# Ultra wide black-hole - neutron star binaries
as a possible source for gravitational waves and short gamma ray bursts

Erez Michaely Department of Physics and Astronomy, University of California,
Log Angeles, CA 90095, USA  Mani L. Bhaumik Institute for Theoretical Physics,
University of California, Log Angeles, CA 90095, USA  Smadar Naoz Department
of Physics and Astronomy, University of California, Log Angeles, CA 90095, USA
Mani L. Bhaumik Institute for Theoretical Physics, University of California,
Log Angeles, CA 90095, USA

###### Abstract

The third observing run of the LIGO/Virgo/KARGA collaboration reported a few
neutron star–black hole (NSBH) merger events. While NSBH mergers have yet to
receive extensive theoretical attention, they may have a promising
electromagnetic signature in the form of short gamma-ray bursts. Here we show
that NSBH dynamical mergers can naturally form from ultra-wide binaries in the
field. Flyby gravitational interactions with other neighbors in the galaxy in
these ultra-wide systems may result in high eccentricity that drives the
binary into a merger. We show that this process can result in a merger rate at
the order of $\sim 10$ Gpc-3 yr-1 ($\sim 5$ Gpc-3 yr-1) for elliptical
(spiral) galaxies. This channel predicts higher merger rate with higher
velocity dispersion of the host-galaxy, delay time distribution which
shallower than uniform but steeper that $1/t$, higher merger rate for lower BH
to NS mass ratio.

gravitational waves () — stars: kinematics and dynamics — stars: neutron —
stars: black holes

##  1 Introduction

With the recent third catalog of observed gravitational waves signals was
published (The LIGO Scientific Collaboration et al., 2021a), a total of $90$
GW candidates were detected, which allows a significant improvement of the
inferred merger rates for each of the three classes of binary systems, Binary
Black-hole (BBH), Binary neutron stars (BNS) and black-hole (BH) neutron star
(NS) binary (BHNS). The LIGO-VIRGO-KARGA collaboration (The LIGO Scientific
Collaboration et al., 2021b) report that the for the BBH class the merger rate
is $17.3-45{\rm Gpc^{-3}yr^{-1}}$, for the BNS the inferred rate is
$13-1900{\rm Gpc^{-3}yr^{-1}}$ and for the BHNS the corresponding rate is
$7.4-320{\rm Gpc^{-3}yr^{-1}}.$

The formation of GW sources can roughly be divided into four groups:

(1) Isolated stellar binary: In this channel, massive stellar binaries evolve
together, in the absence of additional dynamical interaction (e.g., Dominik et
al., 2015; de Mink & Belczynski, 2015; Belczynski et al., 2016; Eldridge et
al., 2017; Giacobbo et al., 2018; Olejak et al., 2020), or via stable mass
transfer (e.g. Gallegos-Garcia et al., 2021). Fraction of the close massive
binaries evolve through one or two common envelope process resulting in a
possible formation of short period binary. Some fraction of these system are
sufficiently close to merge by the emission of GW within Hubble time. A well
known challenge to this avenue is the common envelope phase which may cause
the binary to merge before the formation of the compact objects. However, a
proposed avenue that may overcome this challenge may lay in the form of
chemically homogeneous stellar evolution which may avoid mass transfer (e.g.,
Marchant et al., 2016; de Mink & Mandel, 2016; Mandel & de Mink, 2016).

(2) Dynamical mergers in dense environments. Dense environments are efficient
in producing many binary exotica due to the high stellar density. Galactic
centers, or globular clusters and even low mass young clusters and open
clusters are heavily populated with stellar and compact objects. In these
environments, stars and compact objects, both singles and binaries experience
strong gravitational interactions with individual stars or with higher
multiplicity systems. These interactions tend to harden hard binaries, and
possibly unbind soft binaries (Heggie, 1975), whilst change their
eccentricities. As a results some fraction of these binaries may merge via GW
emission in a Hubble time (e.g., Antonini & Perets, 2012; Samsing et al.,
2014; Rodriguez et al., 2016, 2018, 2021; Fragione & Kocsis, 2018; Banerjee,
2018; Hamers et al., 2018; Leigh et al., 2018; Kremer et al., 2020; Ye et al.,
2020; Stephan et al., 2019; Hoang et al., 2020; Wang et al., 2020; Rastello et
al., 2021; Fragione, 2021). Furthermore, it was suggested that AGN disks may
assist with the merger of BH binaries (Bartos et al., 2017; Stone et al.,
2017, e.g.,).

(3) Secular evolution in hierarchical triple systems. This channel considers
the long term evolution of triple body system via the Eccentric Kozai-Lidov
(EKL) mechanism (e.g., Lidov, 1962; Kozai, 1962; Naoz, 2016, see latter for
review).

The EKL mechanism drive the inner binary to high eccentricity, which boost the
GW emission power of the inner binary and lead to a merger. Triples are hosted
either in the field of the host galaxy (Antonini et al., 2016, 2017; Silsbee &
Tremaine, 2017; Liu & Lai, 2018; Vigna-Gómez et al., 2021, e.g.,) or in dense
environments (Britt et al., 2021; Miller & Hamilton, 2002; Antonini et al.,
2014; Kimpson et al., 2016; Samsing et al., 2018; Hamilton & Rafikov, 2019;
Martinez et al., 2020, e.g.,). Moreover, at the center of galaxies, the third
object can be the supermassive BH (e.g., Antonini & Perets, 2012; Stephan et
al., 2016, 2019; Petrovich & Antonini, 2017; Hoang et al., 2018; Fragione et
al., 2019; Wang et al., 2020).

(4) Mergers from ultra wide systems in the field. Recently, it was shown that
isolated ultra-wide systems (with semi-major axis (SMA) $>100\rm AU$), either
binaries or triples, in the field of the host galaxy, may be driven to
extremely high eccentricities to allow strong interaction between the two
binary objects (Michaely & Perets, 2016, 2019, 2020; Michaely, 2020; Michaely
& Shara, 2021). For ultra-wide systems, the field is collisional environment
due to consecutive frequent flyby interactions with random field stars. These
interactions may excite the eccentricity (in the case of binaries) or outer
eccentricity (in case of triples), with the result that mergers can occur via
increased GW emission (binaries) or three-body instabilities (triples).

Here we focus on this latter channel and study the BHNS formation channel. In
particular we present a novel scenario that can lead to a BHNS merger via GW
and a possibility to a EM counterpart.

In the last years the majority of the literature focused on the BBH formation
channels. However, recently a few studies have focused on identifying the
possible origins of these BHNS mergers, (e.g., Hoang et al., 2020; Ye et al.,
2020; Mandel & Smith, 2021, see former study’s introduction for an overview of
the field).

BHNS mergers are uniquely interesting due to the possibility of an
electromagnetic (EM) counterpart signal caused by the disruption of the NS
from the BH tidal forces. This might result in a short-Gamma ray burst (sGRB,
e.g., Berger, 2014) and other EM transients. sGRBs are intense, non-repeating
and short flashes of gamma rays caused by the disruption of a NS due to tidal
forces from its companion (either a NS or a BH, in this manuscript we focus on
the BH-NS binaries). Unlike, long duration GRBs (longer than 2 seconds), which
are associated with massive stars or core-collapse supernovae, that are
observed solely in star forming galaxies, sGRBs are observed in both, star
forming galaxies and elliptical galaxies. Overall about $20\%$ of sGRBs are
found in early-type (elliptical and $S0$ galaxies) (e.g., Leibler & Berger,
2010; Berger, 2014). Here we show that our novel channel naturally predicts
$\sim 1\%-50\%$ of all BH-NS mergers to be accompanied with sGRB transient.

This paper is organized as follows. In section 2 we preset the physical
picture, derive the governing rate equations and calculate the merger rates.
We discuss the results and summarize the manuscript in section 3.

##  2 Physical Picture

###  2.1 The Collisional Nature of Ultra-Wide Binaries in the Field

This section describes the merger scenario and calculates the merger rate for
a Milky-Way (MW)-like galaxy and an elliptical galaxy. Although low in stellar
density, the galactic field is considered collisional for ultra-wide systems.
These wide systems interact relatively frequently with random flyby stars in
the field because of their large interaction cross-section proportional to
their SMA. Hence, an ultra-wide system encounters many random interactions
with flyby field stars.

These interactions are typically impulsive by nature, i.e., the interaction
timescale $t_{{\rm int}}\equiv b/v_{{\rm enc}}$ (where $b$ denotes the closest
approach to the binary’s center of mass, and $v_{{\rm enc}}$ denotes the
relative velocity between the random flyby star and the binary) is much
shorter than the orbital period time, $P$. Moreover, these interactions change
the binary orbit characteristics, namely the orbital energy (decrease/increase
the SMA), but more significantly, torque the system and change the binary
eccentricity $e$ (e.g., Lightman & Shapiro, 1977; Merritt, 2013). After many
impulsive interactions the wide binary might be driven into a sufficiently
high eccentricity, and as a result to sufficiently small pericenter distance,
$q$, which allows the binary to merge via GW emission within Hubble time (or
less). Once a BH-NS mergers via GW we might expect an EM transient associated
with the mergers, namely a sGRB.

###  2.2 Relevant Timescales

As mentioned above in this scenario, the impulse interaction timescale
$t_{{\rm int}}\equiv b/v_{{\rm enc}}$ needed to be shorter than the binary
orbital period $P$. Specifically we require that $t_{{\rm int}}=\alpha P$,
with $\alpha\ll 1$. This gives upper bound to the closest approach distance,
$b_{\rm{impulse}}=\alpha Pv_{\rm enc}$ and hence limits the average time
between encounters. For example, consider a binary with total mass of
$10M_{\odot}+1.5M_{\odot}=11.5M_{\odot}$, SMA of $a\sim 10^{4}{\rm AU}$ which
translates to an orbital period of $P\approx 3.2\times 10^{5}{\rm yr}$ and
velocity encounter of $v_{{\rm enc}}=50{\rm kms^{-1}}$ we can restrict
$b_{\rm{impulse}}$ for $\alpha=0.1$. Thus, we get $b_{\rm impulse}=\alpha
t_{{\rm int}}\times v_{{\rm enc}}\sim 3.3\times 10^{5}{\rm AU}$.

Additionally, we consider the average time between two flybys of the binary
system and a random stellar perturber, $t_{{\rm enc}}=1/f=\left(n_{*}\sigma
v_{{\rm enc}}\right)^{-1}$ where $n_{*}$ denotes the stellar number density,
$\sigma=\pi b^{2}$ is the geometric cross-section of the binary and the
stellar fly-by, and $v_{{\rm enc}}$ is the relative velocity of the fly-by and
the binary center of mass.

Lastly, we consider the GW timescale (e.g., Peters, 1964)

$t_{{\rm
merger}}\approx\frac{a^{4}}{\beta^{\prime}}\times\left(1-e^{2}\right)^{7/2}$
(1)

where

$\beta^{\prime}=\frac{85}{3}\frac{G^{3}m_{{\rm BH}}m_{{\rm NS}}\left(m_{{\rm
BH}}+m_{{\rm NS}}\right)}{c^{5}}$

$G$ is Newton’s constant and $c$ is the speed of light. Thus, we focus on the
merger of a wide binary with semi-major axis, $a>10^{2}{\rm AU}$.

###  2.3 Analytic Description of the BH-NS merger

Consider an ensemble of isotropic wide binaries, each consisting of a BH with
mass $m_{{\rm BH}}=10$ and a NS with mass $m_{{\rm NS}}=1.5M_{\odot}$. We
assume all binaries to have the same SMA, and a thermal distribution of
orbital eccentricities, $f(e)de=2ede$. We derive the merger probability from
this ensemble and find its dependence on the SMA of the binaries, $a$ for a
given environment, i.e. stellar density $n_{*}$ and velocity dispersion.

We start by calculating the fraction of system that merge from this ensemble
given thermal distribution of eccentricity using Equation (1).

Given the merger time $t_{{\rm merger}}=T$ and the SMA of the binary $a$ we
define a critical eccentricity $e_{c}$

$e_{c}=\left[1-\left(\frac{\beta^{\prime}T}{a^{4}}\right)^{2/7}\right]^{1/2},$
(2)

for which all systems with eccentricities equal or greater than $e_{c}$ merge
within time $T$. Below we will adopt $T$ as the average encounter time. Thus,
given a thermal distribution of eccentricities we find the fraction of systems
that merger within $T$ and lost from the ensemble to be:

$F_{q}=\int_{e_{c}}^{1}2ede=1-e_{c}^{2}=\left(\frac{\beta^{\prime}T}{a^{4}}\right)^{2/7}\
.$ (3)

All other binaries outside the loss-cone are susceptible for interaction with
a random flyby and can be perturbed to change their angular momentum and
replenish the loss cone.

After a single random flyby interaction the relative velocity vector of the
binary changes. The new direction of the velocity vector maps a cone around
the original relative velocity vector. The average _fraction_ of the phase-
space region into which the binaries in the ensemble are perturbed after a
single fly-by is termed the smear cone. The smear cone is computed by the
ratio of the size of the cone with all possible directions (e.g., Kaib &
Raymond, 2014; Michaely & Perets, 2016, 2019). In order to calculate the smear
cone one should first find the size of which , and defined by the following:

$\theta=\frac{\left\langle\Delta v\right\rangle}{v_{k}}\ ,$ (4)

where $v_{k}$ is the Keplerian velocity of the binary and $\left\langle\Delta
v\right\rangle$ is the average change of $v_{k}$ from a single perturbation.
The value of $v_{k}$ is calculated at the average separation of a Keplerian
orbit is $\left\langle r\right\rangle=a\left(1+1/2e^{2}\right)$ and we
approximate $e\rightarrow 1$, yielding

$v_{k}\sim\left(\frac{GM}{3a}\right)^{1/2}\ ,$

where $M\equiv\left(m_{{\rm BH}}+m_{{\rm NS}}\right)$ is the total mass of the
binary.

The average change of the Keplerian velocity, $\left\langle\Delta
v\right\rangle$ is given by (e.g., Hills, 1981)

$\left\langle\Delta v\right\rangle\simeq\frac{3Gam_{p}}{v_{{\rm enc}}b^{2}},$
(5)

where $v_{{\rm enc}}$ is the velocity of the fly-by star with respect to the
binary center of mass, $m_{p}$ is the perturber mass. Therefore, the square of
the angular size of the smear cone cause by the impulse of the fly-by on the
binary is (using Equations (4) - (5))

$\theta^{2}=\frac{9G^{2}a^{2}m_{p}^{2}}{\left(v_{{\rm
enc}}b^{2}\right)^{2}}\frac{3a}{GM}=\frac{27Ga^{3}m_{p}^{2}}{M\left(v_{{\rm
enc}}b^{2}\right)^{2}}\ ,$ (6)

and for $\theta\ll 1$ we get the average size of the cone post interaction.
The smear-cone is the fraction of that size with all possible directions, i.e.
over the $4\pi$ sphere to be after a single passage of the perturber (e.g.,
Kaib & Raymond, 2014; Michaely & Perets, 2016, 2019)

$F_{s}=\frac{\pi\theta^{2}}{4\pi}=\frac{27}{4}\left(\frac{m_{p}}{M}\right)^{2}\left(\frac{GM}{av_{{\rm
enc}}^{2}}\right)\left(\frac{a}{b}\right)^{4}\ .$ (7)

For a given binary, the smear cone size in Equation (7) depends on the
perturber and the environment, specifically, its mass, $m_{p}$, closest
approach, $b$ and the encounter velocity, $v_{{\rm enc}}.$

The ratio between the smear and loss cones, is the fraction of the loss cone
filled after a single flyby. Namely, for cases where $F_{s}\geq F_{q}$ the
entire ensemble regain its thermal eccentricity distribution.

$\frac{F_{s}}{F_{q}}=\frac{27}{4}\left(\frac{m_{p}}{M}\right)^{2}\left(\frac{GM}{av_{{\rm
enc}}^{2}}\right)\left(\frac{a}{b}\right)^{4}\left(\frac{a^{4}}{\beta^{\prime}T}\right)^{2/7}\
.$ (8)

The are two conditions for the loss cone to be continuously full is (1) that
the kick is sufficiently strong to fill the loss cone, namely, $F_{s}\geq
F_{q}$, and (2) the replenishment is as fast as the orbits are lost from the
ensemble. The replenishment rate is determined by the interaction rate which
can easily by calculate by

$f\equiv\frac{1}{t_{{\rm enc}}}=n_{*}\sigma\left(b\right)v_{{\rm enc}}.$ (9)

The first condition, the demand that the flyby kick is sufficiently strong,
sets a limit on the closest approach,

$b_{\rm kick}^{2}\leq v_{\rm
enc}^{-1}\left(\frac{27G}{4}a^{29/7}\frac{m_{p}^{2}}{M}\left(\frac{n_{*}\pi}{\beta^{\prime}}\right)^{2/7}\right)^{7/12}.$
(10)

We set the close approach to be the minimum of the two distance scales, the
first $b_{\rm impulse}$ is required because we focus only on the impulsive
regime, the second $b_{\rm kick}$ is required to ensure each fly-by is
sufficiently strong to fill the loss cone. Specifically,

$b=min\left(b_{\rm kick},b_{\rm impulse}\right),$ (11)

see Figure 1 for visualization.

Next, we can calculate the average encounter time, $t_{\rm enc}$ from Equation
(9). We chose the merger time to the average time between encounters, which is
a function of the closest approach, $T\equiv t_{\rm enc}\left(b\right)$. Note
that the merger time $T$ is dependent on the closest approach $b$, which is
determined from Equation (11).

Figure 1: The two distance scales $b_{\rm kick}$ (dashed red) and $b_{\rm
impulse}$ (solid black). Each fly-by with $b<b_{\rm kick}$ will fill the loss
cone and each one with $b<b_{\rm impulse}$ will interact impulsively with the
wide BHNS binary. We are interested in the grey area which insures both
conditions. Calculated for binary with $m_{{\rm BH}}=10M_{\odot}$ and $m_{{\rm
NS}}=1.5$ with $v_{{\rm enc}}=60{\rm kms^{-1}}$. The stellar density number is
$n_{*}=0.1{\rm pc^{-3}}$.

Now we are set to find the critical SMA from which the loss cone is
continuously full, under the impulse approximation, as a function of the local
environment characteristics. Thus, for $F_{s}=F_{q}$ and $1/t_{{\rm
enc}}=n_{*}\sigma v_{{\rm enc}}$, we find:

$a_{{\rm crit}}=\left[\frac{4}{27}\frac{M\beta^{\prime
2/7}T\left(b\right)^{-12/7}}{Gm_{p}^{2}n_{*}^{2}\pi^{2}}\right]^{7/29}.$ (12)

Another way to express the critical SMA is without the specific explicit
function of the geometric cross section, $\sigma$

$a_{{\rm crit}}=\left(\frac{GM}{n_{*}^{2}\sigma(b)^{2}v_{\rm
enc}^{2}}\right)^{1/3}\ .$ (13)

Binaries with SMA than equal to the $a_{{\rm crit}}$ have the highest
probability to merger via GW emmision. Binaries with SMA smaller than $a_{{\rm
crit}}$ are harder to perturb, therefore the probability rate decreases.
Binaries with SMA larger than $a_{{\rm crit}}$ also descreas in probability
due to the smaller loss cone, additionally, these binaries are more likely to
become unbind during the interaction, e.g. Michaely & Perets (2019)]. This
interplay leads to a sharp transition in the merger probability, as
highlighted in Figure 2, see below.

In other words, the full loss cone regime is for all systems with $a>a_{{\rm
crit}}$. The loss rate of systems from the full loss-cone, ($\dot{L}_{{\rm
full}}$), is given by the rate they leave the loss cone

$\dot{L}_{{\rm full}}=\frac{F_{q}}{P}\ .$ (14)

On the other hand, tighter binaries are less susceptible for change due to a
fly-by, this is evident from equation (7). Therefore for $a<a_{{\rm crit}}$ we
expect that the loss cone will not be full all the time, in this “empty loss
cone” the loss rate depends on the rate of orbits being kicked into the loss
cone is just the interaction rate, $f$:

$\dot{L}_{{\rm empty}}=F_{q}f=\frac{F_{q}}{t_{{\rm enc}}}$ (15)

From the critical SMA we can calculate the merger probability for each regime,
for $a<a_{{\rm crit}}$ (empty loss cone) and for $a>a_{{\rm crit}}$ (full loss
cone). We note here that by considering the critical SMA we implicitly include
the dependency on the impact parameter, $b$.

In what follows we calculate the time dependent loss probability of the two
regimes, the empty and full cones. We note that $F_{q}$ is the fraction of
systems that are lost from the ensemble and therefore $\left(1-F_{q}\right)$
represents the fraction of binaries that survive as wide binaries at the
relevant timescale, after a single flyby with the relevant timescale.
Therefore, $\left(1-F_{q}\right)$ is a monotonically decreasing function of
time. Hence, the surviving fraction of binaries after time $t$ and the
relevant timescale for the empty loss cone $f$ is $\left(1-F_{q}\right)^{tf}$
(e.g., Kaib & Raymond, 2014; Michaely & Perets, 2016, 2019). Therefore the
probability for a wide binary merger, which complements this expression to
unity is

$L_{a<a_{{\rm crit}}}=1-\left(1-F_{q}\right)^{tf}$ (16)

where $t$ is the time since birth of the binary. As one can expect the
probability only depends on the size of the loss cone and the rate of
interactions. For the limit of $tfF_{q}\ll 1$ we can expand Equation (16) and
take the leading term, to find the loss probability to be approximated by

$L_{a<a_{{\rm crit}}}=tf(b)F_{q}.$ (17)

For the full loss cone regime the limiting factor is not the value of $f$, but
rather the orbital period $P$. Therefore, the full expression for the loss
probability for $a>a_{crit}$ is

$L_{a>a_{{\rm crit}}}=1-\left(1-F_{q}\right)^{t/P}.$ (18)

For the limit $F_{q}\cdot t/P\ll 1$ we can approximate the probability by

$L_{a>a_{{\rm crit}}}=tF_{q}~{}\frac{1}{P(a)}.$ (19)

Figure 2: The merger probability of BHNS with $m_{{\rm BH}}=10M_{\odot}$ and
$m_{{\rm NS}}=1.5$ with fly-by mass $m_{p}=0.6M_{\odot}$ and $v_{{\rm
enc}}=50{\rm kms^{-1}}$. The stellar density number is $n_{*}=0.1{\rm
pc^{-3}}$. The probability it calculated after $t=10{\rm Gyr}$ since the BHNS
was formed. The peak probability is achieved at $a=a_{{\rm crit}}$. For
binaries with tighter orbits the merger probability drops because it is harder
to change their orbital parameters. For binaries with $a>a_{{\rm crit}}$ the
probability decreases because the loss cone dependency with $a$, see text.

Until this point we neglect the ”ionization’ of wide binaries in the field.
This process reduce the available number of wide binaries. To account for this
we use the half-life time approximate relation given by (Bahcall et al.,
1985), the half-life time of a wide binary is

$t_{1/2}=0.00233\frac{v_{{\rm enc}}}{Gm_{p}n_{*}a}.$ (20)

Correcting for this we get for Equation (17) and Equation (19) the following
approximations:

$L_{a<a_{{\rm crit}}}=\tau f(b)F_{q}\left(1-e^{-t/\tau}\right)$ (21)

and

$L_{a>a_{{\rm crit}}}=\tau\frac{F_{q}}{P(a)}\left(1-e^{-t/\tau}\right),$ (22)

where $\tau=t_{1/2}/\ln 2$ is the mean lifetime of the binary. For a more
detailed derivation see equations (11) and (12) in Michaely & Perets (2019)
and in the Appendix.

###  2.4 Volumetric merger rates

In this subsection we calculate the GW signal rate of the local universe from
wide BH-NS binaries. In order to do that we need (1) to approximate the
fraction of wide BH-NS binaries out of the stellar population. (2) To model
the stellar density of the host galaxy.

First we estimate the fraction of wide BH-NS systems from the stellar
population. The initial binary is comprised out of two stars with masses
$M_{1}$ and $M_{2}$. As a prove of concept we assume that all stars with zero
age main sequence (ZAMS) masses greater than $20M_{\odot}$ turn into BHs
without any natal-kicks (Belczynski et al., 2016; Mandel, 2016, and others).
We emphasize that this is a simplifying assumption, it is currently unknown
the mapping of ZAMS, metallicity and the stellar spin to the remnant BH mass.
Additionally, we assume that all stars with ZAMS between $8-10M_{\odot}$ turn
into NS via electron capture Supernova (SN) and born without natal kicks
(Nomoto, 1982, 1984). For Kroupa (Salpeter) initial mass function (IMF) we
find that $f_{{\rm BH}}\approx 2\times 10^{-3}\ \left(5\times 10^{-4}\right)$
(Kroupa, 2001; Salpeter, 1955). The companion mass is calculated given a mass
ratio distribution, $Q\equiv M_{2}/M_{1}$ with $Q\propto M_{{\rm 1}}^{-2}$
(Moe & Di Stefano, 2016) with the boundaries $Q_{{\rm
outer}}\in\left(0.3,1\right)$, this translate to $f_{{\rm NS}}\approx 0.13\
\left(0.12\right)$. We assume that the binary fraction $f_{{\rm binary}}=1$
(Duchêne & Kraus, 2013). Next we estimate the fraction of binaries with SMA
greater than $a>1000{\rm AU}$ to be $f_{{\rm wide}}\approx 0.2$ (Michaely &
Perets, 2019)

$f_{{\rm BH-NS}}=f_{{\rm BH}}\times f_{{\rm NS}}\times f_{{\rm binary}}\times
f_{{\rm wide}}\approx 5.2\times 10^{-5}$ (23)

and for Salpeter IMF we get $f_{{\rm BH-NS}}\approx 1.2\times 10^{-5}$.

Next, we model the stellar density function for a Milky-Way like spiral galaxy
and for an elliptical galaxy. For the spiral case we follow the model used in
(Michaely & Perets, 2016). The number of stars in region $dr$ at a distance
$r$ from the center is

$dN(r)=n_{*}\left(r\right)~{}2\pi~{}r~{}h~{}dr$ (24)

where the scale height is $h$. We continue with modeling the Galactic stellar
density in the Galactic disk as follows

$n_{*}\left(r\right)=n_{0}e^{-\left(r-r_{\odot}\right)/R_{l}}\ ,$ (25)

where $n_{0}=0.1{\rm pc}^{-3}$ is the stellar density in the solar
neighborhood, $R_{l}=2.6{\rm kpc}$ (Jurić et al., 2008) is the galactic length
scale and $r_{\odot}=8{\rm kpc}$ is the distance of the Sun from the Galactic
center. The mass of the perturber is taken to be $0.6M_{\odot}$ which is the
average mass of a star in the galaxy. The encounter velocity is set to be the
velocity dispersion of the flat rotation curve of the galaxy, namely
$\hat{\sigma}=50{\rm kms^{-1}}$.

For the elliptical galaxies, we use the density profile function from
(Hernquist, 1990), given the average stellar mass of $0.6M_{\odot}$ we get

$\tilde{n}_{*}\left(r\right)=\frac{M_{{\rm galaxy}}}{2\pi
r}\frac{r_{*}}{\left(r+r_{*}\right)^{3}}$ (26)

where $r_{*}=1{\rm kpc}$ is the scale length of the galaxy, $M_{{\rm
galaxy}}=10^{11}M_{\odot}$ is the total stellar (not including dark matter)
mass of the galaxy, we focus on $r=0.5-30\rm{kpc}$. The elliptical galaxy
velocity dispersion is taken to be $\hat{\sigma}=160{\rm kms^{-1}}.$

The merger rate is calculated by integrating the merger probability for all
SMA, with log-uniform distribution, $f_{a}$, and the entire host galaxy

$\Gamma_{{\rm MW}}=\int_{0.5{\rm kpc}}^{15{\rm kpc}}\int_{100{\rm AU}}^{5\cdot
10^{4}{\rm AU}}\frac{L\left(a,r\right)}{T_{\rm int}}f_{a}f_{{\rm BH-NS}}dadr\
$ (27)

where $T_{\rm int}$ is the integration time, for $T_{\rm int}=10\rm{Gyr}$ the
average merger rate, $\Gamma$ is for the lifetime of the host galaxy. The
integral limits are taken to be the relevant size of the MW galaxy and the all
the binaries SMA we consider. The function $L\left(a,r\right)$ is taken from
eq. (21) and (22). Thus, for the MW-like galaxy we get

$\Gamma_{{\rm MW}}\approx 0.06{\rm Myr}^{-1}\ ,$ (28)

where, we use Belczynski et al. (2016) in order to calculate the merger rate,
$R$ per ${\rm Gpc^{3}}$ by using the following equation

${R_{{\rm MW}}}=10^{3}\rho_{{\rm gal}}\times\Gamma_{{\rm MW}}\approx 0.66{\rm
Gpc^{-3}yr^{-1}}$ (29)

while $\rho_{{\rm gal}}$ is local density of the MW-like galaxies with the
value of $\rho_{{\rm gal}}=0.0116{\rm Mpc^{-3}}$(e.g. Kopparapu et al. (2008))
and $\Gamma_{{\rm MW}}$ is given in the units of ${\rm Myr^{-1}.}$

For the elliptical case we compute:

$\Gamma_{{\rm elliptical}}=\int_{0.5{\rm kpc}}^{30{\rm kpc}}\int_{100{\rm
AU}}^{5\cdot 10^{4}{\rm AU}}\frac{L\left(a,r\right)}{T_{\rm int}}f_{a}f_{{\rm
BHNS}}dadr$ (30)

and get $\Gamma_{{\rm elliptical}}\approx 0.39{\rm Myr^{-1}}$.

The number density of elliptical galaxies in the local universe elliptical is
crudely approximated by $\sim 0.1{\rm Mpc^{-3}}$ (Samsing et al., 2014) and
get

${R_{{\rm elliptical}}}\approx 39{\rm Gpc^{-3}yr^{-1}.}$ (31)

Adding the contributions of each type of galaxy, we get

$R_{\rm tot}=R_{\rm elliptical}+R_{\rm MW}\approx 40\rm Gpc^{-3}yr^{-1}.$ (32)

##  3 Discussion and summary

###  3.1 Caveats and assumptions

The mathematical model we present here is simplistic and based on several
assumptions.

BH masses: As mentioned in Sec. 2.4, we make simplifying assumptions on the
remnant mass of the BH. The mapping of initial mass of the star, metallicity
and spin to final BH mass is not well understood. The stellar evolution
physics, e.g. wind mass loss, nuclear physics and supernova physics (Baes et
al., 2007; Zhu et al., 2010; Fryer et al., 2012). Hence we present our results
with the prefactor of $f_{\rm BH}=2\times 10^{-3}$. Different assumption will
lead to different fraction of BH from the initial populations.

BH Natal kicks: The calculated merger rates depends strongly on the binary
fraction estimation. This estimation is stringently depended on the natal
kicks upon formation of BHs. These natal kicks are poorly constrained (e.g.,
Repetto et al., 2012, 2017). However, there are evidence that BH are formed by
failed SN (e.g., Ertl et al., 2015). In the failed SN scenario large amount of
fallback is accreted on the newly formed compact object and suppresses any
natal kicks. Moreover, many theoretical studies of GW merger rate assume not
natal-kicks for BH (e.g., Belczynski et al., 2007, 2008, 2016).

NS Natal kicks: From the observations of pulsar it is well known that NS are
born with natal kicks. However, in the narrow range of masses between
$8-10M_{\odot}$, stars are believed to undergo e-electron capture SN. In this
work we limit ourselves only to this narrow range of stars and assume no natal
kicks for the newly born NS.

SMA and eccentricity distributions The underlying assumption of the results
presented here are the distributions of the SMA and eccentricities of the wide
systems. We use Opik law, namely, log-uniform distribution for the SMA of the
systems, and use thermal eccentricity distribution for the eccentricity of the
binaries.

###  3.2 Model Predictions and Signatures

In this study we explore a novel scenario that may lead to a merger between BH
and a NS or a NS disruption. In what follows we present some of the model
features.

####  3.2.1 Velocity dispersion

A unique signature of the model presented here is the dependence on the host
galaxy velocity dispersion. The velocity dispersion sets the rate of
encounters and the timescale of binary ionization. Following the model
presented in the previous section we can calculate the scaling relation
between the merger rate and the hot galaxy velocity dispersion. We distinguish
between the regime where $b=b_{\rm{kick}}$ and the regime where
$b=b_{\rm{impulsive}}$. For simplicity we start with the non-ionized case
Equation (17) and Equation (19). The scaling relation is calculated by
considering each term dependency on $v_{\rm enc}$. The frequency $f\propto
v_{\rm enc}$ from equation (9). The loss cone, $F_{q}\propto T^{2/7}a^{-8/7}$,
noting that $a\propto v_{\rm enc}^{-2/3}$ from equation (13) we get
$F_{q}\propto v_{\rm enc}^{10/21}$. Therefore we get:

$L_{\rm impulse,empty}=tfF_{q}\propto v_{\rm enc}^{31/21}$ (33)

The integrated rate, $\Gamma$, is the integral of (33) over all SMA, $\int
Lf_{a}da$.Hence, the scaling relation is

$\Gamma\propto L\frac{1}{a_{\rm crit}}a_{\rm crit}\propto v_{\rm enc}^{31/21}$
(34)

Including the ionization process, the merger probability changes to Eq. (21),
where

$\tau\propto\frac{v_{\rm enc}}{a}\ .$ (35)

For the case of $-t/\tau\rightarrow 0$ we get $\tau\propto v_{\rm enc}^{0}$
for the other extreme, $-t/\tau\rightarrow\infty$ we get $\tau\propto v_{\rm
enc}^{5/3}$. For the regime where $b=b_{\rm kick}$, the calculation is similar
but with different scaling. Specifically,

$L_{\rm kick,empty}=tfF_{q}\propto v_{\rm enc}^{0}.$ (36)

As a result the $v_{\rm enc}$ dependency disappears in the non-ionization
case. Once we include ionization we get the following:

$\Gamma\propto\int\tau Lf_{a}da(1-e^{-t/\tau})\propto v_{\rm
enc}(1-e^{-t/\tau}),$ (37)

where

$\tau\propto v_{\rm enc}\ .$ (38)

In Figure 3 we depict the merger rate scaling relation as a function of the
velocity encounter for a range of velocity dispersions. One can see the
crossover in scaling around $v_{\rm enc}\approx 70~{}\rm{km~{}s^{-1}}$ between
the impulsive and the kick regimes.

Currently we cannot identify BHNS mergers for a specific galaxy, however, it
is possible to assign a host galaxy to a sGRB coincident with GW detection.

Figure 3:  Average merger rate for a MW galaxy-like as a function of
unrealistic velocity dispersion. Rate is calculated after $10^{10}\rm yrs$
from star formation accounting for binary ionization. Binaries consist of
$m_{\rm{NS}}=1.5M_{\odot}$ and $m_{\rm{BH}}=10M_{\odot}$. For $v_{\rm
enc}\lesssim 40\quad(\gtrsim 150)\quad\rm{kms^{-1}}$, the closest approach
$b=b_{\rm impulse}\quad(b=b_{\rm kick})$ for all SMA (grey area), see text.
The white area is a mix of both regimes. The rate for an elliptical galaxy
shows a similar behavior and is omitted here to avoid clutter.

####  3.2.2 Delay time distribution

The BHNS delay time distribution (DTD), $d\Gamma/dt$, is the binary merger
rate as a function of time since star formation. In this case we ignore the
stellar evolution time, which is much quicker ($\sim 10^{6}\rm yr$) than
galaxy lifetime timescale ($\sim 10^{10}\rm yr$). One can calculate the
expected DTD of GW sources, $\Gamma$, from this channel numerically,
specifically we compute equations (27) and (30) for different integration
times, $T_{\rm int}$. If we ignore the ionization process, we expect a
constant DTD, because the merger probability from equations (17) and (19) are
linear in $t$. However, in reality wide systems are continuously ionized in
the field due to the flyby interactions, and there are less available binaries
to merger via GW. The time dependence is given by Equations (21) and (22). We
solve these equations numerically for different times after formation. We note
that the total rate also depends on the SMA and local stellar density, which
make the analytical dependence of $t$ very complicated, therefore we only find
the DTD numerically. In Figure 4 we present the DTD for MW-like galaxy, we
present the best power law fit in order to compare to the classical $1/t$ we
expect from SNe (Maoz et al., 2014) and sGRB (Berger, 2014). We find for a MW-
like galaxy the DTD is proportional to $\sim t^{-0.41}$. The exact value of
the power law depends on the host galaxy characteristics (velocity dispersion
and the distribution of the stellar density), as well as the distribution of
the SMA of the wide binaries.

Figure 4:  The Delay time distribution (DTD) for a MW-like galaxy with
velocity dispersion of $\sigma=40\rm{kms^{-1}}$. Blue circles are the model
calculations. Black solid line is the best power law fit. Note that this fit
is specific for MW-galaxy adopted here, see text for more details. The GW
merger span for the entire lifetime of the galaxy, which suggests these
mergers occur in early and late type of galaxies.

####  3.2.3 Mass ratio

Here we present the model dependency on the mass ratio, $Q\equiv m_{\rm
NS}/m_{\rm BH}$. Unlike Michaely & Perets (2019) that considered BBH, here one
of the companions is a NS and therefore has a constant mass. It is important
to note that the merger rate dependency on $Q$ might be a source of confusion.
In our case, NS-BH binary, a change is $Q$ infers also a change in the total
binary mass $M$, due to the constant NS mass. Moreover, in what follows we do
not assume the BH mass distribution but rather just present the merger rate
assuming all BHs have the same mass. In Figure 5 we present the merger rate as
a function of mass ration $Q$. We find that for the BHNS case a higher merger
rate for lower $Q$, as can seen in Figure 5.

Figure 5:  Average merger rate for a MW galaxy-like (red dots) and elliptical
galaxy (blue dots). The velocity dispersion is $\sigma=40\rm kms^{-1}$ for the
MW-like galaxy and $\sigma=160\rm kms^{-1}$ for the elliptical. Rate is
calculated after $10^{10}\rm yrs$ from star formation, for binaries with
$m_{\rm{NS}}=1.5M_{\odot}$ and varying $m_{\rm{BH}}=10M_{\odot}$.

####  3.2.4 Spin distribution

Because the origin of these mergers are from extremely wide binaries we do not
expect these compact objects to have any correlation in their respective
spins. Therefore, one should not expect any spin alignment between the BH and
the NS in the moment of merger. Similar to other dynamically induced mergers
the $\xi_{{\rm eff}}$ is expected to be close to zero.

###  3.3 sGRB

Any BHNS merger is a potential for sGRB (Berger, 2014). A merger will not
result in a disruption of the NS if the NS enters the BH event horizon in its
entirety and therefore EM radiation is not created and omitted. The physical
quantity that governs the outcome of the merger, whether or not it will
produce an EM transient is the NS equation of state (EOS) which is still
unconstrained. Fragione (2021) showed that for the fraction of BH-NS mergers
that produce a sGRB, under some assumptions of their EOS is $\sim 1-50\%$.
Accounting for this fraction and our merger rates we get that the sGRB from
$R_{\rm{tot}}$:

$\Gamma_{\rm{sGRB}}\approx 0.5-20\rm{Gpc^{3}yr^{-1}}.$ (39)

Additionally, unlike GW mergers detection, which we have very poor
localization, and currently cannot assign a host galaxy to the detection, we
are able in principle to identify the host galaxy, and check this model
assumptions. For example, as mentioned above, sGRB are observed in both star
forming galaxies and in ellipticals. This can be explained when considering
the DTD of this model, see Figure 4. The DTD extends from star formation till
$10^{10}\rm{yr}$, thus we expect these mergers and sGRBs to in both star
forming and elliptical galaxies.

###  3.4 Summary

In this paper we present a novel dynamical channel for the formation of field
wide BH-NS binaries mergers via GWs by random flyby interactions with passing
stars. We find that the volumetric merger rate is up to $R_{\rm tot}\approx
40\rm Gpc^{3}yr^{-1}$. Additionally, we report the following signatures: (1)
the merger rate increase with the host galaxy velocity dispersion.

(2) The DTD is function of the galaxy characteristics and shallower than
uniform but steeper that $1/t$. For a MW-like spiral galaxy is $\propto
t^{-0.41}$.

(3) The merger rate decrease with increasing mass ration of the binary.

(4) The effective spin distribution is expected to be zero due to the fact
that the merger is originating from initially wide binary system, which we do
not expect to have any correlations in a the spins.

(5) EM radiation, in the form of sGRB is expected to occur from $1-50\%$ of
the merger we calculate, which translates in for this scenario up to
$0.4-20\rm{Gpc^{3}yr^{-1}}$.

## Acknowledgments

EM and SN thank the Bhaumik Institute for Theoretical Physics and Howard and
Astrid Preston for their generous support. We also acknowledges the partial
support from NASA ATP 80NSSC20K0505. Here we present the explicit function
form of the rate equation. We distinguish between two different regimes, the
impulsive regime and the kick regime.

Impulsive regime. In this case $b_{\rm impulsive}<b_{\rm kick}$. We remind the
reader that $b_{\rm impulsive}=\alpha Pv_{\rm enc}$, hence the explicit form
of the critical SMA is

$a_{\rm crit}=\left(\frac{\left(GM\right)^{3/2}}{8\pi^{4}n_{*}v_{\rm
enc}^{3}\alpha^{2}}\right)^{2/9}$ (1)

and the corresponding $f$ is

$f(a)=n_{*}\alpha^{2}P(a)^{2}v_{\rm
enc}^{3}=n_{*}\alpha^{2}\left(\frac{4\pi^{2}a^{3}}{GM}\right)v_{\rm enc}^{3}.$
(2)

The loss cone $F_{q}$ is given by (3), hence the loss rate function for the
impulsive regime, without accounting for binary ionization, is given by the
following two equations, the first for the empty loss cone regime:

$L_{a<a_{\rm
crit}}=tfF_{q}=tn_{*}\alpha^{2}\left(\frac{4\pi^{2}a^{3}}{GM}\right)v_{\rm
enc}^{3}\left(\frac{\beta}{f(a)a^{4}}\right)^{2/7}\propto a$ (3)

and for the full loss cone regime

$L_{a>a_{\rm
crit}}=t\frac{F_{q}}{P(a)}=t\left(\frac{\beta}{f(a)a^{4}}\right)^{2/7}\left(\frac{GM}{4\pi^{2}a^{3}}\right)\propto
a^{-5}.$ (4)

However, in order to calculate the rate including the ionization that the
binaries experience in the field one should take into account that half
lifetime of the binaries from equation (20). This introduces a nontrivial
dependency on the SMA, $a$ and we get

$L_{a<a_{\rm crit}}=\tau fF_{q}\left(1-e^{-t/\tau}\right)=\tau
n_{*}\alpha^{2}\left(\frac{4\pi^{2}a^{3}}{GM}\right)v_{\rm
enc}^{3}\left(\frac{\beta}{f(a)a^{4}}\right)^{2/7}\left(1-e^{-t/\tau}\right)$
(5)

and for the full loss cone

$L_{a>a_{\rm
crit}}=\tau\frac{F_{q}}{P(a)}\left(1-e^{-t/\tau}\right)=\tau\left(\frac{\beta}{f(a)a^{4}}\right)^{2/7}\left(\frac{GM}{4\pi^{2}a^{3}}\right)\left(1-e^{-t/\tau}\right).$
(6)

Kick regime. In this case $b_{\rm impulsive}>b_{\rm kick}$. In this case
$b_{\rm kick}$ is given by equation (10), and the corresponding $a_{\rm crit}$
and $f$ are given by

$a_{\rm
crit}=\left(\frac{GM}{n_{*}^{2}\pi^{2}}\right)^{6/47}\left(\frac{4M}{27Gm_{p}^{2}}\left(\frac{\beta}{n_{*}\pi}\right)^{2/7}\right)^{7/47}$
(7)

and

$f=n_{*}\sigma(b_{\rm kick})v_{\rm
enc}=n_{*}\pi\left(\frac{27G}{4}a^{29/7}\frac{m_{p}^{2}}{M}\left(\frac{n_{*}\pi}{\beta^{\prime}}\right)^{2/7}\right)^{7/12}.$
(8)

Next we write the rate equations for both the empty and full cone regimes,
without binary ionization to get

$L_{a<a_{\rm
crit}}=tfF_{q}=tn_{*}\pi\left(\frac{27G}{4}a^{29/7}\frac{m_{p}^{2}}{M}\left(\frac{n_{*}\pi}{\beta^{\prime}}\right)^{2/7}\right)^{7/12}\left(\frac{\beta}{f(a)a^{4}}\right)^{2/7}\propto
a^{7/12}$ (9)

and

$L_{a>a_{\rm
crit}}=t\frac{F_{q}}{P(a)}=t\left(\frac{\beta}{f(a)a^{4}}\right)^{2/7}\left(\frac{GM}{4\pi^{2}a^{3}}\right)\propto
a^{-29/6}.$ (10)

Next we write the same rate equation with the correction for the binary
ionization:

$L_{a<a_{\rm crit}}=\tau
n_{*}\pi\left(\frac{27G}{4}a^{29/7}\frac{m_{p}^{2}}{M}\left(\frac{n_{*}\pi}{\beta^{\prime}}\right)^{2/7}\right)^{7/12}\left(\frac{\beta}{f(a)a^{4}}\right)^{2/7}\left(1-e^{-t/\tau}\right)$
(11)

and

$L_{a>a_{\rm
crit}}=t\frac{F_{q}}{P(a)}=\tau\left(\frac{\beta}{f(a)a^{4}}\right)^{2/7}\left(\frac{GM}{4\pi^{2}a^{3}}\right)\left(1-e^{-t/\tau}\right).$
(12)

## References

  * Antonini et al. (2016) Antonini, F., Chatterjee, S., Rodriguez, C. L., et al. 2016, ApJ, 816, 65, doi: 10.3847/0004-637X/816/2/65
  * Antonini et al. (2014) Antonini, F., Murray, N., & Mikkola, S. 2014, ApJ, 781, 45, doi: 10.1088/0004-637X/781/1/45
  * Antonini & Perets (2012) Antonini, F., & Perets, H. B. 2012, ApJ, 757, 27, doi: 10.1088/0004-637X/757/1/27
  * Antonini et al. (2017) Antonini, F., Toonen, S., & Hamers, A. S. 2017, ApJ, 841, 77, doi: 10.3847/1538-4357/aa6f5e
  * Baes et al. (2007) Baes, M., Sil’chenko, O. K., Moiseev, A. V., & Manakova, E. A. 2007, A&A, 467, 991, doi: 10.1051/0004-6361:20066758
  * Bahcall et al. (1985) Bahcall, J. N., Hut, P., & Tremaine, S. 1985, ApJ, 290, 15, doi: 10.1086/162953
  * Banerjee (2018) Banerjee, S. 2018, MNRAS, 473, 909, doi: 10.1093/mnras/stx2347
  * Bartos et al. (2017) Bartos, I., Haiman, Z., Marka, Z., et al. 2017, Nature Communications, 8, 831, doi: 10.1038/s41467-017-00851-7
  * Belczynski et al. (2008) Belczynski, K., Kalogera, V., Rasio, F. A., et al. 2008, ApJS, 174, 223, doi: 10.1086/521026
  * Belczynski et al. (2016) Belczynski, K., Repetto, S., Holz, D. E., et al. 2016, ApJ, 819, 108, doi: 10.3847/0004-637X/819/2/108
  * Belczynski et al. (2007) Belczynski, K., Taam, R. E., Kalogera, V., Rasio, F. A., & Bulik, T. 2007, ApJ, 662, 504, doi: 10.1086/513562
  * Berger (2014) Berger, E. 2014, ARA&A, 52, 43, doi: 10.1146/annurev-astro-081913-035926
  * Britt et al. (2021) Britt, D., Johanson, B., Wood, L., Miller, M. C., & Michaely, E. 2021, MNRAS, 505, 3844, doi: 10.1093/mnras/stab1570
  * de Mink & Belczynski (2015) de Mink, S. E., & Belczynski, K. 2015, ApJ, 814, 58, doi: 10.1088/0004-637X/814/1/58
  * de Mink & Mandel (2016) de Mink, S. E., & Mandel, I. 2016, MNRAS, 460, 3545, doi: 10.1093/mnras/stw1219
  * Dominik et al. (2015) Dominik, M., Berti, E., O’Shaughnessy, R., et al. 2015, ApJ, 806, 263, doi: 10.1088/0004-637X/806/2/263
  * Duchêne & Kraus (2013) Duchêne, G., & Kraus, A. 2013, ARA&A, 51, 269, doi: 10.1146/annurev-astro-081710-102602
  * Eldridge et al. (2017) Eldridge, J. J., Stanway, E. R., Xiao, L., et al. 2017, PASA, 34, e058, doi: 10.1017/pasa.2017.51
  * Ertl et al. (2015) Ertl, T., Janka, H.-T., Woosley, S. E., Sukhbold, T., & Ugliano, M. 2015, ArXiv e-prints.  https://arxiv.org/abs/1503.07522
  * Fragione (2021) Fragione, G. 2021, ApJ, 923, L2, doi: 10.3847/2041-8213/ac3bcd
  * Fragione et al. (2019) Fragione, G., Grishin, E., Leigh, N. W. C., Perets, H. B., & Perna, R. 2019, MNRAS, 488, 47, doi: 10.1093/mnras/stz1651
  * Fragione & Kocsis (2018) Fragione, G., & Kocsis, B. 2018, Phys. Rev. Lett., 121, 161103, doi: 10.1103/PhysRevLett.121.161103
  * Fryer et al. (2012) Fryer, C. L., Belczynski, K., Wiktorowicz, G., et al. 2012, ApJ, 749, 91, doi: 10.1088/0004-637X/749/1/91
  * Gallegos-Garcia et al. (2021) Gallegos-Garcia, M., Berry, C. P. L., Marchant, P., & Kalogera, V. 2021, ApJ, 922, 110, doi: 10.3847/1538-4357/ac2610
  * Giacobbo et al. (2018) Giacobbo, N., Mapelli, M., & Spera, M. 2018, MNRAS, 474, 2959, doi: 10.1093/mnras/stx2933
  * Hamers et al. (2018) Hamers, A. S., Bar-Or, B., Petrovich, C., & Antonini, F. 2018, ApJ, 865, 2, doi: 10.3847/1538-4357/aadae2
  * Hamilton & Rafikov (2019) Hamilton, C., & Rafikov, R. R. 2019, MNRAS, 488, 5512, doi: 10.1093/mnras/stz2026
  * Heggie (1975) Heggie, D. C. 1975, MNRAS, 173, 729, doi: 10.1093/mnras/173.3.729
  * Hernquist (1990) Hernquist, L. 1990, ApJ, 356, 359, doi: 10.1086/168845
  * Hills (1981) Hills, J. G. 1981, AJ, 86, 1730, doi: 10.1086/113058
  * Hoang et al. (2018) Hoang, B.-M., Naoz, S., Kocsis, B., Rasio, F. A., & Dosopoulou, F. 2018, ApJ, 856, 140, doi: 10.3847/1538-4357/aaafce
  * Hoang et al. (2020) Hoang, B.-M., Naoz, S., & Kremer, K. 2020, ApJ, 903, 8, doi: 10.3847/1538-4357/abb66a
  * Jurić et al. (2008) Jurić, M., Ivezić, Ž., Brooks, A., et al. 2008, ApJ, 673, 864, doi: 10.1086/523619
  * Kaib & Raymond (2014) Kaib, N. A., & Raymond, S. N. 2014, ApJ, 782, 60, doi: 10.1088/0004-637X/782/2/60
  * Kimpson et al. (2016) Kimpson, T. O., Spera, M., Mapelli, M., & Ziosi, B. M. 2016, MNRAS, 463, 2443, doi: 10.1093/mnras/stw2085
  * Kopparapu et al. (2008) Kopparapu, R. K., Hanna, C., Kalogera, V., et al. 2008, ApJ, 675, 1459, doi: 10.1086/527348
  * Kozai (1962) Kozai, Y. 1962, AJ, 67, 591
  * Kremer et al. (2020) Kremer, K., Spera, M., Becker, D., et al. 2020, ApJ, 903, 45, doi: 10.3847/1538-4357/abb945
  * Kroupa (2001) Kroupa, P. 2001, MNRAS, 322, 231, doi: 10.1046/j.1365-8711.2001.04022.x
  * Leibler & Berger (2010) Leibler, C. N., & Berger, E. 2010, ApJ, 725, 1202, doi: 10.1088/0004-637X/725/1/1202
  * Leigh et al. (2018) Leigh, N. W. C., Geller, A. M., McKernan, B., et al. 2018, MNRAS, 474, 5672, doi: 10.1093/mnras/stx3134
  * Lidov (1962) Lidov, M. L. 1962, Planet. Space Sci., 9, 719, doi: 10.1016/0032-0633(62)90129-0
  * Lightman & Shapiro (1977) Lightman, A. P., & Shapiro, S. L. 1977, ApJ, 211, 244, doi: 10.1086/154925
  * Liu & Lai (2018) Liu, B., & Lai, D. 2018, ApJ, 863, 68, doi: 10.3847/1538-4357/aad09f
  * Mandel (2016) Mandel, I. 2016, MNRAS, 456, 578, doi: 10.1093/mnras/stv2733
  * Mandel & de Mink (2016) Mandel, I., & de Mink, S. E. 2016, MNRAS, 458, 2634, doi: 10.1093/mnras/stw379
  * Mandel & Smith (2021) Mandel, I., & Smith, R. J. E. 2021, ApJ, 922, L14, doi: 10.3847/2041-8213/ac35dd
  * Maoz et al. (2014) Maoz, D., Mannucci, F., & Nelemans, G. 2014, ARA&A, 52, 107, doi: 10.1146/annurev-astro-082812-141031
  * Marchant et al. (2016) Marchant, P., Langer, N., Podsiadlowski, P., Tauris, T. M., & Moriya, T. J. 2016, A&A, 588, A50, doi: 10.1051/0004-6361/201628133
  * Martinez et al. (2020) Martinez, M. A. S., Fragione, G., Kremer, K., et al. 2020, ApJ, 903, 67, doi: 10.3847/1538-4357/abba25
  * Merritt (2013) Merritt, D. 2013, Classical and Quantum Gravity, 30, 244005, doi: 10.1088/0264-9381/30/24/244005
  * Michaely (2020) Michaely, E. 2020, MNRAS, doi: 10.1093/mnras/staa3623
  * Michaely & Perets (2016) Michaely, E., & Perets, H. B. 2016, MNRAS, 458, 4188, doi: 10.1093/mnras/stw368
  * Michaely & Perets (2019) —. 2019, ApJ, 887, L36, doi: 10.3847/2041-8213/ab5b9b
  * Michaely & Perets (2020) Michaely, E., & Perets, H. B. 2020, Monthly Notices of the Royal Astronomical Society, doi: 10.1093/mnras/staa2720
  * Michaely & Shara (2021) Michaely, E., & Shara, M. M. 2021, MNRAS, 502, 4540, doi: 10.1093/mnras/stab339
  * Miller & Hamilton (2002) Miller, M. C., & Hamilton, D. P. 2002, ApJ, 576, 894, doi: 10.1086/341788
  * Moe & Di Stefano (2016) Moe, M., & Di Stefano, R. 2016, ArXiv e-prints.  https://arxiv.org/abs/1606.05347
  * Naoz (2016) Naoz, S. 2016, ARA&A, 54, 441, doi: 10.1146/annurev-astro-081915-023315
  * Nomoto (1982) Nomoto, K. 1982, ApJ, 253, 798, doi: 10.1086/159682
  * Nomoto (1984) —. 1984, ApJ, 277, 791, doi: 10.1086/161749
  * Olejak et al. (2020) Olejak, A., Fishbach, M., Belczynski, K., et al. 2020, ApJ, 901, L39, doi: 10.3847/2041-8213/abb5b5
  * Peters (1964) Peters, P. C. 1964, Physical Review, 136, 1224, doi: 10.1103/PhysRev.136.B1224
  * Petrovich & Antonini (2017) Petrovich, C., & Antonini, F. 2017, ApJ, 846, 146, doi: 10.3847/1538-4357/aa8628
  * Rastello et al. (2021) Rastello, S., Mapelli, M., Di Carlo, U. N., et al. 2021, MNRAS, 507, 3612, doi: 10.1093/mnras/stab2355
  * Repetto et al. (2012) Repetto, S., Davies, M. B., & Sigurdsson, S. 2012, MNRAS, 425, 2799, doi: 10.1111/j.1365-2966.2012.21549.x
  * Repetto et al. (2017) Repetto, S., Igoshev, A. P., & Nelemans, G. 2017, MNRAS, 467, 298, doi: 10.1093/mnras/stx027
  * Rodriguez et al. (2018) Rodriguez, C. L., Amaro-Seoane, P., Chatterjee, S., et al. 2018, Phys. Rev. D, 98, 123005, doi: 10.1103/PhysRevD.98.123005
  * Rodriguez et al. (2016) Rodriguez, C. L., Chatterjee, S., & Rasio, F. A. 2016, Phys. Rev. D, 93, 084029, doi: 10.1103/PhysRevD.93.084029
  * Rodriguez et al. (2021) Rodriguez, C. L., Kremer, K., Chatterjee, S., et al. 2021, Research Notes of the American Astronomical Society, 5, 19, doi: 10.3847/2515-5172/abdf54
  * Salpeter (1955) Salpeter, E. E. 1955, ApJ, 121, 161, doi: 10.1086/145971
  * Samsing et al. (2014) Samsing, J., MacLeod, M., & Ramirez-Ruiz, E. 2014, ApJ, 784, 71, doi: 10.1088/0004-637X/784/1/71
  * Samsing et al. (2018) —. 2018, ApJ, 853, 140, doi: 10.3847/1538-4357/aaa715
  * Silsbee & Tremaine (2017) Silsbee, K., & Tremaine, S. 2017, ApJ, 836, 39, doi: 10.3847/1538-4357/aa5729
  * Stephan et al. (2016) Stephan, A. P., Naoz, S., Ghez, A. M., et al. 2016, MNRAS, 460, 3494, doi: 10.1093/mnras/stw1220
  * Stephan et al. (2019) —. 2019, ApJ, 878, 58, doi: 10.3847/1538-4357/ab1e4d
  * Stone et al. (2017) Stone, N. C., Metzger, B. D., & Haiman, Z. 2017, MNRAS, 464, 946, doi: 10.1093/mnras/stw2260
  * The LIGO Scientific Collaboration et al. (2021a) The LIGO Scientific Collaboration, the Virgo Collaboration, the KAGRA Collaboration, et al. 2021a, arXiv e-prints, arXiv:2111.03606.  https://arxiv.org/abs/2111.03606
  * The LIGO Scientific Collaboration et al. (2021b) —. 2021b, arXiv e-prints, arXiv:2111.03634.  https://arxiv.org/abs/2111.03634
  * Vigna-Gómez et al. (2021) Vigna-Gómez, A., Toonen, S., Ramirez-Ruiz, E., et al. 2021, ApJ, 907, L19, doi: 10.3847/2041-8213/abd5b7
  * Wang et al. (2020) Wang, H., Stephan, A. P., Naoz, S., Hoang, B.-M., & Breivik, K. 2020, arXiv e-prints, arXiv:2010.15841.  https://arxiv.org/abs/2010.15841
  * Ye et al. (2020) Ye, C. S., Fong, W.-f., Kremer, K., et al. 2020, ApJ, 888, L10, doi: 10.3847/2041-8213/ab5dc5
  * Zhu et al. (2010) Zhu, G., Blanton, M. R., & Moustakas, J. 2010, ApJ, 722, 491, doi: 10.1088/0004-637X/722/1/491


# A new Renormalized Volume Type Invariant

Wu Jinyang<EMAIL_ADDRESS>225N MacLean Hall, Department of Mathematics,
University of Iowa, Iowa City, IA, 52242

(Date: July 9, 2023)

###### Abstract.

In this paper, we define a new conformal invariant on complete non-compact
hyperbolic surfaces that can be conformally compactified to bounded domains in
$\mathbb{C}$. We study and compute this invariant up to one-connected
surfaces. Our results give a new geometric criterion for choosing canonical
representations of bounded domains in $\mathbb{C}$.

The author acknowledges support from the Erwin and Peggy Kleinfeld Graduate
Fellowship fund.

###### Contents

  1. 1 Introduction
  2. 2 Technical Tools
  3. 3 One-Connected Domains
  4. 4 Rigidity of One-Connected Domains
  5. 5 Appendix

##  1\. Introduction

Let $(M,g_{M})$ be a complete non-compact hyperbolic surface of finite
topological type that can be conformally compactified to a bounded domain in
$\mathbb{C}$. Consider all possible choices of $\Omega\subset\mathbb{C}$ such
that $M$ can be conformally compactified into, which are conformally
equivalent to each other. In this article, we study a new canonical
representation through proper geometric consideration.

By the Riemann mapping theorem, all simply connected domains that are not
$\mathbb{C}$ are conformally equivalent to a round disk, which can be viewed
as a canonical model. It is well known that any one-connected region is
conformally equivalent to a ring
$\\{z\in\mathbb{C};\operatorname{\beta}<\absolutevalue{z}<1\\}$, where
$0\leq\operatorname{\beta}<1$, for a reference, see [Ahl53]. Note that when
$\operatorname{\beta}=0$, the domain is simply a punctured disk. For multi-
connected domains, see [Ahl53], for some classification results. These results
lead to a complete characterization of the moduli space of such surfaces. For
example, all $n$-connected bounded domains with non-degenerate boundary
components form a $3n-6$-dimensional space. In this article, we purpose a new
geometric point of view.

We set notations for our construction. Let $M$ be given as before and
$\Omega\subset\mathbb{C}$ be a fixed conformal compactification of $M$. Then
there exists a smooth function $u$ on $\Omega$ such that $(M,e^{-2u}g_{M})$ is
isometric to $(\Omega,g_{E})$. Let $v=e^{-u}$. Then $u$ and $v$ satisfy the
following equations,

$\begin{cases}\Delta u=e^{2u}&\text{ in }\Omega,\\\ u=+\infty&\text{ on
}\partial\Omega,\end{cases}$ (1)

and

$\begin{cases}v\Delta v=\absolutevalue{\nabla v}^{2}-1&\text{ in }\Omega,\\\
v=0&\text{ on }\partial\Omega.\end{cases}$ (2)

Furthermore, $v$ has asymptotics near $C^{3,\alpha}$ boundary components of
$\partial\Omega$ HanQing2018BbfL,SHEN2021109228,

$v(z)=d(z)-\frac{1}{2}\kappa(y)d(z)^{2}+c_{3}(y)d(z)^{3}+O(d^{3+\alpha}(z)),$
(3)

$d(z)$ is the distance from $z$ to the boundary $\partial\Omega$,
$y\in\partial\Omega$ is the point where $d(z)=\absolutevalue{z-y}$, and
$\kappa(y)$ is the curvature at $y$. $c_{3}(y)$ is the first global term in
(3), which depends on the global geometry of $\Omega$. We remark that when
$\partial\Omega$ is smooth, $v$ is the special boundary defining function as
in [Gra00, Lemma 2.1].

We consider a scaling free quantity following the work of [SW21]. Assuming
that the outermost boundary component $\mathcal{C}$ of $\Omega$ is
$C^{3,\alpha}$, consider for $\mathcal{C}$ the following functional,

$\lambda(\Omega,v)=-\int_{\mathcal{C}}\differential{l}\cdot\int_{\mathcal{C}}c_{3}(y)\differential{l(y)},$
(4)

where the line integral is taken under the Euclidean metric.

The functional $\lambda$ was first studied by [SW21] [SW21] in a slightly
different setting. By our notation, [SW21] have proven the following.

######  Theorem 1.1 ([SW21, Theorem 1.1, 1.2, 1.3]).

Let $\Omega$ be a bounded $C^{3,\alpha}$ domain, and $v$ be a solution to (2)
on $\Omega$, then

  1. a)

$\lambda(\Omega,v)\geq 0$, with equality if and only if $\Omega$ is a round
disk;

  2. b)

if $\Omega$ is multi-connected, then

$\lambda(\Omega,v)>\frac{2\pi^{2}}{3}.$ (5)

Theorem 1.1 can be viewed as a “rigidity and gap theorem” and may also be
interpreted as a type of positive mass theorem.

If $M$ is multi-connected, each boundary component can be realized as the out-
most boundary via a proper conformal transformation. Even with the fixed
boundary component, (4) is not conformally invariant. We define the following
renormalized volume type invariant that is also conformally invariant for the
pair $(M,g_{M})$.

######  Definition 1.2.

Let $(M,g_{M})$ be a complete non-compact hyperbolic surface of finite
topological type that can be conformally compactified to a bounded domain in
$\mathbb{C}$, define

$\operatorname{\Lambda}(M,g_{M})=\inf_{\Omega}\left\\{\textstyle\lambda(\Omega,v)\left|\begin{aligned}
&(M,v^{2}g_{M})\text{ is isometric to }(\Omega,g_{E})\text{ and the}\\\
&\text{outermost boundary component of $\Omega$ is
}C^{3,\alpha}\end{aligned}\right.\right\\}$ (6)

By the Riemann mapping theorem, for any $M$ we may find a compactification
such that its outermost boundary is smooth. Therefore $\Lambda$ is well
defined.

Our main result is the following.

######  Theorem 1.3.

Let $(M,g_{M})$ be a complete non-compact hyperbolic one-connected surface
with conformal compactification $(\Omega_{0},g_{E})$, where
$\Omega_{0}\subset\mathbb{R}^{2}$ is bounded. Then

$\operatorname{\Lambda}(M,g_{M})=\frac{2\pi^{2}}{3}\biggl{[}\biggl{(}\frac{\pi}{\ln\operatorname{\beta}}\biggr{)}^{2}+1\biggr{]},$
(7)

where $0\leq\operatorname{\beta}<1$ is the exponent of the modulus of
continuity of $\Omega_{0}$, so that $\Omega_{0}$ is biholomorphic to
$B_{1}-\overline{B}_{\operatorname{\beta}}$.

Moreover, for any bounded $\Omega$ biholomorphic to $\Omega_{0}$ with
$C^{3,\alpha}$ outermost boundary component, we have

$\lambda(\Omega,v)\geq\frac{2\pi^{2}}{3}\biggl{[}\biggl{(}\frac{\pi}{\ln\operatorname{\beta}}\biggr{)}^{2}+1\biggr{]},$
(8)

and equality holds if and only if $\Omega$ is the image of
$B_{1}-\overline{B}_{\operatorname{\beta}}$ under a Möbius transformation.

Our definition gives a new type of conformal invariant for $2$-dimensional
hyperbolic CCE. For general dimensions, the renormalized volume is a preferred
conformal invariant. However it is topological in dimension $2$ for hyperbolic
CCE; by [YCQ08, Corollary 3.5] and [PP01, Appendix A.1], the renormalized
volume of a conformally compact hyperbolic surface $(M,g_{M})$ equals to
$-2\pi\chi(M)$. 1.2 gives an alternative invariant to work on. Our
construction focus on the geometric meaning of the next term in corresponding
expansion (3). We are hopeful that our invariant can be helpful to replace
renormalized volume in dimension $2$.

Theorem 1.3 extends [SW21]’s work, in which they essentially compute values of
$\operatorname{\Lambda}$ when $M$ is simply connected, and separate this case
with multi-connected domains. Theorem 1.3 further characterizes one-connected
$M$. Theorem 1.3 shows $\operatorname{\Lambda}$ can be achieved, and when
$\operatorname{\Lambda}$ is achieved, we determine its unique geometric model
as $B_{1}-\overline{B}_{\operatorname{\beta}}$ up to Möbius transformations.
In particular, when $\operatorname{\beta}=0$, Theorem 1.3 also gives the sharp
case of (5) in Theorem 1.1 .

An interesting side result of Theorem 1.3 is a new geometric interpretation of
the exponent of modulus of continuity $\operatorname{\beta}$.

For CCE manifolds in dimension $4$, there is also some interest in studying
particular compactifications, see [FG01], [CG18],[CGQ20] and [Cha+21] for an
incomplete list.

Our approach is partially inspired by [SW21], but there are several
differences. [SW21] use conformal transformations and compare the boundary
integral (4) of the underlying domain with that of the punctured disk,
techniques involving Schwarzian derivative are then applied. In our work, we
use more refined models. Specifically, we use Fourier series to derive
inequalities and establish rigidity results.

Note that by [LM93], solutions to (1) on $C^{3,\alpha}$ domains exist and are
unique. Then the functional $\lambda$ can be defined on bounded $C^{3,\alpha}$
domains $\Omega\subset\mathbb{C}$ without specifying the solution to (2) on
$\Omega$. On punctured domains, we prove the following extension to the
existence result and include the proof in the appendix.

######  Proposition 1.4.

Let $\Omega$ be a smooth domain, there exists a solution to (1) on
$\Omega-\\{p_{1},\cdots,p_{n}\\}$ and it is unique among all solutions $u$
satisfying the following growth condition,

$u(x)\geq-\log\bigl(-r_{l}\cdot\absolutevalue{x-p_{l}}\cdot\log\bigl(\frac{\absolutevalue{x-p_{l}}}{r_{l}}\bigr{missing})\bigr{missing}),$
(9)

near $p_{l}$ for all $l=1,\cdots,k$, for $r_{l}>0$ such that
$B_{r_{l}}(p_{l})\supset\Omega$.

We list some new directions. It is worthwhile to consider
$\operatorname{\Lambda}(M,g_{M})$ for general multi-connected domains. The
resulting compactification will be a new canonical classification of bounded
domains in $\mathcal{C}$, see [Ahl53] for some known classifications. We are
looking for correct models for multi-connected cases. For conformally compact
hyperbolic surfaces of nonzero genus, and conformally compact hyperbolic
manifolds in general, defining $\operatorname{\Lambda}(M,g_{M})$ for such
spaces would be interesting.

This paper is organized as follows. In section 2 we prove some technical
lemmas. The proof of Theorem 1.3 is split iton two parts. In section 3, we
prove the inequality in (8). In section 4, we study when equality in (8)
holds.

The author would like to thank Biao Ma for introducing the work of [SW21]. The
author would also like to thank Hao Fang, Lihe Wang and Biao Ma for helpful
discussions, and Hao Fang for all the guidance on writing this paper.

##  2\. Technical Tools

In this section, we prove some technical lemmas for future use. Notice that by
[Rud66], the results in this subsection are standard for model cases.

The first lemma gives a sufficient condition to guarantee the existence of a
square root.

######  Lemma 2.1.

Let $f\colon B_{1}-\overline{B}_{\operatorname{\beta}}\to\mathbb{C}$ be a
holomorphic function. If $f_{z}\neq 0$ on
$B_{1}-\overline{B}_{\operatorname{\beta}}$ and has the Laurent expansion

$f_{z}=\sum_{k=-{2n}}^{\infty}a_{k}z^{k}$ (10)

for some integer $n>0$, with $a_{-2n}\neq 0$, and

$\absolutevalue{a_{-2n}}\geq\sup_{B_{1}-\overline{B}_{\operatorname{\beta}}}\absolutevalue{f_{z}-a_{-2n}z^{-2n}},$
(11)

then there exists a holomorphic function $g\colon
B_{1}-\overline{B}_{\operatorname{\beta}}\to\mathbb{C}$ such that
$g^{2}=1/f_{z}$.

###### Proof.

By [Rud66, Theorem 13.11], to show the existence of the desired holomorphic
function $g$, it suffices to check that for $\gamma\colon[0,2\pi]\to
B_{1}-\overline{B}_{\operatorname{\beta}}$ with $\gamma=re^{i\theta}$,
$\operatorname{\beta}<r<1$,

$\frac{1}{2\pi
i}\int_{\gamma}\frac{(1/f_{z})_{z}}{1/f_{z}}\differential{\gamma}=\frac{-1}{2\pi
i}\int_{\gamma}\frac{f_{zz}}{f_{z}}\differential{\gamma}\in 2\cdot\mathbb{Z}.$
(12)

By direct computation,

$\frac{-1}{2\pi
i}\int_{\gamma}\frac{f_{zz}}{f_{z}}\differential{\gamma}=\frac{-1}{2\pi}\int_{0}^{2\pi}z\cdot\frac{\sum_{k=-2n}^{\infty}a_{k}kz^{k-1}}{\sum_{k=-2n}^{\infty}a_{k}z^{k}}\differential{\theta}.$
(13)

By $a_{-2n}\neq 0$ and (11), we have

$\begin{split}z\cdot\frac{f_{zz}}{f_{z}}&=\frac{-2na_{-2n}+\sum_{k=-2n+1}^{\infty}a_{k}kz^{2n+k}}{a_{-2n}+\sum_{k=-2n+1}^{\infty}a_{k}z^{2n+k}}\\\
&=\bigl{(}-2na_{-2n}+\sum_{k=1}^{\infty}a_{k-2n}(k-2n)z^{k}\bigr{)}\cdot\frac{1}{a_{-2n}}\\\
&\quad\cdot\biggl{[}1-\bigl{(}\frac{1}{a_{-2n}}\cdot\sum_{k=1}^{\infty}a_{k-2n}z^{k}\bigr{)}+\bigl{(}\frac{1}{a_{-2n}}\cdot\sum_{k=1}^{\infty}a_{k-2n}z^{k}\bigr{)}^{2}+\cdots\biggr{]},\end{split}$
(14)

which is a holomorphic function on $B_{1}-\overline{B}_{\beta}$, with the
constant term $-2n$.

In other words, we have

$z\cdot\frac{f_{zz}}{f_{z}}=-2n+\sum_{k=1}^{\infty}b_{k}z^{k}.$ (15)

Integrate from $0$ to $2\pi$, the $b_{k}z^{k}$ terms all vanish. Thus, we
conclude

$\frac{-1}{2\pi i}\int_{\gamma}\frac{f_{zz}}{f_{z}}\differential{\gamma}=2n.$

Therefore, by [Rud66], we have shown the existence of $g$ such that
$g^{2}=1/f_{z}$. ∎

Next, we approximate the holomorphic function $f$ by a sequence of functions
$f_{n}$ such that each $1/(f_{n})_{z}$ admits a “square root”. We use
“$\rightrightarrows$” to denote uniform convergence.

######  Lemma 2.2.

Let $f\colon B_{1}-\overline{B}_{\operatorname{\beta}}\to\mathbb{C}$ be a
holomorphic function. If $f_{z},f\neq 0$ on
$B_{1}-\overline{B}_{\operatorname{\beta}}$. Then there exist two sequences of
holomorphic functions

$f_{n},g_{n}\colon
B_{1-1/n}-\overline{B}_{\operatorname{\beta}+1/n}\to\mathbb{C}$ (16)

such that

$g_{n}^{2}=\frac{1}{(f_{n})_{z}}$ (17)

and on any $K\Subset B_{1}-\overline{B}_{\operatorname{\beta}}$,

$\displaystyle f_{n}$ $\displaystyle\rightrightarrows f,$
$\displaystyle\partialderivative{r}\biggl{|}\frac{1}{(f_{n})_{z}}\biggr{|}$
$\displaystyle\rightrightarrows\partialderivative{r}\biggl{|}\frac{1}{f_{z}}\biggr{|},$
$\displaystyle\partialderivative[order=2]{r}\biggl{|}\frac{1}{(f_{n})_{z}}\biggr{|}$
$\displaystyle\rightrightarrows\partialderivative[order=2]{r}\biggl{|}\frac{1}{f_{z}}\biggr{|},$
(18)

as $n\to\infty$.

###### Proof.

Consider the Laurent expansion of $f$

$f=\sum_{k=-\infty}^{\infty}a_{k}z^{k}.$ (19)

We can define $f_{n}$

$f_{n}=\frac{1}{\sqrt{2n+1}}z^{-(2n+1)}+\sum_{k=-{2n}}^{\infty}a_{k}z^{k},$
(20)

so that

$(f_{n})_{z}=\sqrt{2n+1}z^{-(2n+2)}+\sum_{k=-{2n}}^{\infty}a_{k}k\cdot
z^{k-1}.$ (21)

We have $f_{n}$ converges uniformly to $f$ on any $K\Subset
B_{1}-\overline{B}_{\operatorname{\beta}}$. Since $f,f_{z}\neq 0$ on
$B_{1}-\overline{B}_{\operatorname{\beta}}$, by passing on to a subsequence
$f_{n_{k}}$ if necessary, we may assume $f_{n_{k}},(f_{n_{k}})_{z}\neq 0$ on
$B_{1-1/n}-\overline{B}_{\operatorname{\beta}+1/n}$ and (11) is satisfied for
$f_{n_{k}}$ on $B_{1-1/n}-\overline{B}_{\operatorname{\beta}+1/n}$. The
existence of $g_{n_{k}}$ with $g_{n_{k}}^{2}=1/[(f_{n_{k}})]_{z}$ directly
follows from 2.1.

We prove the convergence (18) by brute force. Fix $K\Subset
B_{1}-\overline{B}_{\operatorname{\beta}}$. For simplicity’s sake, let

$h=\frac{1}{f_{z}},\qquad h_{n}=\frac{1}{(f_{n})_{z}}-\frac{1}{f_{z}}.$ (22)

Notice that on $K$, we have

$\displaystyle\biggl{|}\frac{1}{(f_{n})_{z}}-\frac{1}{f_{z}}\biggr{|}$
$\displaystyle=\absolutevalue{h_{n}}\rightrightarrows 0$ (23)
$\displaystyle\biggl{|}\partial_{z}\biggl{(}\frac{1}{(f_{n})_{z}}-\frac{1}{f_{z}}\biggr{)}\biggr{|}$
$\displaystyle=\absolutevalue{\partial_{z}h_{n}}\rightrightarrows 0$ (24)
$\displaystyle\biggl{|}\partial_{z}\partial_{z}\biggl{(}\frac{1}{(f_{n})_{z}}-\frac{1}{f_{z}}\biggr{)}\biggr{|}$
$\displaystyle=\absolutevalue{\partial_{zz}h_{n}}\rightrightarrows 0$ (25)

as $n\to\infty$. Let $r=\absolutevalue{z}$, we have by (23), (24), (25),

$\displaystyle\bigl{|}\absolutevalue{h}^{2}-\absolutevalue{h+h_{n}}^{2}\bigr{|}$
$\displaystyle\leq
2\absolutevalue{h_{n}}\absolutevalue{h}+\absolutevalue{h_{n}}\absolutevalue{h_{n}}\rightrightarrows
0$ (26)
$\displaystyle\biggl{|}\partialderivative{r}\bigl{(}\absolutevalue{h}^{2}-\absolutevalue{h+h_{n}}^{2}\bigr{)}\biggr{|}$
$\displaystyle\leq
2\absolutevalue{\partial_{z}h_{n}}\absolutevalue{h}+2\absolutevalue{\partial_{z}h}\absolutevalue{h_{n}}+2\absolutevalue{\partial_{z}h_{n}}\absolutevalue{h_{n}}\rightrightarrows
0$ (27)
$\displaystyle\biggl{|}\partialderivative[order=2]{r}\bigl{(}\absolutevalue{h}^{2}-\absolutevalue{h+h_{n}}^{2}\bigr{)}\biggr{|}$
$\displaystyle\leq
2\absolutevalue{\partial_{zz}h}\absolutevalue{h_{n}}+2\absolutevalue{\partial_{zz}h_{n}}\absolutevalue{h}+4\absolutevalue{\partial_{z}h}\absolutevalue{\partial_{z}h_{n}}$
(28)
$\displaystyle\quad+2\absolutevalue{\partial_{zz}h_{n}}\absolutevalue{h_{n}}+2\absolutevalue{\partial_{z}h_{n}}\absolutevalue{\partial_{z}h_{n}}\rightrightarrows
0$

as $n\to\infty$. Here we use the equality
$\absolutevalue{\partial_{rr}h}=\absolutevalue{\partial_{zz}h}$ which follows
easily from examining the Cauchy-Riemann Equations and letting
$\partial_{r}=\partial_{x}$:

$\partial_{xx}h=(\real h)_{xx}+i(\imaginary h)_{xx}=\bigl{(}(\real
h)_{x}+i(\imaginary h)_{x}\bigr{)}_{x}=\partial_{zz}h.$ (29)

Finally, (27) and (28) imply that

$\displaystyle\absolutevalue{h}\partial_{r}\absolutevalue{h}-\absolutevalue{h+h_{n}}\partial_{r}\absolutevalue{h+h_{n}}$
$\displaystyle\rightrightarrows 0$ (30)
$\displaystyle\absolutevalue{h}\partial_{rr}\absolutevalue{h}+\partial_{r}\absolutevalue{h}\cdot\partial_{r}\absolutevalue{h}-\absolutevalue{h+h_{n}}\partial_{rr}\absolutevalue{h+h_{n}}-\partial_{r}\absolutevalue{h+h_{n}}\cdot\partial_{r}\absolutevalue{h+h_{n}}$
$\displaystyle\rightrightarrows 0.$ (31)

Combining with (26) and the assumption that
$\absolutevalue{1/f_{z}}=\absolutevalue{h}>0$ on $K$, we get

$\displaystyle\absolutevalue{h}-\absolutevalue{h+h_{n}}$
$\displaystyle\rightrightarrows 0$ (32)
$\displaystyle\partialderivative{r}\bigl{(}\absolutevalue{h}-\absolutevalue{h+h_{n}}\bigr{)}$
$\displaystyle\rightrightarrows 0$ (33)
$\displaystyle\partialderivative[order=2]{r}\bigl{(}\absolutevalue{h}-\absolutevalue{h+h_{n}}\bigr{)}$
$\displaystyle\rightrightarrows 0$ (34)

as $n\to\infty$. Completing the proof. ∎

##  3\. One-Connected Domains

In this section, we prove our main theorem. In the simply connected case, it
has been shown by [SW21].

######  Lemma 3.1.

Let $f\colon\Omega\to\mathbb{C}$ be an injective holomorphic map. Then

$f^{\ast}g_{E}=\absolutevalue{f_{z}}^{2}g_{E}.$ (35)

In particular, $f$ is conformal.

###### Proof.

Write $f=u+iv$. For a vector equaling to
$a\partialderivative{}/{x}+b\partialderivative{}/{y}$, it follows from the
Cauchy Riemann equations that

$\begin{split}f_{\ast}\biggl{(}a\partialderivative{x}+b\partialderivative{y}\biggr{)}&=\biggl{(}a\partialderivative{u}{x}+b\partialderivative{u}{y}\biggr{)}\partialderivative{x}+\biggl{(}a\partialderivative{v}{x}+b\partialderivative{v}{y}\biggr{)}\partialderivative{y}\\\
&=\biggl{(}a\partialderivative{u}{x}+b\partialderivative{u}{y}\biggr{)}\partialderivative{x}+\biggl{(}-a\partialderivative{u}{y}+b\partialderivative{u}{x}\biggr{)}\partialderivative{y}.\\\
\end{split}$

We can see that $f_{\ast}(\partialderivative{}/{x})$ and
$f_{\ast}(\partialderivative{}/{y})$ are orthogonal, and

$\biggl{|}f_{\ast}\biggl{(}a\partialderivative{x}+b\partialderivative{y}\biggr{)}\biggr{|}=(a^{2}+b^{2})\absolutevalue{f_{z}}^{2}.\qed$

Let $u$ be a solution to (1) on $\Omega$. Let $f\colon
B_{1}-\overline{B}_{\operatorname{\beta}}\to\Omega$ be a biholomorphic
function, and $u_{\operatorname{\beta}}$ be the solution to (1) for
$\Omega_{\operatorname{\beta}}=B_{1}-\overline{B}_{\operatorname{\beta}}$,
then $(\Omega,(f^{-1})^{\ast}e^{2u_{\operatorname{\beta}}}g_{E})$ is complete
with Gaussian curvature $-1$, hence

$u=(u_{\operatorname{\beta}}-\ln\absolutevalue{f_{z}})\circ f^{-1}$ (36)

is a solution to (1) in $\Omega$. For $v=e^{-u}$ and
$v_{\beta}=e^{-u_{\beta}}$, this relation becomes

$v=(v_{\operatorname{\beta}}\cdot\absolutevalue{f_{z}})\circ(f^{-1}).$ (37)

######  Remark 3.2.

We quote the following explicit solution from [SW21]. The solution to (2) on
$B_{1}-\overline{B}_{\operatorname{\beta}}$ is explicitly written as follows,

$v_{\operatorname{\beta}}(r)=-r\cdot\biggl{(}\frac{1}{\pi}\ln\operatorname{\beta}\biggr{)}\cdot\sin\frac{\ln
r}{1/\pi\cdot\ln\operatorname{\beta}}.$ (38)

At $\partial B_{1}$, it has expansion

$v_{\operatorname{\beta}}=d-\frac{1}{2}d^{2}-\frac{1}{6}\biggl{[}\biggl{(}\frac{\pi}{\ln\operatorname{\beta}}\biggr{)}^{2}+1\biggr{]}d^{3}+O(d^{4}),$
(39)

and the first global term of $v_{\operatorname{\beta}}$ near $\partial B_{1}$
is

$c_{\operatorname{\beta},3}=-\frac{1}{6}\biggl{[}\biggl{(}\frac{\pi}{\ln\operatorname{\beta}}\biggr{)}^{2}+1\biggr{]}.$
(40)

The following lemma is proved in [SW21], here we include it for completeness.

######  Lemma 3.3.

Let $f\colon B_{1}-\overline{B}_{\operatorname{\beta}}\to\Omega$ be a
biholomorphic function. Let $c_{3},c_{\operatorname{\beta},3}$ be the first
global terms of the solutions $v,v_{\beta}$ to (2) on
$\Omega,B_{1}-\overline{B}_{\operatorname{\beta}}$ near $\mathcal{C},\partial
B_{1}$ respectively, where $\mathcal{C}$ is the outermost boundary component
of $\Omega$. If $f$ takes $\partial B_{1}$ to $\mathcal{C}$, then

$\int_{\mathcal{C}}-6c_{3}\differential{l}=\int_{\partial
B_{1}}\frac{-6c_{\operatorname{\beta},3}}{\absolutevalue{f_{z}}}+2\pi\fint_{\partial
B_{1}}\partialderivative[order=2]{r}\frac{1}{\absolutevalue{f_{z}}}-2\pi\fint_{\partial
B_{1}}\partialderivative{r}\frac{1}{\absolutevalue{f_{z}}}.$ (41)

###### Proof.

By [SW21], the $c_{3}$ term in the expansion of $v$ can be expressed, in a way
independent of the distance function $d(x)$, as

$-6c_{3}=\partial_{N}\Delta v,$ (42)

where $N$ is the outward pointing normal and $\Delta$ is the Laplacian under
metric $(\Omega,g_{E})$.

Let $w$ and $z$ be the standard euclidean coordinate on $\Omega$ and
$B_{1}-\overline{B}_{\operatorname{\beta}}$ respectively. Then

$\begin{split}\int_{\mathcal{C}}-6c_{3}\differential{l}&=\int_{f(\partial
B_{1})}\partial_{N}\Delta_{w}v\differential{l}=\int_{\partial
B_{1}}f^{\ast}(\partial_{N}\Delta_{w}v\cdot\differential{l})\\\
&=\int_{\partial B_{1}}((f^{-1})_{\ast}\partial_{N})(\Delta_{w}v\circ
f)f^{\ast}\differential{l}\\\ &=\int_{\partial
B_{1}}\frac{1}{\absolutevalue{f_{z}}}\partialderivative{r}\bigl{[}(\Delta_{w}v)\circ
f\bigr{]}\absolutevalue{f_{z}}\differential{l}\\\ &=\int_{\partial
B_{1}}\frac{1}{\absolutevalue{f_{z}}}\partialderivative{r}\bigl{[}\Delta_{\absolutevalue{f_{z}}^{2}g_{E}}(v_{\beta}\cdot\absolutevalue{f_{z}})\bigr{]}\absolutevalue{f_{z}}\differential{l}\end{split}$

The computation of Laplacian in conformal metrics is simple in $2$-dimension.
Let $(x^{1},x^{2})$ be the standard Euclidean coordinate with
$\tilde{g}=\absolutevalue{f_{z}}^{2}\cdot g_{E}$. We have

$\begin{split}\Delta_{\tilde{g}}(v_{\beta}\cdot\absolutevalue{f_{z}})&=\frac{1}{\sqrt{\det\tilde{g}}}\partialderivative{x^{i}}\biggl{(}\tilde{g}^{ij}\sqrt{\det\tilde{g}}\cdot\partialderivative{(v_{\beta}\cdot\absolutevalue{f_{z}})}{x^{j}}\biggr{)}\\\
&=\frac{1}{\absolutevalue{f_{z}}^{2}}\Delta_{g_{E}}(v_{\beta}\cdot\absolutevalue{f_{z}}),\end{split}$
(43)

and so

$\begin{split}\int_{\mathcal{C}}-6c_{3}\differential{l}&=\int_{\partial
B_{1}}\frac{1}{\absolutevalue{f_{z}}}\partialderivative{r}\biggl{[}\frac{1}{\absolutevalue{f_{z}}^{2}}\Delta_{g_{E}}(v_{\beta}\cdot\absolutevalue{f_{z}})\biggr{]}\absolutevalue{f_{z}}\differential{l}\\\
&=\int_{\partial
B_{1}}\partialderivative{r}\biggl{[}\frac{1}{\absolutevalue{f_{z}}^{2}}\Delta_{g_{E}}(v_{\beta}\cdot\absolutevalue{f_{z}})\biggr{]}\differential{l}.\\\
\end{split}$ (44)

We can compute the terms involving with $v_{\beta}$ on $\partial B_{1}$
explicitly:

$\partial_{r}v_{\operatorname{\beta}}=-1,\qquad\Delta_{z}v_{\operatorname{\beta}}=-2,\qquad\partial_{r}\Delta_{z}v_{\operatorname{\beta}}=-6c_{\operatorname{\beta},3}=\biggl{[}\biggl{(}\frac{\pi}{\ln\operatorname{\beta}}\biggr{)}^{2}+1\biggr{]}$
(45)

Thus we get,

$\begin{split}\int_{\mathcal{C}}-6c_{3}\differential{l}&=\int_{\partial
B_{1}}\biggl{(}\partialderivative{r}\frac{1}{\absolutevalue{f_{z}}}\biggr{)}\Delta
v_{\operatorname{\beta}}+\int_{\partial
B_{1}}\frac{1}{\absolutevalue{f_{z}}}\partialderivative{r}\Delta
v_{\operatorname{\beta}}-2\int_{\partial
B_{1}}\partialderivative[order=2]{r}v_{\operatorname{\beta}}\partialderivative{r}\frac{1}{\absolutevalue{f_{z}}}\\\
&\quad-2\int_{\partial
B_{1}}\partialderivative{r}v_{\operatorname{\beta}}\partialderivative[order=2]{r}\frac{1}{\absolutevalue{f_{z}}}+\int_{\partial
B_{1}}\partialderivative{r}v_{\operatorname{\beta}}\Delta\frac{1}{\absolutevalue{f_{z}}}+\int_{\partial
B_{1}}v_{\operatorname{\beta}}\partialderivative{r}\Delta\frac{1}{\absolutevalue{f_{z}}}\\\
&=-6c_{\operatorname{\beta},3}\int_{\partial
B_{1}}\frac{1}{\absolutevalue{f_{z}}}+2\int_{\partial
B_{1}}\partialderivative[order=2]{r}\frac{1}{\absolutevalue{f_{z}}}-\int_{\partial
B_{1}}\Delta\frac{1}{\absolutevalue{f_{z}}}\\\ &=\int_{\partial
B_{1}}\frac{-6c_{\operatorname{\beta},3}}{\absolutevalue{f_{z}}}+2\pi\fint_{\partial
B_{1}}\partialderivative[order=2]{r}\frac{1}{\absolutevalue{f_{z}}}-2\pi\fint_{\partial
B_{1}}\partialderivative{r}\frac{1}{\absolutevalue{f_{z}}}.\qed\end{split}$
(46)

The following theorem is an improvement of 5 in [SW21].

######  Theorem 3.4.

Let $\Omega$ be a $1$-connected $C^{3,\alpha}$ domain. If $\Omega$ is
biholomorphic to $B_{1}-\overline{B}_{\operatorname{\beta}}$, then

$\lambda(\Omega,v)\geq\frac{2\pi^{2}}{3}\biggl{[}\biggl{(}\frac{\pi}{\ln\operatorname{\beta}}\biggr{)}^{2}+1\biggr{]}.$
(47)

###### Proof.

Let $f\colon B_{1}-\overline{B}_{\operatorname{\beta}}\to\Omega$ be a
biholomorphic map that takes $\partial B_{1}$ to $\mathcal{C}$, the outermost
boundary component of $\Omega$. By 3.3, we have

$\int_{\mathcal{C}}-6c_{3}\differential{l}=\int_{\partial
B_{1}}\frac{-6c_{\operatorname{\beta},3}}{\absolutevalue{f_{z}}}+2\pi\fint_{\partial
B_{1}}\partialderivative[order=2]{r}\frac{1}{\absolutevalue{f_{z}}}-2\pi\fint_{\partial
B_{1}}\partialderivative{r}\frac{1}{\absolutevalue{f_{z}}}.$ (48)

Let $\\{f_{n}\\},\\{g_{n}\\}$ be the sequences constructed in 2.2. Then for
any $\operatorname{\beta}<r<1$,

$\displaystyle r^{2}\fint_{\partial
B_{r}}\partialderivative[order=2]{r}\frac{1}{\absolutevalue{f_{z}}}-r\fint_{\partial
B_{r}}\partialderivative{r}\frac{1}{\absolutevalue{f_{z}}}$
$\displaystyle=r^{2}\fint_{\partial
B_{r}}\partialderivative[order=2]{r}\frac{1}{\absolutevalue{(f_{n})_{z}}}-r\fint_{\partial
B_{1}}\partialderivative{r}\frac{1}{\absolutevalue{(f_{n})_{z}}}$ (49)
$\displaystyle\quad+r^{2}\fint_{\partial
B_{r}}\partialderivative[order=2]{r}\biggl{(}\frac{1}{\absolutevalue{f_{z}}}-\frac{1}{\absolutevalue{(f_{n})_{z}}}\biggr{)}$
$\displaystyle\quad+r\fint_{\partial
B_{r}}\partialderivative{r}\biggl{(}\frac{1}{\absolutevalue{(f_{n})_{z}}}-\frac{1}{\absolutevalue{f_{z}}}\biggr{)}.$

By 2.2, the last two terms on the right hand side of (49) converge to $0$ as
$n\to\infty$. The first two terms on the right hand side of (49) can be
rewritten as

$r\fint_{\partial
B_{r}}\partialderivative[order=2]{r}\frac{1}{\absolutevalue{(f_{n})_{z}}}-\fint_{\partial
B_{r}}\partialderivative{r}\frac{1}{\absolutevalue{(f_{n})_{z}}}=r\fint_{\partial
B_{r}}\partialderivative[order=2]{r}\absolutevalue{g_{n}}^{2}-\fint_{\partial
B_{r}}\partialderivative{r}\absolutevalue{g_{n}^{2}}.$ (50)

Consider the Laurent expansion of $g_{n}$,

$g_{n}=\sum_{k=-\infty}^{\infty}b^{n}_{k}z^{k},$ (51)

which leads to,

$\fint_{\partial B_{r}}\absolutevalue{g_{n}}^{2}=\frac{1}{2\pi
r}\int_{\partial B_{r}}\absolutevalue{g_{n}^{2}}=\frac{1}{2\pi
r}\int_{\partial
B_{r}}g_{i}\cdot\overline{g}_{i}=\sum_{k=-\infty}^{\infty}\absolutevalue{b^{n}_{k}}^{2}r^{2k}.$
(52)

Therefore,

$\begin{split}r^{2}\fint_{\partial
B_{r}}\partialderivative[order=2]{r}g_{n}^{2}-r\fint_{\partial
B_{r}}\partialderivative{r}\absolutevalue{g_{n}^{2}}&=\biggl{(}r^{2}\partialderivative[order=2]{r}-r\partialderivative{r}\biggr{)}\sum_{k=-\infty}^{\infty}\absolutevalue{b^{n}_{k}}^{2}r^{2k}\\\
&=\sum_{k=-\infty}^{\infty}\absolutevalue{b^{n}_{k}}^{2}2k(2k-1)r^{2k}-\absolutevalue{b^{n}_{k}}^{2}2kr^{2k}\\\
&=\sum_{k=-\infty}^{\infty}\absolutevalue{b^{n}_{k}}^{2}2k(2k-2)r^{2k}\geq
0.\end{split}$ (53)

We conclude that for $\operatorname{\beta}<r<1$,

$r^{2}\fint_{\partial
B_{r}}\partialderivative[order=2]{r}\frac{1}{\absolutevalue{f_{z}}}-r\fint_{\partial
B_{r}}\partialderivative{r}\frac{1}{\absolutevalue{f_{z}}}\geq 0.$ (54)

Let $r\to 1$, we conclude that

$\int_{\mathcal{C}}-6c_{3}\differential{l}=\int_{\partial
B_{1}}\frac{-6c_{\operatorname{\beta},3}}{\absolutevalue{f_{z}}}+2\pi\fint_{\partial
B_{1}}\partialderivative[order=2]{r}\frac{1}{\absolutevalue{f_{z}}}-2\pi\fint_{\partial
B_{1}}\partialderivative{r}\frac{1}{\absolutevalue{f_{z}}}\geq\int_{\partial
B_{1}}\frac{-6c_{\operatorname{\beta},3}}{\absolutevalue{f_{z}}}.$ (55)

Finally, by Hölder’s inequality,

$\begin{split}\lambda(\Omega,v)&\geq\frac{1}{6}\biggl{[}\biggl{(}\frac{\pi}{\ln\operatorname{\beta}}\biggr{)}^{2}+1\biggr{]}\cdot\int_{\partial
B_{1}}\frac{1}{\absolutevalue{f_{z}}}\differential{l}\cdot\int_{\partial
B_{1}}\absolutevalue{f_{z}}\differential{l}\\\
&\geq\frac{1}{6}\biggl{[}\biggl{(}\frac{\pi}{\ln\operatorname{\beta}}\biggr{)}^{2}+1\biggr{]}\biggl{(}\int_{\partial
B_{1}}\frac{1}{\absolutevalue{f_{z}}^{1/2}}\cdot\absolutevalue{f_{z}}^{1/2}\biggr{)}^{2}\\\
&\geq\frac{2\pi^{2}}{3}\biggl{[}\biggl{(}\frac{\pi}{\ln\operatorname{\beta}}\biggr{)}^{2}+1\biggr{]}.\qed\end{split}$
(56)

The first part of a) in Theorem 1.1 can be recovered by the same proof with
little modification. First note that for a simply connected domain $\Omega$,
and biholomorphic $f\colon B_{1}\to\Omega$. 3.3 remains true after we change
$c_{\operatorname{\beta},3}$ to the first global term $c_{B_{1},3}$ of the
solution $v_{B_{1}}$ to (2) on $B_{1}$ near $\partial B_{1}$. $v_{B_{1}}$
takes the following form,

$\displaystyle v_{B_{1}}$ $\displaystyle=\frac{1-r}{2},$ $\displaystyle
c_{B_{1},3}$ $\displaystyle=0.$ (57)

Then for the $1$-connected domain $\Omega$,

$\int_{\mathcal{C}}-6c_{3}\differential{l}=2\pi\fint_{\partial
B_{1}}\partialderivative[order=2]{r}\frac{1}{\absolutevalue{f_{z}}}-2\pi\fint_{\partial
B_{1}}\partialderivative{r}\frac{1}{\absolutevalue{f_{z}}}.$ (58)

which by (54), the right hand side is $\geq 0$. Thus we conclude that for any
simply connected domain $\Omega$,

$\lambda(\Omega,v)\geq 0.$ (59)

The other part of a) in Theorem 1.1 will be recovered in the following
section.

##  4\. Rigidity of One-Connected Domains

In this section, we prove a rigidity theorem for one-connected domains.

Let $f\colon B_{1}-\overline{B}_{\operatorname{\beta}}\to\Omega$ be a
holomorphic function with $f_{z},f\neq 0$. Consider

$\displaystyle A(t)$ $\displaystyle=\frac{1}{2\pi e^{t}}\cdot\int_{\partial
B_{e^{t}}(0)}\frac{1}{\absolutevalue{f_{z}}}$ $\displaystyle B(t)$
$\displaystyle=A_{tt}(t)-2A_{t}(t),$ (60)

for $t\in(\ln\operatorname{\beta},0)$.

######  Lemma 4.1.

Let $r=e^{t}$, then

$B(t)=r^{2}\fint_{\partial
B_{r}(0)}\partialderivative[order=2]{r}\frac{1}{\absolutevalue{f_{z}}}-r\fint_{\partial
B_{r}(0)}\partialderivative{r}\frac{1}{\absolutevalue{f_{z}}}$ (61)

for $t\in(\ln\operatorname{\beta},0)$.

###### Proof.

For convenience, take

$\phi(r)=\frac{1}{2\pi r}\int_{\partial
B_{r}(0)}\frac{1}{\absolutevalue{f_{z}}}$ (62)

so that $\phi(r)=A(t)$. For fixed $r\in(\operatorname{\beta},1)$, we have

$\partialderivative{t}A=e^{t}\cdot\biggl{(}\partialderivative{r}\phi\biggr{)}(r)=\frac{r}{2\pi
r}\int_{\partial
B_{r}}\partialderivative{r}\frac{1}{\absolutevalue{f_{z}}}\differential{l},$
(63)

and

$\begin{split}\partialderivative[order=2]{t}A&=e^{t}\cdot\biggl{(}\partialderivative{r}\phi\biggr{)}(r)+e^{2t}\biggl{(}\partialderivative[order=2]{r}\phi\biggr{)}(r)\\\
&=\frac{r}{2\pi r}\int_{\partial
B_{r}}\partialderivative{r}\frac{1}{\absolutevalue{f_{z}}}\differential{l}+\frac{r^{2}}{2\pi
r}\int_{\partial
B_{r}}\partialderivative[order=2]{r}\frac{1}{\absolutevalue{f_{z}}}\differential{l}.\end{split}$
(64)

Thus the claim follows. ∎

By (61) and 3.3, we see that

$\lim_{t\to 0}2\pi\cdot B(t)=\int_{\mathcal{C}}-6c_{3}-\int_{\partial
B_{1}}\frac{-6c_{\operatorname{\beta},3}}{\absolutevalue{f_{z}}},$ (65)

which in a rough sense, measures the difference between $\lambda(\Omega,v)$
and
$\lambda(B_{1}-\overline{B}_{\operatorname{\beta}},v_{\operatorname{\beta}})$.
4.2 gives a description of the according holomorphic functions $f$ in case
when $B(0)$ vanishes.

######  Proposition 4.2.

Let $A(t)$ and $B(t)$ be defined by (60). Then

$B(t)\geq 0,$ (66)

for all $t\in(\ln\operatorname{\beta},0)$.

If for some $t_{0}\in(\ln\operatorname{\beta},0)$, $B(t_{0})=0$, or
$\lim_{t\to 0}B(t)=0$, then

$B(t)=0$ (67)

for all $t\in(\ln\operatorname{\beta},0)$ and

$f(z)=C_{1}+C_{2}z\quad\text{or}\quad f(z)=C_{1}+\frac{C_{2}}{z+C_{3}}$ (68)

###### Proof.

The non-negativeness of $B$ follows from (53) and (61).

Let $f\colon B_{1}-\overline{B}_{\operatorname{\beta}}$ be a holomorphic
function with $f,f_{z}\neq 0$. Define sequences $f_{n},g_{n}$ by 2.2. Consider
Laurent expansions of $g_{n}$ and $1/f_{z}$,

$\displaystyle g_{n}$ $\displaystyle=\sum_{k=-\infty}^{\infty}b_{k}^{n}z^{k},$
$\displaystyle\frac{1}{f_{z}}$
$\displaystyle=\sum_{k=-\infty}^{\infty}d_{k}z^{k}.$ (69)

Since $g_{n}^{2}=1/(f_{n})_{z}$ converges uniformly to $1/f_{z}$ on compact
subsets of $B_{1}-\overline{B}_{\operatorname{\beta}}$, for all $k_{0}$,

$d_{k_{0}}=\lim_{n\to\infty}\sum_{k=-\infty}^{\infty}b_{k}^{n}b_{k_{0}-k}^{n}.$
(70)

Moreover, we have

$\limsup_{n\to\infty}\absolutevalue{g_{n}}^{2}\leq\absolutevalue{f}$ (71)

on $B_{1}-\overline{B}_{\operatorname{\beta}}$. By Cauchy’s Integral formula,
we observe that

$\limsup_{n\to\infty}\absolutevalue{b_{k}^{n}}<\infty.$ (72)

By (53) and (61), we have

$e^{-t}B(t)=\lim_{n\to\infty}\sum_{k=-\infty}^{\infty}\absolutevalue{b_{k}^{n}}^{2}2k(2k-2)r^{2k}.$
(73)

In particular, for any $k\neq 0,1$,

$\lim_{n\to\infty}\absolutevalue{b_{k}^{n}}^{2}\leq r^{-2k}\cdot e^{-t}B(t)$
(74)

for all $t\in(\ln\operatorname{\beta},0)$, $r=e^{t}$. By the assumption,
either $B(t)=0$ for some $t$ or $\lim_{t\to 0}B(t)=0$, it follows that,

$\lim_{n\to\infty}\absolutevalue{b_{k}^{n}}^{2}=0.$ (75)

for $k\neq 0,1$.

We are now ready to compute $d_{k_{0}}$ explicitly. When $k_{0}=0$, we have

$d_{0}=\lim_{n\to\infty}\sum_{k=-\infty}^{-2}b_{k}^{n}b_{-k}^{n}+(b_{0}^{n})^{2}+2b_{1}^{n}\cdot
b_{-1}^{n}+\sum_{k=2}^{\infty}b_{k}^{n}b_{-k}^{n},$ (76)

where by Cauchy’s inequality,

$\begin{split}\lim_{n\to\infty}\sum_{k=-\infty}^{-2}\absolutevalue{b_{k}^{n}b_{-k}^{n}}&\leq\lim_{n\to\infty}\sum_{k=-\infty}^{-2}\absolutevalue{b_{k}^{n}}^{2}r^{2k}+\absolutevalue{b_{-k}^{n}}^{2}r^{-2k}\\\
&\leq\lim_{n\to\infty}\sum_{k=-\infty}^{\infty}\absolutevalue{b_{k}^{n}}^{2}2k(2k-2)r^{2k}=e^{-t}B(t).\end{split}$
(77)

By the assumption of this proposition, either $B(t)=0$ for some $t$ or
$\lim_{t\to 0}B(t)=0$, thus (77) implies

$\lim_{n\to\infty}\sum_{k=-\infty}^{-2}\absolutevalue{b_{k}^{n}b_{-k}^{n}}=0$
(78)

The last term in (76) also converges to $0$ by the same calculation. Combined
with (75), (76) becomes

$d_{0}=\lim_{n\to\infty}(b_{0}^{n})^{2}.$ (79)

The other coefficients $d_{k_{0}}$ of the Laurent coefficients of $1/f_{z}$
can be computed following the same idea as in the calculation (78) and (75).
We skip the details here.

When $k_{0}=1$, we have

$\begin{split}d_{1}&=\lim_{n\to\infty}\sum_{k=-\infty}^{-1}b_{k}^{n}b_{1-k}^{n}+b_{0}^{n}b_{1}^{n}+b_{1}^{n}b_{0}^{n}+\sum_{k=2}^{\infty}b_{k}^{n}b_{1-k}^{n}\\\
&=\lim_{n\to\infty}2b_{0}^{n}b_{1}^{n}.\end{split}$ (80)

When $k_{0}=2$, we have

$\begin{split}d_{2}&=\lim_{n\to\infty}\sum_{k=-\infty}^{-1}b_{k}^{n}b_{1-k}^{n}+b_{0}^{n}b_{2}^{n}+b_{1}^{n}b_{1}^{n}+b_{2}^{n}b_{0}^{n}+\sum_{k=3}^{\infty}b_{k}^{n}b_{1-k}^{n}\\\
&=\lim_{n\to\infty}b_{1}^{n}b_{1}^{n}.\end{split}$ (81)

$d_{0},d_{1},d_{2}$ are the only nonzero coefficients.

For $k_{0}>2$, we have

$\begin{split}d_{k_{0}}&=\lim_{n\to\infty}\sum_{k=-\infty}^{-1}b_{k}^{n}b_{k_{0}-k}^{n}+b_{0}^{n}b_{k_{0}}^{n}+b_{1}^{n}b_{k_{0}-1}^{n}+\sum_{k=2}^{k_{0}-2}b_{k}^{n}b_{k_{0}-k}^{n}\\\
&\quad+b_{k_{0}-1}^{n}b_{1}^{n}+b_{k_{0}}^{n}b_{0}^{n}+\sum_{k=k_{0}+1}^{\infty}b_{k}^{n}b_{k_{0}-k}^{n}\\\
&=0.\end{split}$ (82)

and for $k_{0}<0$, we have

$\begin{split}d_{k_{0}}&=\lim_{n\to\infty}\sum_{k=-\infty}^{k_{0}-2}b_{k}^{n}b_{k_{0}-k}^{n}+b_{k_{0}-1}^{n}b_{1}^{n}+b_{k_{0}}^{n}b_{0}^{n}+\sum_{k=k_{0}+1}^{-1}b_{k}^{n}b_{k_{0}-k}^{n}\\\
&\quad+b_{0}^{n}b_{k_{0}}^{n}+b_{1}^{n}b_{k_{0}-1}^{n}+\sum_{k=2}^{\infty}b_{k}^{n}b_{k_{0}-k}^{n}\\\
&=0.\end{split}$ (83)

From (79), (80) and (81), we see that $(b_{0}^{n})^{2}$, $b_{0}^{n}b_{1}^{n}$
and $(b_{1}^{n})^{2}$ all converge as $n\to\infty$. Therefore the limit of
$b_{0}^{n}$, $b_{1}^{n}$ as $n\to\infty$ exist up to a sign. Taking a
subsequence if necessary, we may assume that the limits exist and let

$\displaystyle b_{0}$ $\displaystyle=\lim_{n\to\infty}b_{0}^{n},$
$\displaystyle b_{1}$ $\displaystyle=\lim_{n\to\infty}b_{1}^{n}.$ (84)

Then $f_{z}$ is of the form

$f_{z}=\frac{1}{(b_{0}+b_{1}z)^{2}}.$ (85)

Suggesting that

$f=\begin{cases}C+b_{0}^{-2}z&\text{ if }b_{1}=0\\\
C-b_{1}^{-2}\cdot\bigl{(}b_{0}/b_{1}+z\bigr{)}^{-1}&\text{ if }b_{1}\neq
0,\end{cases}$ (86)

for some constant $C\in\mathbb{C}$.

It is easy to check that when (86) holds, $B(t)=0$ for all $t\in(\ln r,0)$. ∎

######  Theorem 4.3.

Let $\Omega$ be a one-connected domain with $C^{3,\alpha}$ outermost boundary.
If $\Omega$ is biholomorphic to $B_{1}-\overline{B}_{\operatorname{\beta}}$,
and

$\lambda(\Omega,v)=\frac{2\pi^{2}}{3}\biggl{[}\biggl{(}\frac{\pi}{\ln\operatorname{\beta}}\biggr{)}^{2}+1\biggr{]},$
(87)

then $\Omega$ can be obtained from $B_{1}-\overline{B}_{\operatorname{\beta}}$
after a Möbius transformation.

###### Proof.

By (61) and the proof of Theorem 3.4, the equality holds exactly when
$\lim_{t\to 0}B(t)=0$, which by 4.2, the biholomorphic map $f\colon
B_{1}-\overline{B}_{\operatorname{\beta}}\to\Omega$ has to be a Möbius
transformation. ∎

The second part of a) in Theorem 1.1 can be proven as an easy corollary of
4.2. To see that, by Theorem 3.4 and standard comparison theorem,
$\lambda(\Omega,v)=0$ implies that $\Omega$ must be a simply connected domain.
Let $f\colon B_{1}\to\Omega$ be a biholomorphic map and define $A(t),B(t)$
accordingly, then 4.2 implies that $f$ is a Möbius transformation, hence
$f(B_{1})=\Omega$ must be a disk.

##  5\. Appendix

By [SW21], a solution to (1) on $B_{r}(p)-\\{p\\}$ is given by

$u_{B_{r}(p)-\\{p\\}}(x)=-\log(-\frac{\absolutevalue{x-p}}{r}\cdot\log\bigl(\frac{\absolutevalue{x-p}}{r}\bigr{missing}))-2\ln
r.$ (88)

A solution to $B_{1}(p)-\overline{B}_{r}(p)$ is given by

$u_{B_{1}(p)-\overline{B}_{r}(p)}(x)=-\log\biggl(\frac{\absolutevalue{x-p}}{\pi}\cdot\log(1/r)\cdot\sin\bigl(\frac{\pi\log(1/\absolutevalue{x-p})}{\log(1/r)}\bigr{missing})\biggr{missing}).$
(89)

Then the growth condition (9) is equivalent to $u\geq
u_{B_{r_{l}}(p_{l})-\\{p_{l}\\}}$ for all $l=1,\cdots,n$.

######  Theorem 5.1.

Let $\Omega$ be a bounded domain with smooth boundary. Let
$p_{1},\cdots,p_{n}\in\Omega$ be a collection of points. Then there exists
$w\in C^{\infty}(M)$ such that

$\begin{cases}\Delta w=e^{2w}&\text{ in }\Omega-\\{p_{1},\cdots,p_{n}\\}\\\
w=\infty&\text{ on }\partial\Omega\cup\\{p_{1},\cdots,p_{n}\\},\end{cases}$
(90)

and (9) is satisfied at $\\{p_{1},\cdots,p_{n}\\}$.

###### Proof.

We follow the same idea as in [LM93]. Let $u_{\Omega}$ be the solution to (1)
on $\Omega$. Let $A_{m}$ be an exhaustion of $\Omega$ by compact subsets with
smooth boundary. Take $\Omega_{m}=A_{m}-\bigcup_{l=1}^{n}B_{1/m}(p_{l})$. Let
$w_{m}$ be the solution to (1) on $\Omega_{m}$. By the Maximum principle,
$w_{m}$ is a decreasing sequence. Additionally, it is bounded below by
$u_{B_{r_{l}}(p_{l})-\\{p_{l}\\}}$ for some $r$ sufficiently large, with
$l=1,2,\cdots,n$.

Fix $m$, by the interior $L^{p}$ estimate, [Gil83, Theorem 9.11], for $k>m$,
we have

$\norm{w_{k}}_{2,p;\Omega_{m}}\leq
C(p,\Omega_{m},\Omega_{m+1})\bigl{(}\norm{w_{k}}_{p;\Omega_{m+1}}+\norm{e^{w_{k}}}_{p;\Omega_{m+1}}\bigr{)}$
(91)

where $\norm{w_{k}}_{p;\Omega_{m+1}},\norm{e^{w_{k}}}_{p;\Omega_{m+1}}$ is
bounded independent of $k$, since

$\max(u_{B_{r_{l}}(p_{l})-\\{p_{l}\\}},u_{\Omega})\leq w_{k}\leq w_{m+1}$ (92)

for all $l$. This shows that $w_{k}$ is a bounded sequence in
$W^{2,p}(\Omega_{m})$. Hence is bounded in $C^{1,\alpha}(\Omega_{m})$.

By the interior Schauder estimate, [Gil83, Corollary 6.3], we have

$\absolutevalue{v_{k}}_{2,\alpha;\Omega_{m}}\leq
C(\alpha,\Omega_{m},\Omega_{m+1})\bigl{(}\absolutevalue{v_{k}}_{0;\Omega_{m+1}}+\absolutevalue{e^{2v_{k}}}_{0,\alpha;\Omega_{m+1}}\bigr{)},$
(93)

Hence $v_{k}$ is bounded in $C^{2,\alpha}(\Omega_{m})$ independent of $k$. By
taking a subsequence if necessary, $w_{m}$ converges in $C^{2}(\Omega_{m})$ to
some $w$ such that

$\Delta w=e^{2w}$ (94)

in $\Omega_{m}$ and $\max(u_{\Omega},u_{B_{r_{l}}(p_{l})-\\{p_{l}\\}})\leq
w\leq w_{m+1}$.

Pushing $m$ to $\infty$, we get a smooth function $w$ satisfying $\Delta
w=e^{2w}$ and

$\max(u_{\Omega},u_{B_{r_{l}}(p_{l})-\\{p_{l}\\}})<w.$ (95)

Since $u_{\Omega}=\infty$ on $\partial\Omega$, and
$u_{B_{r_{l}}(p_{l})-\\{p_{l}\\}}=\infty$ on $p_{l}$ for all $l$, we see that
$w$ is the desired function. ∎

Following the work of [LM93], we shall establish the uniqueness of solution
$u$ to (1) on $\Omega-\\{p_{1},\cdots,p_{n}\\}$ satisfying (9) in two steps.
First we study the asymptotic behavior of a solution near
$\\{p_{1},\cdots,p_{n}\\}$. Then we apply maximum principle to the difference
of two solutions.

In the following, we assume without loss of generality that

$B_{2}(p_{l})\subset\Omega-\\{p_{1},\cdots,p_{n}\\}$ (96)

for all $l=1,\cdots,n$.

######  Proposition 5.2.

Let $u$ be a solution to (1) on $\Omega-\\{p_{1},\cdots,p_{n}\\}$. If $u$
satisfies (9) at $\\{p_{1},\cdots,p_{n}\\}$, then there exists $C$ such that

$\absolutevalue{u-\log\big(-\absolutevalue{x-p_{l}}\cdot\log(\absolutevalue{x-p_{l}})\bigr{missing})}<C$
(97)

in a neighborhood of $p_{l}$ for all $l=1,\cdots,n$.

###### Proof.

Fix any $l$, assume without loss of generality that $p_{l}=0$. By the maximum
principle,

$u_{B_{r_{l}}-\\{0\\}}\leq u\leq u_{B_{1}-\overline{B}_{r}}$ (98)

on $B_{1}-\overline{B}_{r}$, for any $0<r<1$. For $x\in
B_{1}-\overline{B}_{r}$, let $\absolutevalue{x}$ be the distance to $p_{l}$.
Take $r=\absolutevalue{x}^{k}$, then

$u_{B_{1}-\overline{B}_{r}}(x)=-\log\biggl(\frac{-k\absolutevalue{x}}{\pi}\cdot\log(\absolutevalue{x})\cdot\sin\bigl(\frac{\pi}{k}\bigr{missing})\biggr{missing})$
(99)

Pushing $k\to\infty$, we see that

$u\leq-\log\big(-\absolutevalue{x}\cdot\log(\absolutevalue{x})\bigr{missing})$
(100)

near $p_{l}$. Now, the difference between the upper bound and the lower bound
of $u$ equals to

$\displaystyle-\log(-\absolutevalue{x}\log(\absolutevalue{x}))+\log(-r\absolutevalue{x}\log(\absolutevalue{x}/r))=\log\biggl(\frac{r\log(\absolutevalue{x}/r)}{\log(\absolutevalue{x})}\biggr{missing}),$
(101)

which converges to $0$ as $\absolutevalue{x}\to 0$. Proving the claim. ∎

###### Proof of 1.4.

Existence of solution follows from Theorem 5.1. We follow the same idea as in
[LM93]. Let $u_{1},u_{2}$ be two solutions satisfying (9) at
$\\{p_{1},\cdots,p_{n}\\}$. By adding a constant if necessary, we may assume
that $u_{1},u_{2}\geq 2\log 2$, then for these two functions after
modification, there exists some constant $C$ such that

$\Delta u_{i}=Ce^{2u_{i}}$ (102)

on $\Omega$ for $i=1,2$.

For any $1/2<s<1$, by 5.2,

$\frac{su_{1}}{u_{2}}=\frac{su_{1}/(-\log(-\absolutevalue{x-p_{l}}\cdot\log(\absolutevalue{x-p_{l}})))}{u_{2}/(-\log(-\absolutevalue{x-p_{l}}\cdot\log(\absolutevalue{x-p_{l}})))}\to
s<1$ (103)

near $p_{l}$ for all $l=1,2,\cdots,n$. By [HS18], near other non-degenerate
boundary components, $su_{1}/u_{2}$ also converges to $s<1$.

We claim that $su_{1}\leq u_{2}$. If not, let $x_{0}$ be where $su_{1}-u_{2}$
attains maximum. Then $(su_{1}-u_{2})(x_{0})>0$, and

$0\geq\Delta(su_{1}-u_{2})(x_{0})=sCe^{2u_{1}}-Ce^{2u_{2}}\geq
e^{2su_{1}}-e^{2u_{2}}>0.$ (104)

where the second inequality comes from [LM93, Lemma 4.3]. This lead to a
contradiction.

Pushing $s\to 1$, we get $u_{1}\leq u_{2}$. Switching $u_{1}$ and $u_{2}$, we
get the inequality from other direction. Proving the claim. ∎

## References

  * [Ahl53] Lars V.(Lars Valerian) Ahlfors  “Complex analysis : an introduction to the theory of analytic functions of one complex variable.”, International series in pure and applied mathematics  New York: McGraw-Hill, 1953
  * [Rud66] Walter Rudin  “Real and complex analysis.”, McGraw-Hill series in higher mathematics  New York: McGraw-Hill, 1966
  * [Gil83] David Gilbarg  “Elliptic partial differential equations of second order / David Gilbarg, Neil S. Trudinger.”, Grundlehren der mathematischen Wissenschaften ; 224  Berlin ; New York: Springer-Verlag, 1983
  * [LM93] Alan C. Lazer and P.. McKenna  “On a problem of Bieberbach and Rademacher”  In _Nonlinear Analysis-theory Methods & Applications_ 21, 1993, pp. 327–335
  * [Gra00] Robin C. Graham  “Volume and area renormalizations for conformally compact Einstein metrics”  In _Proceedings of the 19th Winter School "Geometry and Physics"_ Palermo: Circolo Matematico di Palermo, 2000, pp. 31–42  URL: http://eudml.org/doc/221675
  * [FG01] Charles Fefferman and C. Graham  “Q-Curvature and Poincare Metrics”  In _Math. Res. Lett._ 9, 2001  DOI: 10.4310/MRL.2002.v9.n2.a2
  * [PP01] S.J. Patterson and Peter Perry  “The divisor of Selberg’s zeta function for Kleinian groups. Appendix A by Charles Epstein”  In _Duke Mathematical Journal_ 106, 2001  DOI: 10.1215/S0012-7094-01-10624-8
  * [YCQ08] P. Yang, S.-Yu.. Chang and J. Qing  “On the renormalized volumes for conformally compact Einstein manifolds”  In _Journal of Mathematical Sciences_ 149.6, 2008, pp. 1755–1769  DOI: 10.1007/s10958-008-0094-0
  * [CG18] Sun-Yung A. Chang and Yuxin Ge  “Compactness of conformally compact Einstein manifolds in dimension 4”  In _Advances in Mathematics_ 340, 2018, pp. 588–652  DOI: https://doi.org/10.1016/j.aim.2018.10.010
  * [HS18] Qing Han and Weiming Shen  “Boundary behaviors for Liouville’s equation in planar singular domains”  In _Journal of functional analysis_ 274.6 SAN DIEGO: Elsevier Inc, 2018, pp. 1790–1824
  * [CGQ20] Sun-Yung A. Chang, Yuxin Ge and Jie Qing  “Compactness of conformally compact Einstein 4-manifolds II”  In _Advances in Mathematics_ 373, 2020, pp. 107325  DOI: https://doi.org/10.1016/j.aim.2020.107325
  * [Cha+21] Sun-Yung A. Chang, Yuxin Ge, Xiaoshang Jin and Jie Qing  “On compactness conformally compact Einstein manifolds and uniqueness of Graham-Lee metrics, III”, 2021  arXiv:2107.03075 [math.DG]
  * [SW21] Weiming Shen and Yue Wang  “Rigidity and gap theorems for Liouville’s equation”  In _Journal of Functional Analysis_ 281.10, 2021, pp. 109228  DOI: https://doi.org/10.1016/j.jfa.2021.109228