# Impermanent Loss Conditions: An Analysis
of Decentralized Exchange Platforms

Matthias Hafner University of Zurich, Center for Cryptoeconomics, and Swiss
Economics Helmut Dietl University of Zurich

###### Abstract

Decentralized exchanges are widely used platforms for trading crypto assets.
The most common types work with automated market makers (AMM), allowing
traders to exchange assets without needing to find matching counterparties.
Thereby, traders exchange against asset reserves managed by smart contracts.
These assets are provided by liquidity providers in exchange for a fee. Static
analysis shows that small price changes in one of the assets can result in
losses for liquidity providers. Despite the success of AMMs, it is claimed
that liquidity providers often suffer losses. However, the literature does not
adequately consider the dynamic effects of fees over time. Therefore, we
investigate the impermanent loss problem in a dynamic setting using Monte
Carlo simulations. Our findings indicate that price changes do not necessarily
lead to losses. Fees paid by traders and arbitrageurs are equally important.
In this respect, we can show that an arbitrage-friendly environment benefits
the liquidity provider. Thus, we suggest that AMM developers should promote an
arbitrage-friendly environment rather than trying to prevent arbitrage.

###### keywords:

Decentralized Finance Decentralized Exchanges Automated Market Makers Platform
Economics

##  1 Introduction

Decentralized exchanges (DEXs) are essential for many blockchain applications
as they enable users to access different services and investors to access
various (crypto) assets. In comparison to centralized exchanges, DEXs are
simple to access, do not require any customer identity verification, and can
be seamlessly integrated into other decentralized financial (DeFi) services at
almost no cost. The Block Research reports that the combined trading volume of
all DEXs exceeded USD 1,000 billion in 2021.[1]

DEXs function without an orderbook. Traders are not matched, but rather
exchange their assets against a liquidity pool. This liquidity pool is filled
with assets by so-called liquidity providers, who receive fees in return for
providing liquidity. The prices at which traders can exchange their assets
against the liquidity pool are adjusted by automated market makers (AMMs).
AMMs are simple mathematical formulas or more complex algorithms that
determine quantities and prices for the assets in the liquidity pools. These
formulas or algorithms guarantee that liquidity pools never run out of any of
their assets, adjusting prices accordingly. If, for example, a liquidity pool
consists of USDC and ETH and the number of ETH decreases, the AMM increases
the price of ETH in USDC. AMMs are usually programmed as smart contracts on
blockchains.

Due to these automated quantity and price adjustments, the total value of the
assets in the liquidity pool can be lower than their value would have been had
the liquidity providers kept these assets in their wallets. This difference is
called impermanent loss and is described by liquidity providers’ profit
function.[2, 3] Many practitioners argue that impermanent loss is a major
issue. As a result of this belief, many DEXs updated their protocols
accordingly (e.g., Bancor’s Impermanent Loss Protection).[3] However, the
growth and market share of Uniswap, the world’s largest DEX (which did not
address the problem directly), does not indicate impermanent loss to be the
most relevant issue for liquidity providers.[1]

The peer-reviewed literature on impermanent losses is limited. The earliest
contributions are from Angeris et al., published in 2021.[4, 5] Assuming no
fees, they formally analyze Uniswap V2 and show that the liquidity providers’
profits are negatively related to changes in market prices and that AMMs
closely track the reference market price. Evans (2021) shows that these
results also hold for generalized models.[6] Our analysis builds on this
analysis. In addition, there are some working papers focusing on impermanent
loss. Angeris et. al (2022) compute the profit functions of liquidity
providers in liquidity pools with constant function (automated) market
makers.[5] Other authors demonstrate approximate hedging techniques.[7, 8, 9]
Aigner and Dhaliwal (2021) describe the risk profile of a liquidity provider
and compute the impermanent loss function for Uniswap V2 and show that in a
static setting, small changes in the relative market price between two assets
result in an impermanent loss. In particular, they show that the impermanent
loss is an inversely U-shaped function of the relative price changes of the
underlying assets/tokens. In a similar setting, Labadie (2022) shows that the
impermanent loss increases faster than linear and disappears after a price
reversion.[10] Lehar and Parlour (2023) describe the impermanent loss as a
function of fee gains and adverse rebalancing of the AMM. They show that
liquidity provision is endogenously determined by this trade-off.[11]

We build on the literature partly to get a better understanding of the profit
dynamics of liquidity providers. In contrast to previous literature, however,
we explicitly include accumulated trader and arbitrageur fees over time in the
profit function of liquidity providers. We use an agent-based model that
allows us to consider dynamic effects between participants over time without
making too many restrictions regarding their interactions. We simulate the
behavior of liquidity providers, traders, and arbitrageurs under various
scenarios to compute the resulting profits and losses of liquidity providers
based on a constant product AMM (see, e.g., Uniswap’s V2). This approach
allows us to identify under which conditions liquidity providers will incur
impermanent losses and how the inclusion of fees changes standard results.

The remainder of the paper is structured as follows. Section 2 introduces the
basic economics of AMMs and the relevant literature. Section 3 describes the
setup of the agent-based model and explains the simulation. Section 4 presents
and discusses the results. Section 5 concludes.

##  2 The Economics of Decentralized Exchanges

As shown in Figure 1, DEXs are three-sided platforms. Liquidity providers
interact with the platform by supplying assets to the liquidity pool in
exchange for a fee. The liquidity pool depicted in Figure 1 consists of two
assets denoted A and B, for example, USDC and WETH. Traders can exchange one
of the two assets for the other by paying a fee. Arbitrageurs also interact
with the platform whenever they perceive price differences between the
exchange rate of A and B at the DEX and that of other markets. If, for
example, asset B is more valuable (relative to Asset A) at another exchange,
arbitrageurs will buy B from the DEX by sending asset A and a fee to this DEX
in exchange for asset B. At the same time, these arbitrageurs will sell B for
A at another exchange to earn a risk-free profit.

Figure 1: Illustration of the interactions of different market players in a
DEX.

Based on the amounts of the two assets/tokens in the liquidity pool (denoted
by $R_{A}$ and $R_{B}$, respectively), the AMM determines the price at which
traders and arbitrageurs can exchange assets with the liquidity pool. If
traders or arbitrageurs send an amount of asset A (denoted by $\Delta_{A}$) to
the liquidity pool they will receive an amount of asset/token B (denoted by
$\Delta_{B}$) in return according to the fee and the price determined by the
AMM. Of course, traders and arbitrageurs can also buy B by sending the
required amount of A to the liquidity pool. The liquidity pool was filled with
asset A and B by liquidity providers in advance. For this liquidity provision
they receive fees from the DEX. Of particular interest is the AMM’s formula or
algorithm that determines the exchange rates offered to traders and
arbitrageurs. These algorithms adjust exchange rates based on the relative
demand and supply of all assets in the liquidity pool. If there is more demand
for asset B in relation to A, the AMM will increase the exchange rate. Traders
and arbitrageurs will have to send more units of A to receive one unit of B.
In addition, most AMMs set prices such that pools never run out of any of
their assets/tokens. In this paper, we focus on AMMs that use a “constant
product” function such as the one implemented in the Uniswap V2 protocol for
the determination of the exchange rate. We focus on this type because it was
not only the first, but currently, is also the most common type. Note,
however, that there also exist other exchange rate formulas and algorithms, in
particular for stable swaps, i.e., for asset pairs whose prices are (almost)
identical.[12, 4, 13] Uniswap’s updated version (V3) is also based on the
constant product function. It allows liquidity providers to define a price
range at which the liquidity pool can offer trades. For simplicity, we did not
model this additional feature.

In line with previous research we describe the two token/asset constant
product function by

$R_{A}*R_{B}=k$ (1)

where $k$ is a constant value and $R_{i}$ are the reserves, i.e., the actual
amount of asset $i$ in the liquidity pool.[4, 14]

Assuming no fees, describing the new reserves in $t+1$ as the old reserves in
$t$ plus/minus the traded amounts or $\Delta_{A}$ and $\Delta_{B}$ (trade
asset B for asset A), it can be shown that (see also Appendix B)

$(R_{A}-\Delta_{A})*(R_{B}+\Delta_{B})=R_{A}*R_{B}\,.$ (2)

Consequently, the amount of asset A received ($\Delta_{A}$) in exchange for a
certain amount of asset B sent to the liquidity pool ($\Delta_{B}$) is defined
by solving equation (2) for $\Delta_{A}$ (see also Krishnamachari et al., 2021
for a general solution for more than two assets and unequal weights)[15]:

$\Delta_{A}=\frac{R_{A}*\Delta_{B}}{R_{B}+\Delta_{B}}\,.$ (3)

The resulting exchange rate offered by the liquidity pool for the demanded
asset A ($\epsilon_{AMM}$) describes how much of asset A a trader will receive
for a certain amount of B ($\Delta_{A}=\epsilon_{AMM}*\Delta_{B}$):

$\epsilon_{AMM}=\frac{\Delta_{A}}{\Delta_{B}}=\frac{R_{A}}{R_{B}+\Delta_{B}}\,.$
(4)

This exchange rate of A ($\epsilon_{A}$) is a decreasing function of the
amount of asset B ($\Delta_{B}$) to be exchanged. Consequently, trades with
larger volumes will result in lower exchange rates; i.e., if traders or
arbitrageurs want larger amounts of A, they have to send more units of B per
unit of A to the liquidity pool.

Note that the reference price of asset A offered by the AMM ($p_{AMM}$) which
describes how much of asset B a trader has to pay (send) to receive a certain
amount of A ($\Delta_{A}*p_{AMM}=\Delta_{B}$) is the inverse of the exchange
rate

$p_{AMM}=\frac{1}{\epsilon_{A}}=\frac{R_{B}+\Delta_{B}}{R_{A}}\,.$ (5)

Figure 2 illustrates the exchange rates for a trade $\Delta_{B}$ $\rightarrow$
$\Delta_{A}$.

Figure 2: Illustration of the price function of a constant product function
AMM.[15]

The curve ($R_{B}=k/R_{A}$) in Figure 2 represents all combinations of
reserves $R_{A}$ and $R_{B}$ in the liquidity pool for which $R_{A}*R_{B}=k$
holds. For every state of the reserves ${R_{A},R_{B}}$ and any amount
$\Delta_{B}$ to be exchanged, the curve describes how much the trader receives
of the other asset $\Delta_{A}$.

Assume, for example, the liquidity pool consists of 50 units of A and 100
units of B. If a trader wants to receive 10 units of A in exchange for B, she
has to send 25 units of B to the DEX. After this exchange, the liquidity pool
consists of 125 units of B and 40 units of A, and the constant product
function (2) holds because $100*50=125*40$. As every trade changes the
reserves within the liquidity pool, the price changes with every trade. Since
the price is defined as $\Delta_{B}/\Delta_{A}$, the spot price ($p_{AMM}$),
which represents the price offered for an indefinite small amount, is defined
by the slope of the constant product function[15]:

$p_{AMM}=-\left(\frac{\partial R_{B}}{\partial
R_{A}}\right)=\frac{k}{R_{A}^{2}}\,.$ (6)

Since $k=R_{A}*R_{B}$, the spot price is the ratio of the two reserves, i.e.,

$p_{AMM}=R_{B}/R_{A}\,.$ (7)

Assuming no fees, Angeris et al. (2021) show that the liquidity providers’
total relative gain (${\delta}^{LP}$) depends solely on the market price
development[4]

$\delta^{LP}=\sqrt{p_{m}^{T}/p_{m}^{1}}\,.$ (8)

As a result, impermanent loss ($IL$) can formally be described as

$IL=\frac{W_{1}*\delta^{LP}}{W_{1}*\delta^{Ref}}-1=\frac{2*\sqrt{p_{m}^{T}/p_{m}^{1}}}{1+p_{m}^{T}/p_{m}^{1}}-1$
(9)

with $\delta^{Ref}$ being the relative gain of the reference portfolio and
$W_{1}$ the initial wealth of the liquidity provider.

According to (9) it follows that without any fees a market price change will
always result in an impermanent loss for the liquidity provider. Because
liquidity providers earn fees for their services, their overall return has to
take these fees into account. Since this overall return depends on complex
interactions with arbitrageurs and traders, we develop an agent-based model to
simulate these interactions.

##  3 The Agent-Based Model

We simulate the interactions between market prices, traders, arbitrageurs,
liquidity providers, and the AMM of a liquidity pool for the WETH/USDC asset
pair. This simulation is based on historical on-chain data collected from
Uniswap V2 during the period from May 2021 to May 2022.[16]. Traders exchange
WETH for USDC (or vice versa) with the liquidity pool. The AMM updates the
exchange rate according to a constant product function as described in Section
2. Whenever the spot exchange rate determined by the AMM deviates from the
market exchange rate, arbitrageurs will trade against the liquidity pool to
earn risk-free profits. During the simulation, liquidity providers do not
interact with the liquidity pool.

###  3.1 General Procedure

The simulation consists of the steps depicted in the flowchart of Figure 3.
One loop represents one simulation period. This loop is repeated iteratively
for a period of one year. Each loop consists of calculations which can be
classified into four blocks: state calculations, arbitrageur calculations,
trader calculations, and AMM calculations. Arbitrageurs will check whether
there is an arbitrage opportunity after each state update and after each new
trade. Therefore, individual calculation blocks can run multiple times within
one iteration.

Figure 3: Process flowchart of the ABM.

###  3.2 Agent Decisions and other Calculations

Initial configuration—The simulation begins with an initial configuration that
populates the “world” with representative traders and arbitrageurs as well as
information on the asset market prices and the number of assets in the pool.
We feed the initial configuration with effective values of Uniswap’s WETH/USDC
pool (trades/fees/pool size) and Binance (market price) in 2021. This data
comprises 1.3 million trades conducted over a period of one year (see Appendix
A for more details). We have hereby excluded any MEV trades, since we account
for arbitrage trades endogenously.[17, 18]

State calculations—After these initial configurations, the simulation process
starts with the state calculations. First, market prices of the relevant
assets are updated.

Arbitrageurs’ actions—After this price update, arbitrageurs anticipate the AMM
behavior and calculate Uniswap’s spot price (see also AMM calculations) and
compare it with the market price. In the baseline simulation, we assume almost
perfect market conditions for arbitrageurs; i.e., if spot prices deviate from
market prices by more than the trading fee plus a small transaction cost,
arbitrageurs will trade with the AMM. In particular, they will choose the
amount that maximizes their profit and send it to the AMM. If, for example,
the spot price for asset A is significantly lower than its market price,
arbitrageurs will (i) buy asset B on a competing exchange, (ii) send it to the
AMM, (iii) receive asset A, and (iv) sell it on a competing exchange. Thus, in
the no-fee case, arbitrageurs’ profit function is

$\pi=(p_{A}*\Delta_{A}-p_{B}*\Delta_{B})\,.$ (10)

It can be shown that solving this maximization problem yields the optimal
amount of $A$[4]

$\Delta_{A}^{*}=(R_{A}-\sqrt{(R_{A}*R_{B})/p_{m}})_{+}$ (11)

where $(x)_{+}=max\\{x,0\\}$ for $x\in\mathbb{R}$. See also Appendix D for the
derivation and the solution including trading fees. Accordingly, arbitrageurs
will trade in the opposite direction if the spot price of A at Uniswap is
significantly above its (external) market price.

AMM calculations—Whenever agents (arbitrageurs or traders) decide to trade,
they send one of the two assets to the liquidity pool. These trades trigger
AMM calculations according to the constant product function for two assets
($R_{A}*R_{B}=k$). Based on the asset type and amount provided by the traders
or arbitrageurs, the AMM will calculate how much they will receive in return
and how much will be captured as fees. In addition, due to the accumulated
fees, the AMM will also update the constant $k$ for the next trade.

Traders’ actions—In a next step, the trader summits an order to the AMM and
exchanges an amount of A tokens ($-\Delta_{A}$) for B tokens ($\Delta_{B}$) or
vice versa. This decision is based on the historical trading amount of
Unsiwap’s WETH/USDC pool. After traders’ actions and before the next
iterations begins, a next round of arbitrageur calculations takes place.

###  3.3 Experiments

To analyze which factors impact the profits of liquidity providers, we conduct
four experiments. First, we test if our model behaves as intended, by testing
hypothesis regarding liquidity providers’ profits in terms of trading activity
and the size of the liquidity pool. Then, we test the impact of ETH price
changes on liquidity providers’ profits by adjusting the ETH price growth.
Finally, we test the impact of transaction costs and arbitrageur competition
on their profits.

Trading activity—An increase in trading volume increases accumulated trading
fees over time, which should increase profits of liquidity providers. Thus, we
adjust the trading volume by restricting the number of trades. More
specifically, we create a subset that only consists of each nth trade and then
analyze the impact on profits.

Liquidity—Liquidity providers’ income consists of fees from traders and
arbitrageurs. Thus, the larger the liquidity pool, the lower the ratio between
fees and liquidity, which reduces liquidity providers’ profits. Thus, we
adjust the liquidity by increasing the initial pool size, i.e., increasing the
initial WETH/USDC reserves.

Growth—Literature predicts that without fees, a positive or negative trend in
the market prices results in impermanent loss for the liquidity provider.[2]
Thus, we adjust the trend of the ETH price to analyze the impact on profit.
More specifically, we simulate a price feed using geometric Brownian motion to
match various expected growth rates $g$.[19] In addition, we adjust traded
quantities such that trading volumes remain unchanged:

$p_{t}=p_{t-1}*e^{(g-0.5*\sigma^{2})*\Delta t+\sigma*\sqrt{\Delta
t}*z_{t}}\text{.}$ (12)

Arbitrageur Transaction Costs—High arbitrageur activity is generally believed
to have a negative impact on liquidity providers’ profits. For example,
Milionis et al. (2023) show that any arbitrage profits will result in losses
for liquidity providers.[20] Arbitrageurs trade whenever spot prices deviate
by more than the sum of fees and transaction costs. Thus, we adjust
arbitrageurs’ transaction costs ($\tau$) –– which are a proxy for arbitrageur
competition –– and analyze their impact on profits:

$p_{m}>p_{AMM}*1/(1-fees-\tau)\,.$ (13)

##  4 Results and Discussions

This section presents the results from the experiments in three steps. First,
we replicate standard results regarding trading activity and pool size from
previous studies. Second, we discuss the effect of different changes in market
price. Finally, we analyze how competition among arbitrageurs affects
liquidity providers’ profits.

###  4.1 Standard Results

Based on the baseline scenario (see also Appendix A), we first test whether we
can replicate standard results from previous literature, i.e., that an
increase in trading activity or a decrease in the provided liquidity decreases
the profitability of liquidity providers. Thus, we run various simulations
using adjusted trading volume and adjusted initial liquidity pool sizes.
Unsurprisingly and in line with theory, the simulations show that higher
trading activity results in more accumulated fees, which increases liquidity
providers’ profits. In addition, an increase in liquidity decreases
profitability because fees are distributed over a larger amount of invested
liquidity.

###  4.2 Market Price Changes

To understand the effect of the fee, we analyze the effect of market price
changes on liquidity providers’ profits. We first analyze whether we can
replicate results from literature that analyze impermanent loss under the no-
fee assumption. We then conduct the same analysis but introduce protocol fees
and compare results.

No-fee experiment—Based on the baseline scenario, we test whether we can
replicate previous analysis that assumes fees to be zero.[2] We can replicate
this scenario by using our baseline scenario but assuming all fees and other
transaction costs to be zero. Thus, we run various experiments assuming
different market trends for the simulated ETH price path. More specifically,
we vary the yearly price trend from -90% (i.e., a price decrease from 2,765
USD to 277 USD) to +90% (i.e., a price increase from 2,765 USD to 5,254 USD).
For all the runs, we simulate liquidity providers’ profits. We find the
impermanent loss figure described by several papers (see also Figure 4). Thus,
the results are expected and in line with previous literature and theory.[4]
However, as explained earlier, to get the full picture, it is essential to
take into account fees as well

Figure 4: Simulation of liquidity providers’ relative profits (without fees)
assuming different market trends.

Fee experiment—In this experiment, we analyze whether results change if we
assume positive trading fees. Again, we run various experiments assuming
different market trends for the simulated ETH price path as in the no-fee
experiment.

Figure 5: Simulation of liquidity providers’ relative profits (with fees)
assuming different market trends.

Figure 5 depicts the liquidity providers’ returns (y-axis) assuming different
market price trends (x-axis). According to the simulation, liquidity providers
make a profit relative to a holding strategy as long as market prices do not
drop by more than 75% or increase by more than 300% within a year. Thus, a
decrease or an increase in the market price trend only has a significant
effect on liquidity providers’ profits for exceptionally strong price
movements.

The reason for this result is the impact of arbitrageurs and traders on the
profits of liquidity providers: on one hand, arbitrageurs’ trades cause a
change in the assets in the liquidity pool, leading to a drop in the profits
of liquidity providers (the “rebalancing losses”). On the other hand, traders
and arbitrageurs pay a fee for each trade, which increases liquidity
providers’ profits (the “fee gains”). The overall result depends heavily on
the size of these two effects: if the rebalancing losses are larger than the
fee gains, then the net result will be negative (the liquidity providers earn
less from fees than they lose from rebalancing). The fee effect itself depends
strongly on the ratio between trading volume and liquidity (see also standard
results, above). Assuming the ratio of our baseline model of around 2/100 (250
million USDC / 11.9 billion USDC) the rebalancing effect outweighs the fee
effect only in case of exceptionally strong price movements.

###  4.3 Arbitrageurs’ Transaction Cost Changes

Finally, we analyze the impact of arbitrageurs’ transaction costs on the
returns of liquidity providers. It is commonly presumed that high arbitrageur
activity adversely affects the profitability of liquidity providers. However,
arbitrageurs also incur fees, causing a non-straightforward effect. We analyze
the effect of arbitrageurs by varying their barriers to trade with the
liquidity pool (i.e., their transactions costs). More specifically, we assume
transaction costs from 0 to 5% and simulate the profits of liquidity
providers. The results are provided in Figure 6.

Figure 6: Simulation of liquidity providers’ relative profits assuming
different transaction costs for arbitrageurs.

The findings are quite remarkable: our simulations reveal that as transaction
costs for arbitrageurs rise, the profits of liquidity traders decrease. In
other words, increased activity by arbitrageurs boosts the profits of
liquidity providers, which contradicts the popular belief that arbitrage
activity harms liquidity providers. The dynamic nature of the setting accounts
for this result. Although arbitrageurs reduce liquidity providers’ profits
through rebalancing losses, they also positively contribute by inducing fee
gains. It can be shown that rebalancing losses are independent of the number
of arbitrage trades and remain consistent over time as shown in equation (8).
Conversely, gains from fees increase with arbitrage activity. Therefore,
increasing arbitrageur activity increases liquidity providers’ profits. This
discovery differs from static analysis, since fee gains caused by arbitrageurs
are often overlooked or modelled as a single trade. Our simulation shows that
these fee by arbitrageurs are significant. Hence, it is vital to consider the
fee gains from arbitrageurs in addition to rebalancing losses and trader fee
gains to analyze liquidity providers’ profits.

##  5 Conclusion

In this paper, we investigate the impermanent loss problem in decentralized
exchanges with automated market makers. We model the dynamic interactions
between liquidity providers, traders, and arbitrageurs, and explicitly
incorporate accumulated fees of traders and arbitrageurs into the profit
function of liquidity providers. We use an agent-based model to simulate one
of the most common AMM smart contract types (constant product AMM) and the
behavior of their participants under various scenarios.

Our analysis demonstrates that impermanent losses are less severe than
previously believed. While static analyses indicate that small price increases
can result in losses for liquidity providers, our experiments, which
incorporate dynamic effects from the accumulation of fees over time, suggest
that price changes themselves do not necessarily lead to losses. Our analysis
reveals that impermanent losses only occur when price increases are permanent,
unexpected, and sufficiently large. A liquidity provider’s return is
positively influenced by trading fees and negatively impacted by (permanent)
price changes and liquidity volume. In a competitive environment, liquidity
provision is determined by expected price changes and trading volumes.
Consequently, impermanent losses occur in the event of unexpected changes in
these variables and if accumulated fees do not offset the losses.

In addition, we show that the size of the impermanent loss decreases with the
intensity of competition among arbitrageurs because fee gains from
arbitrageurs are significant. Hence, it is vital to consider the fee gains
from arbitrageurs in addition to rebalancing losses and trader fee gains. We
conclude that developers of AMMs should prioritize fostering an arbitrage-
friendly environment with low transaction fees, rather than attempting to
prevent them.

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  * [20] Milionis, J., Moallemi, C. C., Roughgarden, T. “Automated Market Making and Arbitrage Profits in the Presence of Fees.” _arXiv preprint_ (2023) https://doi.org/10.48550/arXiv.2305.14604.

##  Appendix A Data

Variable |  Historical value |  Initial value
---|---|---
Initial pool size |  Ø240.6m USDC |  250m USDC
Trading volume per year (Q) |  11.9bn USDC |  11.9bn USDC (100%)
Growth of the market price (g) |  0% |  0%
Market price return volatility ($\sigma$) |  1.2 |  1
Number of trades (N) |  1.31m |  1.31m
Time period (T) |  1 year |  1 year
Protocol fee (fee) |  0.3% |  0.3%
Table 1: Historical data of our reference dataset and initial values.

##  Appendix B Introduction to Automated Market Makers

The generalized form of constant product AMM formulas is defined by

$\prod_{i=1}^{n}R_{i}^{w_{i}}=k\,,$ (14)

where $k$ is a constant value, $R_{i}$ are the reserves, i.e., the actual
amount of asset $i$ in the liquidity pool, and $w_{i}$ is the weight that
determines the ratio of the reserves in the pool. For the purpose of
simplicity, we focus on the simplest form with two assets and equal weights,
which is also used by Uniswap V2. Note that our main results also hold for
more complex functions. In line with previous research we describe the two
token/asset constant product function by[14, 4]

$R_{A}*R_{B}=k\,.$ (15)

In order that (15) holds, each interaction of traders or arbitrageurs with the
liquidity pool must, therefore, adjust the reserves $R_{A}$ and$R_{B}$ such
that $k$ remains constant:

$R_{A}^{t}*R_{B}^{t}=k\,,$ (16) $R_{A}^{t+1}*R_{B}^{t+1}=k\,.$ (17)

Thus, a trade must satisfy

$R_{A}^{t+1}*R_{B}^{t+1}=R_{A}^{t}*R_{B}^{t}\,.$ (18)

Assuming no fees, describing the new reserves in $t+1$ as the old reserves in
$t$ plus or minus the traded amounts $\Delta_{A}$ and $\Delta_{B}$ (trade
asset B for asset A), we can express (18) as

$(R_{A}-\Delta_{A})*(R_{B}+\Delta_{B})=R_{A}*R_{B}\,.$ (19)

##  Appendix C Derivation of the Spot Market Price and Exchange Rate (Fees)

Calculations including fees with $\gamma=1-fee$:

$(R_{A}-\Delta_{A})*(R_{B}+\gamma*\Delta_{B})=R_{A}*R_{B}$ (20)

Amount A

$\Delta_{A}=\frac{R_{A}*\gamma*\Delta_{B}}{R_{B}+\gamma*\Delta_{B}}$ (21)

The resulting exchange rate offered by the liquidity pool for the demanded
asset A ($\epsilon_{AMM}$), which describes how much of asset A a trader
receives for a certain amount of B ($\Delta_{A}*\epsilon_{AMM}=\Delta_{B}$),
is

$\epsilon_{AMM}=\frac{\Delta_{A}}{\Delta_{B}}=\frac{\gamma*R_{A}}{R_{B}+\gamma*\Delta_{B}}\,.$
(22)

This exchange rate of A ($\epsilon_{A}$) is a decreasing function of the
amount of asset type B ($\Delta_{B}$) to be exchanged. Consequently, trades
with larger volumes result in a lower exchange rate; i.e., if traders or
arbitrageurs want larger amounts of A, they have to send more units of B per
unit of A to the liquidity pool. Note that the reference price of asset A
offered by the AMM ($p_{AMM}$), which describes how much of asset B a trader
has to pay (send) to receive a certain amount of A
($\Delta_{A}*p_{AMM}=\Delta_{B}$), is the inverse of the exchange rate:

$p_{AMM}=\frac{1}{\epsilon_{A}}=\frac{R_{B}+\gamma*\Delta_{B}}{\gamma*R_{A}}\,.$
(23)

For infinite small trades, this results in

Spot exchange rate:

$\epsilon_{AMM}=\frac{\Delta_{A}}{\Delta_{B}}=\gamma*\frac{R_{A}}{R_{B}}$ (24)

Spot price:

$m_{AMM}=\frac{1}{\epsilon_{A}}=\frac{1}{\gamma}*\frac{R_{B}}{R_{A}}$ (25)

##  Appendix D Derivation of the Arbitrageur’s Maximization Problem

If the market price for asset A ($p_{A}$) is significantly larger than its
spot price $m_{AMM}$, arbitrageurs will (i) buy asset B ($\Delta_{B}$) on a
competing exchange, (ii) send it to the AMM, (iii) receive asset A
($\Delta_{A}$), and (iv) sell it on the competing exchange. In this case,
arbitrageurs profit function is

$\pi=p_{A}*\Delta_{A}-p_{B}*\Delta_{B}\,.$ (26)

Denominated in the reference market prices, this results in the following
maximization problem:

$\displaystyle\underset{\Delta_{A}}{\max}$
$\displaystyle\quad\pi=p_{m}*\Delta_{A}-\Delta_{B}$ (27) subject to
$\displaystyle\quad(R_{A}-\Delta_{A})*(R_{B}+\gamma*\Delta_{B})=R_{A}*R_{B},$
$\displaystyle\quad\Delta_{A},\Delta_{B}>0\,.$

In a first step, we put the constraint (AMM’s constant product function) into
the maximization problem: to find the optimal $\Delta_{A}^{*}$ we solve AMM’s
constant product formula (20) for $\Delta_{B}$ and put the solution (26) into
the investors’ profit function (28). The resulting profit function depends
solely on quantity $\Delta_{A}$ and not anymore on $\Delta_{B}$:

$\Delta_{B}=\frac{1}{\gamma}*\left(\frac{R_{A}*R_{B}}{R_{A}-\Delta_{A}}-R_{B}\right)\,,$
(28)

$\pi=p_{m}*\Delta_{A}-\frac{1}{\gamma}*\left(\frac{R_{A}*R_{B}}{R_{A}-\Delta_{A}}-R_{B}\right)\,.$
(29)

Note that in this format, the objective function resembles the problem of a
profit-maximizing firm that operates in a competitive goods market selling
good $\Delta_{A}$:

$\pi(\Delta_{A})=\text{Revenues}(\Delta_{A})-\text{Costs}(\Delta_{A})\ $ (30)

with

$Revenues=p_{m}*\Delta_{A}$ (31)

and

$\text{Costs}=\frac{1}{\gamma}*\left(\frac{R_{A}*R_{B}}{R_{A}-\Delta_{A}}-R_{B}\right)\,.$
(32)

To optimize profits, investors seek the quantity $\Delta_{A}^{*}$ for which
marginal revenues (MR) equal marginal costs (MC).

Thus, in a second step, we calculate marginal costs and revenues

$MR=MC\,,$ (33)

$p_{m}=\frac{1}{\gamma}*\frac{R_{A}*R_{B}}{(R_{A}-\Delta_{A})^{2}}\,.$ (34)

In a third step, we solve for the optimal quantity $\Delta_{A}^{*}$:

$\Delta_{A}=R_{A}\pm\sqrt{\frac{R_{A}*R_{B}}{\gamma*p_{m}}}\,.$ (35)

Solution for positive $\Delta_{A}$, $\Delta_{B}$, $R_{A}$, $R_{B}$, and
$p_{m}$

$\Delta_{A}^{*}=\left(R_{A}-\sqrt{\frac{R_{A}*R_{B}}{\gamma*p_{m}}}\right)$
(36)

and

$\Delta_{B}^{*}*\gamma=\sqrt{R_{A}*R_{B}*\gamma*p_{m}}-R_{B}\,.$ (37)

Furthermore, it can be shown that

$\Delta_{A}^{*}=R_{A}*\left(1-\sqrt{\frac{p_{AMM}}{\gamma*p_{m}}}\right)$ (38)

and

$\Delta_{B}^{*}=\frac{1}{\gamma}*R_{B}*\left(\sqrt{\frac{\gamma*p_{m}}{p_{\text{AMM}}}}-1\right)\,.$
(39)

Note, if the AMM asks for a fee ($fee>0$) arbitrageurs trade less and with
smaller amounts. Including fees, arbitrageurs face larger marginal costs since
they receive less for any amount sent to the protocol; see $1/\gamma$ in (28).


# On the boundary behaviour of the squeezing function near linearly convex
boundary points

Ninh Van Thu, Nguyen Thi Lan Huong and Nguyen Quang Dieu1,2 Ninh Van Thu
School of Applied Mathematics and Informatics, Hanoi University of Science and
Technology, No. 1 Dai Co Viet, Hai Ba Trung, Hanoi, Vietnam
<EMAIL_ADDRESS>Nguyen Thi Lan Huong  Department of Mathematics
Hanoi University of Mining and Geology
18 Pho Vien, Bac Tu Liem, Hanoi, Vietnam<EMAIL_ADDRESS>Nguyen
Quang Dieu  1 Department of Mathematics, Hanoi National University of
Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam  2 Thang Long Institute of
Mathematics and Applied Sciences, Nghiem Xuan Yem, Hoang Mai, HaNoi, Vietnam
<EMAIL_ADDRESS>
###### Abstract.

The purpose of this article is twofold. The first aim is to prove that if
there exist a sequence $\\{\varphi_{j}\\}\subset\mathrm{Aut}(\Omega)$ and
$a\in\Omega$ such that $\lim_{j\to\infty}\varphi_{j}(a)=\xi_{0}$ and
$\lim_{j\to\infty}\sigma_{\Omega}(\varphi_{j}(a))=1$, where $\xi_{0}$ is a
linearly convex boundary point of finite type, then $\xi_{0}$ must be strongly
pseudoconvex. Then, the second aim is to investigate the boundary behaviour of
the squeezing function of a general ellipsoid.

###### Key words and phrases:

Hyperbolic complex manifold, exhausting sequence, $h$-extendible domain

###### 2020 Mathematics Subject Classification:

Primary 32H02; Secondary 32M05, 32F18.

##  1\. introduction

Let $\Omega$ be a domain in $\mathbb{C}^{n}$ and $p\in\Omega$. For a
holomorphic embedding $f\colon\Omega\to\mathbb{B}^{n}:=\mathbb{B}(0;1)$ with
$f(p)=0$, we set

$\sigma_{\Omega,f}(p):=\sup\left\\{r>0\colon B(0;r)\subset
f(\Omega)\right\\},$

where $\mathbb{B}^{n}(z;r)\subset\mathbb{C}^{n}$ denotes the ball of radius
$r$ with center at $z$. Then the squeezing function
$\sigma_{\Omega}:\Omega\to\mathbb{R}$ is defined as

$\sigma_{\Omega}(p):=\sup_{f}\left\\{\sigma_{\Omega,f}(p)\right\\}.$

(See Definition in [DGZ12].) Note that the squeezing function is invariant
under biholomorphisms and $0<\sigma_{\Omega}(z)\leq 1$ for any $z\in\Omega$.
Moreover, by definition one sees that $\Omega$ is biholomorphically equivalent
to the unit ball $\mathbb{B}^{n}$ if $\sigma_{\Omega}(z)=1$ for some
$z\in\Omega$.

It is well-known that if $p$ is a strongly pseudoconvex boundary point, then
$\lim\limits_{\Omega\ni z\to p\in\partial\Omega}\sigma_{\Omega}(z)=1$ (cf.
[DGZ16, DFW14, KZ16]). Conversely, motivated by Problem $4.1$ in [FW18], let
us consider the following problem.

######  Problem 1.

If $\Omega$ is a bounded pseudoconvex domain with smooth boundary, and if
$\lim\limits_{j\to\infty}\sigma_{\Omega}(\eta_{j})=1$ for some sequence
$\\{\eta_{j}\\}\subset\Omega$ converging to $p\in\partial\Omega$, then is the
boundary of $\Omega$ strongly pseudoconvex at $p$?

In the case that $\partial\Omega$ is pseudoconvex of D’Angelo finite type near
$\xi_{0}$, the answer to this problem is affirmative for the following cases:

  * •

$\\{\eta_{j}\\}\subset\Omega$ converges to $\xi_{0}$ along the inner normal
line to $\partial\Omega$ at $\xi_{0}$ (for details, see [JK18] for $n=2$ and
[MV19] for general case).

  * •

$\\{\eta_{j}\\}\subset\Omega$ converges nontangentially to $\xi_{0}$ (see
[Nik18]).

  * •

$\\{\eta_{j}\\}\subset\Omega$ converges
$\left(\frac{1}{m_{1}},\ldots,\frac{1}{m_{n-1}}\right)$-nontangentially to an
$h$-extendible boundary point $\xi_{0}$ (see [NN20, Definition $3.4$]), where
$(1,m_{1},\ldots,m_{n-1})$ is the _multitype of $\partial\Omega$ at $\xi_{0}$_
and the _$h$ -extendiblility at $\xi_{0}$_ means that the Catlin multitype and
D’Angelo multitype of $\partial\Omega$ at $\xi_{0}$ coincide (see [Yu95,
Definition $3.3$]).

Now we consider the case that $\\{\eta_{j}\\}\subset\Omega$ be a sequence
converging
$\left(\frac{1}{m_{1}},\ldots,\frac{1}{m_{n-1}}\right)$-nontangentially to
$\xi_{0}$. Then, the condition that
$\lim\limits_{j\to\infty}\sigma_{\Omega}(\eta_{j})=1$ ensures that the unit
ball $\mathbb{B}^{n}$ is biholomorphically equivalent to some model $M_{P}$
given by

$M_{P}=\left\\{z\in\mathbb{C}^{n}\colon\mathrm{Re}(z_{n})+P(z^{\prime})<1\right\\},$

where $P$ is a
$\left(\frac{1}{m_{1}},\ldots,\frac{1}{m_{n-1}}\right)$-homogeneous polynomial
on $\mathbb{C}^{n-1}$ (see [Yu95, Definition $3.1$]). Therefore,
$m_{1}=m_{2}=\cdots=m_{n-1}=1$, or $\xi_{0}$ is strongly pseudoconvex
([NN20]). Unfortunately, the point $\xi_{0}$ may not be strongly psudoconvex
when $\\{\eta_{j}\\}\subset\Omega$ does not converge
$\left(\frac{1}{m_{1}},\ldots,\frac{1}{m_{n-1}}\right)$-nontangentially to
$\xi_{0}$. For instance, the following example points out that
$\lim\limits_{j\to\infty}\sigma_{\Omega}(\eta_{j})=1$ for some
$\\{\eta_{j}\\}\subset\Omega$ converging to a weakly pseudoconvex boundary
point (see also Example 4.1 for general case).

######  Example 1.1.

Let
$E_{1,2}:=\\{(z_{1},z_{2})\in\mathbb{C}^{2}\colon|z_{2}|^{2}+|z_{1}|^{4}<1\\}$.
Consider the sequence $\displaystyle
a_{n}=\Big{(}\sqrt[4]{\frac{2}{n}-\frac{2}{n^{2}}},1-\frac{1}{n}\Big{)}\to(0,1)$
as $n\to\infty$. Denote by $\rho(z):=|z_{2}|^{2}-1+|z_{1}|^{4}$ a defining
function for $E_{1,2}$ and denote by $\sigma(z_{1})=|z_{1}|^{4}$ a
$(\frac{1}{4})$-weighted homogeneous polynomial. Then, a computation shows
that

$\displaystyle\rho(a_{n})=\Big{|}1-\frac{1}{n}\Big{|}^{2}-1+\Big{|}\sqrt[4]{\frac{2}{n}-\frac{2}{n^{2}}}\Big{|}^{4}=-\frac{2}{n}+\frac{1}{n^{2}}+\frac{2}{n}-\frac{2}{n^{2}}=-\dfrac{1}{n^{2}}<0$

Therefore, $\displaystyle\mathrm{dist}(a_{n},\partial
E_{1,2})\approx|\rho(a_{n})|=\frac{1}{n^{2}}$,
$\displaystyle|\mathrm{Re}(a_{n2})-1|=\Big{|}-\frac{1}{n}\Big{|}=\frac{1}{n}$,
and
$\displaystyle\sigma(a_{n1})=\sigma(\sqrt[4]{\frac{2}{n}-\frac{2}{n^{2}}})=\left(\sqrt[4]{\frac{2}{n}-\frac{2}{n^{2}}}\right)^{4}=\frac{2}{n}-\frac{2}{n^{2}}\approx\frac{2}{n}$.
This implies that $\\{a_{n}\\}$ does not converge
$(\frac{1}{4})$-nontangentially to the boundary point $p=(0,1)$.

Let us consider the automorphism $\psi_{n}\in\mathrm{Aut}(E_{1,2})$, given by

$\psi_{n}(z)=\left(\frac{(1-|a_{n2}|^{2})^{1/4}}{(1-\bar{a}_{n2}z_{2})^{1/2}}z_{1},\frac{z_{2}-a_{n2}}{1-\bar{a}_{n2}z_{2}}\right),$

and hence $\psi_{n}(a_{n})=(b_{n},0)$, where
$b_{n}=\dfrac{a_{n1}}{(1-|a_{n2}|^{2})^{1/4}}=\dfrac{\sqrt[4]{\frac{2}{n}-\frac{2}{n^{2}}}}{\sqrt[4]{\frac{2}{n}-\frac{1}{n^{2}}}}\to
1$ as $n\to\infty$. Since $\psi_{n}(a_{n})$ converges to the strongly
pseudoconvex boundary point $(1,0)$ of $\partial E_{1,2}$, it follows that
$\sigma_{E_{1,2}}(a_{n})=\sigma_{E_{1,2}}(\psi_{n}(a_{n}))\to 1$ as
$n\to\infty$. However, the point $(0,1)$ is weakly pseudoconvex. $\Box$

To give a statement of our result, let us recall that $\partial\Omega$ is
linearly convex near $\xi_{0}\in\partial\Omega$ if there exists a
neighbourhood $U$ of $\xi_{0}$ such that, for all $z\in\partial\Omega\cap U,$
the intersection

$(z+T^{10}_{z}\partial\Omega)\cap(\Omega\cap U)=\emptyset.$

In [Ni09], the first author proved a characterization of linearly convex
domains in $\mathbb{C}^{n}$ by their noncompact automorphism groups.

The first aim of this paper is the following theorem.

######  Theorem 1.1.

Let $\Omega$ be a bounded domain in $\mathbb{C}^{n}$ with smooth pseudoconvex
boundary. Assume that $\xi_{0}$ is a boundary point of $\Omega$ of D’Angelo
finite type such that $\partial\Omega$ is linearly convex at $\xi_{0}$ and
there exists a sequence $\\{\varphi_{j}\\}\subset\mathrm{Aut}(\Omega)$ such
that $\eta_{j}:=\varphi_{j}(a)\to\xi_{0}$ as $j\to\infty$ for some
$a\in\Omega$. If $\lim\limits_{j\to\infty}\sigma_{\Omega}(\eta_{j})=1$, then
$\partial\Omega$ is strongly pseudoconvex at $\xi_{0}$.

######  Remark 1.1.

Thanks to the linear convexity of $\partial\Omega$ near a boundary orbit
accumulation point $\xi_{0}$ and the condition that
$\lim\limits_{j\to\infty}\sigma_{\Omega}(\eta_{j})=1$, the scaling method can
be applied to implies that $\mathbb{B}^{n}$ is biholomorphically equivalent to
a model $M_{P}$, where is a real nondegenerate plurisubharmonic polynomial of
degree less than or equal to the type of $\partial\Omega$ at $\xi_{0}$.
Moreover, since $\eta_{j}=\varphi_{j}(a)$ for some
$\\{\varphi_{j}\\}\subset\mathrm{Aut}(\Omega)$ and $a\in\Omega$, the scaling
method yields $\Omega$ is biholomorphically equivalent to a model $M_{P}$ (see
[Ni09, Theorem $1.1$]), that is, $\Omega$ is biholomorphically equivalent to
the unit ball $\mathbb{B}^{n}$. Consequently, the point $\xi_{0}$ is strongly
pseudoconvex, as desired.

Now we move to the second part of this paper. First of all, let us fix
positive integers $m_{1},\ldots,m_{n-1}$ and let $P(z^{\prime})$ be a
$(1/m_{1},\ldots,1/m_{n-1})$-homogeneous polynomial given by

$P(z)=\sum_{wt(K)=wt(L)=1/2}a_{KL}{z^{\prime}}^{K}\bar{z}^{\prime L},$

where $a_{KL}\in\mathbb{C}$ with $a_{KL}=\bar{a}_{LK}$, satisfying that
$P(z^{\prime})>0$ whenever $z^{\prime}\neq 0$. Here and in what follows,
$z^{\prime}:=(z_{1},\ldots,z_{n-1})$ and
$wt(K):=\sum_{j=1}^{n-1}\frac{k_{j}}{2m_{j}}$ denotes the weight of any multi-
index $K=(k_{1},\ldots,k_{n-1})\in\mathbb{N}^{n-1}$ with respect to
$\Lambda:=(1/m_{1},\ldots,1/m_{n-1})$. Then the general ellipsoid $D_{P}$ in
$\mathbb{C}^{n}\;(n\geq 1)$, defined in [NNTK19] by

$\begin{split}D_{P}&:=\\{(z^{\prime},z_{n})\in\mathbb{C}^{n}\colon|z_{n}|^{2}+P(z^{\prime})<1\\}.\end{split}$

Throughout this paper, we assume that the domain $D_{P}$ is a WB-domain, i.e.,
$D_{P}$ is strongly pseudoconvex at every boundary point outside the set
$\\{(0^{\prime},e^{i\theta})\colon\theta\in\mathbb{R}\\}$ (cf. [AGK16]).

For any $s,r\in(0,1]$, inspired by [Liu18, Lemma $2.5$] let us define
$D^{s}_{P}$ and $D^{s,r}_{P}$ respectively by

$\begin{split}D^{s}_{P}&:=\\{z\in\mathbb{C}^{n}\colon|z_{n}-b|^{2}+sP(z^{\prime})<s^{2}\\};\\\
D^{s,r}_{P}&:=\\{z\in\mathbb{C}^{n}\colon|z_{n}-b|^{2}+\dfrac{s}{r}P(z^{\prime})<s^{2}\\},\end{split}$

where $s=1-b$. We note that $D_{P}^{s,1}=D_{P}^{s}$ and the property that
$\lim\psi_{j}^{-1}(D^{s}_{P})=D_{P}$ for a certain family
$\\{\psi_{j}\\}\subset\mathrm{Aut}(D_{P})$ (cf. Lemma 2.1 in Section 4) plays
a key role in the proofs of our main theorems below.

Indeed, we prove the following theorem.

######  Theorem 1.2.

Let $\Omega$ be a subdomain of $D_{P}$ such that
$D^{s}_{P}\subset\Omega\subset D_{P}$ for some $s\in(0,1]$. Then, for any
$r\in(0,1)$ there exists $\gamma_{0}>0$ depending on $r$ such that

$\sigma_{\Omega}(z)\geq\gamma_{0},\;\forall\;z\in D^{s,r}_{P}\cap\Omega.$

######  Remark 1.2.

Notice that the point $p=(0^{\prime},1)$ is $(P,s)$-extreme for each domain
$\Omega_{j}$ (cf. [NNC21, Definition $1.1$] for the notion of $(P,s)$-extreme
points) and the convergence of a sequence of points in $D_{P}^{s,r}$ to $p$ is
exactly the $\Lambda$-nontangential convergence introduced in [NN20,
Definition $3.4$].

Now we consider the case that $\\{a_{j}\\}\subset\Omega\cap U$ converges
$\Lambda$-tangentially to $p=0$ in the sense that for any $0<r<1$ there exists
$j_{r}\in\mathbb{N}$ such that $a_{j}\not\in D_{s,r}$ for all $j\geq j_{r}$.
Then, the following theorem shows that the squeezing function converges to $1$
provided that all $\partial\Omega_{j}$ share a small neighborhood of the point
$(0^{\prime},1)$ with $\partial D_{P}$ to which the sequence of points
converges $\Lambda$-tangentially. More precisely, we prove the following
theorem.

######  Theorem 1.3.

Let $\\{\Omega_{j}\\}$ be a sequence of subdomains of $D_{P}$ such that
$\Omega_{j}\cap U=D_{P}\cap U$, $j\geq 1$, for a fixed neighborhood $U$ of the
origin in $\mathbb{C}^{n}$. Let $\\{\eta_{j}\\}\subset D_{P}\cap U$ be a
sequence that converges $\Lambda$-tangentially to $(0^{\prime},1)$ in $D_{P}$.
Then, $\lim_{j\to\infty}\sigma_{\Omega_{j}}(\eta_{j})=1$.

The organization of this paper is as follows: In Sections 2, we recall a
definition and results needed later. Next, in Section 3 we give a proof of
Theorem 1.1. Finally, the proofs of Theorem 1.2 and Theorem 1.3 are given in
Section 4.

##  2\. Several technical lemmas

First of all, we recall the following definition (see [GK87] or [DN09]).

######  Definition 2.1.

Let $\\{\Omega_{i}\\}_{i=1}^{\infty}$ be a sequence of open sets in a complex
manifold $M$ and $\Omega_{0}$ be an open set of $M$. The sequence
$\\{\Omega_{i}\\}_{i=1}^{\infty}$ is said to converge to $\Omega_{0}$ (written
$\lim\Omega_{i}=\Omega_{0}$) if and only if

  1. (i)

For any compact set $K\subset\Omega_{0},$ there is an $i_{0}=i_{0}(K)$ such
that $i\geq i_{0}$ implies that $K\subset\Omega_{i}$; and

  2. (ii)

If $K$ is a compact set which is contained in $\Omega_{i}$ for all
sufficiently large $i,$ then $K\subset\Omega_{0}$.

Next, to give proofs of Theorem 1.2 and Theorem 1.3, we need the following
lemma which is a generalization of [Liu18, Lemma $2.5$].

######  Lemma 2.1.

Let $\\{\psi_{j}\\}\subset\mathrm{Aut}(D_{P})$ be a sequence of automorphisms

$\psi_{j}(z,w)=\left(\dfrac{\sqrt[2m_{1}]{1-|a_{j}|^{2}}}{\sqrt[m_{1}]{1+\bar{a}_{j}z_{n}}}z_{1},\ldots,\dfrac{\sqrt[2m_{n-1}]{1-|a_{j}|^{2}}}{\sqrt[m_{n-1}]{1+\bar{a}_{j}z_{n}}}z_{n-1},\dfrac{z_{n}+a_{j}}{1+\bar{a}_{j}z_{n}}\right),$

where $a_{j}\in(0,1)$ with $\lim a_{j}=1$. Then, for any $s\in(0,1)$ we have
$\psi_{j}^{-1}(D_{P}^{s})\to D_{P}$ as $j\to\infty$. In addition, for any
$0<\epsilon<1/2$ and any neighborhood $U$ of $(0^{\prime},1)$ in
$\mathbb{C}^{n}$ there exists an integer $j_{0}\geq 1$ such that
$\overline{D_{P}}\setminus
B((0^{\prime},-1),\epsilon))\subset\psi_{j}^{-1}(\overline{D_{P}}\cap U)$ for
all $j\geq j_{0}$.

###### Proof.

A computation shows that

$\displaystyle\hskip
14.22636pt\left|\dfrac{z_{n}+a_{j}}{1+a_{j}z_{n}}-b\right|^{2}+sP\left(\dfrac{\sqrt[2m_{1}]{1-|a_{j}|^{2}}}{\sqrt[m_{1}]{1+\bar{a}_{j}w}}z_{1},\ldots,\dfrac{\sqrt[2m_{n-1}]{1-|a_{j}|^{2}}}{\sqrt[m_{n-1}]{1+\bar{a}_{j}z_{n}}}z_{n-1}\right)<s^{2}$
$\displaystyle\Leftrightarrow\left|\dfrac{z_{n}+a_{j}}{1+a_{j}z_{n}}-b\right|^{2}+s\dfrac{1-|a_{j}|^{2}}{|1+a_{j}z_{n}|^{2}}P(z)<s^{2}$
$\displaystyle\Leftrightarrow\left|w-\frac{b(1-a_{j})}{1+a_{j}-2a_{j}b}\right|^{2}+\dfrac{(1-b)(1+a_{j})}{1+a_{j}-2a_{j}b}P(z)<\dfrac{1+a_{j}-2b}{1+a_{j}-2a_{j}b}+\left|\frac{b(1-a_{j})}{1+a_{j}-2a_{j}b}\right|^{2}.$

Moreover, by a straightforward calculation, one has that

$\displaystyle\lim\limits_{j\to\infty}\frac{b(1-a_{j})}{1+a_{j}-2a_{j}b}=0,\;\lim\limits_{j\to\infty}\dfrac{(1-b)(1+a_{j})}{1+a_{j}-2a_{j}b}=1,\;\lim\limits_{j\to\infty}\dfrac{1+a_{j}-2b}{1+a_{j}-2a_{j}b}=1.$

This yields $\psi_{j}^{-1}(D_{P}^{s})\to D_{P}$ as $j\to\infty$. ∎

##  3\. Squeezing function for linearly convex domains

Throughout this section, the domain $\Omega\subset\mathbb{C}^{n}$ and the
boundary point $\xi_{0}\in\partial\Omega$ are assumed to satisfy the
hypothesis of Theorem 1.1, namely $\partial\Omega$ is linearly convex, of
finite type $2m$ near a point $\xi_{0}$ of $\partial\Omega$. We may also
assume that $\xi_{0}=0$. There exists a neighbourhood $U$ of $\xi_{0}=0$ in
$\mathbb{C}^{n}$ such that $\Omega\cap U$ is linearly convex and is defined by
a smooth function

$\rho(z^{\prime},z_{n})=\mathrm{Re}(z_{n})+h\big{(}\mathrm{Im}(z_{n}),z^{\prime}\big{)},$

where $h$ is a function of class $C^{\infty}$. We may also assume that there
exists a real positive number $\epsilon_{0}$ such that for every
$-\epsilon_{0}<\epsilon<\epsilon_{0}$, the level sets $\\{\rho(z)=\epsilon\\}$
are linearly convex.

For each $\epsilon\in(0,\epsilon_{0}/2),\ \eta\in\Omega\cap U$ with
$|\rho(\eta)|<\epsilon_{0}/2$ and each unit vector
$v\in\mathbb{S}^{n-1}:=\\{v\in\mathbb{C}^{n}\colon|v|=1\\},$ we set

$\tau(\eta,v,\epsilon):=\sup\\{r>0:\rho(\eta+\lambda v)-\rho(\eta)<\epsilon\
\text{for all }\ \lambda\in\mathbb{C}\ \text{with}\ |\lambda|<r\\}.$

Then, it is easy to see that $\tau(\eta,v,\epsilon)$ is the distance from
$\eta$ to $S_{\eta,\epsilon}:=\\{\rho(z)=\rho(\eta)+\epsilon\\}$ along the
complex line $\\{\eta+\lambda v\colon\lambda\in\mathbb{C}\\}$.

To every point $\eta\in\Omega\cap U$ and every sufficiently small positive
constant $\epsilon$ we associate

  1. (1)

A holomorphic coordinate system $(z_{1},z_{2},\ldots,z_{n})$ centered at
$\eta$ and preserving orthogonality,

  2. (2)

Points $p_{1},p_{2},\ldots,p_{n}$ on the hypersurface $S_{\eta,\epsilon}$ and,

  3. (2)

Positive real numbers
$\tau_{1}(\eta,\epsilon),\tau_{2}(\eta,\epsilon),\ldots,\tau_{n}(\eta,\epsilon)$.

The construction proceeds as follows. We first set

$e_{n}:=\frac{\nabla\rho(\eta)}{|\nabla\rho(\eta)|}\ \text{and}\
\tau_{n}(\eta,\epsilon):=\tau(\eta,e_{n},\epsilon).$

Working with sufficiently small $\epsilon$, there exists a unique point
$p_{n}$ in $S_{\eta,\epsilon}$ where this distance is achieved. Choose a
parameterization of the complex line from $\eta$ to $p_{n}$ such that
$z_{n}(0)=\eta$ and $p_{n}$ lies on the positive $\mathrm{Re}(z_{n})$ axis. By
the choice of real axis for $z_{n}$, we have $\frac{\partial r}{\partial
x_{n}}(\eta)=1$ and thus, if $U$ is small enough,

$\frac{\partial r}{\partial x_{n}}(z)\approx 1\ \text{ for all }\ z\in U.$

We also have

(1) $\tau_{n}(\eta,\epsilon)\approx\epsilon,$

where the constant is independent of $\eta$ and $\epsilon$. Now consider the
orthogonal complement $H_{n}$ of the span of the coordinate $z_{n}$ in
$\mathbb{C}^{n}.$ For any $\gamma\in H_{n}\cap\mathbb{S}^{n-1}$, compute
$\tau(\eta,\gamma,\epsilon)$. Because of the assumption of finite type, the
largest such distance is finite and is achieved at a vector $e_{1}\in
H_{1}\cap\mathbb{S}^{n-1}$. Set
$\tau_{1}(\eta,\epsilon):=\tau(\eta,e_{1},\epsilon).$ Let $p_{1}\in
S_{\eta,\epsilon}$ be a point such that
$p_{1}=\eta+\tau_{1}(\eta,\epsilon)e_{1}$. The coordinate $z_{1}$ is defined
by parameterizing the complex line from $\eta$ to $p_{1}$ in such a way that
$z_{1}(0)=\eta$ and $p_{1}$ lies on the positive $\mathrm{Re}(z_{1})$ axis.
For the next step, define $H_{1}$ as the orthogonal complement of the span of
$z_{1}$ and $z_{n}$ and repeat the above construction. Continuing this
process, we obtain $n$ coordinate functions $z_{k}$, vectors $e_{k}$, the
numbers $\tau_{k}(\eta,\epsilon)$ and the distinguished points $p_{k}\ (1\leq
k\leq n)$. Let $z_{k}=x_{k}+iy_{k}\ (1\leq k\leq n)$ denote the underlying
real coordinates.

We assume that $\xi_{0}$ is an accumulating point for a sequence of
automorphisms of $\Omega$. Let $\\{\eta_{j}\\}\subset\Omega$ be a sequence
converging to $\xi_{0}$. Moreover, we may assume that $\eta_{j}\in\Omega\cap
U$ for all $j$. Let us set $\epsilon_{j}:=-\rho(\eta_{j})$ for all $j$. Then,
by argument as above, we construct the new coordinates
$(z^{j}_{1},\ldots,z^{j}_{n})$, the positive numbers
$\tau_{j,1},\ldots,\tau_{j,n}$, and the points $p_{1}^{j},\ldots,p_{n}^{j}$
associated with $\eta_{j}$ and $\epsilon_{j}$.

The change of coordinates from the canonical system to the system
$(z^{j}_{1},\ldots,z^{j}_{n})$ is the composition of a translation $T_{j}$ and
of a unitary transform $A_{j}$. In addition, we may assume that $(A_{j}\circ
T_{j})^{-1}$ is defined in a fixed neighborhood of the origin and thus the
corresponding defining function $\rho_{j}$ is defined by

$\rho_{j}:=\rho\circ(A_{j}\circ T_{j})^{-1},$

which is given in a fixed neighborhood of $0$ by

$\begin{split}\rho_{j}(z)=-\epsilon_{j}+\mathrm{Re}(\sum_{k=1}^{n}a_{k}^{j}z_{k})&+\sum_{2\leq|\alpha|+|\beta|\leq
2m}C^{j}_{\alpha\beta}{z^{\prime}}^{\alpha}{z^{\prime}}^{\beta}+O(|z|^{2m+1}),\end{split}$

where $\alpha=(\alpha_{1},\ldots,\alpha_{n-1}),\
|\alpha|=\alpha_{1}+\cdots+\alpha_{n-1}$ and
${z^{\prime}}^{\alpha}=z_{1}^{\alpha_{1}}.\ldots.z_{n-1}^{\alpha_{n-1}}$. We
note that $O(|z|^{2m+1})$ is independent of $j$.

Let $\rho\circ A$ be the limit of $\rho_{j}$ when $j$ goes to infinity, where
$A$ is a unitary transform and this convergence is $\mathcal{C}^{\infty}$ on a
fixed compact neighborhood of $\xi_{0}$. Then, for every $j$ less than or
equal to $n$ and for every multi-index $\alpha$ and $\beta$ satisfying
$2\leq|\alpha|+|\beta|\leq 2m$, there exist two complex numbers $a_{j}$ and
$C_{\alpha\beta}$ such that

$\lim_{j\to\infty}a^{j}_{k}=a_{k}\ \text{and}\
\lim_{j\to\infty}C^{j}_{\alpha\beta}=C_{\alpha\beta}.$

Now let us consider the dilation

$\Lambda_{j}(z):=(\tau_{j,1}z_{1},\ldots,\tau_{j,n}z_{n})$

and the function

$\tilde{\rho}_{j}=\frac{1}{\epsilon_{j}}\rho_{j}\circ\Lambda_{j}.$

Therefore, the defining function $\tilde{\rho}_{j}$ has the following form

$\begin{split}\tilde{\rho}_{j}(z)=-1+\frac{1}{\epsilon_{j}}\mathrm{Re}\big{(}\sum_{j=1}^{n}a_{j}^{j}\tau_{j,j}z_{j}\big{)}&+\frac{1}{\epsilon_{j}}\sum_{2\leq|\alpha|+|\beta|\leq
2m}C^{j}_{\alpha\beta}\tau_{j}^{\alpha+\beta}{z^{\prime}}^{\alpha}{z^{\prime}}^{\beta}+O((\epsilon_{j})^{1/2m}|z|^{2m+1}),\end{split}$

where
$\tau_{j}^{\alpha+\beta}=\tau_{j,1}^{\alpha_{1}+\beta_{1}}.\ldots.\tau_{j,n-1}^{\alpha_{n-1}+\beta_{n-1}}$.
Furthermore, it follows from [Ni09, Prop.3.1] that the functions
$\tilde{\rho}_{j}$ are smooth and plurisubharmonic, and after taking a
subsequence, we may assume that $\\{\tilde{\rho}_{j}\\}$ that converges
uniformly on compacta of $\mathbb{C}^{n}$ to a smooth plurisubharmonic
function $\tilde{\rho}$ of the form

$\tilde{\rho}(z)=-1+\mathrm{Re}\big{(}\sum_{k=1}^{n}b_{k}z_{k}\big{)}+P(z^{\prime}),$

where $P$ is a plurisubharmonic polynomial of degree less than or equal to
$2m$.

In what follows, let us denote by $\Gamma_{j}:=\Lambda^{-1}_{j}\circ
A_{j}\circ T_{j}$ for all $j$. Then, one can deduce that
$\\{\Gamma_{j}(\Omega\cap U)\\}$ converges to the following model

$\widetilde{M}_{P}:=\left\\{z\in\mathbb{C}^{n}\colon\tilde{\rho}(z)=-1+\mathrm{Re}\big{(}\sum_{k=1}^{n}b_{k}z_{k}\big{)}+P(z^{\prime})<0\right\\},$

which is clearly biholomorphically equivalent to

$M_{P}:=\left\\{z\in\mathbb{C}^{n}\colon\hat{\rho}(z):=\mathrm{Re}(z_{n})+P(z^{\prime})<0\right\\}.$

Let us consider a sequence of the biholomorphisms $F_{j}\colon
f_{j}(\Omega\cap U)\to\Gamma_{j}(\Omega\cap U)$ defined by
$F_{j}=\Gamma_{j}\circ f_{j}^{-1}$. Since $F_{j}(0)=0\in\widetilde{M}_{P}$, it
follows that our sequence $\\{F_{j}\\}$ is not compactly divergence. Moreover,
the normality of $\\{F_{j}\\}$ is ensured by [Ni09, Lemma 4.1].

Now we are ready to prove Theorem 1.1.

###### Proof of Theorem 1.1.

Let $\\{\eta_{j}\\}\subset\Omega$ be a sequence given in Theorem 1.1, that is,
$\lim\limits_{j\to\infty}\eta_{j}=\xi_{0}$ and
$\lim\limits_{j\to\infty}\sigma_{\Omega}(\eta_{j})=1$. Firstly, let us set
$\delta_{j}=2(1-\sigma_{\Omega}(\eta_{j}))$ for all $j$. Then by our
assumption, for each $j$, there exists an injective holomorphic map
$f_{j}:\Omega\to\mathbb{B}^{n}$ such that $f_{j}(\eta_{j})=(0^{\prime},0)$ and
$\mathbb{B}(0;1-\delta_{j})\subset f_{j}(\Omega)$. Then by [DN09, Proposition
$2.2$] and the hypothesis of Theorem 1.1, without loss of generality we may
assume that for each compact subset $K\Subset\mathbb{B}^{n}$ and each
neighborhood $U$ of $\xi_{0}$, there exists an integer $j_{0}$ such that
$f_{j}^{-1}(K)\subset\Omega\cap U$ for all $j\geq j_{0}$, i.e.
$f_{j}(\Omega\cap U)$ converges to $\mathbb{B}^{n}$.

Next, it follows from [Ni09, Lemma 4.1] that the sequence $\Gamma_{j}\circ
f_{j}^{-1}\colon f_{j}(\Omega\cap U)\to\Gamma_{j}(\Omega\cap U)$ is normal and
its limit is a holomorphic mapping from $\mathbb{B}^{n}$ to
$\widetilde{M}_{P}$. Moreover, by Montel’s theorem the sequence
$f_{j}\circ\Gamma_{j}^{-1}\colon\Gamma_{j}(\Omega\cap U)\to f_{j}(\Omega\cap
U)\subset\mathbb{B}^{n}$ is also normal. In addition, our the sequence
$\\{\Gamma_{j}\circ f_{j}^{-1}\\}$ is not compactly divergent since
$\Gamma_{j}\circ f_{j}^{-1}(0,0^{\prime})=(0,0^{\prime})$. Then by [DN09,
Proposition $2.1$], after taking some subsequence of $\\{\Gamma_{j}\circ
f^{-1}_{j}\\}$, we may assume that such a subsequence converges uniformly on
every compact subset of $\mathbb{B}^{n}$ to a biholomorphism $F$ from
$\mathbb{B}^{n}$ onto $\widetilde{M}_{P}$, which is clearly equivalent to
$M_{P}$.

On the other hand, by [Ni09, Theorem $1.1$] $\Omega$ is also biholomorphically
equivalent to $M_{P}$, and hence $\Omega$ is biholomorphically equivalent to
$\mathbb{B}^{n}$. Therefore, $\partial\Omega$ is strongly pseudoconvex at
$\xi_{0}$ ($\xi_{0}$ is of the D’Angelo type $2$), which ends our proof. ∎

##  4\. Proofs of Theorem 1.2 and Theorem 1.3

This section is devoted to proofs of Theorem 1.2 and Theorem 1.3.

###### Proof of Theorem 1.2.

Let $q=(q^{\prime},q_{n})\in D^{s,r}_{P}$ near $p=(0^{\prime},1)$. By the
invariance of $D_{s,r},D_{P}$ under the rotation
$(z^{\prime},z_{n})\mapsto(z^{\prime},e^{i\theta}z_{n})$ for
$\theta\in\mathbb{R}$ satisfying that $\mathrm{Im}(e^{i\theta}q_{n})=0$,
without loss of generality we may assume that $\mathrm{Im}(q_{n})=0$ for every
$j\geq 1$.

We now consider the automorphism $\psi_{a}\in\mathrm{Aut}(D_{P})$, given by

$\psi_{a}(z)=\left(\dfrac{\sqrt[2m_{1}]{1-|a|^{2}}}{\sqrt[m_{1}]{1+\bar{a}z_{n}}}z_{1},\ldots,\dfrac{\sqrt[2m_{n-1}]{1-|a|^{2}}}{\sqrt[m_{n-1}]{1+\bar{a}z_{n}}}z_{n-1},\dfrac{z_{n}+a}{1+\bar{a}z_{n}}\right),$

where $a=\mathrm{Re}(q_{n})=q_{n}\in(0,1)$. Then, Lemma 2.1 yields

$\begin{split}\lim\limits_{a\to
1}\psi_{a}^{-1}(D_{s,r})=D_{P,r};\;\lim\limits_{a\to
1}\psi_{a}^{-1}(\Omega)=D_{P},\end{split}$

where $\displaystyle
D_{P,r}:=D_{P/r}=\Big{\\{}z\in\mathbb{C}^{n}\colon|z_{n}|^{2}+\frac{1}{r}P(z^{\prime})<1\Big{\\}}$.
Moreover, we have that
$\psi_{a}^{-1}(q)=\Big{(}\dfrac{q_{1}}{\lambda^{1/2m_{1}}},\ldots,\dfrac{q_{n-1}}{\lambda^{1/2m_{n-1}}},0\Big{)}\in
D_{P,r}\cap\\{z_{n}=0\\}$, where $\lambda=1-|a|^{2}$ and
$D_{P,r}\cap\\{z_{n}=0\\}\Subset D_{P}\cap\\{z_{n}=0\\}$. Therefore, by Lemma
$2.1$ in [NNC21] and again by the invariance of the squeezing function under
biholomorphisms, we conclude that

$\sigma_{\Omega}(q)=\sigma_{\psi_{a}^{-1}(\Omega)}(\psi_{a}^{-1}(q))>\delta/d>0,\;\forall\;q\in
E^{r}\cap B(0;\epsilon_{0}),$

where $d$ denotes the diameter of $D_{P}$ and
$\delta:=\mathrm{dist}(Z_{\rho}(P),Z_{1}(P))/2$ with
$Z_{\rho}(P)=\\{z^{\prime}\in\mathbb{C}^{n-1}\colon P(z^{\prime})=r\\}$. ∎

###### Proof of Theorem 1.2.

For each $j\geq 1$, choose $\theta_{j}\in\mathbb{R}$ such that that
$\mathrm{Im}(e^{i\theta_{j}}\eta_{jn})=0$. Since $\mathrm{Im}(\eta_{jn})\to 0$
as $j\to\infty$, one has that $\theta_{j}\to 0$ as $j\to\infty$. Moreover, by
shrinking $U$ if necessary we may also assume that
$\rho_{\theta_{j}}(\Omega_{j})\cap U=D_{P}\cap U$ for all $j\geq 1$.
Therefore, by the invariance of $D_{P}$ under the rotation
$\rho_{\theta_{j}}\colon(z^{\prime},z_{n})\mapsto(z^{\prime},e^{i\theta_{j}}z_{n})$
for $j\geq 1$, without loss of generality we may assume that
$\mathrm{Im}(\eta_{jn})=0$ for every $j\geq 1$ and $\\{\Omega_{j}\\}$ is a
sequence of subdomains of $D_{P}$ such that $\Omega_{j}\cap U=D_{P}\cap U$ for
all $j\geq 1$.

We now consider the sequence of automorphisms
$\\{\psi_{j}\\}\subset\mathrm{Aut}(D_{P})$, given by

$\psi_{j}(z)=\left(\dfrac{\sqrt[2m_{1}]{1-|a_{j}|^{2}}}{\sqrt[m_{1}]{1+\bar{a}_{j}z_{n}}}z_{1},\ldots,\dfrac{\sqrt[2m_{n-1}]{1-|a_{j}|^{2}}}{\sqrt[m_{n-1}]{1+\bar{a}_{j}z_{n}}}z_{n-1},\dfrac{z_{n}+a_{j}}{1+\bar{a}_{j}z_{n}}\right),$

where $a_{j}=\mathrm{Re}(\eta_{jn})=\eta_{jn}\in\mathbb{R}$ for all $j\geq 1$.

Let us set $b_{j}=\psi_{j}^{-1}(\eta_{j})$ for all $j\geq 1$. Then, a
straightforward computation shows that

$b_{j}=\psi_{j}^{-1}(\eta_{j})=\Big{(}\dfrac{\eta_{j1}}{\lambda_{j}^{1/2m_{1}}},\ldots,\dfrac{\eta_{j(n-1)}}{\lambda_{j}^{1/2m_{n-1}}},0\Big{)}\in
D_{P}\cap\\{z_{n}=0\\},$

where $\lambda_{j}=1-|a_{j}|^{2}$ for all $j\geq 1$.

Since $\\{\eta_{j}\\}$ converges $\Lambda$-tangentially to $(0^{\prime},1)$,
it follows that there exists a sequence $\\{\rho_{j}\\}\subset(0,1)$ with
$\rho_{j}\to 1$ as $j\to\infty$ such that

$|\eta_{jn}-1-s|^{2}+\dfrac{s}{\rho_{j}}P(\eta_{j}^{\prime})>s^{2},\;\forall
j\geq 1.$

This implies that

$\begin{split}P(b_{j}^{\prime})&=\dfrac{1}{\lambda_{j}}P(\eta_{j}^{\prime})\geq\dfrac{2\rho_{j}(1-a_{j})}{1-a_{j}^{2}}-\dfrac{\rho_{j}}{s}\dfrac{|1-a_{j}|^{2}}{1-a_{j}^{2}}\\\
&\geq\dfrac{2\rho_{j}}{1+a_{j}}-\dfrac{\rho_{j}}{s}\dfrac{(1-a_{j})}{1+a_{j}}\end{split}$

for all $j\geq 1$. Therefore, we obtain that $P(b_{j}^{\prime})\to 1$ as
$j\to\infty$, and hence by passing to a subsequence if necessary, we may
assume that $\psi_{j}^{-1}(\eta_{j})$ converges to some strongly pseudoconvex
boundary point $p\in\partial D_{P}\cap\\{z_{n}=0\\}$. Since $a_{j}\to 1$ as
$j\to\infty$, Lemma 2.1 yields

$\begin{split}\lim\limits_{j\to\infty}\psi_{j}^{-1}(\Omega_{j})=\lim\limits_{j\to\infty}\psi_{j}^{-1}(\Omega_{j}\cap
U)=\lim\limits_{j\to\infty}\psi_{j}^{-1}(D_{P}\cap U)=D_{P}.\end{split}$

In addition, for any $\epsilon>0$ sufficiently small there exists $j_{0}\geq
1$ such that

$\psi_{j}^{-1}(\overline{\Omega_{j}})\setminus
B((0^{\prime},-1),\epsilon)=\overline{D_{P}}\setminus
B((0^{\prime},-1),\epsilon)$

for any $j\geq j_{0}$. Hence, since $\sigma_{D_{P}}(b_{j})\to 1$ as
$j\to\infty$ and by Theorem $3.1$ in [KZ16], one concludes that
$\sigma_{\Omega_{j}}(\eta_{j})=\sigma_{\psi_{j}^{-1}(\Omega_{j})}(b_{j})\to 1$
as $j\to\infty$. ∎

We close this section with an example, which is a generalization of Example
1.1.

######  Example 4.1.

Fix positive integers $m_{1},\ldots,m_{n-1}$ and denote by
$\Lambda:=(1/m_{1},\ldots,1/m_{n-1})$. Let us consider a general ellipsoid
$D_{P}$ in $\mathbb{C}^{n}\;(n\geq 2)$, defined by

$\begin{split}D_{P}:=\\{(z^{\prime},z_{n})\in\mathbb{C}^{n}\colon|z_{n}|^{2}+P(z^{\prime})<1\\},\end{split}$

where $P(z^{\prime})$ is a $(1/m_{1},\ldots,1/m_{n-1})$-homogeneous polynomial
given by

$P(z^{\prime})=\sum_{wt(K)=wt(L)=1/2}a_{KL}{z^{\prime}}^{K}\bar{z^{\prime}}^{L},$

where $a_{KL}\in\mathbb{C}$ with $a_{KL}=\bar{a}_{LK}$, satisfying that
$P(z^{\prime})>0$ whenever $z^{\prime}\neq 0$. Suppose that the domain $D_{P}$
is a WB-domain, i.e., $\partial D_{P}$ is strongly pseudoconvex at every
boundary point outside the set
$\\{(0^{\prime},e^{i\theta})\colon\theta\in\mathbb{R}\\}$ (cf. [AGK16]).

Now let us denote by $\rho(z):=|z_{n}|^{2}-1+P(z^{\prime})$ a local defining
function for $D_{P}$ and consider a sequence
$\\{a_{j}=(a_{j}^{\prime},a_{jn})\\}\subset D_{P}$ which converges
$\Lambda$-tangentially to $p:=(0^{\prime},1)$. Since $D_{P}$ is invariant
under the map $z^{\prime}\mapsto z^{\prime};z_{n}\mapsto e^{i\theta}z_{n}$ and
$\sigma_{D_{P}}$ is invariant under biholomorphisms, we may assume that
$\mathrm{Im}(a_{jn})=0$ for all $j$. Since $\mathrm{dist}(a_{j},\partial
D_{P})\approx-\rho(a_{j})\approx 1-|a_{jn}|^{2}-P(a_{j}^{\prime})$ and
$\\{a_{j}\\}$ converges $\Lambda$-tangentially to $p$, it follows that
$P(a_{j}^{\prime})\geq\;c_{j}\;\mathrm{dist}(a_{j},\partial D_{P})$ for some
sequence $\\{c_{j}\\}\subset\mathbb{R}$ with $0<c_{j}\to+\infty$. This implies
that $P(a_{j}^{\prime})\geq c_{j}^{\prime}(1-|a_{jn}|^{2}-P(a_{j}^{\prime}))$
for some sequence $\\{c_{j}^{\prime}\\}\subset\mathbb{R}$ with
$0<c_{j}^{\prime}\to+\infty$ and hence

$P(a_{j}^{\prime})\geq\dfrac{c_{j}^{\prime}}{1+c_{j}^{\prime}}(1-|a_{jn}|^{2}),\forall
j\geq 1.$

Let us denote by $\psi_{j}$ the automorphism of $D_{P}$, given by

$\psi_{j}(z)=\left(\frac{(1-|a_{jn}|^{2})^{1/2m_{1}}}{(1-\bar{a}_{jn}z_{n})^{1/m_{1}}}z_{1},\ldots,\frac{(1-|a_{jn}|^{2})^{1/2m_{n-1}}}{(1-\bar{a}_{jn}z_{n})^{1/m_{n-1}}}z_{n-1},\frac{z_{n}-a_{jn}}{1-\bar{a}_{jn}z_{n}}\right),$

and hence $\psi_{j}(a_{j})=(b_{j},0)$, where

$b_{j}=\left(\frac{a_{j1}}{(1-|a_{jn}|^{2})^{1/2m_{1}}},\ldots,\frac{a_{j(n-1)}}{(1-|a_{jn}|^{2})^{1/2m_{n-1}}}\right).$

Thanks to the boundedness of $\\{b_{j}\\}$, without loss of generality we may
assume that $b_{j}\to b\in\mathbb{C}^{n-1}$ as $j\to\infty$. In addition, we
have that
$P(b_{j})=\dfrac{1}{1-|a_{jn}|^{2}}P(a_{j}^{\prime})\geq\dfrac{c_{j}^{\prime}}{1+c_{j}^{\prime}},\forall
j\geq 1$. Therefore, we arrive at the situation $b_{j}\to b$ with $P(b)=1$ and
thus $\psi_{j}(a_{j})$ converges to the strongly pseudoconvex boundary point
$(b,0)$ of $\partial D_{P}$, which implies that
$\sigma_{D_{P}}(a_{j})=\sigma_{D_{P}}(\psi_{j}(a_{j}))\to 1$ as $j\to\infty$
even the boundary point $p$ is weakly pseudoconvex.

######  Acknowledgement.

Part of this work was done while the first author was visiting the Vietnam
Institute for Advanced Study in Mathematics (VIASM). He would like to thank
the VIASM for financial support and hospitality.

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