# Managing Portfolio for Maximizing Alpha and Minimizing Beta

Soumyadip Sarkar
<EMAIL_ADDRESS>
###### Abstract

Portfolio management is an essential component of investment strategy that
aims to maximize returns while minimizing risk. This paper explores several
portfolio management strategies, including asset allocation, diversification,
active management, and risk management, and their importance in optimizing
portfolio performance. These strategies are examined individually and in
combination to demonstrate how they can help investors maximize alpha and
minimize beta. Asset allocation is the process of dividing a portfolio among
different asset classes to achieve the desired level of risk and return.
Diversification involves spreading investments across different securities and
sectors to minimize the impact of individual security or sector-specific
risks. Active management involves security selection and risk management
techniques to generate excess returns while minimizing losses. Risk management
strategies, such as stop-loss orders and options strategies, aim to minimize
losses in adverse market conditions. The importance of combining these
strategies for optimizing portfolio performance is emphasized in this paper.
The proper implementation of these strategies can help investors achieve their
investment goals over the long-term, while minimizing exposure to risks. A
call to action for investors to utilize portfolio management strategies to
maximize alpha and minimize beta is also provided.

###### Index Terms:

Portfolio Management, Asset Allocation, Diversification, Active Management,
Risk Management

##  I Introduction

Portfolio management is the process of selecting and managing a portfolio of
investments to achieve specific investment objectives. It involves
constructing a portfolio of assets that align with an investor’s goals, risk
tolerance, and investment horizon. Portfolio management includes asset
allocation, diversification, active management, and risk management techniques
to maximize portfolio returns while minimizing risk. The primary goal of
portfolio management is to generate optimal returns while reducing the impact
of market fluctuations and minimizing the risk of loss.

Maximizing alpha and minimizing beta are important concepts in portfolio
management because they provide investors with a measure of the performance of
their portfolio compared to a benchmark, typically an index of the broader
market. Alpha represents the excess returns generated by a portfolio relative
to the benchmark, while beta represents the degree to which the portfolio
moves in line with the market.

Maximizing alpha is important because it reflects the portfolio’s ability to
generate excess returns that are not explained by the overall performance of
the market. This is achieved by selecting securities or using investment
strategies that outperform the benchmark. By generating excess returns,
investors can increase their portfolio’s overall returns and potentially
outperform the market over time.

Minimizing beta is important because it reflects the portfolio’s ability to
reduce the impact of market fluctuations and reduce overall risk. By
minimizing beta, investors can reduce the impact of market downturns and
potentially experience less volatility in their portfolio’s returns.

In combination, maximizing alpha and minimizing beta can help investors
achieve their investment objectives, whether that be long-term capital
appreciation or generating income. By achieving excess returns while
minimizing risk, investors can increase their chances of meeting their goals
while reducing the impact of market fluctuations. Overall, maximizing alpha
and minimizing beta are important concepts in portfolio management that can
help investors achieve optimal returns while managing risk.

The paper examines several strategies for portfolio management that can be
used to optimize alpha and minimize beta. These strategies include:

####  I-1 Asset allocation

Asset allocation is the process of dividing a portfolio among different asset
classes such as stocks, bonds, and cash. By diversifying across different
asset classes, investors can reduce their overall risk while potentially
increasing returns.

####  I-2 Diversification

Diversification involves investing in a variety of assets within each asset
class to reduce the impact of individual security or sector-specific risks.
Diversification helps to minimize the overall risk of the portfolio.

####  I-3 Active management

Active management involves selecting individual securities or using investment
strategies to try to generate excess returns compared to a benchmark. Active
management strategies can include fundamental analysis, technical analysis,
and quantitative analysis.

####  I-4 Risk management

Risk management involves using strategies to minimize the risk of loss in the
portfolio. This may include techniques such as stop-loss orders or options
strategies.
The paper examines each of these strategies in-depth, highlighting their
importance and contribution towards achieving the objective of maximizing
alpha and minimizing beta. Additionally, the paper explores how these
strategies can be combined to construct portfolios that generate excess
returns while reducing market risk. The study provides valuable insights to
investors and portfolio managers seeking to enhance portfolio performance and
achieve their investment objectives.

##  II Asset Allocation

Asset allocation is the process of dividing an investment portfolio among
different asset classes such as stocks, bonds, and cash. The purpose of asset
allocation is to spread an investor’s money across different asset classes
that have different levels of risk and return, in order to achieve a desired
level of diversification and balance risk and reward.

The goal of asset allocation is to optimize the risk and return profile of the
portfolio, based on the investor’s goals, risk tolerance, and investment
horizon. By allocating assets across different asset classes, investors can
potentially earn higher returns while reducing overall risk.

The allocation of assets is typically based on the investor’s goals,
investment horizon, and risk tolerance. Investors with a longer investment
horizon and higher risk tolerance may allocate more of their portfolio to
stocks, which historically have higher returns but also higher volatility.
Investors with a shorter investment horizon or lower risk tolerance may
allocate more of their portfolio to fixed-income securities, which offer lower
returns but also lower volatility.

Asset allocation requires ongoing monitoring and adjustment, as market
conditions and the investor’s circumstances change. Investors should
periodically review their portfolio and make adjustments to their asset
allocation as needed to ensure that it remains aligned with their goals and
risk tolerance.

###  II-A Strategic Asset Allocation

Strategic asset allocation is a long-term investment strategy that involves
allocating a portfolio across different asset classes based on the investor’s
goals, risk tolerance, and investment horizon. This strategy is based on the
belief that the performance of different asset classes varies over time and
that by diversifying across asset classes, investors can achieve optimal
returns while reducing overall risk.

Strategic asset allocation involves setting target allocations for different
asset classes and periodically rebalancing the portfolio to maintain those
targets. The target allocation is typically based on the investor’s goals and
risk tolerance, as well as historical performance data for different asset
classes.

For example, a strategic asset allocation plan might target a 60% allocation
to stocks, 30% allocation to bonds, and 10% allocation to cash. If the stock
market performs well and the value of the stock holdings in the portfolio
increases, the portfolio may become overweighted in stocks, and the investor
may need to rebalance the portfolio by selling some of the stocks and buying
more bonds or cash to bring the portfolio back to its target allocation.

Strategic asset allocation is a passive investment strategy that does not
involve active management of individual securities. Instead, it focuses on
diversification across different asset classes and maintaining a long-term
investment horizon. This strategy may not provide the highest returns in the
short-term but may provide a more stable and predictable return over the long-
term.

Overall, strategic asset allocation is an important strategy in portfolio
management, as it can help investors achieve their long-term investment
objectives by balancing risk and return across different asset classes.

###  II-B Tactical Asset Allocation

Tactical asset allocation is an active investment strategy that involves
adjusting a portfolio’s allocation to different asset classes based on short-
term market conditions and economic forecasts. This strategy seeks to take
advantage of opportunities to generate excess returns by temporarily shifting
investments to asset classes that are expected to outperform or reducing
exposure to asset classes that are expected to underperform.

Unlike strategic asset allocation, which is a long-term investment strategy,
tactical asset allocation is focused on short-term adjustments to the
portfolio’s allocation. This strategy requires active management and
monitoring of the portfolio and market conditions.

For example, if an investment manager believes that the stock market is
overvalued and likely to decline, they may reduce the portfolio’s allocation
to stocks and increase its allocation to bonds or cash. Conversely, if the
manager believes that the stock market is undervalued and likely to rise, they
may increase the portfolio’s allocation to stocks.

Tactical asset allocation is based on the belief that market conditions and
economic forecasts can provide valuable information for generating excess
returns. However, this strategy requires a high level of skill and expertise,
as it is difficult to predict short-term market movements accurately.
Additionally, tactical asset allocation may lead to higher trading costs and
tax implications.

Overall, tactical asset allocation can be an effective strategy for generating
excess returns in the short-term, but it should be used in combination with a
long-term strategic asset allocation plan to balance risk and return over the
long-term.
Asset allocation is a critical component of portfolio management because it
can help investors optimize portfolio returns while reducing overall risk. By
diversifying a portfolio across different asset classes, investors can
potentially achieve a higher return while reducing the risk of losing money.

####  II-B1 Optimizing portfolio returns

Different asset classes have varying levels of risk and potential returns. By
allocating assets across different asset classes, investors can potentially
earn higher returns while diversifying their portfolio’s risk. For example,
stocks have historically provided higher returns than bonds, but also come
with higher risk. By allocating a portion of a portfolio to stocks and a
portion to bonds, an investor can potentially achieve higher returns than they
would by investing solely in one asset class.

####  II-B2 Reducing overall risk

Diversification through asset allocation can also help reduce the overall risk
of a portfolio. When one asset class underperforms, the losses can potentially
be offset by gains in other asset classes. This can help mitigate the impact
of market volatility and reduce the risk of a significant loss.
Asset allocation can also help investors align their portfolio with their
goals and risk tolerance. Investors with a longer investment horizon and
higher risk tolerance may allocate more of their portfolio to stocks, which
historically have higher returns but also higher volatility. Investors with a
shorter investment horizon or lower risk tolerance may allocate more of their
portfolio to fixed-income securities, which offer lower returns but also lower
volatility.

Overall, asset allocation is a critical strategy in portfolio management as it
can help investors optimize returns while reducing overall risk. By
diversifying a portfolio across different asset classes, investors can
potentially achieve their investment objectives while mitigating the impact of
market volatility.

##  III Diversification

Diversification is an investment strategy that involves spreading an
investor’s portfolio across different asset classes, sectors, industries, and
geographic regions to reduce overall risk. The objective of diversification is
to minimize the impact of any individual investment’s performance on the
portfolio’s overall return.

The basic idea behind diversification is that different types of investments
perform differently under different market conditions. By diversifying across
different types of investments, an investor can potentially offset losses in
one area with gains in another. For example, if a portfolio is heavily
invested in one industry, such as technology, and that industry experiences a
downturn, the portfolio’s overall performance will be negatively impacted.
However, if the portfolio is diversified across different sectors and
industries, the losses from the technology sector may be offset by gains in
other areas.

Diversification can be achieved through asset allocation, as discussed
earlier, or through individual security selection. For example, an investor
may choose to invest in stocks across different industries, or in bonds with
different maturities or credit ratings.

Diversification is not a guarantee against losses, but it can help reduce the
risk of large losses in a portfolio. It is important to note that over-
diversification can also have negative consequences, such as lower potential
returns or higher transaction costs. Finding the right balance between
diversification and concentration is key.

###  III-A Asset Class Diversification

Asset class diversification is an investment strategy that involves
diversifying a portfolio across different types of asset classes. Asset
classes are broad categories of investments that have similar characteristics
and behavior. The most common asset classes are stocks, bonds, and cash
equivalents.

Asset class diversification seeks to reduce the risk of loss by spreading
investments across different asset classes. The basic idea behind this
strategy is that different asset classes tend to perform differently under
different market conditions. For example, when stock prices are falling, bond
prices may rise. By holding a mix of stocks, bonds, and cash equivalents, an
investor can potentially reduce the overall risk of their portfolio.

There are several different ways to achieve asset class diversification. One
approach is to use a strategic asset allocation model that establishes target
percentages for each asset class based on an investor’s goals and risk
tolerance. For example, an investor with a longer investment horizon and
higher risk tolerance may allocate more of their portfolio to stocks, while an
investor with a shorter investment horizon or lower risk tolerance may
allocate more of their portfolio to fixed-income securities.

Another approach to asset class diversification is to use mutual funds or
exchange-traded funds (ETFs) that invest in a diversified mix of asset
classes. For example, a target-date fund may hold a mix of stocks, bonds, and
cash equivalents that is appropriate for an investor with a specific
retirement date in mind.

Asset class diversification can help reduce the risk of loss in a portfolio,
but it is important to note that it is not a guarantee against losses. Market
conditions can change quickly, and no investment strategy is foolproof.
Nonetheless, asset class diversification is a sound investment strategy that
can help investors achieve their goals while managing risk.

###  III-B Sector Diversification

Sector diversification is an investment strategy that involves diversifying a
portfolio across different sectors or industries. Sectors are groups of
companies that operate in similar industries, such as technology, healthcare,
energy, or consumer goods.

Sector diversification seeks to reduce risk by spreading investments across
different sectors. The basic idea behind this strategy is that different
sectors tend to perform differently under different market conditions. For
example, the technology sector may perform well during a period of economic
growth, while the healthcare sector may perform well during a recession. By
holding a mix of sectors, an investor can potentially reduce the impact of
market volatility on their portfolio.

There are several different ways to achieve sector diversification. One
approach is to use a tactical asset allocation model that adjusts the
percentage of the portfolio allocated to each sector based on the current
market conditions. For example, if the technology sector is performing well,
an investor may increase their allocation to that sector, while decreasing
their allocation to a sector that is underperforming.

Another approach to sector diversification is to use mutual funds or exchange-
traded funds (ETFs) that invest in a diversified mix of sectors. For example,
a sector-specific ETF may invest in a mix of technology, healthcare, and
consumer goods companies.

Sector diversification can help reduce the risk of loss in a portfolio, but it
is important to note that it is not a guarantee against losses. Market
conditions can change quickly, and no investment strategy is foolproof.
Nonetheless, sector diversification is a sound investment strategy that can
help investors achieve their goals while managing risk.
Diversification is an important investment strategy because it can help
minimize the impact of individual security or sector-specific risks on a
portfolio. Individual security risks refer to risks that are specific to a
particular company, such as the risk of a company going bankrupt or facing
legal challenges. Sector-specific risks refer to risks that are specific to a
particular industry or sector, such as the risk of a recession affecting the
healthcare sector.

By diversifying across different securities or sectors, investors can reduce
their exposure to these risks. For example, if an investor holds a portfolio
of only tech stocks, they are highly exposed to the risks specific to the
technology sector. However, if they diversify their portfolio to include
stocks from other sectors such as healthcare, energy, and consumer goods, they
can reduce the impact of sector-specific risks.

Diversification can also help minimize the impact of individual security
risks. If an investor holds a portfolio of only one or two stocks, they are
highly exposed to the risks specific to those companies. However, if they
diversify their portfolio to include a mix of stocks from different companies,
they can reduce the impact of any one company’s individual risks.

While diversification cannot eliminate all investment risks, it can help
reduce the overall risk of a portfolio. By spreading investments across
different securities, sectors, and asset classes, investors can potentially
achieve higher returns and reduce their exposure to market volatility.
Diversification is a key component of any well-designed investment portfolio
and is essential for managing risk and achieving long-term financial goals.

##  IV Active Management

Active management is an investment strategy in which a portfolio manager or
team of managers seeks to outperform the overall market by actively selecting
individual securities or adjusting the portfolio’s asset allocation. In
contrast to passive management, which seeks to replicate the performance of a
market index, active management involves ongoing analysis and decision-making
to try to beat the market.

Active managers use a variety of strategies to achieve their goal, such as
fundamental analysis, technical analysis, quantitative analysis, and market
timing. Fundamental analysis involves analyzing a company’s financial
statements and other data to identify undervalued or overvalued stocks.
Technical analysis involves studying market trends and patterns to identify
trading opportunities. Quantitative analysis involves using mathematical
models to evaluate securities based on factors such as earnings growth, cash
flow, and dividend yields. Market timing involves making investment decisions
based on an analysis of current market conditions and economic trends.

Active management can be more expensive than passive management, as it
requires more time and resources to conduct ongoing research and analysis.
However, active managers believe that their expertise and insights can lead to
higher returns than those achieved by passive investing.

While active management can potentially generate higher returns than passive
management, it is also associated with higher risks. Active managers may make
incorrect investment decisions or fail to anticipate changes in market
conditions, resulting in lower returns or losses. Additionally, active
management can be influenced by factors such as manager bias, market timing
errors, and high fees.

###  IV-A Quantitative and Qualitative Analysis

Quantitative and qualitative analysis are two methods of evaluating securities
or investment opportunities.

Quantitative analysis involves using numerical and statistical data to
evaluate the performance and characteristics of securities. This type of
analysis is often used by investors who rely on mathematical models and
algorithms to make investment decisions. Quantitative analysis can include
metrics such as earnings growth, price-to-earnings ratios, dividend yields,
and other financial data.

Qualitative analysis, on the other hand, involves evaluating the subjective,
non-numerical aspects of an investment opportunity. This can include
evaluating a company’s management team, corporate culture, competitive
advantages, and other intangible factors that can impact the success of an
investment. Qualitative analysis often involves conducting interviews with
company executives, industry experts, and other stakeholders to gather
insights and perspectives.

Both quantitative and qualitative analysis have strengths and weaknesses.
Quantitative analysis can provide precise, objective data that can be used to
compare and rank different investment opportunities. However, it may not
capture all of the factors that can impact a company’s performance, such as
changes in market conditions or industry trends. Qualitative analysis can
provide a deeper understanding of a company’s operations, culture, and
competitive position. However, it may be subject to bias or personal opinions,
and it can be difficult to quantify the impact of qualitative factors on
investment performance.

To effectively evaluate investment opportunities, many investors use a
combination of quantitative and qualitative analysis. By combining these two
methods, investors can gain a more comprehensive understanding of the risks
and opportunities associated with a particular investment.

###  IV-B Fundamental Analysis

Fundamental analysis is a method of evaluating securities that involves
analyzing a company’s financial and economic fundamentals to determine its
intrinsic value. This type of analysis seeks to identify the underlying
factors that drive a company’s performance, such as its revenue growth,
profitability, cash flow, and competitive position within its industry.

The goal of fundamental analysis is to estimate the fair value of a security
based on its underlying economic and financial factors. To conduct fundamental
analysis, investors typically examine a company’s financial statements, such
as its balance sheet, income statement, and cash flow statement. They also
consider other factors such as management quality, brand strength, and
industry trends.

Some of the key metrics that investors may use in fundamental analysis include
earnings per share, price-to-earnings ratio, return on equity, and debt-to-
equity ratio. These metrics can help investors determine whether a company is
undervalued or overvalued relative to its peers and the overall market.

Fundamental analysis is often used by value investors who seek to identify
stocks that are trading at a discount to their intrinsic value. Value
investors believe that over the long term, the market will eventually
recognize the true value of a company, and its stock price will rise
accordingly. Fundamental analysis can also be used to identify companies that
are poised for long-term growth, based on their financial and competitive
strengths.

However, fundamental analysis has some limitations. It may not capture all of
the factors that can impact a company’s performance, such as changes in market
conditions or emerging competitive threats. Additionally, the valuation of a
company can be subjective, and different analysts may arrive at different
estimates of a company’s intrinsic value.

Overall, fundamental analysis can be a useful tool for investors who are
looking to identify undervalued or high-growth companies. However, it should
be used in conjunction with other types of analysis, such as technical
analysis and qualitative analysis, to gain a comprehensive understanding of an
investment opportunity.

###  IV-C Technical Analysis

Technical analysis is a method of evaluating securities that involves studying
charts and other technical indicators to identify patterns and trends in a
security’s price and volume data. This type of analysis is based on the
premise that historical price and volume data can provide insights into future
market movements.

Technical analysts use a variety of tools and techniques to analyze market
data, including trend lines, moving averages, momentum indicators, and chart
patterns. They look for patterns in the data that suggest changes in market
sentiment or supply and demand dynamics, and use these patterns to make
investment decisions.

One of the key principles of technical analysis is that market trends tend to
persist over time. This means that once a trend is established, it is more
likely to continue than to reverse. Technical analysts use trend lines and
moving averages to identify trends and to determine whether they are likely to
continue or to reverse.

Another key principle of technical analysis is that market participants tend
to behave in predictable ways. Technical analysts use indicators such as
volume, open interest, and sentiment data to gauge the behavior of market
participants and to identify potential turning points in the market.

Technical analysis can be used to analyze a wide range of securities,
including stocks, bonds, commodities, and currencies. It is often used by
short-term traders and day traders who seek to profit from short-term
fluctuations in the market.

However, technical analysis has some limitations. It may not capture all of
the factors that can impact a security’s price, such as changes in market
conditions or news events. Additionally, technical analysis can be subject to
interpretation, and different analysts may arrive at different conclusions
based on the same data.

Overall, technical analysis can be a useful tool for investors who are looking
to profit from short-term market movements. However, it should be used in
conjunction with other types of analysis, such as fundamental analysis and
qualitative analysis, to gain a comprehensive understanding of an investment
opportunity.
Active management refers to the process of actively managing a portfolio of
securities in order to generate excess returns compared to a benchmark index
or other passive investment strategy. This approach typically involves a
combination of fundamental analysis, technical analysis, and other
quantitative and qualitative methods to identify undervalued or high-growth
securities.

One of the key benefits of active management is the potential to generate
excess returns relative to the market. By actively managing a portfolio, an
investor can take advantage of market inefficiencies and mispricings to
generate returns that exceed those of a passive investment strategy.

Additionally, active management can help to minimize risk by allowing an
investor to diversify across multiple asset classes and sectors. By carefully
selecting securities based on their individual risk profiles and their
correlation with other holdings in the portfolio, an active manager can
construct a portfolio that is less volatile and more resilient to market
downturns.

Another benefit of active management is the ability to adapt to changing
market conditions. Unlike a passive investment strategy, which simply tracks a
benchmark index, an active manager can adjust the portfolio holdings in
response to changes in the market environment. This flexibility can help to
mitigate downside risk and capture upside potential as market conditions
evolve.

However, active management also has some drawbacks. It typically involves
higher fees than passive investment strategies, which can erode returns over
time. Additionally, active managers may underperform the market in certain
market conditions or during periods of market volatility.

Overall, the importance of active management in generating excess returns
while minimizing risk depends on the specific investment goals and risk
tolerance of the investor. Active management can be a useful tool for
investors who are willing to accept higher fees in exchange for the potential
to outperform the market and to minimize risk through diversification and
flexibility. However, it may not be appropriate for all investors, and it
should be carefully considered in the context of an overall investment
strategy.

##  V Risk Management

Risk management is the process of identifying, assessing, and controlling
risks that may negatively impact an investment portfolio or business
operation. The goal of risk management is to minimize the likelihood and
impact of potential risks while maximizing the opportunity for returns.

Risk management involves several steps. The first step is to identify
potential risks that may affect the portfolio or business. This may include
risks related to market conditions, economic factors, regulatory changes,
geopolitical events, and other factors that may impact the performance of the
investments or business operations.

Once potential risks have been identified, the next step is to assess the
likelihood and impact of each risk. This involves analyzing the probability of
the risk occurring and the potential magnitude of its impact on the portfolio
or business.

Once the risks have been identified and assessed, risk management strategies
can be implemented to control and mitigate the risks. This may involve
diversifying the portfolio across different asset classes and sectors, hedging
against specific risks through the use of derivatives, and implementing risk
limits or stop-loss orders to limit potential losses.

Effective risk management also requires ongoing monitoring and evaluation of
the portfolio or business operations to ensure that risks are being
effectively managed and controlled. This may involve regular reviews of the
portfolio holdings, monitoring of market conditions and economic indicators,
and ongoing assessment of regulatory and geopolitical risks.

###  V-A Stop-loss Orders

Stop-loss orders are a type of risk management tool that investors can use to
limit potential losses on an investment. A stop-loss order is an order placed
with a broker or trading platform that automatically executes a trade to sell
a security if its price falls below a certain level, known as the stop-loss
price.

For example, if an investor purchases a stock at $50 per share and places a
stop-loss order at $45 per share, the order will automatically execute if the
stock price falls to $45 or below. This can help to limit potential losses by
ensuring that the investor exits the position before the price falls further.

Stop-loss orders can be a useful tool for managing risk, particularly in
volatile or uncertain market conditions. They can help to prevent emotional or
impulsive decision-making by automatically executing a trade when a
predetermined threshold is reached, rather than relying on the investor to
make a decision in the moment.

However, it is important to note that stop-loss orders are not foolproof and
can sometimes result in unexpected losses. For example, in a rapidly declining
market, the stop-loss order may execute at a lower price than the investor
intended, resulting in a larger loss than anticipated. Additionally, stop-loss
orders can be triggered by short-term market fluctuations, which may not
reflect the long-term value of the investment.

Overall, stop-loss orders can be a useful risk management tool when used
appropriately in conjunction with other risk management strategies. Investors
should carefully consider their investment goals and risk tolerance when
deciding whether to use stop-loss orders, and should also regularly monitor
and adjust their stop-loss orders as market conditions evolve.

###  V-B Options Strategies

Options strategies are investment strategies that involve the use of options
contracts to achieve specific investment objectives. Options are financial
derivatives that give the holder the right, but not the obligation, to buy or
sell an underlying asset at a specific price and time.

There are several common options strategies that investors can use to manage
risk and enhance returns. These include:

####  V-B1 Covered call strategy

This involves holding a long position in an asset and selling call options on
that asset. If the price of the asset rises, the holder of the call option can
exercise the option and buy the asset at the strike price, which limits the
potential gains of the holder of the asset. This strategy can be used to
generate additional income from a portfolio.

####  V-B2 Protective put strategy

This involves purchasing a put option on an asset to protect against potential
losses in the event that the price of the asset declines. This strategy can
help investors limit their downside risk while maintaining their upside
potential.

####  V-B3 Long call strategy

This involves purchasing a call option on an asset, giving the holder the
right to buy the asset at the strike price before the expiration date. This
strategy can be used to profit from a price increase in the underlying asset.

####  V-B4 Long put strategy

This involves purchasing a put option on an asset, giving the holder the right
to sell the asset at the strike price before the expiration date. This
strategy can be used to profit from a price decrease in the underlying asset.

####  V-B5 Bull call spread strategy

This involves buying a call option at a lower strike price and selling a call
option at a higher strike price. This strategy can be used to profit from a
price increase in the underlying asset, while limiting potential losses.

####  V-B6 Bear put spread strategy

This involves buying a put option at a higher strike price and selling a put
option at a lower strike price. This strategy can be used to profit from a
price decrease in the underlying asset, while limiting potential losses.

####  V-B7 Butterfly spread strategy

This involves buying a call option and a put option with the same strike price
and selling two options with a strike price between them. This strategy can be
used to profit from a range-bound market, while limiting potential losses.

####  V-B8 Iron condor strategy

This involves buying a call spread and a put spread with different strike
prices. This strategy can be used to profit from a range-bound market, while
limiting potential losses.

####  V-B9 Straddle strategy

This involves buying both a call option and a put option on an asset with the
same strike price and expiration date. This strategy can be used when an
investor expects a significant price move in either direction, as it allows
them to profit from a price increase or decrease.

####  V-B10 Strangle strategy

This involves buying both a call option and a put option on an asset with
different strike prices, but with the same expiration date. This strategy can
be used when an investor expects a significant price move in either direction,
but is unsure which way the price will move.
Overall, options strategies can be a useful tool for managing risk and
enhancing returns, but they can also be complex and involve significant risk.
Investors should carefully consider their investment goals, risk tolerance,
and knowledge of options before using options strategies in their portfolio.

The importance of risk management in minimizing losses cannot be overstated.
Risk management is the process of identifying, assessing, and controlling
risks that can affect an investment portfolio. By implementing effective risk
management strategies, investors can reduce the likelihood and impact of
potential losses.

There are many types of risks that can affect an investment portfolio,
including market risk, credit risk, liquidity risk, and operational risk.
Market risk is the risk of losses due to fluctuations in the overall market,
while credit risk is the risk of losses due to defaults or credit downgrades
of individual securities. Liquidity risk is the risk of losses due to an
inability to sell a security quickly and at a fair price, while operational
risk is the risk of losses due to internal or external operational failures.

To minimize losses due to these and other risks, investors can use a variety
of risk management strategies. Stop-loss orders, for example, can help limit
potential losses by automatically selling a security when its price falls
below a certain level. Options strategies, such as protective puts, can also
be used to limit losses while maintaining upside potential. Diversification
can also help to reduce the impact of individual security or sector-specific
risks on a portfolio.

By taking a proactive approach to risk management, investors can minimize the
impact of potential losses on their portfolios, which can help to protect
their long-term investment goals. It’s important to remember, however, that
risk cannot be eliminated entirely and that all investments carry some level
of risk. Therefore, it’s essential to carefully assess and manage risk to
ensure that it aligns with an investor’s individual investment objectives and
risk tolerance.

##  VI Conclusion

In this paper, we examined several strategies for portfolio management with
the goal of maximizing alpha and minimizing beta. These strategies included:

####  VI-1 Asset Allocation

The process of dividing an investment portfolio among different asset classes,
such as stocks, bonds, and cash, to achieve a specific investment objective.

####  VI-2 Strategic Asset Allocation

A long-term investment strategy that involves establishing a target mix of
assets and periodically rebalancing the portfolio to maintain that mix.

####  VI-3 Tactical Asset Allocation

A short-term investment strategy that involves adjusting the portfolio’s asset
allocation to take advantage of market opportunities or to manage risk.

####  VI-4 Diversification

The process of spreading investments across different securities or asset
classes to reduce the impact of individual security or sector-specific risks.

####  VI-5 Active Management

The process of actively managing a portfolio to generate excess returns while
minimizing risk.

####  VI-6 Fundamental Analysis

The process of evaluating securities based on their financial and economic
characteristics, such as revenue, earnings, and growth potential.

####  VI-7 Technical Analysis

The process of evaluating securities based on their price and trading patterns
to identify trends and potential price movements.

####  VI-8 Risk Management

The process of identifying, assessing, and controlling risks that can affect
an investment portfolio.
These strategies can be used in combination or individually to optimize
portfolio returns while minimizing risk.

Combining strategies for portfolio management is essential for optimizing
portfolio performance. Each strategy has its own strengths and weaknesses, and
by combining them, investors can benefit from the strengths of each while
minimizing their weaknesses.

For example, asset allocation provides a framework for diversifying a
portfolio among different asset classes, but it does not address the timing of
trades or the selection of individual securities. Tactical asset allocation,
on the other hand, can help to take advantage of short-term market
opportunities, but it does not provide a long-term investment plan. By
combining these strategies, investors can achieve a balanced approach to
portfolio management that considers both short-term and long-term investment
objectives.

Similarly, active management can help to generate excess returns while
minimizing risk, but it requires in-depth analysis and research of individual
securities. Fundamental analysis and technical analysis can provide valuable
insights into the potential performance of individual securities, but they may
not account for market trends or other external factors that can impact the
performance of a portfolio. By combining active management with fundamental
and technical analysis, investors can benefit from a more comprehensive
approach to security selection and portfolio management.

Finally, risk management is essential for protecting the long-term value of a
portfolio. Stop-loss orders and options strategies can help to limit losses
due to market fluctuations, but they may not account for broader market trends
or other macroeconomic factors that can impact a portfolio. By combining risk
management strategies with asset allocation and active management, investors
can benefit from a more holistic approach to portfolio management that
considers both individual securities and broader market trends.

In conclusion, combining strategies for portfolio management is essential for
optimizing portfolio performance. By taking a balanced approach that considers
both short-term and long-term investment objectives, investors can achieve
their investment goals while minimizing risk.

As an investor, the ultimate goal is to maximize returns while minimizing
risk. The strategies discussed in this paper, including asset allocation,
diversification, active management, and risk management, can help you achieve
this goal.

By utilizing these strategies, you can build a well-diversified portfolio that
takes advantage of market opportunities while limiting exposure to individual
security and sector-specific risks. You can also benefit from a more
comprehensive approach to security selection and risk management, which can
help you generate excess returns while minimizing losses.

Therefore, it is essential to take the time to understand these portfolio
management strategies and implement them into your investment approach.
Consulting with a financial advisor or portfolio manager can also be
beneficial in designing and implementing a portfolio management strategy that
is tailored to your individual investment goals and risk tolerance.

Investing can be a challenging and complex process, but by utilizing these
portfolio management strategies, you can make informed decisions and achieve
your investment goals over the long-term. So, don’t hesitate to take action
and start utilizing these strategies today to maximize alpha and minimize beta
in your investment portfolio.

## References

  * [1] Brinson, G. P., Hood, L. R., & Beebower, G. L. (1986). Determinants of portfolio performance. Financial Analysts Journal, 42(4), 39-44.
  * [2] Elton, E. J., Gruber, M. J., & Blake, C. R. (1996). The adequacy of investment performance measurement methods: a status report. Journal of Investing, 5(1), 7-19.
  * [3] Markowitz, H. (1952). Portfolio selection. The Journal of Finance, 7(1), 77-91.
  * [4] Malkiel, B. G. (2003). The efficient market hypothesis and its critics. Journal of Economic Perspectives, 17(1), 59-82.
  * [5] Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. The Journal of Finance, 19(3), 425-442.
  * [6] Bodie, Z., Kane, A., & Marcus, A. J. (2014). Investments. McGraw-Hill Education.
  * [7] Fabozzi, F. J. (2002). Handbook of portfolio management. John Wiley & Sons.
  * [8] Hull, J. C. (2018). Options, futures, and other derivatives. Pearson.
  * [9] Malkiel, B. G. (2016). A random walk down Wall Street: The time-tested strategy for successful investing. WW Norton & Company.
  * [10] Swensen, D. F. (2017). Pioneering portfolio management: An unconventional approach to institutional investment. Free Press.


does not. We show that this is always pathological, as it always generates a
curvature singularity.

######  Lemma 5.7.

Assume $(\Sigma^{2},g_{\Sigma})$ is a metric polygon with a point
$p\in\partial\Sigma^{2}$ that is not a vertex of the polygon, but is a
Lipschitz point of $\varphi^{1}$ or $\varphi^{2}$ as functions of $x$, $y$.
Then $g_{\Sigma}$ has a curvature singularity at $p$.

###### Proof.

We have that $p\in\partial\Sigma^{2}$ is a point along an edge so that, even
though there is no direction change, there is a speed change with speeds
$s_{0}$, $s_{1}$ on either side of $p$.

First we scale the metric and the polygon, and take a limit. Letting
$n\rightarrow\infty$, we scale the $(\varphi^{1},\varphi^{2})$-plane by a
factor of $n$ and scale $g_{\Sigma}$ by a factor of $n^{2}$. Taking a pointed
Hausdorff limit of the polygon $\Sigma^{2}$ and taking a pointed Gromov-
Hausdorff limit of $g_{\Sigma}$ we obtain a metric polygon that is a half-
plane. The boundary values do not change under this simultaneous scaling, so
the limiting polygon $(\Sigma^{2},g_{\Sigma})$ is a half-plane with a single
Lipschitz point.

Using the boundary-matching method of the proof of Lemma 5.1, set

(107) $\displaystyle\begin{aligned}
&\tilde{\varphi}{}^{1}\;=\;\frac{s_{0}}{2}\left(x-\sqrt{x^{2}+y^{2}}\right)+\frac{s_{1}}{2}\left(x+\sqrt{x^{2}+y^{2}}\right),\quad\tilde{\varphi}{}^{2}\;=\;\frac{1}{2}y^{2}\end{aligned}$

where $s_{0}\neq{}s_{1}$ are the labels. The boundary-matching method combined
with the Improved Liouville Theorem, as executed in Theorem 5.6, shows that
the functions $\varphi^{1}$, $\varphi^{2}$ are equal to
$\tilde{\varphi}{}^{1}$, $\tilde{\varphi}{}^{2}$ up to a summand of $Cy^{2}$,
$C>0$. After possible scaling the moment functions and translating the polygon
so $p=(0,0)$, we see that

(108) $\displaystyle\begin{aligned}
&\varphi^{1}\;=\;\frac{s_{0}}{2}\left(x-\sqrt{x^{2}+y^{2}}\right)+\frac{s_{1}}{2}\left(x+\sqrt{x^{2}+y^{2}}\right)+Cxy^{2},\\\
&\varphi^{2}\;=\;\frac{1}{2}y^{2}.\end{aligned}$

Computing $K_{\Sigma}$ using (40), we obtain

(109) $\displaystyle\begin{aligned}
K_{\Sigma}&\;=\;\frac{s_{1}-s_{0}}{2}\cdot\frac{s_{1}(x-\sqrt{x^{2}+y^{2}})+s_{0}(x+\sqrt{x^{2}+y^{2}})}{\left(s_{1}(x-\sqrt{x^{2}+y^{2}})+s_{0}(x+\sqrt{x^{2}+y^{2}})\right)^{3}}.\end{aligned}$

This expression has an infinite singularity at the origin. Because this
curvature singularity exists after the blow-up process, by continuity it
certainly existed before—recall the blowup process systematically decreases
curvature. This concludes the proof that $g_{\Sigma}$ has a curvature
singularity at $p$. ∎

##  6\. Consideration of the asymptotic conditions

Our classification requires only that the polygon be topologically closed. We
show this is true under either of the asymptotic conditions

  * A1)

The action fields decay slowly (or not at all):
$\sqrt{|\mathcal{X}^{1}|^{2}+|\mathcal{X}^{2}|^{2}}\;>\;C_{1}r^{-1+\epsilon}$
for some $\epsilon>0$ and $C_{1}>\infty$, when $r$ is sufficiently large.

  * A2)

Curvature decays quickly: $|\mathop{\rm Rm}|<(2-\epsilon)r^{-2}$, for any
$\epsilon>0$, when $r$ is sufficiently large.

The boundary is connected (by condition (B)), and must include some point of
closure, or else by Theorem 1.1 the manifold is flat. From the literature,
specifically [8] or [20], we know the local structure near points of closure
on $\Sigma^{2}$: such a point is locally part of a of a vertex or part of a
segment. Then we take a limit along a segment as we approach a non-included
point. Because the map $\Phi:M^{4}\rightarrow\Sigma^{2}$ is smooth, this can
only occur if we approach infinity in $M^{4}$ but remain finite in the
coordinate plane. However the value of $\varphi^{i}$ can be determined by its
gradient $|\nabla\varphi^{i}|=|\mathcal{X}^{i}|$, so $\varphi^{i}$ can only
remain finite if the norm $|\mathcal{X}^{i}|$ gets too small too fast.

###  6.1. The asyptotic conditions (A1) and (A2)

######  Proposition 6.1.

Assume $(M^{4},J,g,\mathcal{X}^{1},\mathcal{X}^{2})$ is a ZSC toric Kähler
manifold of finite topology, and assume all manifold ends satisfy (A1) or
(A2). Then its reduction $(\Sigma^{2},g_{\Sigma})$ is topologically closed.

###### Proof.

We show that $\Sigma^{2}$ includes its boundary. Conditions (A1) and (A2)
involve the distance $r$. It is ordinarily very difficult to find geodesics
without further information about the metric. But by (D) all edges of
$\Sigma^{2}$ are geodesics. The idea is to show that, under (A1) or (A2), if a
boundary segment has infinite metric-length, it must have infinite coordinate-
length, too. Thus the boundary cannot have any excluded points.

First we rule out the case of no edges. But then
$span\\{\mathcal{X}^{1},\,\mathcal{X}^{2}\\}$ is everywhere rank 2 so the
moment map $\Phi:M^{4}\rightarrow\Sigma^{2}$ is a Riemannian submersion. Then
$(\Sigma^{2},g_{\Sigma})$ is a complete Riemannian manifold, so by Corollary
1.2 it is flat.

Therefore $span\\{\mathcal{X}^{1},\,\mathcal{X}^{2}\\}$ has rank $1$ or $0$
somewhere, meaning the volume function
$\mathcal{V}=|\mathcal{X}^{1}|^{2}|\mathcal{X}^{2}|^{2}-\left<\mathcal{X}^{1},\mathcal{X}^{2}\right>{}^{2}$
has zeros. We first establish the local structure of any such zero. The local
structure can be found, for instance, in [8] or in the book [20]. To wit, at a
point $p\in{}M^{4}$ if the rank of the distribution lowers to $1$, then a
neighborhood of $p$ maps to a half-disk in the
$(\varphi^{1},\varphi^{2})$-plane with the image of $p$ being on the closed
half-disk boundary. If the rank reduces to $0$, a neighborhood of $p$ reduces
to a wedge with the image of $p$ being at the vertex. Therefore at least some
edges exist. Using (A1) or (A2) we prove that any such edge is closed.

Pick a line segment $l\subset\Sigma^{2}$; after an affine transformation we
may assume $l$ is oriented along the $\varphi^{1}$-axis. Over such a line
segment is a 2-dimensional submanifold $\Phi^{-1}(l)=L^{2}\in{}M^{4}$ on which
the Killing field $\mathcal{X}^{1}$ is not identically zero. The momentum
function $\varphi^{1}$ satisfies $\nabla\varphi^{1}=J\mathcal{X}^{1}$. The
manifold $L^{2}\subset{}M^{4}$ is a totally geodesic, holomorphic submanifold
(as it is a zero-set of the holomorphic killing field $\mathcal{X}^{2}$). See
Figure 15.

Figure 15. A non-closed segment $l$ and its overlying 2-manifold
$L^{2}=\Phi^{-1}(l)$. An open point is at metric infinity, but finite
$\varphi^{1}$.

If the segment $l\subset\Sigma^{2}$ is not closed and if a point $p$ moves
along $l$ toward a point of closure, then by continuity, all points in the
pre-image $\Phi^{-1}(p)$ must have no cluster points, meaning the set
$\Phi^{-1}(p)$ must travel infinitely far away. Because $l$ and $L^{2}$ are
totally geodesic, $l$ and $L^{2}$ must both be unbounded in the metric sense.

Either of the hypotheses (A1) or (A2) can show this to be impossible. If (A1)
is true, then $|\mathcal{X}^{1}|>\min\\{C_{0},C_{1}r^{-1+\epsilon}\\}$.
Because $L^{2}$ is holomorphic and $\mathcal{X}^{1}$ is a Killing field on
$L^{2}$, also $\nabla\varphi^{1}=J\mathcal{X}^{1}$ is tangent to $L^{2}$. Thus
$\nabla\varphi^{1}=\frac{\partial\varphi^{1}}{\partial{}r}dr$ where $r$ is any
rotationally-invariant distance function along $L^{2}$. Then using
$|\nabla\varphi^{1}|=|\mathcal{X}^{1}|>C_{1}r^{-1+\epsilon}$ we have

(110) $\displaystyle\begin{aligned}
&\frac{\partial\varphi^{1}}{\partial{}r}\;=\;\nabla\varphi^{1}\;\geq\;C_{1}r^{-1+\epsilon},\quad\text{so}\quad\varphi^{1}(r)\;=\;O(r^{\epsilon})\end{aligned}$

Since $\epsilon>0$ we have $\varphi^{1}\rightarrow\infty$ as
$r\rightarrow\infty$, the sought-for contradiction.

Next assume (A2) holds. Again $\mathcal{X}^{1}$ is a Killing field on $L^{2}$,
so in particular it obeys the Jacobi equation

(111)
$\nabla_{\frac{\partial}{\partial{}r}}\nabla_{\frac{\partial}{\partial{}r}}\mathcal{X}^{1}+\mathop{\rm
Rm}\left(\mathcal{X}^{1},\frac{\partial}{\partial{}r}\right)\frac{\partial}{\partial{}r}\;=\;0.$

Because $|\mathcal{X}^{1}|>0$ (or else there would reach a vertex and the
segment would be closed) and because $|\mathop{\rm Rm}|<(2-\epsilon)r^{-2}$,
the usual Rauch comparison says that $|\mathcal{X}^{1}|>f$ where $f$ is a non-
negative solution of the comparison Riccati equation
$f_{rr}-(2-\epsilon)r^{-2}f=0$. This gives lower boundary
$f=C_{0}r^{\frac{1-\sqrt{9-4\epsilon}}{2}}$ for $|\mathcal{X}^{1}|$. Then
because $|\nabla\varphi^{1}|=|\partial\varphi^{1}/\partial{}r|$ on $L^{2}$ (as
$\varphi^{1}$ only changes in the radial direction, not in the direction of
the Killing field), we have

(112)
$\left|\frac{\partial\varphi^{1}}{\partial{}r}\right|\;=\;|\mathcal{X}^{1}|\;\geq\;C_{0}r^{\frac{1-\sqrt{9-4\epsilon}}{2}}.$

Integrating, we have $\varphi^{1}=O(r^{\frac{3-\sqrt{9-4\epsilon}}{2}})$. The
exponent $\frac{3-\sqrt{9-4\epsilon}}{2}$ is positive, so again
$\varphi^{1}\rightarrow\infty$ as $r\rightarrow\infty$, giving the desired
contradiction. ∎

######  Proposition 6.2.

Assume $(M^{4},J,g,\mathcal{X}^{1},\mathcal{X}^{2})$ is a ZSC toric Kähler
manifold, and that all ends are asymptotically spheroidal. Then its metric
reduction $(\Sigma^{2},g_{\Sigma})$ is topologically closed. Its polygon
$\Sigma^{2}$ does not have parallel rays, and is not the half-plane.

###### Proof.

$\Sigma^{2}$ is topologically closed by Proposition 6.1. What remain is to
understand the moment diagram at infinity. But the level-sets of the distance
function are $\mathbb{S}^{3}$, and these level-sets inherit the toric
structure. But any isometric toric structure on $\mathbb{S}^{3}$ has two
singular orbits. These singular orbits are not the zero-set of any one single
vector field, and therefore the rays of $\partial\Sigma^{2}$ they represent
are not parallel. (This asymptotic structure is in the first image of in
Figure 16). ∎

Figure 16.  On the left is the “asymptotically spherical” structure, which is
asymptotically a wedge. On the right is the “asymptotically toroidal”
structure, which has no edges and asymptotically fills the plane.

######  Proposition 6.3.

Assume $(M^{4},J,g,\mathcal{X}^{1},\mathcal{X}^{2})$ is a toric Kähler
manifold with non-negative scalar curvature, and at least one end is
asymptotically toroidal. Then its metric reduction $(\Sigma^{2},g_{\Sigma})$
is both topologically closed and geodesically complete. Consequently,
$(M^{4},g)$ is a flat manifold.

###### Proof.

$\Sigma^{2}$ is topologically closed by Proposition 6.1. What remains is to
understand the moment diagram at infinity.

But the level-sets of the distance function have a free action of $T^{2}$, so
far enough away, the polygon $\Sigma^{2}$ has no edges. Because the polygon is
convex and closed, it has no edges at all, and is therefore complete.
Therefore the metric is flat by Theorem 3.4). ∎

######  Corollary 6.4 (cf. Corollary 1.9).

Assume $(M^{4},J,g,\mathcal{X}^{1},\mathcal{X}^{2})$ is a scalar-flat toric
Kähler manifold of finite topology that satisfies asymptotic condition (A1) or
(A2). Let $(\Sigma^{2},g_{\Sigma})$ be its reduction.

Then $(\Sigma^{2},g_{\Sigma})$ is either an infinite closed strip or else
satisfies conditions (A)-(F), and there are five possibilities:

  * i)

$(M^{4},g)$ is flat,

  * ii)

$\Sigma^{2}$ is an infinite closed strip in the $\varphi^{1}$-$\varphi^{2}$
plane,

  * iii)

$(M^{4},g)$ is the exceptional half-plane instanton,

  * iv)

$\Sigma^{2}$ has parallel rays, and for given boundary values the metric
belongs to a 1-parameter family of possibilities,

  * v)

or $M^{4}$ is asymptotically spheroidal (never toroidal), and is

    * a)

Asymptotically locally Euclidean—and for any set of labels there is precisely
one such metric—or

    * b)

Asymptotically spheriodal and asymptotically equivalent to a Taub-NUT, chiral
Taub-NUT, or exceptional Taub-NUT; after labels are determined, the metric
belongs to a 2-parameter family of possibilities.

###### Proof.

We verify that $\Sigma^{2}$ obeys conditions (A)-(F). By Proposition 6.1 the
polygon is closed; this verifies condition (A). For condition (B), by
convexity, if the polygon boundary were disconnected the polygon could only be
the strip, so condition (B) holds. Conditions (C), (D), and (E) follow
directly from the fact that $(\Sigma^{2},g_{\Sigma})$ is the reduction of a
Kähler manifold. Condition (C) is the fact that
$\Phi:M^{4}\rightarrow\Sigma^{2}$ is generically a Riemannian submersion.
Condition (D), the assertion that $\nabla\varphi^{i}/|\nabla\varphi^{i}|$ is
covariant constant along polygon edges, is simply the fact that the pre-image
of any edge in $(M^{4},g_{\Sigma})$ is totally geodesic, as it is the zero-set
of a Killing field. Condition (E) is the fact that
$[\nabla\varphi^{1},\nabla\varphi^{2}]=0$ on $M^{4}$ and therefore on
$\Sigma^{2}$. Condition (F), the pseudo-ZSC condition, follows from
$(M^{4},g)$ being ZSC.

The fact that the boundary numbers $s_{0},\dots,s_{d}$ on $\Sigma^{2}$ are all
uniquely determined follows either the definition in §1.3 or at the beginning
of §5.

Since $\Sigma^{2}$ is topologically closed and has finitely many facets
(because $M^{4}$ has finite topology), there are four possibilities. First
$\Sigma^{2}$ has no edges, so $(\Sigma^{2},g_{\Sigma})$ is flat. Second
$\Sigma^{2}$ is the closed half-plane, so $(\Sigma^{2},g_{\Sigma})$ is a half-
plane instanton, as shown in Section 5.5.

Third $\Sigma^{2}$ might have parallel rays. We may assume the rays are
parallel to the $\varphi^{1}$-axis. Then the level-sets $\varphi^{1}=Const$
for large $Const$ have an action of $\mathcal{X}^{1}$ with no zeros so this
action splits off a circle factor, and an action of $\mathcal{X}^{2}$ with
disconnected zero locus so the other factor is $\mathbb{S}^{2}$. The
asymptotic structure is therefore
$\mathbb{R}\times\mathbb{S}\times\mathbb{S}^{2}$.

Finally $\Sigma^{2}$ may be the general case. In this case $\Sigma^{2}$ is
asymptotically a wedge, as shown in the first image in Figure 16. Consider
level-sets of a buseman $r$; because the rays are not parallel there is no
single Killing field that has zeros along both rays. The structure of the
level-sets, therefore, is a solid torus times a segment, with torus cycles
identified to a point over top of the two segment endpoints. Thus each level
set of $r$ (for $r$ sufficiently large) is a union of two solid tori, so is a
lens space. This is the asymptotically spheroidal case.

The assertion that $(M^{4},J,g)$ belongs to either a 1- or 2-parameter family
of possibilities, depending on $\Sigma^{2}$, now follows from Theorems 1.6,
1.7, and 1.8.

Finally we check the asymptotic possibilities for case (v). In the quarter-
plane, $\varphi^{1}$ and $\varphi^{2}$ proceeding to $\infty$ is the same as
$x^{2}+y^{2}$ proceeding to $\infty$. Referring to Lemma 5.1 and adding on the
possible multiples of $y^{2}$, the momentum functions asymptotically approach

(113)
$\displaystyle\varphi^{1}\approx-\frac{s_{0}}{2}\left(-x+\sqrt{x^{2}+y^{2}}\right)\,+\,C_{1}y^{2}$
$\displaystyle\varphi^{2}\approx\frac{s_{d}}{2}\left(x+\sqrt{x^{2}+y^{2}}\right)\,+\,C_{2}y^{2}$

This is because all of the intermediate interpolation functions proceed to a
constant so cease to affect the metric, and only the first and last summands
continue to make any difference when $x^{2}+y^{2}$ are large. But, up to
constant multiples, these are the momentum equations from (18), the Taub-NUT,
Generalized Taub-NUTs, and Exceptional Taub-NUT metrics. Asymptotically,
therefore, the metrics in case (v) approach one of these. ∎

###  6.2. The Insufficiency of the ALE-ALF-ALG-ALH schema

The advantage of the concepts “asymptotically spherical” and “asymptotically
toroidal” over the traditional ALE-F-G-H classification is that the
traditional classification requires manifold ends to be modeled on fiber
bundles over standard base manifolds, and these models are much too inflexible
to represent the natural diversity of typical manifold ends. For example, the
chiral Taub-NUT metrics (discovered by Donaldson) have quadratically decaying
curvature and 3-dimensional blowdowns. But quotienting out by the collapsing
field at infinity yields (at best!) a cone over an orbifold version of
$\mathbb{S}^{2}$, whereas the ALF model requires the quotient to be a
manifold. Therefore, no matter how closely a chiral Taub-NUT may resemble a
standard Taub-NUT in the larger 4-manifold setting, it will never be close to
a model end as prescribed in the ALF model.

To describe the Cherkis-Kapustin framework briefly, a model end is a product
$[R,\infty)\times{}N^{3}$ where $N^{3}$ is the total space of a fiber bundle
$\pi:N^{3}\rightarrow{}B$ with fiber $F$, and which has metric

(114) $\displaystyle\begin{aligned}
g_{Model}\;=\;dr^{2}\,+\,r^{2}g_{B}\,+\,g_{F}\end{aligned}$

A manifold end fits into the schema (see [6], [11]) if it “close” in an
appropriate asymptotic sense to a finite quotient of one of the following
models:

  * i)

ALE if $N^{3}=B^{3}=\mathbb{S}^{3}$, the fiber is a point, and $g_{B}$ is the
standard round metric.

  * ii)

ALF if $N^{3}=\mathbb{S}^{3}$ and $\pi:N^{3}\rightarrow{}B^{2}$ is the Hopf
fibration, with fiber $F^{1}=\mathbb{S}^{1}$. The metric $g_{B}$ is the round
metric on the 2-sphere of radius $1/2$ and the fiber metric gives $S^{1}$ a
radius of 1.

  * iii)

ALG if $N^{3}$ is a torus bundle over a circle (in other words, it is a
solvemanifold) and $\pi:N^{3}\rightarrow{}B^{1}$ is the projection onto its
base circle. The metric $g_{B}$ gives a circle and $g_{F}$ gives a flat
2-torus.

  * iv)

ALH if $N^{3}$ is the 3-torus and $\pi:N^{3}\rightarrow{}B^{0}$ is the
projection onto a point; the metric $g_{F}$ gives a flat 3-torus.

The paper [25] explains, in exacting detail, the asymptotics of the chiral
Taub-NUT metrics. The asymptotics are ALF-like in many respects, but
definitely cannot be modeled in the form of (114). The basic reason is that
the level-sets $N^{3}$ might not have a collpasing field at infinity that can
possibly be modeled on a fiber-bundle over a manifold, or even a fibration
over a manifold. The base of the projection $\pi:N^{3}\rightarrow{}B$ is often
an orbifold, and a “bad” orbifold at that. Worse, the fiber $F$ might have
2-dimensional closure (eg. an irrational subgroup of the torus), so $B$ does
not have a manifold structure of any kind. One could quotient by the closure
of the fibers, but then the base would be a closed line segment which is also
not part of the model.

That these situations actually happen very naturally gives a concrete
illustration of the basic insufficiency of the ALF model, and the need for the
terms “asymptotically spherical” (if not “asymptocially toroidal,” for which
there are no examples in the scalar-flat Kähler setting as we have proved in
Corollary 3.5 and Proposition 6.3). Some very naturally occurring metrics with
close resemblance to the classic Taub-NUT simply cannot be placed into a form
that fits the ALF model.

##  7\. Examples

###  7.1. Non-pathological examples

####  7.1.1. The LeBrun metrics on $\mathcal{O}(-k)$

Let $e_{1},e_{2},e_{3}$ be the left-invariant frames and
$\sigma^{1},\sigma^{2},\sigma^{3}$ the corresponding left-invariant coframes
on $\mathbb{S}^{3}$, normalized in the usual way so
$d\sigma^{i}=-\epsilon^{i}{}_{jk}\sigma^{j}\wedge\sigma^{k}$. From [16] a
scalar-flat metric on $\mathcal{O}(-k)$ is

$g=\frac{1}{\left(1-\frac{1}{r^{2}}\right)\left(1+\frac{k-1}{r^{2}}\right)}dr^{2}+r^{2}\left(1-\frac{1}{r^{2}}\right)\left(1+\frac{k-1}{r^{2}}\right)(\sigma^{1})^{2}+r^{2}\Big{(}(\sigma^{2})^{2}+(\sigma^{3})^{2}\Big{)}$

where $r\in[1,\infty)$. The Kähler form is

(115) $\omega\;=\;-rdr\wedge\sigma^{1}+r^{2}\sigma^{2}\wedge\sigma^{3}.$

From $d(\sigma^{2}\wedge\sigma^{3})=0$ and
$d\sigma^{1}=-2\sigma^{2}\wedge\sigma^{3}$, we easily verify $d\omega=0$. To
find commuting Killing fields we must recall the Euler coordinates on
$\mathbb{S}^{3}$ and corresponding polar coordinates on
$\mathbb{R}^{4}\approx\mathbb{C}^{2}$, given by

(116)
$(r,\psi,\theta,\varphi)\;\longmapsto\;\left(r\cos(\theta/2)e^{-\frac{i}{2}(\psi+\varphi)},\;r\sin(\theta/2)e^{-\frac{i}{2}(\psi-\varphi)}\right)$

with ranges $r\in[0,\infty)$, $\psi,\varphi\in[0,4\pi)$, and
$\theta\in[0,\pi]$. We choose toric structure

(117)
$\mathcal{X}^{1}\;=\;2\frac{\partial}{\partial\psi},\quad\mathcal{X}^{2}\;=\;2\frac{\partial}{\partial\varphi}.$

In terms of coordinates, the left-invariant 1-forms are

(118)
$\displaystyle\sigma^{1}=\frac{1}{2}(d\psi+\cos(\theta)d\varphi),\quad\sigma^{2}=\frac{1}{2}(\sin(\psi)d\theta-\cos(\psi)\sin(\theta)d\varphi),$
$\displaystyle\sigma^{3}=\frac{1}{2}(\cos(\psi)d\theta+\sin(\psi)\sin(\theta)d\varphi).$

With these expressions we can plug the fields (117) in to (115) to get

(119) $\displaystyle
d\varphi^{1}\;=\;\omega\left(2\frac{\partial}{\partial\psi},\;\cdot\,\right)\;=\;rdr\;=\;d\left(\frac{1}{2}r^{2}\right)$
$\displaystyle
d\varphi^{2}\;=\;\omega\left(2\frac{\partial}{\partial\varphi},\;\cdot\,\right)\;=\;r\cos(\theta)\,dr-\frac{1}{2}r^{2}\sin(\theta)\,d\theta\;=\;d\left(\frac{1}{2}r^{2}\cos\theta\right)$

therefore our symplectic coordinates are
$(\varphi^{1},\varphi^{2},\theta_{1},\theta_{2})$ where
$\varphi^{1}=\frac{1}{2}r^{2}$, $\varphi^{2}=\frac{1}{2}r^{2}\cos\theta$,
$\theta_{1}=\frac{1}{2}\psi$, and $\theta_{2}=\frac{1}{2}\varphi$. The
coordinate ranges are $\varphi^{1}\geq 1/2$,
$-\varphi^{1}\leq\varphi^{2}\leq\varphi^{1}$, and
$\theta_{1},\theta_{2}\in[0,2\pi)$; see Figure 17. The polygon metric is then

(120)
$(g_{\Sigma}^{ij})\;=\;\left(\begin{array}[]{cc}\frac{(2\varphi^{1}-1)(2\varphi^{1}+(k-1))}{2\varphi^{1}}&\frac{(2\varphi^{1}-1)(2\varphi^{1}+(k-1))\varphi^{2}}{2(\varphi^{1})^{2}}\\\
\frac{(2\varphi^{1}-1)(2\varphi^{1}+(k-1))\varphi^{2}}{2(\varphi^{1})^{2}}&2\varphi^{1}+\frac{(2\varphi^{1}(k-2)-(k-1))(\varphi^{2})^{2}}{2(\varphi^{1})^{3}}\end{array}\right).$

The volumetric normal coordinates, using $y=\sqrt{\det{}g_{\Sigma^{ij}}}$ and
(31) for $J_{\Sigma}$, are

(121)
$y\;=\;\frac{1}{\varphi^{1}}\sqrt{(2\varphi^{1}-1)(2\varphi^{1}+k-1)((\varphi^{1})^{2}-(\varphi^{2})^{2})},\quad
x\;=\;\frac{1}{2}(k-2)\frac{\varphi^{2}}{\varphi^{1}}+2\varphi^{2}.$

Inverting this gives the $\varphi^{i}$ as functions of $x$ and $y$:

(122) $\displaystyle\varphi^{1}$
$\displaystyle\;=\;\frac{1}{2}\left(1\;-\;\frac{k}{2}\;+\;\frac{1}{2}\left(\sqrt{\left(x-\frac{k}{2}\right)^{2}+y^{2}}+\sqrt{\left(x+\frac{k}{2}\right)^{2}+y^{2}}\right)\right),$
$\displaystyle\varphi^{2}$
$\displaystyle\;=\;\frac{1}{2}\left(x+\left(\frac{1}{2}-\frac{1}{k}\right)\left(\sqrt{\left(x-\frac{k}{2}\right)^{2}+y^{2}}-\sqrt{\left(x+\frac{k}{2}\right)^{2}+y^{2}}\right)\right).$

One may check directly that
$y(\varphi^{i}_{xx}+\varphi^{i}_{yy})-\varphi^{i}_{y}=0$. The “oultine map” is
the restriction of $(\varphi^{1},\varphi^{2})$ to $\\{y=0\\}$. We compute

$(\varphi^{1},\varphi^{2})\Big{|}_{y=0}=\begin{cases}\left(\frac{1}{4}(2-k-2x),\;\frac{1}{4}(-2+k+2x)\right),\quad&x<-\frac{k}{2}\\\
\left(\frac{1}{2},\;\frac{1}{k}x\right),\quad&-\frac{k}{2}\leq{}x\leq\frac{k}{2}\\\
\left(\frac{1}{4}(2-k+2x),\;\frac{1}{4}(2-k+2x)\right),\quad&x>\frac{k}{2}\end{cases}$

Figure 17.  Polygons and computed labels for the LeBrun metrics on
$\mathcal{O}(-k)$, before and after an affine change of variables.

Finally we compute the “parameterization speeds” or labels. Using (73),

(123) $\displaystyle
s\;=\;\sqrt{(\varphi^{1}_{x})^{2}+(\varphi^{2}_{x})^{2}}\Big{|}_{y=0}\;=\;\begin{cases}\frac{1}{\sqrt{2}}\quad&x<-\frac{k}{2}\\\
\frac{1}{k}&-\frac{k}{2}<x<\frac{k}{2}\\\
\frac{1}{\sqrt{2}}\quad&x>\frac{k}{2}\end{cases}$

Finally note that this polygon is not in the right position for the outline
matching of Section 5.1, in the sense that its terminal rays are not parallel
to the first quadrant axes. This is rectified with the affine transformation

(124) $\left(\begin{array}[]{c}\varphi^{1}\\\
\varphi^{2}\end{array}\right)\;\longmapsto\;\left(\begin{array}[]{cc}k&-k\\\
k&k\end{array}\right)\left(\begin{array}[]{c}\varphi^{1}\\\
\varphi^{2}\end{array}\right).$

Lastly we see how to use the polygon markings to reconstruct the manifold. We
can use either polygon of Figure 17. We choose the polygon on the right.
Choose a torus action with $\mathcal{X}_{1}$, $\mathcal{X}_{2}$ being standard
generators with coordinate ranges $\theta_{1},\theta_{2}\in[0,2\pi{}k)$. The
diagonal action, with its zero-set being the diagonal segment, has range
$[0,2\sqrt{2}\pi{}k)$. Referring to the discussion in Section 1.3, the cone
angle along each ray is the range divided by the label, so we see cone angle
$2\pi$ along the two rays, but cone angle $2\pi{}k$ along the diagonal
segment. This is rectified by taking a $k$-to-1 torus quotient along the
diagonal. Each of the generators still has range $[0,2\pi{}k)$ so the cone
angles along the rays remains $2\pi$, but now the diagonal segment’s cone
angle is also $2\pi$, so we have a manifold.

####  7.1.2. Polygons for $\mathbb{S}^{2}\times\mathbb{H}^{2}$

Up to isometry there is essentially just one Killing field on $\mathbb{S}^{2}$
(the rotational field), but on the hyperbolic plane $\mathbb{H}^{2}$ there are
three: the elliptic field (which fixes a point), the parabolic field (which
fixes a point on the ideal circle), and the hyperbolic field (which fixes two
points on the ideal circle). The parabolic and hyperbolic fields have no
zeros, so if we think of these fields as coming from actions of
$\mathbb{R}^{1}$ on $\mathbb{H}^{2}$ then we can mod-out by integer
translations to obtain $\mathbb{S}^{1}$ action on $\mathbb{H}^{2}/\mathbb{Z}$;

Given the elliptic field on $\mathbb{H}^{2}$, the moment functions for the
product metric are

(125) $\displaystyle\begin{aligned}
&\varphi^{1}=\frac{1}{2}\left(-1+\sqrt{x^{2}+y^{2}}+\sqrt{(x-1)^{2}+y^{2}}\right)\\\
&\varphi^{2}=\frac{1}{2}\left(1-\sqrt{x^{2}+y^{2}}+\sqrt{(x-1)^{2}+y^{2}}\right).\end{aligned}$

Given the parabolic field, the moment functions for the product metric are

(126) $\displaystyle\begin{aligned}
&\varphi^{1}=\frac{1}{2}\left(\sqrt{x^{2}+y^{2}}\right),\quad\varphi^{2}=\frac{1}{2}\left(1+\frac{x}{\sqrt{x^{2}+y^{2}}}\right).\end{aligned}$

The hyperbolic field, on the other hand, produces a unique situation. Its
polygon is the infinite strip and, notably, its volumetric function
$z:\Sigma^{2}\rightarrow\overline{}H{}^{2}$ is ramified, having a 2-to-1
branch point. On any subdomain of $\Sigma^{2}$ on which $z$ is injective we
can still perform the constructions of this paper, and obtain moment functions

(127) $\displaystyle\begin{aligned}
&\varphi^{1}\;=\;\frac{1}{\sqrt{2}}\sqrt{1-\left(x^{2}+y^{2}\right)^{2}+\sqrt{\left(1-\left(x^{2}+y^{2}\right)^{2}\right)^{2}\,+\,4y^{2}}}\\\
&\varphi^{2}\;=\;\frac{1}{\sqrt{2}}\sqrt{-1+\left(x^{2}+y^{2}\right)^{2}+\sqrt{\left(1-\left(x^{2}+y^{2}\right)^{2}\right)^{2}\,+\,4y^{2}}}.\end{aligned}$

These moment functions are singular. The map
$(\varphi^{1},\varphi^{2}):\overline{H}{}^{2}\rightarrow\Sigma^{2}$, with
these moment functions, brings $\overline{H}{}^{2}$ to precisely one-half of
the polygon.

Figure 18. Three polygons that produce scalar-flat metrics on
$\mathbb{S}^{2}\times\mathbb{H}^{2}$, along with a depiction of the three
corresponding rotational structures on $\mathbb{H}^{2}$: elliptic, parabolic,
and hyperbolic.

This construction gives rise to orbifold and conifold metrics as well, which
we obtain by adjusting the speeds. Setting $s_{0},s_{2}>0$ and using

(128) $\displaystyle\begin{aligned}
&\varphi^{1}=\frac{s_{0}}{2}\left(-x+\sqrt{x^{2}+y^{2}}\right)+\frac{s_{2}}{2}\left(x-1+\sqrt{(x-1)^{2}+y^{2}}\right)\\\
&\varphi^{2}=\frac{1}{2}\left(1+\frac{x}{\sqrt{x^{2}+y^{2}}}\right).\end{aligned}$

the polygon is the semi-open half-strip, depicted in the second image of
Figure 18 or in Figure 19, but with parameterization speeds $s_{0}$, and
$s_{2}$. When $s_{0}=s_{2}$ is rational, the construction gives a “good
orbifold” on $\mathbb{S}^{2}$; when $s_{0}\neq{}s_{2}$ the construction can
give a “bad orbifold.” When $s_{0}$ or $s_{2}$ are irrational, the
construction gives a conifold.

Figure 19.  Illustration of how polygon labels indicate topology. A cross
product of smooth $\mathbb{S}^{2}$ with $\mathbb{H}^{2}$, and a cross product
of a bad orbifold with $\mathbb{H}^{2}$. The latter is a topological product,
but, in the scalar-flat case, is never a metric product.

####  7.1.3. A two-ended instanton

Consider the moment functions

(129) $\displaystyle\begin{aligned}
\varphi^{1}&\;=\;\frac{1}{2}\left(x+\sqrt{x^{2}+y^{2}}\right)+\frac{1}{2}\left(-1+\frac{x}{\sqrt{x^{2}+y^{2}}}\right)+\frac{M(1-k)}{2}y^{2}\\\
\varphi^{2}&\;=\;\frac{1}{2}\left(-x+\sqrt{x^{2}+y^{2}}\right)+\frac{1}{2}\left(1-\frac{x}{\sqrt{x^{2}+y^{2}}}\right)+\frac{M(1+k)}{2}y^{2}\end{aligned}$

where $M\geq 0$ and $k\in[-1,1]$ are constants. Each term
$x(x^{2}+y^{2})^{-\frac{1}{2}}$ creates a “jump” singularity for the moment
functions along the line $\\{y=0\\}$, and produces a non-closed segment on the
polygon boundary, which should be considered a segment of infinite
parameterization speed. Figure 20 depicts these functions and the semi-open
polygon they generate.

Figure 20. Graphs of $\varphi^{1}$ and $\varphi^{2}$ over the upper half-
plane, and the image of the map $(x,y)\mapsto(\varphi^{1},\varphi^{2})$.

A full exploration of this two-ended manifold
$(M^{4},g,J,\mathcal{X}^{1},\mathcal{X}^{2})$ would require a lot of space, so
we state the overall properties without recording the justifications.

If $M=0$, one end is asymptotically Euclidean (AE) and the other end is
cuspidal, whose structure is similar to the cusp-like middle image in Figure
18. This manifold is scalar-flat and has bounded sectional curvature. It is
not Ricci-flat for any values of $M\geq 0$, $k\in[-1,1]$. It has finite
energy; in the case $M=0$ we have $\int|\mathop{\rm Rm}|^{2}dVol=96\pi^{2}$.

If $M\neq 0$ then this manifold is still two-ended and complete. Sectional
curvature remains bounded. The cuspidal end remains cuspidal, but when
$k\in(-1,1)$ the AE end becomes spheroidal with cubic volume growth—it is
technically not ALF unless $k=0$. The energy $\int|\mathop{\rm Rm}|^{2}$
remains finite. In the $k=-1,1$ case, the end is no longer ALF-like. It has
quartic volume growth, and its asymptotic structure is that of the exceptional
Taub-NUT [25].

####  7.1.4. A reduced manifold $(\Sigma^{2},g_{\Sigma})$ which is not a
polygon

Consider the function

(130)
$\displaystyle{\bf{u}}(\varphi^{1},\varphi^{2})\;=\;-\frac{1}{2}\log\left(1-\left(\varphi^{1}\right)^{2}-\left(\varphi^{2}\right)^{2}\right)$

defined on the open disk
$\Sigma^{2}=\\{\left(\varphi^{1}\right)^{2}+\left(\varphi^{2}\right)^{2}<1\\}$.
Interpreting this as a symplectic potential, we obtain the “polygon” metric
$g_{\Sigma,ij}={\bf{u}}_{ij}d\varphi^{i}\otimes{}d\varphi^{j}$ where
${\bf{u}}_{ij}$ given by
${\bf{u}}_{ij}\triangleq\frac{\partial^{2}{\bf{u}}}{\partial\varphi^{i}\partial\varphi^{j}}$.
Explicitly this is

(133)
$\displaystyle{\bf{u}}_{ij}\;=\;\left(\begin{array}[]{cc}\frac{1+\left(\varphi^{1}\right)^{2}-\left(\varphi^{2}\right)^{2}}{\left(1-\left(\varphi^{1}\right)^{2}-\left(\varphi^{2}\right)^{2}\right)^{2}}&\frac{\varphi^{1}\varphi^{2}}{\left(1-\left(\varphi^{1}\right)^{2}-\left(\varphi^{2}\right)^{2}\right)^{2}}\\\
\frac{\varphi^{1}\varphi^{2}}{\left(1-\left(\varphi^{1}\right)^{2}-\left(\varphi^{2}\right)^{2}\right)^{2}}&\frac{1-\left(\varphi^{1}\right)^{2}+\left(\varphi^{2}\right)^{2}}{\left(1-\left(\varphi^{1}\right)^{2}-\left(\varphi^{2}\right)^{2}\right)^{2}}\end{array}\right).$

Figure 21. The reduction $\Sigma^{2}$ of a complete Kähler manifold with
bounded curvature, whose symplectic potential is given by (130). This is not a
polygon.

The reduced manifold $(\Sigma^{2},g_{\Sigma})$ of Figure 21 with potential
${\bf{u}}$ is a complete 2-manifold, and is the reduction of a complete toric
Kähler manifold $(M^{4},J,g,\mathcal{X}^{1},\mathcal{X}^{2})$ on which the
symplectomorphic fields are never colinear or zero. Setting
$({\bf{u}}^{ij})\triangleq({\bf{u}}_{ij})^{-1}$, the Abreu equation (which is
(10) of [1]) gives scalar curvature
$R=-\frac{1}{2}\frac{\partial^{2}{\bf{u}}^{ij}}{\partial\varphi^{i}\partial\varphi^{j}}$.
The scalar curvature of $(M^{4},g)$ is not signed; at the origin $R=+8$ and
asymptotically $R$ decreases to $-3$. Computing the full Riemann tensor, one
finds that all sectional curvatures are bounded.

###  7.2. Pathological examples

####  7.2.1. Non-convex polygons

The outline-matching programme of Lemma 5.1 does not require that the polygons
be convex. Indeed we can create non-convex shapes and even shapes with self-
intersecting outlines as in Figure 22. Although we will not prove it here, all
such shapes create pathologies. In particular their polygon metrics will
always be singular.

Figure 22. Examples of outline matching with pathological results: moment
functions with non-convex outline and moment functions with self-intersecting
outline.

####  7.2.2. Removed Edges

To extend the theory of this paper to the case of non-closed polygons, one
would have to contend with many potential pathologies. Removing boundary
segments or rays can result in a non-pathological instanton, as the 2-ended
instanton of 7.1.3, but can create pathologies.

Figure 23. Pathological examples.

Moment functions for the quarter-plane with a removed ray are

(134) $\displaystyle\begin{aligned}
\varphi^{1}\;=\;x+\sqrt{x^{2}+y^{2}},\quad\varphi^{2}=y^{2}.\end{aligned}$

Moment functions for pathological half-strip are

(135) $\displaystyle\begin{aligned}
\varphi^{1}\;=\;y^{2},\quad\varphi^{2}\;=\;\frac{1}{2}\left(1-\frac{x}{\sqrt{x^{2}+y^{2}}}\right).\end{aligned}$

Moment functions for the half-plane with a removed segment are

(136) $\displaystyle\begin{aligned}
\varphi^{1}\;=\;x+\frac{x}{\sqrt{x^{2}+y^{2}}},\quad\varphi^{2}\;=\;y^{2}.\end{aligned}$

These examples do have associated scalar-flat instantons
$(M^{4},g,J,\mathcal{X}^{1},\mathcal{X}^{2})$, but all three have curvature
singularities.

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