetry that can be induced by the microscopic
tuning of one or the other sub-cavity’s length, we can change the slope of the
asymptote towards which the maxima moving further away from $\Delta\nu=0$
would tend. On Fig. 5, we see that, for the same first sub-cavity microscopic
length detuning, we obtain a double-peak whose right maximum drifts faster,
and whose left maximum drifts slower, for a macroscopic length ratio
$L_{2}=2L_{1}$ compared to the case where $L_{1}=2L_{2}$. In this way, by
choosing a macroscopic asymmetry between sub-cavities, it is possible to
control the frequency shift that a microscopic detuning would generate. Note
that the microscopic detuning of a macroscopically asymmetric three-mirror
cavity would also result in a change of relative height between maxima as we
see on Fig. 5(c). This is a projection effect that originates from the drift
of maximum transmitted power along the extremum position traces.

(a) Initial lengths of the sub-cavities: $L_{1}=$0.5\text{\,}\mathrm{m}$$,
$L_{2}=$1\text{\,}\mathrm{m}$$.

(b) Initial lengths of the sub-cavities: $L_{1}=$2\text{\,}\mathrm{m}$$,
$L_{2}=$1\text{\,}\mathrm{m}$$.

(c) Transmitted field power as a function of detuning from (a) and (b)
horizontal cuts. The symmetric configuration
$L_{1}=L_{2}=$1\text{\,}\mathrm{m}$$ is represented in orange dashed line as a
reference.

Figure 5: Illustration of peak drift rotation from initial sub-cavities
lengths $L_{1}$ and $L_{2}$ of which the first is modulated by a microscopic
detuning $\Delta L_{1}$. Mirrors reflectivity are set to
$R_{1}=R_{2}=R_{3}=0.9$ and $L_{2}=$1\text{\,}\mathrm{m}$$ for each plot. On
plots (a) and (b), the transmitted field is represented as a function of input
field detuning and first sub-cavity microscopic length detuning. On plot (c),
some specific $L_{1}/L_{2}$ ratio are represented for the same first sub-
cavity detuning.

Although setting microscopic and macroscopic asymmetry between sub-cavities
would result in an asymmetrical change of the positions of the maxima and a
drift of the maximum transmitted power along extremum position traces, it is
possible to act only on the latter by changing the ratio between the first and
the third mirror reflectivity.

On Fig. 6(a) and Fig. 6(b), we can see that setting the first and third
mirror’s reflectivity coefficients so that $R_{1}>R_{3}$ or $R_{1}<R_{3}$ lead
to a drift of maximum power along resonance lines. On Fig. 6(c), we can see
that this result in variation of height ratio between maxima, which, for this
particular case, strictly invert the resonance pattern, as we invert $R_{1}$
and $R_{3}$.

(a) Mirror reflectivities are fixed to $R_{1}=R_{2}=0.9$ and $R_{3}=0.99$.

(b) Mirror reflectivities are fixed to $R_{1}=0.99$ and $R_{2}=R_{3}=0.9$.

(c) Transmitted field power as a function of detuning from (a) and (b)
horizontal cuts.

Figure 6: Power asymmetry between double-peak maxima as a function of the
ratio $R_{1}/R_{3}$; (a) and (b) represent the transmitted field as a function
of input field detuning and the first sub-cavity’s microscopic length
detuning. For each configuration, a horizontal cut at $\Delta
L_{1}=$-50\text{\,}\mathrm{n}\mathrm{m}$$ is represented on plot (c).

On the other hand, the absolute reflectivity of the first and third mirrors
are responsible for the intrinsic widths of the maxima. As can be seen on Fig.
7(a), a fully transparent input mirror, which corresponds to a Fabry-Perot
cavity constituted by the middle and third mirrors only, would lead, as
expected, to a single peak. However, as the first mirror’s transmissivity
decreases, the initial peak splits in two maxima to form a double peak. By
lowering even further the first mirror transmissivity, each peak becomes
sharper but remains at the same frequency as represented on Fig. 7(b). On the
contrary, the variation of middle mirror transmissivity leads to a different
double-peak deformation. As shown on Fig. 8, starting from a configuration
where $t_{2}=1$, which is also equivalent to a Fabry-Perot cavity, and
decreasing its transmissivity would bring each maximum together until they
completely merge for low $t_{2}$ values. Those results shows that the
first/third mirrors and middle mirror play complementary roles on the relative
position of the peaks and their intrinsic width. It is also interesting to
note that, as we have set the microscopic length of each sub-cavity to be
resonant, the equivalent Fabry-Perot cavity constituted by the first and third
mirrors is anti-resonant. This can be seen on Fig. 8 as the transmitted field
power is minimized for input field detuning $\Delta\nu=0$ at $t_{2}=1$.

(a) Transmitted field as a function of input field detuning and first mirror
transmissivity. The horizontal cuts at $t_{1}=0.5$, $t_{1}=0.3$, $t_{1}=0.1$
and $t_{1}=0.01$ are represented on plot (7(b)) opposite.

(b) Normalized transmitted field power as a function of input field detuning
for different values of the transmissivity coefficient of first mirror.

Figure 7: Transmitted field as a function of input mirror transmission $t_{1}$
and input field detuning. Other parameters are fixed to $R_{2}=R_{3}=0.9$ and
$L_{1}=L_{2}=$1\text{\,}\mathrm{m}$$. Figure 8: Transmitted field as a
function of middle mirror transmissivity $t_{2}$ and input field detuning.
Other parameters are fixed to $R_{1}=R_{3}=0.9$ and
$L_{1}=L_{2}=$1\text{\,}\mathrm{m}$$. A horizontal cut at $t_{2}=1$ would
correspond to a Fabry-Perot cavity of length $2\text{\,}\mathrm{m}$ for which
the free spectral range is about $75\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$.

As we have seen, the double peak shape can be modulated in strongly different
ways thanks to the mirrors’ reflectivities and sub-cavities’ lengths, however
it does not show off for all three-mirror cavity configuration. A necessary
condition to obtain the double peak pattern is that each sub-cavity has to be
close to its own resonance. In this way, the ratio between intrinsic sub-
cavities’ free spectral range governs the ratio between the number of double
and single peaks [9]. Moreover, as we can see in Fig. 9, single peaks
appearing between double peaks are highly distorted compared to those of a
two-mirror Fabry-Perot cavity.

Figure 9: Transmitted field power as a function of input-field detuning, for a
three-mirror configuration with sub-cavities of respective lengths
$L_{1}=$1\text{\,}\mathrm{m}$$ and $L_{2}=$3\text{\,}\mathrm{m}$$ (solid
line), and two separate Fabry-Perot configurations with respective lengths
$L_{\text{tot}}=$1\text{\,}\mathrm{m}$=L_{1}$ and
$L_{\text{tot}}=$3\text{\,}\mathrm{m}$=L_{2}$ (dashed lines).

To conclude this part, from these simulations we show that the shape of the
double peak is almost completely adaptable by designing a three-mirror cavity
for which the double peak appearance conditions are met. However, the
modulation of the double-peak shape goes beyond the scope of the design.
Indeed, as the macroscopic distances separating the mirrors of an already
implemented cavity could be microscopically detuned in real time, it could be
possible to let the resonance behavior of the system be adaptable and not only
fixed by design as is the case with Fabry-Perot cavities. Conversely, this
important increase of versatility comes with the counterpart of more complex
stability behavior, this is what we present in the next part.

##  3 Stability

###  3.1 Preliminary considerations

Up to this point, we have described the particular resonance behavior of a
linear three-mirror cavity, but we simplified our approach by considering
infinite flat mirrors. Still ignoring diffraction or other loss sources, we
are now going to focus on stability behavior and how it dictates their
geometries. The stability properties studied here are purely geometrical, so
we will not consider the coupling by the intermediate mirror and restrict our
study to the stability of two cavities sharing a common mirror and a common
beam geometry. In what follows we will thus describe a three-mirror cavity as
made of two half-cavities or equivalently two cavities. It is clear that those
two (half-)cavities are simple two-mirror Fabry-Perot cavities sharing a
common mirror.

A stable three-mirror cavity geometry can be of four types, ignoring the fact
that the substrate of the intermediate mirror can lie in either half-cavity.
Those geometries are illustrated in Fig. 10: (A) a flat central mirror and two
concave end mirrors; (B) a flat end mirror followed by a concave intermediate
mirror and a concave end mirror; (C) a biconcave first cavity followed by a
concave end mirror, and (D) two convex-concave cavities. The second geometry
(B) is in fact the same as the third one (C), the flat-concave first cavity
being one half of a biconcave cavity where the flat mirror lies at the
position of the beam waist. We will not study here the latter geometry (D) as
it does not allow for designing cavities far from marginal stability. We are
interested in three-mirror cavities where the two half-cavities have different
lengths. One half-cavity of length $L_{1}$ is built with mirrors $M_{1}$ and
$M_{2}$, and the second half-cavity of length $L_{2}$ is composed of mirrors
$M_{2}$ and $M_{3}$. Each mirror $M_{i}$ has a radius of curvature $\rho_{i}$
and a flat surface anti-reflective backside.

Figure 10: Three-mirror stable cavities. (A) Concave-plano-concave, (B) plano-
concave half-cavity and convex-concave second half, (C) biconcave half-cavity
and convex-concave second half-cavity, (D) two convex-concave half-cavities.

The design of a stable three-mirror cavity is not as simple as it may seem if
one wants both halves of the cavity to be far from marginal stability. The
difficulty arises from the fact that a stable three-mirror cavity cannot have
a beam waist in both half-cavities, one of the cavity imposes the beam
geometry and the second half must then be designed to be adapted to this
imposed geometry. The stability criterion of a two-mirror cavity can either be
expressed (i) using the $g$ parameters as defined for example in [13]:

$g_{1,2}=1-\frac{L}{\rho_{1,2}}$ (14)

where $L$ is the cavity length and $\rho_{1,2}$ are the mirrors’ radii of
curvature, the relation:

$0<g_{1}g_{2}<1$ (15)

ensuring the stability of the cavity, or (ii) using the Gouy phase accumulated
in a cavity. If the Gouy phase is too small compared to the width of the
resonance peaks of the cavity, then high-order modes can easily couple to the
fundamental mode, making the cavity unstable. In fact, both descriptions of
stability contain the same information, as the stability of a cavity
essentially depends on the Rayleigh range of the beam compared to the lengths
of the cavities.

In all the following calculations we will use the $g$ parameters of the three
mirrors. The stability of the first cavity will be described using $g_{1}$ and
$g_{2}$ parameters, while the second cavity will be described using
$g_{2}^{*}$ and $g_{3}$ parameters. Usually $g_{2}^{*}\neq g_{2}$ because of
the effect of transmission by the anti-reflective back side of the
intermediate mirror. Indeed, the second cavity “sees” the intermediate mirror
$M_{2}$ as a mirror of radius of curvature $\rho_{2}^{*}$ that is equal to:

$\rho_{2}^{*}\approx\frac{-\rho_{2}}{n}$ (16)

using a first order approximation of Snell’s law where $n$ is the index of
refraction of mirror $M_{2}$ substrate.

In the process of designing a three-mirror non-symmetrical cavity, one half-
cavity necessarily dictates the design of the other half. We will show in
Section 3.2 that in configuration (A) on Fig. 10 of concave plano-concave
cavity, whatever the cavity which is designed first, it is not possible to
obtain a very stable second cavity if the length ratio $L_{1}/L_{2}$ is far
from unity. However configuration (C) on Fig. 10, with a biconcave cavity and
a convex-concave cavity, offers stable solutions for ratio $L_{1}/L_{2}$
farther from unity for some configurations as shown in Section 3.3. Finally we
will show in Section 3.4 the impact on the overall cavity stability of
designing the intermediate mirror as a lens-mirror component.

###  3.2 Concave-plano-concave cavity

The flat central mirror has a flat anti-reflective side, that side lays inside
effect of refraction at this anti-reflective side by assuming that the mirror
thickness is much smaller than the beam’s Rayleigh range so the beam radius of
curvature at the anti-reflective side location is very large and its change
has a small influence on the stability of the system. Let us start by fixing
the geometry of the first half-cavity by imposing its stability factor. We
choose to express it as:

$g_{1}g_{2}=\frac{1}{2}+\varepsilon$ (17)

where $\varepsilon\in\left]-1/2,1/2\right[$. The intermediate mirror being
flat, $g_{2}=1$ and we can express $\rho_{1}$ the radius of curvature of the
concave mirror forming the first cavity as:

$\rho_{1}=\frac{2L_{1}}{1-2\varepsilon}$ (18)

The Rayleigh range $z_{R}$ depends on the beam waist size $w_{0}$ as
$z_{R}=\pi w_{0}^{2}/\lambda$, so it depends on the cavity geometry. The
relationship between the Rayleigh range, the input mirror radius of curvature
$\rho_{1}$ and the first half-cavity length $L_{1}$ is [13]:

$z_{R}^{2}=L_{1}\left(\rho_{1}-L_{1}\right)$ (19)

so:

$z_{R}^{2}=L_{1}^{2}\frac{1+2\varepsilon}{1-2\varepsilon}$ (20)

This Rayleigh range value is the same in the second half-cavity because both
cavities share the same beam waist size at the same location. The radius of
curvature $\rho_{3}$ of the concave mirror forming the second cavity of length
$L_{2}$ is thus:

$\rho_{3}=L_{2}+\frac{z_{R}^{2}}{L_{2}}=L_{2}+\frac{L_{1}^{2}}{L_{2}}\left(\frac{1+2\varepsilon}{1-2\varepsilon}\right)$
(21)

The two half-cavities have different lengths, $L_{2}=mL_{1}$ with
$m\in\mathbb{R^{+*}}$ $\rho_{3}$ can be expressed as:

$\rho_{3}=L_{2}\left(1+\frac{1}{m^{2}}\left(\frac{1+2\varepsilon}{1-2\varepsilon}\right)\right)$
(22)

and finally the stability factor of the second half-cavity is:

$g_{2}g_{3}=g_{3}=1-\frac{1}{1+\frac{1}{m^{2}}\left(\frac{1+2\varepsilon}{1-2\varepsilon}\right)}$
(23)

Values of this stability factor for different values of $m=L_{2}/L_{1}$ are
shown on Fig. 11.

Figure 11: $g_{2}g_{3}$ vs. $g_{1}g_{2}$ for different values of $m$ in a
concave-plano-concave cavity.

We see that as soon as the value of $m$ becomes either large or small compared
to one, the stability factor $g_{2}g_{3}$ rapidly either increases towards 1
or decreases to 0, unless the value of $\varepsilon$ is adapted to compensate
this behavior. We see that as soon as $m$ exceeds a few units it becomes
impossible to strongly stabilize both cavities simultaneously, so the concave-
plano-concave geometry is not adapted to large length differences.

A simple explanation is this: the intermediate flat mirror imposes its $g=1$.
If the short cavity is designed first, it imposes a short Rayleigh range.
Because the long cavity length is large compared to this Rayleigh range, its
end mirror radius of curvature is close to the cavity length, so the $g$
parameter of this mirror is always close to zero and $g_{2}g_{3}$ of the long
cavity is close to zero. Conversely, if the long cavity is designed first then
the $g_{1}$ parameter of the short cavity is close to one, and so is the
$g_{1}g_{2}$ of the short cavity. It will appear later that building a three-
mirror cavity made of two stable cavities requires that the magnitude of the
intermediate mirror $g_{2}$ parameter seen from the long cavity be large
enough, either positive or negative, to compensate the necessarily low value
of the end mirror $g_{3}$ parameter.

###  3.3 Biconcave cavity and convex-concave cavity

It is equivalent to build first the biconcave cavity and then the convex-
concave one or the other way around. To our knowledge there is no well-known
“standard” geometry for a concave-convex cavity, thus we chose to design first
the biconcave cavity of length $L_{1}$ and stability criterion $g_{1}g_{2}$
and then the convex-concave one of length $L_{2}$ and stability criterion
$g_{2}^{*}g_{3}$ as depicted on configuration (C) of Fig. 10 in Section 3.1.

In the simple case of a symmetric biconcave cavity of fixed length $L_{1}$ and
given stability $g_{1}g_{2}$, there are 2 possible radii of curvature for its
mirrors

$\rho_{1}=\rho_{2}=\frac{L_{1}}{1\pm\sqrt{g_{1}g_{2}}}$ (24)

and conversely for a fixed radius of curvature $\rho_{1}=\rho_{2}$ and given
stability $g_{1}g_{2}$, there are two possible cavity lengths.

Then for the convex-concave cavity of length $L_{2}$, a given radius of
curvature $\rho_{2}$ of the intermediate mirror, with refractive index $n$
substrate inside the cavity, and a given stability $g_{2}^{*}g_{3}$ there is
only one possible radius of curvature $\rho_{3}$

$\rho_{3}=\frac{L_{2}}{1-\frac{\rho_{2}g_{2}^{*}g_{3}}{\rho_{2}+nL_{2}}}$ (25)

and conversely for a fixed radius of curvature $\rho_{3}$, a given radius of
curvature $\rho_{2}$ of the intermediate mirror, with refractive index $n$
substrate inside the cavity, and a given stability $g_{2}^{*}g_{3}$ there is
one possible positive cavity length $L_{2}$.

Now we geometrically have two choices: either build a long biconcave cavity or
a short biconcave cavity.

####  3.3.1 Long biconcave cavity

If we choose to build a biconcave long cavity, it happens to make it nearly
impossible to obtain a stable short convex-concave cavity. Let us illustrate
this point by studying a symmetric biconcave cavity of length
$L_{\text{long}}$, with $\rho_{\text{long}}$ mirror radii of curvature and
calculating its Rayleigh range as a function of the stability criterion
$g_{1}g_{2}$. Equation (19) becomes

$z_{R}^{2}=\frac{L_{\text{long}}}{2}\left(\rho_{\text{long}}-\frac{L_{\text{long}}}{2}\right)$
(26)

and consequently the Rayleigh range is

$z_{R}=\frac{L_{\text{long}}\sqrt{1-g_{1}g_{2}}}{2\left(1-\sqrt{g_{1}g_{2}}\right)}$
(27)

The result is shown in Fig. 12. As expected, for Rayleigh range below half of
the cavity length, the cavity becomes unstable. Thus a long biconcave stable
cavity imposes that the Rayleigh range be of the order of magnitude of its
length, $z_{R}\approx L_{\text{long}}$. The beam waist being at the center of
the long cavity, the Gouy phase accumulated in the short one, $\phi_{Gs}$ is
of the order of:

$\phi_{Gs}=\arctan\left(\frac{L_{\text{long}}/2+L_{\text{short}}}{z_{R}}\right)-\arctan\left(\frac{L_{\text{long}}/2}{z_{R}}\right)$
(28)
$\phi_{Gs}\approx\arctan\left(\frac{1}{2}+\frac{L_{\text{short}}}{L_{\text{long}}}\right)-\arctan\left(\frac{1}{2}\right)$
(29)

If one requires a Gouy phase larger than $None$ then the long cavity must not
be longer than $2.7$ times the short cavity, which strongly limits the leeway
to choose a length ratio.

Figure 12: Rayleigh range in cavity length units of a symmetric biconcave
cavity as a function of the stability criterion.

The same conclusion can be obtained by inspecting the $g_{2}^{*}g_{3}$ value
of the short cavity. As the long cavity is designed to be stable, then

$g_{2}^{*}g_{3}\approx\left(1+\frac{L_{\text{short}}}{L_{\text{long}}}\right)\times\left(1-\frac{L_{\text{short}}}{L_{\text{long}}}\right)$
(30)

which brings us to the same conclusion we drew considering the stability from
a Gouy phase point of view.

Let’s just take as an example a long cavity with
$L_{1}=L_{\text{long}}=$50\text{\,}\mathrm{m}$$ and a short cavity with
$L_{2}=L_{\text{short}}=$5\text{\,}\mathrm{m}$$ and compute the stability
factor $g_{1}g_{2}$ of the long cavity as a function of $g_{2}^{*}g_{3}$ for
different values of $g_{3}$. The result is presented in Fig. 13(a) and shows
that the long cavity can be made stable if $g_{3}\ll 1$, but both cavities
cannot be made simultaneously very stable.

(a) Stability of the long biconcave cavity vs. stability of the short concave-
convex cavity for different values of $g_{3}$

(b) Radius of curvature of the intermediate mirror seen from the long cavity
in metres as a function of short cavity stability for different values of
$g_{3}$.

Figure 13: Geometrical properties of the long cavity as a function of the
stability of the short cavity.

Fig. 13(b) shows the evolution of the intermediate mirror radius of curvature
$\rho_{2}$ as a function of the stability of the short cavity. One sees that
this radius of curvature increases rapidly with decreasing $g_{2}^{*}g_{3}$
whatever the value of $g_{3}$, or equivalently $\rho_{3}$. This means that the
intermediate mirror tends to be flat, under most circumstances, a
configuration which has already been described as not ideal.

####  3.3.2 Short biconcave cavity

In the case where a stable short biconcave symmetrical cavity is designed
first, the Rayleigh range is of the order of its length $L_{\text{short}}$
according to Eq. (24). The beam propagating inside the long cavity can then
accumulate a significant Gouy phase and this long cavity can thus be made
stable.

The same conclusion can be obtained by inspecting the $g_{2}^{*}g_{3}$ value
of the long cavity. As the long cavity is much longer than the Rayleigh range,
then the radius of curvature of the end mirror, $\rho_{3}$, is only a little
longer than the long cavity’s length, which results in $g_{3}\gtrsim 0$. The
reason for this large stability is mainly due to the fact that the
intermediate radius of curvature $\rho_{2}^{*}$ is negative and shorter than
the long cavity length, this results in a large $g_{2}^{*}$ value and as
$g_{3}$ is positive their product has a non negligible value.

In practice, we start by building a short symmetrical biconcave cavity with a
stability $g_{1}g_{2}=0.5$ using Eq. (24). We then build a long convex-concave
cavity adapted to the beam geometry following using Eq. (25). In that case the
only limitation of the long cavity length is the available mirror size for the
end cavity mirror.

###  3.4 Intermediate mirror designed as a lens-mirror component

In the three-mirror cavity design we studied up to this point, the anti-
reflective backside of the intermediate mirror was a flat surface, but this
assumption is not mandatory, it is simply the common approach to usual mirror
design where a flat back surface lies outside the area of interest, which is
in front of the mirror. If we relax this constraint and allow the anti-
reflective surface to be curved, we can build an intermediate mirror that acts
simultaneously as a curved mirror and as a lens. This extra degree of freedom
may allow to design stable three-mirror cavities, even with very large length
ratios.

The possibility to add curvature to the anti-reflective side of the
intermediate mirror allows for changing the radius of curvature of the beam at
the intermediate mirror location as seen through the anti-reflection side that
lies, in the following example, in the first cavity. This modifies the value
of $g_{2}$ and for a given value of $L_{1}$ dictates a value for $g_{1}$ and
thus the stability factor $g_{1}g_{2}$ of the first cavity. As an example let
us study the case of a main stable symmetric biconcave fifty-meter-long cavity
with $g_{2}^{*}=g_{3}=1/\sqrt{2}$, coupled to a shorter, five or three-meter
cavity. The anti-reflective curved surface lies in the short cavity and the
long cavity, designed first, imposes the beam geometry. For each value of the
radius of curvature of the anti-reflective surface $\rho_{AR}$ we compute the
value of $\rho_{1}$ which maximizes $g_{1}g_{2}$.

Figure 14: Stability factor $g_{1}g_{2}$ of the short cavity of length $L_{1}$
as a function of the radius of curvature of the anti-reflective coating. The
long cavity length $L_{2}=50\>\mathrm{m}$.

The result, shown in Fig. (14), demonstrates that there exist some ranges of
radii of curvature of the anti-reflective surface that make the short cavity
very stable. The extension of the region of stability strongly depends on the
ratio of the cavity lengths. It is interesting to remark that the point where
$g_{1}g_{2}=0$ corresponds to a confocal cavity geometry. This can be
understood as follows: by design we constrain the short cavity to be stable,
even marginally. This implies that the $g_{1}$, $g_{2}$ pair always lies
within one of the two stability regions of the cavity. In Fig. 15 the two
regions lie between the $g_{1}=0$ and $g_{2}=0$ lines and the two arches of
the $g_{1}g_{2}=1$ hyperbola. As the radius of curvature of the anti-
reflective coating is continuously changed, the $g_{1},g_{2}$ parameter couple
follows a path within these regions. However for some value of the anti-
reflective coating radius of curvature, the sign of $g_{2}$ changes. This
forces the sign of $g_{1}$ to flip as well. The only common point of the two
stability half regions that allows for switching between two positive $g$
values and two negative ones is precisely the point where $g_{1}=g_{2}=0$,
which represents a confocal cavity.

Figure 15: Location of the $g_{1},g_{2}$ values, in red, for a continuous scan
of the radius of curvature of the anti-reflective side. The stability region
is represented by filed area framed by blue dashed curves.

This result should not hide conceivable technical difficulties to use such
kind of optical components: aberrations or mode coupling are much more
enhanced by tilt and misalignment of curved surfaces than flat ones, and the
technical ability to realize optical surfaces with radii of curvature
precisely matched to requirements and sharing the same optical axis is far
from immediate. Another point that may be of importance is that by changing
the beam geometry in the first cavity one modifies its Gouy phase through
propagation, which may lead to some issues in particular optical designs.

##  4 Conclusion

We have presented a few, non-exhaustive, original properties of three-mirror
linear optical resonators, pointing their uniqueness and their limits in
practical use. The coupling of two Fabry-Perot cavities sharing a common
mirror cannot simply be reduced to the succession of two distinct resonators
but that it creates a coupling of optical properties just like coupled
mechanical or electrical resonators do. It results from this coupling a
splitting of the resonance into a double-peak pattern. We have demonstrated
that the reflectivities of mirrors and their relative spacing play different
roles on the relative height between double-peak maxima, their position and
their intrinsic width. Although it is possible to adjust the double peak shape
in very different ways, special attention must be paid to asymmetrical
configurations as the stability go inversely with the asymmetry for cavity
with a flat central mirror. To some extent, these instabilities can be
compensated by the use of curved anti-reflective surfaces but short biconcave
cavity and long convex-concave one offers more stable solutions.

The study of these original properties may be valuable to fields of laser
optics where shaping the behavior of optical resonators is an issue. In
particular, the double-peak pattern could be controlled in real time by tuning
the spacing between mirrors. This increase of controllability could be of
interest for the optimization of future gravitational-wave detectors using
squeezed states of light which has initially motivated our analysis.

##  5 Acknowledgement

The authors acknowledge the support of the French Agence Nationale de la
Recherche (ANR), under Grant Nos. ANR-18-JSTQ-0002 (project QFilter). The
authors thank Eleonora Capocasa for her helpful comments on this publication.

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#  Rigidity of Polyhedral Surfaces, III

Feng Luo  Department of Mathematics, Rutgers University, New Brunswick, New
Jersey 08854<EMAIL_ADDRESS>To Dennis Sullivan on the occasion of his
seventieth birthday

(Date: Oct. 1, 2010.)

###### Abstract.

This paper investigates several global rigidity issues for polyhedral surfaces
including inversive distance circle packings. Inversive distance circle
packings are polyhedral surfaces introduced by P. Bowers and K. Stephenson in
[2] as a generalization of Andreev-Thurston’s circle packing. They conjectured
that inversive distance circle packings are rigid. Using a recent work of R.
Guo [9] on variational principle associated to the inversive distance circle
packing, we prove rigidity conjecture of Bowers-Stephenson in this paper. We
also show that each polyhedral metric on a triangulated surface is determined
by various discrete curvatures introduced in [12], verifying a conjecture in
[12]. As a consequence, we show that the discrete Laplacian operator
determines a Euclidean polyhedral metric up to scaling.

###### Key words and phrases:

polyhedral metrics, discrete curvatures, rigidity

###### 1991 Mathematics Subject Classification:

Primary 54C40, 14E20; Secondary 46E25, 20C20

The work is supported in part by a NSF Grant.

##  1\. Introduction

###  1.1.

This is a continuation of the study of polyhedral surfaces [12], [13]. The
paper focuses on inversive distance circle packings introduced by Bowers and
Stephenson and several other rigidity issues. Using a recent work of Ren Guo
[9], we prove a conjecture of Bowers-Stephenson that inversive distance circle
packings are rigid. Namely, a Euclidean inversive distance circle packing on a
compact surface is determined up to scaling by its discrete curvature. This
generalizes an earlier result of Andreev [1] and Thurston [17] on the rigidity
of circle packing with acute intersection angles. In [12], using 2-dimensional
Schlaefli formulas, we introduced two families of discrete curvatures for
polyhedral surfaces and conjectured that each of one these discrete curvatures
determines the polyhedral metric (up to scaling in the Euclidean case). We
verify this conjecture in the paper. One consequence is that for a Euclidean
or spherical polyhedral metric on a surface, the cotangent discrete Laplacian
operator determines the metric (up to scaling in the case of Euclidean
metric). The theorems are proved using variational principles and are based on
the work of [9] and [12]. The main idea of the paper comes from reading of
[4], [7] and [15].

###  1.2.

Recall that a Euclidean (or spherical or hyperbolic) polyhedral surface is a
triangulated surface with a metric, called a polyhedral metric, so that each
triangle in the triangulation is isometric to a Euclidean (or spherical or
hyperbolic) triangle. To be more precise, let $\mathbb{E}^{2}$,
$\mathbb{S}^{2}$ and $\mathbb{H}^{2}$ be the Euclidean, the spherical and the
hyperbolic 2-dimensional geometries. Suppose $(S,T)$ is a closed triangulated
surface so that $T$ is the triangulation, $E$ and $V$ are the sets of all
edges and vertices. A $K^{2}$ ($K^{2}$ = $\mathbb{E}^{2}$, or
$\mathbb{S}^{2}$, or $\mathbb{H}^{2}$) polyhedral metric on $(S,T)$ is a map
$l:E\to\mathbb{R}$ so that whenever $e_{i},e_{j},e_{k}$ are three edges of a
triangle in $T$, then

$l(e_{i})+l(e_{j})>l(e_{k}),$

and if $K^{2}=\mathbb{S}^{2}$, in addition to the inequalities above, one
requires

$l(e_{i})+l(e_{j})+l(e_{k})<2\pi.$

Given $l:E\to\mathbb{R}$ satisfying the inequalities above, there is a metric
on the surface $S$, called a polyhedral metric, so that the restriction of the
metric to each triangle is isometric to a triangle in $K^{2}$ geometry and the
length of each edge $e$ in the metric is $l(e)$. We also call
$l:E\to\mathbb{R}$ the edge length function. For instance, the boundary of a
generic convex polytope in the 3-dimensional space $\mathbb{E}^{3}$, or
$\mathbb{S}^{3}$ or $\mathbb{H}^{3}$ of constant curvature $0,1,$ or $-1$ is a
polyhedral surface. The discrete curvature $k$ of a polyhedral surface is a
function $k:V\to\mathbb{R}$ so that $k(v)=2\pi-\sum_{i=1}^{m}\theta_{i}$ where
$\theta_{i}$’s are the angles at the vertex $v$. See figure 1.

Since the discrete curvature is built from inner angles of triangles, we
consider inner angles of triangles as the basic unit of measurement of
curvature. Using inner angles, we introduce three families of curvature like
quantities in [12]. The relationships between the polyhedral metrics and
curvatures are the focus of the study in this paper.

######  Definition 1.1.

([12]) Let $h\in\mathbb{R}$. Given a $K^{2}$ polyhedral metric on $(S,T)$
where $K^{2}$ $=\mathbb{E}^{2}$, or $\mathbb{S}^{2}$ or $\mathbb{H}^{2}$, the
$\phi_{h}$ curvature of a polyhedral metric is the function
$\phi_{h}:E\to\mathbb{R}$ sending an edge $e$ to:

(1.1)
$\phi_{h}(e)=\int_{\pi/2}^{a}\sin^{h}(t)dt+\int_{\pi/2}^{a^{\prime}}\sin^{h}(t)dt$

where $a,a^{\prime}$ are the inner angles facing the edge $e$. See figure 1.

The $\psi_{h}$ curvature of the metric $l$ is the function
$\psi_{h}:E\to\mathbb{R}$ sending an edge $e$ to

(1.2)
$\psi_{h}(e)=\int^{\frac{b+c-a}{2}}_{0}\cos^{h}(t)dt+\int^{\frac{b^{\prime}+c^{\prime}-a^{\prime}}{2}}_{0}\cos^{h}(t)dt$

where $b,b^{\prime},c,c^{\prime}$ are inner angles adjacent to the edge $e$
and $a,a^{\prime}$ are the angles facing the edge $e$. See figure 1.

Figure 1.

The curvatures $\phi_{0}$ and $\psi_{0}$ were first introduced by I. Rivin
[Ri] and G. Leibon [Le] respectively. If the surface $S=\mathbb{S}^{2}$, then
these curvatures are essentially the dihedral angles of the associated
3-dimensional hyperbolic polyhedra at edges. The curvature
$\phi_{-2}(e)=-\cot(a)-\cot(a^{\prime})$ is the discrete (cotangent) Laplacian
operator on a polyhedral surface derived from the finite element approximation
of the smooth Beltrami Laplacian on Riemannian manifolds.

One of the remarkable theorems proved by Rivin [15] is that a Euclidean
polyhedral metric on a triangulated surface is determined up to scaling by its
$\phi_{0}$ discrete curvature. In particular, he proved that an ideal convex
hyperbolic polyhedron is determined up to isometry by its dihedral angles.

We prove,

######  Theorem 1.2.

Let $(S,T)$ be a closed triangulated connected surface. Then for any
$h\in\mathbb{R}$,

(1) a Euclidean polyhedral metric on $(S,T)$ is determined up to isometry and
scaling by its $\phi_{h}$ curvature.

(2) a spherical polyhedral metric on $(S,T)$ is determined up to isometry by
its $\phi_{h}$ curvature.

(3) a hyperbolic polyhedral surface is determined up to isometry by its
$\psi_{h}$ curvature.

We remark that theorem 1.2(1) for $h=0$ was aforementioned Rivin’s theorem.
However, our proof of Rivin’s theorem is different from that in [15] and we
use the variational principle established by Cohen-Kenyon-Propp [5]. Theorem
1.2(3) for $h=0$ was first proved by Leibon [11]. Theorem 1.2(2) for $h=0$ was
proved in [14] and theorem 1.2(2) and (3) for $h\leq-1$ or $h\geq 0$ was
proved in [12].

Take $h=-2$ in theorem 1.2, we obtain,

######  Corollary 1.3.

(1) A connected Euclidean polyhedral surface is determined up to scaling by
its discrete Laplacian operator.

(2) A spherical polyhedral surface is determined by its discrete Laplacian
operator.

Note that for a Euclidean polyhedral surface, $\phi_{h}=\psi_{h}$. There
remain two questions on whether $\phi_{h}$ curvature determines a hyperbolic
polyhedral surface or whether $\psi_{h}$ curvature determines a spherical
polyhedral surface. It seems the results may still be true in these cases.

### 1.3.

Inversive distance circle packings are polyhedral metrics on a triangulated
surface introduced by Bowers and Stephenson in [2]. An expansion of the
discussion of [2] is in [3]. See also [16]. They are generalizations of
Andreev and Thurston’s circle packings. Unlike the case of Andreev and
Thurston where adjacent circles are intersecting, Bowers and Stephenson allow
adjacent circles to be disjoint and measure their relative positions by the
inversive distance. As observed in [2], this relaxation of intersection
condition is very useful for practical applications of circle packing to many
fields, including medical imaging and computer graphics. Based on extensive
numerical evidences, they conjectured the rigidity and convergence of
inversive distance circle packings in [2]. Our result shows that Bowers-
Stephenson’s rigidity conjecture holds. The proof is based on a recent work of
Ren Guo [9] which established a variational principle for inversive distance
circle packings. A very nice geometric interpretation of the variational
principle was given in [8].

We begin with a brief recall of the inversive distance in Euclidean,
hyperbolic and spherical geometries. See [3] for a more detailed discussion.
Let $K^{2}$ be $\mathbb{E}^{2}$, or $\mathbb{H}^{2}$ or $\mathbb{S}^{2}$.
Given two circles $C_{1},C_{2}$ in $K^{2}$ centered at $v_{1},v_{2}$ of radii
$r_{1}$ and $r_{2}$ so that $v_{1},v_{2}$ are of distance $l$ apart, the
inversive distance $I=I(C_{1},C_{2})$ between the circles is given by

(1.3) $I=\frac{l^{2}-r_{1}^{2}-r_{2}^{2}}{2r_{1}r_{2}}$

in the Euclidean plane,

(1.4) $I=\frac{\cosh(l)-\cosh(r_{1})\cosh(r_{2})}{\sinh(r_{1})\sinh(r_{2})}$

in the hyperbolic plane and

(1.5) $I=\frac{\cos(l)-\cos(r_{1})\cos(r_{2})}{\sin(r_{1})\sin(r_{2})}$

in the 2-sphere. See [9] for more details on (1.4) and (1.5). If one considers
$\mathbb{E}^{2}$, $\mathbb{H}^{2}$ and $\mathbb{S}^{2}$ as appeared in the
infinity of the hyperbolic 3-space $\mathbb{H}^{3}$, then $C_{1}$ and $C_{2}$
are the boundary of two totally geodesic hyperplanes $D_{1}$ and $D_{2}$. The
inversive distance $I$ is essentially the hyperbolic distance (or the
intersection angle) between $D_{1}$ and $D_{2}$. In particular, for the
Euclidean plane $\mathbb{E}^{2}$, the inversive distance $I(C_{1},C_{2})$ is
invariant under the inversion and hence the name.

Bowers and Stephenson’s construction of an inversive distance circle packing
with prescribed inversive distance on a triangulated surface $(S,T)$ is as
follows. Fix once and for all a vector $I\in\mathbb{[}-1,\infty)^{E}$, called
the inversive distance.

In the Euclidean case, for any $r\in\mathbb{R}_{>0}^{V}$, called the radius
vector, define the edge length function $l\in\mathbb{R}_{>0}^{E}$ by the
formula

(1.6) $l(e)=\sqrt{r(v)^{2}+r(u)^{2}+2r(v)r(u)I(e)}$

where the end points of the edge $e$ is $\\{u,v\\}$. If $l(e)$’s satisfy the
triangular inequalities that

(1.7) $l(e_{i})+l(e_{j})>l(e_{k})$

for three edges $e_{i},e_{j},e_{k}$ of each triangle in $T$, then the length
function $l:E\to\mathbb{R}$ sending $e$ to $l(e)$ defines a Euclidean
polyhedral metric on $(S,T)$ called the inversive distance circle packing with
inversive distance $I(e)$ at edge $e$. Note that if $I(e)\in[0,1]$ for all
$e$, the polyhedral metric is the circle packing investigated by Andreev and
Thurston where the intersection angle between two circles at the end points of
an edge is $\arccos(I(e))$.

In the hyperbolic geometry, one uses

(1.8) $l(e)=\cosh^{-1}(\cosh(r(v))\cosh(r(u))+I(e)\sinh(r(v))\sinh(r(u))$

as the length of an edge. If (1.7) holds, then the lengths $l(e)$’s define a
hyperbolic inversive distance circle packing with inversive distances $I$ on
$(S,T)$. The spherical inversive distance circle packing is defined similarly
with additional condition on $l(e)$’s that

$l(e_{i})+l(e_{j})+l(e_{k})<2\pi$

for each triangle with edges $e_{i},e_{j},e_{k}$.

The geometric meaning of these polyhedral metrics is the following. In each
metric, if one draws a circle of radius $r(v)$ at each vertex $v$, then
inversive distance of two circles at the end points of an edge $e$ is the
given number $I(e)$.

Our result which solves Bowers-Stephenson’s rigidity conjecture is the
following.

######  Theorem 1.4.

Given a closed triangulated connected surface $(S,T)$ with the set of edges
$E$ and $I\in\mathbb{R}_{\geq 0}^{E}$ considered as the inversive distance,

(1) a hyperbolic inversive distance circle packing metric on $(S,T)$ of
inversive distance $I$ is determined by its discrete curvature
$k:V\to\mathbb{R}$.

(2) an Euclidean inversive distance circle packing metric on $(S,T)$ of
inversive distance $I$ is determined by its discrete curvature
$k:V\to\mathbb{R}$ up to scaling.

Note that for $I\in[0,1]^{E}$, the above result was Andreev-Thurston’s
rigidity for circle packing with intersection angles between $[0,\pi/2]$. It
seems the similar result may be true for $I\in[-1,\infty)^{E}$.

### 1.4.

The paper is organized as follows. In §2, we prove an extension lemma for
angles of triangles. We also establish a criterion for extending a locally
convex function to convex function. In §3, we prove theorem 1.4. Theorem 1.2
is proved in §4.

The following notations and conventions will be used in the paper. We use
$\mathbb{R}$, $\mathbb{R}_{>0}$, $\mathbb{R}_{\geq 0}$, $\mathbb{R}_{<0}$ to
denote the sets of all real numbers, positive real numbers, non-negative real
numbers, and negative real numbers respectively. If $X$ is a set,
$\mathbb{R}^{X}=\\{f:X\to\mathbb{R}$} is the vector space of all functions on
$X$. If $A$ is a subspace of a topological space $X$, then the closure of $A$
in $X$ is denoted by $\bar{A}$.

We thank Ren Guo for comments and careful reading of the manuscript.

##  2\. Convex Extension of Locally Convex Functions

###  2.1. Continuous extension by constants

######  Definition 2.1.

Suppose $A$ is a subspace of a topological space $X$ and $f:A\to Y$ is
continuous. If there exists a continuous function $F:X\to Y$ so that
$F|_{A}=f$ and $F$ is a constant function on each connected component of
$X-A$, then we say $f$ can be extended continuously by constant functions to
$X$.

Note that if each connected component of $X-A$ intersects the closure of $A$,
then the extension function $F$ is uniquely determined by $f$.

The key observation of the paper is the following simple lemma.

######  Lemma 2.2.

Suppose $\Delta$ is a triangle in the Euclidean plane $\mathbb{E}^{2}$, or the
hyperbolic plane $\mathbb{H}^{2}$, or the 2-sphere $\mathbb{S}^{2}$ so that
its edge lengths are $l_{1},l_{2},l_{3}$ and its inner angles are
$\theta_{1},\theta_{2},\theta_{3}$. Assume that $\theta_{i}$’s angle is
opposite to the edge of length $l_{i}$ for each $i$. Consider
$\theta_{i}=\theta_{i}(l)$ as a function of $l=(l_{1},l_{2},l_{3})$.

  1. (1)

If $\Delta$ is Euclidean or hyperbolic, the angle function $\theta_{i}$
defined on
$\Omega=\\{(l_{1},l_{2},l_{3})\in\mathbb{R}^{3}|l_{1}+l_{2}>l_{3},l_{2}+l_{3}>l_{1},l_{3}+l_{1}>l_{2}\\}$
can be extended continuously by constant functions to a function
$\tilde{\theta_{i}}$ on $\mathbb{R}^{3}_{>0}$.

  2. (2)

If $\Delta$ is spherical, the angle function $\theta_{i}$ defined on
$\Omega=\\{(l_{1},l_{2},l_{3})\in\mathbb{R}^{3}|l_{1}+l_{2}>l_{3},l_{2}+l_{3}>l_{1},l_{3}+l_{1}>l_{2},l_{1}+l_{2}+l_{3}<2\pi\\}$
can be extended continuously by constant functions to a function
$\tilde{\theta_{i}}$ on $\mathbb{(}0,\pi)^{3}$.

We call the set $\Omega$ in the lemma the natural domain of the length
vectors.

###### Proof.

In the case (1), the extension function $\tilde{\theta_{i}}$ of $\theta_{i}$
is given by $\tilde{\theta_{i}}=\pi$ when $l_{i}\geq l_{j}+l_{k}$, and
$\tilde{\theta_{i}}=0$ when $l_{j}\geq l_{i}+l_{k}$. It remains to verify the
continuity of $\tilde{\theta_{i}}$ on $\mathbb{R}_{>0}^{3}$. It is based on
the cosine law. Given a point $L=(L_{1},L_{2},L_{3})$ in the boundary
$\bar{\Omega}-\Omega$ of $\Omega$ inside $\mathbb{R}^{3}_{>0}$, we may assume
without loss of generality that $L_{1}=L_{2}+L_{3}$. The continuity of
$\tilde{\theta_{i}}$ follows from

$\lim_{l\to L}\theta_{1}(l)=\pi,\quad\lim_{l\to L}\theta_{j}(l)=0,\quad
j=2,3.$

Indeed, the cosine law says, in the case of $\Delta\subset\mathbb{E}^{2}$,
that

(2.1) $\cos(\theta_{i})=\frac{l_{j}^{2}+l_{k}^{2}-l_{i}^{2}}{2l_{j}l_{k}}.$

One sees easily that when $l$ tends to $L$, then the right-hand-side of (2.1)
tends to $1$ if i=2,3 and $-1$ if $i=1$. This verifies the continuity in the
Euclidean case. In the hyperbolic case, the cosine law says

(2.2)
$\cos(\theta_{i})=\frac{\cosh(l_{j})\cosh(l_{k})-\cosh(l_{i})}{\sinh(l_{j})\sinh(l_{k})}.$

Thus one sees that as $l$ tends to $L=(L_{1},L_{2},L_{3})$ with $L_{j}>0$, the
right-hand-side of (2.2) tends to $1$ if $i=2,3$ and to $-1$ if $i=1$. Thus
$\tilde{\theta_{i}}$ is continuous.

To see (2), recall that the cosine law for spherical triangle says

(2.3)
$\cos(\theta_{i})=\frac{\cos(l_{i})-\cos(l_{j})\cos(l_{k})}{\sin(l_{j})\sin(l_{k})}.$

If $l$ tends to $L$ where $L_{1}=L_{2}+L_{3}$ with $L_{i}\in(0,\pi)$, then
$\lim_{l\to L}\cos(\theta_{1})=-1$ and $\lim_{l\to L}\cos(\theta_{i})=1$ when
$i=2,3$. On the other hand, if $L_{1}+L_{2}+L_{3}=2\pi$ for $L_{i}\in(0,\pi)$,
then the cosine law implies that $\lim_{l\to L}\cos(\theta_{i})=-1$ for all
$i$, i.e., all inner angles are $\pi$ in this case. Thus by setting the
extended function $\tilde{\theta_{i}}$ in $(0,\pi)^{3}$ to be
$\tilde{\theta_{i}}(l)=\pi$ if $l_{i}\geq l_{j}+l_{k}$,
$\tilde{\theta_{i}}(l)=0$ if $l_{j}\geq l_{i}+l_{k}$, and
$\tilde{\theta_{i}}(l)=\pi$ if $l_{i}+l_{j}+l_{k}\geq\pi$, (
$\\{i,j,k\\}=\\{1,2,3\\}$), we see that $\tilde{\theta_{i}}$ is continuous.

∎

###  2.2. Continuous extension of 1-forms and of locally convex functions

We establish some simple facts on extending closed 1-forms and locally convex
functions to convex functions in this subsection.

######  Definition 2.3.

A differential 1-form $w=\sum_{i=1}^{n}a_{i}(x)dx_{i}$ in an open set
$U\subset\mathbb{R}^{n}$ is said to be continuous if each $a_{i}(x)$ is a
continuous function on $U$. A continuous 1-form $w$ is called closed if
$\int_{\partial\tau}w=0$ for each Euclidean triangle $\tau$ in $U$.

By the standard approximation theory, if $w$ is closed and $\gamma$ is a
piecewise smooth null homologous loop in $U$, then $\int_{\gamma}w=0$. If $U$
is simply connected, then the integral $F(x)=\int_{a}^{x}w$ is well defined,
independent of the choice of piecewise smooth paths in $U$ from $a$ to $x$.
The function $F(x)$ is $C^{1}$-smooth so that $\partial F(x)/\partial
x_{i}=a_{i}(x)$.

######  Proposition 2.4.

Suppose $X$ is an open set in $\mathbb{R}^{n}$ and $A\subset X$ is an open
subset bounded by a smooth (n-1)-dimensional submanifold in $X$. If
$w=\sum_{i=1}^{n}a_{i}(x)dx_{i}$ is a continuous 1-form on $X$ so that
$w|_{A}$ and $w|_{X-\bar{A}}$ are closed where $\overline{A}$ is the closure
of $A$ in $X$, then $w$ is closed in X.

###### Proof.

Since closedness is a local property and is invariant under smooth change of
coordinates in $X$, we may assume that $X=\mathbb{R}^{n}$ and
$A=\\{(x_{1},...,x_{n})\in\mathbb{R}^{n}|x_{n}>0\\}$. Take a Euclidean
triangle $\tau\subset X$. To verify $\int_{\partial\tau}w=0$, we may assume
that $\tau$ is not in $\overline{A}$ or $X-A$ since otherwise
$\int_{\partial\tau}w=0$ follows from the assumption and the standard
approximation theorem. In the remaining case, $\tau$ intersects both $A$ and
$X-A$. The plane $x_{n}=0$ cuts the triangle $\tau$ into a triangle
$\gamma_{1}$ and a quadrilateral $\gamma_{2}$ so that $\gamma_{1}$ and
$\gamma_{2}$ are in the closure of $A$ and $X-A$. We can express, in the
singular chain level, $\partial\tau=\partial\gamma_{1}+\partial\gamma_{2}$. By
definition, $\int_{\partial\gamma_{i}}w=0$ for each $i$. Thus
$\int_{\partial\tau}w=\int_{\partial\gamma_{1}}w+\int_{\partial\gamma_{2}}w=0$.
∎

A real analytic codimension-1 submanifold $Y$ in an open set $X$ in
$\mathbb{R}^{n}$ is a smooth submanifold so that locally $Y$ is defined by
$k(x)=0$ for a non-constant real analytic function $k$. Note that if $L$ is a
(compact) line segment in $X$, then either $L\subset Y$ or $L\cap Y$ is a
finite set. This is due to the fact that a non-constant real analytic function
on an open interval has isolated zeros.

Recall that a function $f$ defined on a convex set $X\subset\mathbb{R}^{n}$ is
called convex if for all $p,q\in X$ and all $t\in[0,1]$, $tf(p)+(1-t)f(q)\geq
f(tp+(1-t)q)$. It is called strictly convex if for all $p\neq q$ in $X$ and
all $t\in(0,1)$, $tf(p)+(1-t)f(q)>f(tp+(1-t)q)$. A function $f$ defined in an
open set $U\subset\mathbb{R}^{n}$ is said to be locally convex (or locally
strictly convex ) if it is convex (or strictly convex) in a convex
neighborhood of each point.

######  Proposition 2.5.

Suppose $X\subset\mathbb{R}^{n}$ is an open convex set and $A\subset X$ is an
open subset of $X$ bounded by a codimension-1 real analytic submanifold in
$X$. If $w=\sum_{i=1}^{n}a_{i}(x)dx_{i}$ is a continuous closed 1-form on $X$
so that $F(x)=\int^{x}_{a}w$ is locally convex in $A$ and in $X-\overline{A}$,
then $F(x)$ is convex in $X$.

###### Proof.

Since $X$ is simply connected, the function $F$ is well defined. To verify
convexity, take $p,q\in X$ and consider $f(t)=F(tp+(1-t)q)$ for $t\in[0,1]$.
It suffices to show that $f(t)$ is convex in $t$. Since $F$ is $C^{1}$-smooth,
$f$ is $C^{1}$-smooth. Let $\partial A=\bar{A}-A$ and $L$ be the line segment
from $p$ to $q$. Since $\partial A$ is real analytic, either $L$ intersects
$\partial A$ in a finite set of points, or $L$ is in $\partial A$. In the
first case, let $0=t_{0}<t_{1}<...,t_{n}=1$ be the partition of $[0,1]$ so
that the line segment $tp+(1-t)q$ for $t\in(t_{i},t_{i+1})$ is either in $A$
or in $X-\overline{A}$. By definition, $f(t)$ is convex in $[t_{i},t_{i+1}]$,
i.e., $f^{\prime}(t)$ is increasing in $[t_{i},t_{i+1}]$ for $i=0,...,n-1$.
Since $f^{\prime}(t)$ is continuous in $[0,1]$, this implies that
$f^{\prime}(t)$ is increasing in $[0,1]$, i.e., $f(t)$ is convex in $[0,1]$.
In the second case that $L\subset\partial A$, we take two sequences of points
$p_{m}$ and $q_{m}$ converging to $p$ and $q$ respectively in $X$ so that
$p_{m},q_{m}$ are not in $\partial A$. Then by the case just proved, the
functions $f_{m}(t)=F(tp_{m}+(1-t)q_{m})$ are convex in $t$. Furthermore,
$f_{m}$ converges to $f$. Thus $f$ is convex. ∎

######  Corollary 2.6.

Suppose $X\subset\mathbb{R}^{n}$ is an open convex set and $A\subset X$ is an
open subset of $X$ bounded by a real analytic codimension-1 submanifold in
$X$. If $w=\sum_{i=1}^{n}a_{i}(x)dx_{i}$ is a continuous closed 1-form on $A$
so that $F(x)=\int^{x}_{a}w$ is locally convex on $A$ and each $a_{i}$ can be
extended continuously to $X$ by constant functions to a function
$\tilde{a_{i}}$ on $X$, then
$\tilde{F}(x)=\int^{x}_{a}\sum_{i=1}^{n}\tilde{a_{i}}dx_{i}$ is a
$C^{1}$-smooth convex function on $X$ extending $F$.

We remark that the real analytic assumption in the proposition 2.5 can be
relaxed to $C^{2}$ smooth.

##  3\. A Proof of Bowers-Stephenson’s Rigidity Conjecture

We begin by recalling Guo’s work on a variational principle associated to
inversive distance circle packings and then prove theorem 1.4. We will work in
Euclidean and hyperbolic geometries only.

###  3.1. Guo’s variational principle for inversive distance circle packing

Suppose $\Delta$ is a triangle with vertices $v_{1},v_{2},v_{3}$ and edges
$e_{ij}=v_{i}v_{j}$, $i\neq j$. Fix once and for all an inversive distance
$I_{ij}\in[0,\infty)$ at each edge $e_{ij}$. Then for each assignment of
positive number $r_{i}$ at $v_{i}$ for $i=1,2,3$, let

(3.1) $l_{k}=\sqrt{r_{i}^{2}+r_{j}^{2}+2r_{i}r_{j}I_{ij}}$

for Euclidean geometry and

(3.2)
$l_{k}=\cosh^{-1}(\cosh(r_{i})\cosh(r_{j})+I_{ij}\sinh(r_{i})\sinh(r_{j}))$

for hyperbolic geometry where $\\{i,j,k\\}=\\{1,2,3\\}$.

Let
$\Omega=\\{(x_{1},x_{2},x_{3})\in\mathbb{R}_{>0}^{3}|x_{i}+x_{j}>x_{k},\\{i,j,k\\}=\\{1,2,3\\}\\}$.
If $(l_{1},l_{2},l_{3})$ is in $\Omega$, then we construct a Euclidean
triangle $\Delta$ with length $l_{k}$ of $e_{ij}$ given by (3.1) and a
hyperbolic triangle, still denoted by $\Delta$, with length $l_{k}$ of
$e_{ij}$ given by (3.2). Suppose the angle of the triangle at $v_{i}$ is
$\theta_{i}$ and consider $\theta_{i}$ as a function of $(r_{1},r_{2},r_{3})$.
Guo proved the following theorem in [9].

######  Theorem 3.1.

(Guo [9]) Fix any $(I_{12},I_{23},I_{31})\in[0,\infty)^{3}$.

(1) For Euclidean triangles, let $u_{i}=\ln r_{i}$, then the differential
1-form $w=\sum_{i=1}^{3}\theta_{i}du_{i}$ is closed in the open subset of
$\mathbb{R}^{3}$ where it is defined. The integral $F(u)=\int_{0}^{u}w$ is a
locally concave function in $u=(u_{1},u_{2},u_{3})$ and is strictly locally
concave in $u_{1}+u_{2}+u_{3}=0$. Furthermore, if $c\in\mathbb{R}$ and $F(u)$
is defined, then $F(u+(c,c,c))=F(u)$.

(2) For hyperbolic triangles, let $u_{i}=\ln(\tanh(r_{i}/2))$, then the
differential 1-form $w=\sum_{i=1}^{3}\theta_{i}du_{i}$ is closed in the open
subset of $\mathbb{R}_{<0}^{3}$ where it is defined. Furthermore, the integral
$F(u)=\int_{-(1,1,1)}^{u}w$ is a strictly locally concave function in
$u=(u_{1},u_{2},u_{3})$.

It is also proved in [9] that the open sets where the 1-forms $w$ are defined
in theorem 3.1 are connected and simply connected. Theorem 3.1 is a
generalization of an earlier result obtained in [6]. Guo proved a local and
infinitesimal rigidity theorem for inversive distance circle packing using
theorem 3.1. It says that a Euclidean inversive distance circle packing is
locally determined, up to scaling, by the discrete curvature of the underlying
polyhedral surface. He also proved the local and infinitesimal rigidity for
hyperbolic inversive distance circle packings.

###  3.2. Concave extension of Guo’s action functional

Our main observation is that Guo’s differential 1-forms
$w=\sum_{i=1}^{3}\theta_{i}du_{i}$ can be extended to a closed 1-form on
$\mathbb{R}^{3}$ in the Euclidean case and on $\mathbb{R}^{3}_{<0}$ in the
hyperbolic case so that the integrations of the extended 1-forms are still
concave.

######  Proposition 3.2.

Let $w$ be the 1-forms defined in theorem 3.1.

(a) In the case of Euclidean triangles, the 1-form $w$ can be extended to a
continuous closed 1-form $\tilde{w}$ on $\mathbb{R}^{3}$ so that the
integration $\tilde{F}(u)=\int^{u}_{0}\tilde{w}$ is a $C^{1}$-smooth concave
function.

(b) In the case of hyperbolic triangles, the 1-form $w$ can be extended to a
continuous closed 1-form $\tilde{w}$ on $\mathbb{R}_{<0}^{3}$ so that the
integration $\tilde{F}(u)=\int^{u}_{-(1,1,1)}\tilde{w}$ is a $C^{1}$-smooth
concave function.

We begin by focusing the 1-forms in its radius coordinate
$r=(r_{1},r_{2},r_{3})\in\mathbb{R}_{>0}^{3}$. In this case, the 1-forms are
given by $w=\sum_{i=1}^{3}\theta_{i}\frac{dr_{i}}{r_{i}}$ and
$w=\sum_{i=1}^{3}\theta_{i}\frac{dr_{i}}{\sinh(r_{i})}$. The 1-form $w$ is
defined on the open set $U$ of $\mathbb{R}_{>0}^{3}$ where

(3.3)
$U=\\{(r_{1},r_{2},r_{3})\in\mathbb{R}_{>0}^{3}|l_{i}+l_{j}>l_{k},\\{i,j,k\\}=\\{1,2,3\\}\\},$

where $l_{i}=l_{i}(r_{1},r_{2},r_{3})$ is defined on $\mathbb{R}_{>0}^{3}$.
(Note that for hyperbolic and Euclidean geometries, the sets $U$ are different
due to (3.1) and (3.2)). The extension of the 1-form $w$ is the natural one.
Namely, we replace $\theta_{i}$ in $w$ by $\tilde{\theta_{i}}$ appeared in
lemma 2.1. Thus the extended 1-form is
$\tilde{w}=\sum_{i=1}^{3}\tilde{\theta_{i}}\frac{dr_{i}}{r_{i}}$ or
$\tilde{w}=\sum_{i=1}^{3}\tilde{\theta_{i}}\frac{dr_{i}}{\sinh(r_{i})}.$

It remains to show that $\tilde{w}$ is continuous and closed in
$\mathbb{R}_{>0}^{3}$ so that its pull back to the $u$-coordinate has a
concave integration. To this end, we prove,

######  Lemma 3.3.

Let $\bar{U}$ be the closure of $U$ in $\mathbb{R}_{>0}^{3}$. Then,

(1) $\theta_{i}$ is a constant function on each connected component of
$\bar{U}-U$, and

(2) for each connected component $V$ of $\mathbb{R}_{>0}^{3}-U$, the
intersection $V\cap\bar{U}$ is a connected component of $\bar{U}-U$.

###### Proof.

By (3.3), the boundary $\partial U=\bar{U}-U$ is given by
$\cup_{i=1}^{3}\partial_{i}U$ where
$\partial_{i}U=\\{(r_{1},r_{2},r_{3})\in\mathbb{R}_{>0}^{3}|l_{i}=l_{j}+l_{k},\\{j,k\\}=\\{1,2,3\\}-\\{i\\}\\}$.
Furthermore, $\mathbb{R}_{>0}^{3}-U=\cup_{i=1}^{3}V_{i}$ where
$V_{i}=\\{(r_{1},r_{2},r_{3})\in\mathbb{R}_{>0}^{3}|l_{i}\geq
l_{j}+l_{k},\\{j,k\\}=\\{1,2,3\\}-\\{i\\}\\}.$

First, we note that if $I_{ij}\leq 1$, then $\partial_{k}U=\emptyset$ and
$V_{k}=\emptyset$. Indeed, if $I_{ij}\leq 1$, then by (3.1) and (3.2),

$l_{k}\leq r_{i}+r_{j}.$

But due to $I_{ab}\geq 0$, (3.1) and (3.2), $r_{j}<l_{i}$ and $r_{i}<l_{j}$.
Therefore, $l_{k}<l_{i}+l_{j}$. This implies that $\partial_{k}U=\emptyset$
and $V_{k}=\emptyset$.

Next $\partial_{i}U\cap\partial_{j}U=\emptyset$ and $V_{i}\cap
V_{j}=\emptyset$ for $i\neq j$. Indeed, if
$r\in\partial_{i}U\cap\partial_{j}U$ or $r\in V_{i}\cap V_{j}$, then
$l_{i}\geq l_{j}+l_{k}$ and $l_{j}\geq l_{i}+l_{k}$. Thus $l_{k}=0$. But
$l_{k}>r_{i}>0$.

We claim that if $I_{ij}>1$, then both $V_{k}$ and $\partial_{k}U$ are non-
empty and connected. Assume the claim, then the lemma follows. Indeed, since
$l_{s}>0$ for all indices $s$, it follows, by lemma 2.1, that $\theta_{i}$ is
either $0$ or $\pi$ in $\partial_{s}U$, i.e., (1) holds. Next, $V_{s}$’s are
the connected components of $\mathbb{R}_{>0}^{3}-U$ so that
$V_{s}\cap\bar{U}=\partial_{s}U$. Thus (2) holds.

To see the claim, it suffices to show that there is a smooth function
$f(r_{i},r_{j})$ defined on $\mathbb{R}_{>0}^{3}$ so that its graph is
$\partial_{k}U$ and
$V_{k}=\\{(r_{1},r_{2},r_{3})\in\mathbb{R}_{>0}^{3}|0<r_{3}\leq
f(r_{1},r_{2})\\}$.

To this end, consider the equation

(3.4) $l_{k}=l_{i}+l_{j},$

and let the right-hand-side of (3.4) be $g(r_{k},r_{i},r_{j})$. We will deal
with the Euclidean and hyperbolic geometry separately.

CASE 1 Euclidean triangles. In this case, the function $g(r_{k},r_{i},r_{j})$
is given by

(3.5)
$g(r_{k},r_{i},r_{j})=\sqrt{r_{k}^{2}+r_{j}^{2}+2I_{kj}r_{k}r_{j}}+\sqrt{r_{i}^{2}+r_{k}^{2}+2I_{ik}r_{i}r_{k}}$

Evidently, for a fixed $(r_{i},r_{j})\in\mathbb{R}^{2}_{>0}$,
$g(r_{k},r_{i},r_{j})$ is a strictly increasing function of
$r_{k}\in\mathbb{R}_{>0}$ so that
$g(0,r_{i},r_{j})=r_{i}+r_{j}<\sqrt{r_{i}^{2}+r_{j}^{2}+2I_{ij}r_{i}r_{j}}$
(due to $I_{ij}>1$) and $\lim_{r_{k}\to\infty}g(r_{k},r_{i},r_{k})=\infty$. By
the mean-value theorem, there exists a unique positive number $f(r_{i},r_{j})$
so that
$g(f(r_{i},r_{j}),r_{i},r_{j})=\sqrt{r_{i}^{2}+r_{j}^{2}+2r_{i}r_{j}I_{ij}}=l_{k}$.
The smoothness of $f(r_{i},r_{j})$ follows from the implicit function theorem
applied to (3.4). Indeed,

$\frac{\partial g}{\partial
r_{k}}=\frac{r_{k}+2I_{kj}r_{j}}{l_{i}}+\frac{r_{k}+2I_{ik}r_{i}}{l_{j}}>0.$

Thus, $f(r_{i},r_{j})$ is smooth.

This shows $\partial_{k}U$ is the graph of the smooth function $f$ defined on
$\mathbb{R}_{>0}^{2}$, i.e.,

$\partial_{k}U=\\{(r_{1},r_{2},r_{3})\in\mathbb{R}_{>0}^{3}|r_{k}=f(r_{i},r_{j})\\}.$

Thus it is connected. Since $g(r_{k},r_{i},r_{j})$ is an increasing function
of $r_{k}$, $V_{k}=\\{r\in R_{>0}^{3}|0<r_{k}\leq
f(r_{i},r_{j}),\\{i,j\\}=\\{1,2,3\\}-\\{k\\}\\}$. Thus $V_{k}$ is connected.

CASE 2 hyperbolic triangles. By the same argument as in case 1, it suffices to
show the same properties established in case 1 hold for $g(r_{k},r_{i},r_{j})$
given by

(3.6)
$\cosh^{-1}(\cosh(r_{i})\cosh(r_{k})+I_{ik}\sinh(r_{i})\sinh(r_{k}))+\cosh^{-1}(\cosh(r_{k})\cosh(r_{j})+I_{kj}\sinh(r_{k})\sinh(r_{j})).$

Fix $(r_{i},r_{j})\in\mathbb{R}^{2}_{>0}$. Then the function
$g(r_{k},r_{i},r_{j})$ is clearly strictly increasing in
$r_{k}\in\mathbb{R}_{>0}$ so that
$\lim_{r_{k}\to\infty}g(r_{k},r_{i},r_{j})=\infty$ and due to $I_{ij}>1$,

$g(0,r_{i},r_{j})=r_{i}+r_{j}$ $=\cosh^{-1}(\cosh(r_{i}+r_{j}))$
$=\cosh^{-1}(\cosh(r_{i})\cosh(r_{j})+\sinh(r_{i})\sinh(r_{j}))$
$<\cosh^{-1}(\cosh(r_{i})\cosh(r_{j})+I_{ij}\sinh(r_{i})\sinh(r_{j}))=l_{k}.$

By the mean value theorem, there exists a unique positive number
$f(r_{i},r_{j})$ so that $g(f(r_{i},r_{j}),r_{i},r_{j})=l_{k}$. The smoothness
of $f(r_{i},r_{j})$ follows form the implicit function theorem that

$\frac{\partial g}{\partial
r_{k}}=\frac{\cosh(r_{i})\sinh(r_{k})+I_{ik}\sinh(r_{i})\cosh(r_{k})}{\sqrt{(\cosh(r_{i})\cosh(r_{k})+I_{ik}\sinh(r_{i})\sinh(r_{k}))^{2}-1}}$
$+\frac{\cosh(r_{j})\sinh(r_{k})+I_{jk}\sinh(r_{j})\cosh(r_{k})}{\sqrt{(\cosh(r_{j})\cosh(r_{k})+I_{jk}\sinh(r_{j})\sinh(r_{k}))^{2}-1}}$
$>0.$

By the same argument as in case 1, we see that $\partial_{k}U$, being the
graph of the smooth function $f$, is connected and $V_{k}$, being the region
below the positive function $f$ over $\mathbb{R}^{2}_{>0}$, is also connected.
∎

Now back to the proof of proposition 3.2, for part (1), consider the real
analytic diffeomorphism $u=u(r):\mathbb{R}^{3}_{>0}\to\mathbb{R}^{3}$ where
$u_{i}=\ln r_{i}$. The differential 1-form
$w=\sum_{i=1}^{3}\theta_{i}\frac{dr_{i}}{r_{i}}$ pulls back (via
$r=u^{-1}(r)$) to $w=\sum_{i=1}^{3}\theta_{i}du_{i}$ as appeared in theorem
3.1. By lemma 3.3, the extension
$\tilde{w}=\sum_{i=1}^{3}\tilde{\theta_{i}}du_{i}$ is obtained from $w$ by
extending each coefficient $\theta_{i}$ by constant functions on
$\mathbb{R}^{3}-u^{-1}(U)$. Thus, by corollary 2.6, the function
$\tilde{F}(u)=\int^{u}_{0}\tilde{w}$ is a $C^{1}$-smooth concave function in
$u\in\mathbb{R}^{3}$ so that

(3.7) $\partial\tilde{F}/\partial u_{i}=\tilde{\theta_{i}}.$

The same argument also works for part (2) since $u=u(r)$ with
$u_{i}=\ln\tanh(r_{i})$ is a real analytic diffeomorphism from
$\mathbb{R}_{>0}^{3}$ onto $\mathbb{R}_{<0}^{3}$.

###  3.3. A proof of theorem 1.4 for Euclidean inversive distance circle
packing

Suppose otherwise that there exist two inversive circle packing metrics
$d_{1},d_{2}$ on $(S,T)$ with the same inversive distance
$I\in\mathbb{[}0,\infty)^{E}$ so that their discrete curvatures are the same
and $d_{1}\neq\lambda d_{2}$ for any $\lambda$. Let $a\in\mathbb{R}^{V}$ be
their common discrete curvature.

We will use the notation that if $i\in V$ and $x\in\mathbb{R}^{V}$, then
$x_{i}=x(i)$ below. Let $T^{(2)}$ be the set of all triangles in $T$. If a
triangle $s\in T^{(2)}$ has vertices $i,j,k\in V$, then we denote the triangle
by $s=\\{i,j,k\\}$. For circle packing metrics of radii
$r\in\mathbb{R}_{>0}^{V}$ with a given inversive distance $I$, we use
$u\in\mathbb{R}^{V}$ to denote their logarithm coordinate where $u_{i}=\ln
r_{i}$. Thus, there are two points $p,q$ in $\mathbb{R}^{V}$ as the
logarithmic coordinates of $d_{1}$ and $d_{2}$ so that their discrete
curvatures are $a\in\mathbb{R}^{V}$ and $p-q\neq\lambda(1,1,1,..,1)$ for any
$\lambda$.

We will derive a contradiction by using the locally concave functions $F$ and
its concave extension $\tilde{F}=\int^{u}_{0}\tilde{w}$ appeared in
proposition 3.2 associated to theorem 3.1(1).

Define a $C^{1}$-smooth function $W:\mathbb{R}^{V}\to\mathbb{R}$ by

(3.8) $W(u)=-\sum_{s\in T^{(2)},s=\\{i,j,k\\},i,j,k\in
V}\tilde{F}(u_{i},u_{j},u_{k})+\sum_{i\in V}(2\pi-a_{i})u_{i}.$

The function $W$ is convex since it is a summation of convex functions.
Furthermore, by the definition of $W$, (3.7), the definition of discrete
curvature $(a_{i})$, $p$ and $q$ are both critical points of $W$. Since $W$ is
convex in $\mathbb{R}^{V}$, $p$ and $q$ are both minimal points of $W$.
Furthermore, for all $t\in[0,1]$, $tp+(1-t)q$ are minimal points of $W$. In
particular,

$W(tp+(1-t)q)=W(p)$

for all $t\in[0,1]$. Since

$W(tp+(1-t)q)=\sum_{s\in T^{(2)},s=\\{i,j,k\\},i,j,k\in
V}f_{ijk}(t)+\sum_{i\in E}(2\pi-a_{i})(tp_{i}+(1-t)q_{i})$

where the function

(3.9)
$f_{ijk}(t)=-\tilde{F}(tp_{i}+(1-t)q_{i},tp_{j}+(1-t)q_{j},tp_{k}+(1-t)q_{k})$

is convex, it follows that $f_{ijk}(t)$ is linear in $t\in[0,1]$ for all
triangle $s$ with vertices $i,j,k$. This is due to the simple fact that a
summation of a convex function with a strictly convex function is strictly
convex. By the assumption that $p-q\neq c(1,1,....,1)$ in $\mathbb{R}^{V}$ and
the surface is connected, there exists a triangle $s$ with vertices $i,j,k\in
V$ so that $(p_{i},p_{j},p_{k})-(q_{i},q_{j},q_{k})\neq(c,c,c)$ for all
$c\in\mathbb{R}$. By the given assumption, $(p_{i},p_{j},p_{k})$ and
$(q_{i},q_{j},q_{k})$ are in the domain of definition of $w$ in theorem 3.1.
Thus for $t\in[0,1]$ close to $0$ or $1$, by theorem 3.1 on the local strictly
convexity of $-F(u_{1},u_{2},u_{3})$ on $u_{1}+u_{2}+u_{3}=0$ and
$F(u+(c,c,c))=F(u)$, $f_{ijk}(t)$ is strictly convex in $t$ near $0,1$. This
is a contradiction to the linearity of $f_{ijk}(t)$.

###  3.4. A proof of theorem 1.4 for hyperbolic inversive distance circle
packing

The proof is essentially the same as in §3.3 and is simpler. For any
$r\in\mathbb{R}_{>0}^{V}$, define $u=u(r)\in\mathbb{R}_{<0}^{V}$ by
$u_{i}=\ln\tanh(r_{i}/2))$. For a circle packing with radii
$r\in\mathbb{R}_{>0}^{V}$, let $u=u(r)$ and call it the $u$-coordinate of the
circle packing metric.

We use the same notation as in §3.3. Suppose the result does not hold and let
$p\neq q\in\mathbb{R}_{<0}^{V}$ be the $u$-coordinates of the two distinct
hyperbolic circle packing metrics having the same hyperbolic inversive
distance $I\in\mathbb{R}_{\geq 0}^{E}$ and the same discrete curvature
$a=(a_{i})\in\mathbb{R}^{V}$. Define the action functional $W$ on
$\mathbb{R}_{<0}^{V}$ by the same formula (3.8) where $\tilde{F}$ is the
concave function in proposition 3.2 associated to theorem 3.1(2). Then the
same proof goes through as in §3.3 since in this case, one of $f_{ijk}(t)$ is
strictly convex for $t$ near 0 and 1.

##  4\. 2-dimensional Schlaefli Type Action Functionals and Their Extensions

The following was proved in [12]. The proof is a straight forward calculation.

######  Theorem 4.1.

Suppose $\Delta$ is a triangle in the Euclidean plane $\mathbb{E}^{2}$, or the
hyperbolic plane $\mathbb{H}^{2}$, or the 2-sphere $\mathbb{S}^{2}$ so that
its edge lengths are $l_{1},l_{2},l_{3}$ and its inner angles are
$\theta_{1},\theta_{2},\theta_{3}$ where $\theta_{i}$’s angle is opposite to
the edge of length $l_{i}$. Let $h\in\mathbb{R}$ and let $\Omega$ be the
natural domain for length vectors appeared in lemma 2.2.

  1. (1)

For a Euclidean triangle,

$w_{h}=\sum_{i=1}^{3}\frac{\int^{\theta_{i}}_{\pi/2}\sin^{h}(t)dt}{l_{i}^{h+1}}dl_{i}$

is a closed 1-form on $\Omega$. The integral $\int^{u}_{-(h,h,h)}w_{h}$ is
locally convex in variable $u=(u_{1},u_{2},u_{3})$ where $u_{i}=\ln l_{i}$ for
$h=0$ and $u_{i}=-\frac{l_{i}^{-h}}{h}$ for $h\neq 0$. Furthermore,
$\int^{u}_{-(h,h,h)}w_{h}$ is locally strictly convex in hypersurface
$u_{1}+u_{2}+u_{3}=0$.

  2. (2)

For a spherical triangle,

$w_{h}=\sum_{i=1}^{3}\frac{\int^{\theta_{i}}_{\pi/2}\sin^{h}(t)dt}{\sin^{h+1}(l_{i})}dl_{i}$

is a closed 1-form on $\Omega$. The integral $\int^{u}_{0}w_{h}$ is locally
strictly convex in $u=(u_{1},u_{2},u_{3})$ where $u_{i}=\int^{l_{i}}_{\pi/2}$
$\sin^{-h-1}(t)dt$.

  3. (3)

For a hyperbolic triangle,

$w_{h}=\sum_{i=1}^{3}\frac{\int^{\theta_{i}}_{\pi/2}\sin^{h}(t)dt}{\sinh^{h+1}(l_{i})}dl_{i}$

is a closed 1-form.

  4. (4)

For a hyperbolic triangle,

$w_{h}=\sum_{i=1}^{3}\frac{\int^{\frac{1}{2}(\theta_{i}-\theta_{j}-\theta_{k})}_{0}\cos^{h}(t)dt}{\coth^{h+1}(l_{i}/2)}dl_{i}$

is a closed 1-form. The integral $\int^{u}_{0}w_{h}$ is locally strictly
convex in $u=(u_{1},u_{2},u_{3})$ where $u_{i}=\int^{l_{i}}_{1}$
$\coth^{-h-1}(t/2)dt$.

### 4.1.

Recall that the natural domain $\Omega$ of the edge length vectors is given by
$\Omega=\\{(l_{1},l_{2},l_{3})\in\mathbb{R}_{>0}^{3}|l_{i}+l_{j}>l_{k},\\{i,j,k\\}=\\{1,2,3\\}\\}$
for Euclidean and hyperbolic triangles and
$\Omega=\\{(l_{1},l_{2},l_{3})\in\mathbb{R}_{>0}^{3}|l_{i}+l_{j}>l_{k},l_{1}+l_{2}+l_{3}<2\pi,\\{i,j,k\\}=\\{1,2,3\\}\\}$.
Let $J$ be the natural interval for each individual length $l_{i}$, i.e.,
$J=\mathbb{R}_{>0}$ for Euclidean and hyperbolic triangles and $J=(0,\pi)$ for
spherical triangles. In each case of theorem 4.1, there exists a real analytic
diffeomorphism $g:J\to g(J)$ from $J$ onto the open interval $g(J)$ so that
$u_{i}=g(l_{i})$. To be more precise, $g(t)=\ln t$ in the case of $h=0$ of
theorem 4.1(1), $g(t)=-\frac{t^{-h}}{h}$ ($h\neq 0$) in the case of $h\neq 0$
in theorem 4.1(1), $g(t)=\int^{t}_{\pi/2}\sin^{-h-1}(x)dx$ in the case (2) of
theorem 4.1, $g(t)=\int^{t}_{1}\sinh^{-h-1}(x)dx$ in the case (3) of theorem
4.1 and $g(t)=\int^{t}_{1}\coth^{-h-1}(x)dx$ in the case of (4). The real
analytic diffeomorphism $u(l_{1},l_{2},l_{3})=(u_{1},u_{2},u_{3})$ where
$u_{i}=g(l_{i})$ sends $J^{3}$ onto the open cube $g(J)^{3}$ in
$\mathbb{R}^{3}$.

By lemma 2.2, each of the angle function $\theta_{i}(l):\Omega\to\mathbb{R}$
can be extended by constant functions to a continuous function
$\tilde{\theta_{i}}(l):J^{3}\to\mathbb{R}$. Define a continuous 1-form
$\tilde{w_{h}}$ on $J^{3}$ by replacing $\theta_{i}$ in the definition of
$w_{h}$ in theorem 4.1 by $\tilde{\theta_{i}}$.

######  Lemma 4.2.

The continuous differential 1-form $\tilde{w_{h}}$ is closed in $J^{3}$.

###### Proof.

By proposition 2.4 where we take $X=J^{3}$ and $A=\Omega$, it suffices to show
that $\tilde{w_{h}}$ is closed in each connected component $U$ of
$J^{3}-\overline{\Omega}$. By theorem 4.1 $\tilde{w}|_{A}$ is closed, the
restriction of $\tilde{w_{h}}$ to $U$ is of the form
$\sum_{i=1}^{3}c_{i}du_{i}$ where $u_{i}=g(l_{i})$ and $c_{i}$ is a constant.
Thus $\tilde{w_{h}}|_{U}$ is closed. ∎

######  Proposition 4.3.

The pull back 1-form $(u^{-1})^{*}(\tilde{w_{h}})$ on $g(J)^{3}$ is a closed
1-form. Furthermore, if $F(u)=\int^{u}w_{h}$ is locally convex in $u(\Omega)$
(i.e, in the case (1),(2), (4) of theorem 4.1), then
$\tilde{F}(u)=\int^{u}(u^{-1})^{*}(\tilde{w_{h}})$ is convex in $u$ in
$g(J)^{3}$.

Note that by the construction, if $u\in u(\Omega)$ and
$w_{h}=\sum_{i=1}^{3}\alpha_{i,h}(u)du_{i}$ (as shown in theorem 4.1) then

(4.1) $\frac{\partial\tilde{F}(u)}{\partial u_{i}}=\alpha_{i,h}(u).$

Furthermore, by definition, the $\phi_{h}$ and $\psi_{h}$ curvatures are sum
of two of $a_{i,h}(u)$’s.

###### Proof.

By corollary 2.6 where we take $X=g(J)^{3}$ and $A=u(\Omega)$, it suffices to
show that $u(\Omega)$ is bounded by a real analytic surface in $X$ and
$\tilde{F}(u)$ is convex in $u(\Omega)$ and in each component of
$g(J)^{3}-\overline{u(\Omega)}$.

Since $\Omega$ in $J^{3}$ is bounded by hyperplanes and
$u(l)=(g(l_{1}),g(l_{2}),g(l_{3}))$ is a real analytic diffeomorphism, it
follows that $u(\Omega)$ is bounded by a real analytic surface in $g(J)^{3}$.

By the assumption $\tilde{F}(u)$ is convex in $u(\Omega)$. If $U$ is a
connected component of $g(J)^{3}-\overline{u(\Omega)}$, then $\tilde{F}(u)$ is
linear on $U$ since its partial derivatives are constants on $U$ by the
construction. Thus by corollary 2.6, the result follows. ∎

##  5\.  A Proof of Theorem 1.2

The argument is essentially the same as that in §3.3. Recall that $E$ is the
set of all edges in the triangulated surface $(S,T)$. If $x\in\mathbb{R}^{E}$
and $i\in E$, we use $x_{i}$ to denote $x(i)$. If $s\in T^{(2)}$ is a triangle
with edges $i,j,k\in E$, we denote it by $s=\\{i,j,k\\}$.

###  5.1. A proof of theorem 1.2(3)

Suppose otherwise that there exist two distinct hyperbolic polyhedral metrics
on $(S,T)$ so that their $\psi_{h}$ curvatures are the same. Let
$a=(a_{i})\in\mathbb{R}^{E}$ be their common $\psi_{h}$ curvature.

Recall that a polyhedral metric on $(S,T)$ is given by its edge length map
$l:E\to\mathbb{R}_{>0}$. In using the variational principle in theorem 4.1(4),
the natural variable is given by $u:E\to\mathbb{R}$ where $u(e)=g(l(e))$ with
$g(t)=\int^{t}_{1}\coth^{h+1}(s/2)ds$. We call it the $u$-coordinate of the
polyhedral metric $l$ and we will use the $u$-coordinate to set up the
variational principle. Therefore there are two distinct points (as
$u$-coordinates) $p\neq q\in g(\mathbb{R}_{>0})^{E}$ so that their
corresponding $\psi_{h}$ curvatures are the same $a\in\mathbb{R}^{E}$. We will
derive a contradiction by using the locally strictly convex functions $F$ and
its convex extension $\tilde{F}$ introduced in proposition 4.3 (associated to
theorem 4.1(4)).

Define a $C^{1}$-smooth function $W:g(\mathbb{R}_{>0})^{E}\to\mathbb{R}$ by

$W(u)=\sum_{s\in T^{(2)},s=\\{i,j,k\\},i,j,k\in
E}\tilde{F}(u_{i},u_{j},u_{k})-\sum_{i\in E}a_{i}u_{i}.$

The function $W$ is convex since it is a summation of convex functions.
Furthermore, by the definition of $W$, (4.1), the definition of $\psi_{h}$ and
$(a_{i})$, $p$ and $q$ are both critical points of $W$. Since $W$ is convex,
$p$ and $q$ are both minimal points of $W$. Furthermore, for all $t\in[0,1]$,
$tp+(1-t)q$ are minimal points of $W$. In particular,

$W(tp+(1-t)q)=W(p)$

for all $t\in[0,1]$. Since

$W(tp+(1-t)q)=\sum_{i,j,k\in E,\\{i,j,k\\}\in T^{(2)}}f_{ijk}(t)-\sum_{i\in
E}a_{i}(tp_{i}+(1-t)q_{i})$

where the function

(5.1)
$f_{ijk}(t)=\tilde{F}(tp_{i}+(1-t)q_{i},tp_{j}+(1-t)q_{j},tp_{k}+(1-t)q_{k})$

is convex, it follows that $f_{ijk}(t)$ is linear in $t\in[0,1]$. Since $p\neq
q$, there exists a triangle with edges $i,j,k\in E$ so that
$(p_{i},p_{j},p_{k})\neq(q_{i},q_{j},q_{k})$. Thus for $t\in[0,1]$ close to
$0$ or $1$, by theorem 4.1 on the local strictly convexity, $f_{ijk}(t)$ is
strictly convex in $t$ near $0,1$. This is a contradiction to the linearity of
$f_{ijk}(t)$.

###  5.2. A proof of theorem 1.2(2)

The proof is exactly the same as above using the extended convex function
$\tilde{F}$ in proposition 4.3 associated to theorem 4.1(2).

###  5.3. A proof of theorem 1.2(1)

The proof is the same as that in §5.1 using the similarly defined function
$W$. To be more precise, let $g(t)=-\frac{t^{-h}}{h}$ for $h\neq 0$ and
$g(t)=\ln t$. By the same set up as in §5.1, we conclude that $f_{ijk}(t)$
given by (5.1) is linear in $t$. We claim this implies that the two Euclidean
polyhedral metrics $u^{-1}(p)$ and $u^{-1}(q)$ differ by a scalar
multiplication. There are two cases to be discussed depending on $h=$ or
$h\neq 0$.

CASE 1. $h=0$. In this case, $p-q\neq c(1,1,...,1)$ in $\mathbb{R}^{E}$ for
any constant $c$. By the connectivity of the surface $S$, there exists a
triangle with edges $i,j,k\in E$ so that
$(p_{i},p_{j},p_{k})-(q_{i},q_{j},q_{k})\neq(c,c,c)$ for any constant $c$. On
the other hand, by theorem 4.1(1), the action functional $F$ is strictly
locally convex in the hyperplane $u_{1}+u_{2}+u_{3}=0$ and $F(u+(c,c,c)=F(u)$
for a scalar $c$ and $u\in u(\Omega)$. In particular, this implies that the
function $f_{ijk}(t)$ is strictly convex in $t\in[0,1]$ for $t$ close to $0$
or $1$. This contradicts the linearity of the function $f_{ijk}(t)$.

CASE 2. $h\neq 0$. In this case, $p\neq cq$ for any constant $c$. In
particular, there exists a triangle with three edges $i,j,k\in E$ so that
$(p_{i},p_{j},p_{k})\neq c(q_{i},q_{j},q_{k})$ for any $c\in\mathbb{R}$. By
theorem 4.1(1), the function $f_{ijk}(t)$ is strictly convex in $t\in[0,1]$
for $t$ close to $0$ or $1$. This contradicts the linearity of the function
$f_{ijk}(t)$.

Thus the two polyhedral metrics differ by a scaling.

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