# The VIX index under scrutiny of machine learning techniques and neural
networks

Ali Hirsa
IEOR Department
Data Science Institute
Columbia University
<EMAIL_ADDRESS>
Joerg Osterrieder††footnotemark:
School of Engineering
Zurich University of Applied Sciences
Winterthur, Switzerland
<EMAIL_ADDRESS>
Branka Hadji Misheva
School of Engineering
Zurich University of Applied Sciences
Winterthur, Switzerland
<EMAIL_ADDRESS>
Wenxin Cao
IEOR Department
Columbia University
New York, NY 10027
<EMAIL_ADDRESS>
Yiwen Fu
IEOR Department
Columbia University
New York, NY 10027
<EMAIL_ADDRESS>
Hanze Sun
IEOR Department
Columbia University
New York, NY 10027
<EMAIL_ADDRESS>Kin Wai Wong
IEOR Department
Columbia University
New York, NY 10027
<EMAIL_ADDRESS>

###### Abstract

The CBOE Volatility Index, known by its ticker symbol VIX, is a popular
measure of the market’s expected volatility on the S&P 500 Index, calculated
and published by the Chicago Board Options Exchange (CBOE). It is also often
referred to as the fear index or the fear gauge. The current VIX index value
quotes the expected annualized change in the S&P 500 index over the following
30 days, based on options-based theory and current options-market data.
Despite its theoretical foundation in option price theory, CBOE’s Volatility
Index is prone to inadvertent and deliberate errors because it is weighted
average of out-of-the-money calls and puts which could be illiquid. Many
claims of market manipulation have been brought up against VIX in recent
years. This paper discusses several approaches to replicate the VIX index as
well as VIX futures by using a subset of relevant options as well as neural
networks that are trained to automatically learn the underlying formula. Using
subset selection approaches on top of the original CBOE methodology, as well
as building machine learning and neural network models including Random
Forests, Support Vector Machines, feed-forward neural networks, and long
short-term memory (LSTM) models, we will show that a small number of options
is sufficient to replicate the VIX index. Once we are able to actually
replicate the VIX using a small number of S&P options we will be able to
exploit potential arbitrage opportunities between the VIX index and its
underlying derivatives. The results are supposed to help investors to better
understand the options market, and more importantly, to give guidance to the
US regulators and CBOE that have been investigating those manipulation claims
for several years.

00footnotetext: Financial support by the Swiss National Science Foundation
within the project “Mathematics and Fintech - the next revolution in the
digital transformation of the Finance industry” is gratefully acknowledged by
the corresponding author. This research has also received funding from the
European Union’s Horizon 2020 research and innovation program FIN-TECH: A
Financial supervision and Technology compliance training programme under the
grant agreement No 825215 (Topic: ICT-35-2018, Type of action: CSA).
Furthermore, this article is based upon work from COST Action 19130 Fintech
and Artificial Intelligence in Finance, supported by COST (European
Cooperation in Science and Technology), www.cost.eu (Action Chair: Joerg
Osterrieder).
The authors are grateful to Stephan Sturm, management committee members of the
COST (Cooperation in Science and Technology) Action Fintech and Artificial
Intelligence in Finance as well as speakers and participants of the 5th
European COST Conference on Artificial Intelligence in Finance and Industry,
which took place at Zurich University of Applied Sciences, Switzerland, in
September 2020.

_K_ eywords VIX $\cdot$ Machine Learning $\cdot$ Deep Learning $\cdot$ VIX
futures

##  1 Introduction

The VIX is based on S&P 500 option quotes every 15 seconds and is intended to
provide an indication of the fair market price of the expected volatility of
the S&P 500 index over the next 30 days [14]. Although it is not directly
tradeable, there are many derivatives on the index, including both options and
futures. For many years, the VIX has been questioned since it is vulnerable to
liquidity and price fluctuations in the underlying options market as well as
the related derivatives market [1]. Thus, through replicating the VIX index
using approaches different that the official CBOE methodology, we want to
investigate whether it can be represented and influenced by a small subset of
options. Compared to selecting hundreds of options by the official
methodology, we analyze the replication performance by using a fraction of it.
If there exists a way to replicate the index, with a high accuracy, using a
subset of options, this might imply that the VIX is impacted by a small number
of options and it might be possible to exploit potential arbitrage
opportunities between the VIX and its derivatives.

To begin with, we perform the replication on both a daily and an intraday
scale based on the official methodology and formulas outlined in the VIX White
Paper (for more details see [14]). This represents our benchmark model. Next,
we replicate the index with a subsets of options. After analysing the results,
we select the subset with the best performance as the input of our neural
network models, including feed-forward neural networks and LSTM models. Our
findings show that the VIX index can be replicated using a subset of options
and a neural network which learns the dependencies between the input and
output features, i.e. learns the official formula. Furthermore, we outline how
one can exploit potential arbitrage opportunities without having to trade
hundreds of S&P 500 options ([14]).

The remainder of the paper is organized as follows: Section 2 provides a
literature review of relevant studies that discuss different perspective on
the claim of VIX manipulation. Section 3 describes the data and methodology we
use to replicate VIX, as well as the subset selection approaches to show the
significance and influence of a number of options. In Section 4, we describe
our design and implementation of a basic neural network and LSTM models, as
well as the replication results. Finally, Section 5 discusses the conclusion
of our findings and potential future extensions.

##  2 Literature Review

The VIX and its derivatives have been extensively discussed in the literature.
In Osterrieder et al. (2019) [1], the author gives a thorough analysis from
both a theoretical and empirical perspective. By deriving the VIX formula and
empirically examining the individual components of the formula, the authors
conclude that the current formula is not accurate enough and thus opens the
door for potential manipulators. Indeed, futures manipulation has long been
studied in the literature. Kumar and Seppi (1992) [2] investigate the
susceptibility of futures markets to price manipulation in a two-period model
with asymmetric information and cash-settled futures contracts. They interpret
manipulation as a form of endogenous noise trading which can arise in multi-
period security markets. In Pimbley (2018) [3], the authors examines the VIX
formula in detail and give explanations of how the approximations of the CBOE
method promote susceptibility to errors and manipulations. The paper also
discusses how to remove or limit these approximations and gives a review of
numerical results of the improvements.

On the other side of the discussion, the literature also offers papers which
do not support the argument of manipulation of the VIX. In Saha (2018) [4],
the authors construct a regression model with explanatory variables that are
exogenous to the index and find that the movements in the daily levels of the
index are explained by market fundamentals instead of manipulation. By
examining the futures on its expiration days, the paper demonstrates that the
VIX closing values and settlement prices on those days are consistent with
normal market forces and not artificial. Also, the analysis conducted by Wei
(2019) [5] suggests that manipulation techniques are not easily applied in a
real-world setting. He finds that the total trading volume needs to be
extremely large in order to manipulate the VIX. Furthermore, reasons other
than manipulation may explain the trading activity on any particular day.

In addition, there is also substantial research which aims at studying the VIX
and its underlying by using novel techniques. In Osterrieder (2019) [1], the
authors propose a machine learning approach, specifically an LSTM model with
one layer to predict the VIX and shows that ten options are sufficient to
achieve a very high accuracy.

Our paper contributes the literature by:providing a thorough review of the VIX
methodology provided by CBOE and further expanding the topic by using machine
learning and deep learning techniques (including feature selection, basic
neural network models and LSTM models) to replicate VIX as well as VIX futures
and studying their characteristics.

##  3 Constructing the VIX Index

In this section, we describe how we replicated VIX using the original
methodology and formulas documented in the CBOE VIX White Paper. In addition,
the replication of VIX was done with both daily end-of-day (EOD) data and
intraday minute-by-minute data, and our results are satisfactory despite some
minor discrepancies caused by small differences between our replication and
the original methodology.

###  3.1 Data

The data consists of all the S&P 500 (SPX) options in the market. There are
two sets of data: an EOD SPX options quote dataset from January 2017 to March
2018, and a minute-by-minute SPX options quote dataset from January 2018 to
February 2018. The former is used for the daily VIX replication while the
latter is used for the intraday replication. The EOD dataset contains the
closing price released at 3:15 p.m. Chicago time. The VIX index is available
based on 15-second intervals, as per the CBOE methodology.

###  3.2 Methodology for replicating the VIX Index

Our replication methodology follows the procedures and rules documented in the
CBOE VIX White Paper. The components of the VIX Index are near-term and next-
term SPX put and call options with more than 23 days and less than 37 days to
expiration. All the details can be found in the official CBOE VIX white paper
([14]). There are two differences between our replication and the official
methodology. The first difference is the risk-free interest rate used in the
calculation of the forward price. Due to its small influence during the
computation, it is set to a constant of 1.3% in our intraday implementation
and 1% in our daily replication with the purpose of providing an easier
comparison of results. Both values are determined as the approximate mean of
the risk-free rate for the corresponding time intervals. The second difference
is that our implementation does not include the VIX Index Filtering Algorithm.
As described in the VIX White Paper, CBOE uses this filtering algorithm to
address large potential jumps in the VIX index. Due to its small impact (on
average), we are not including this algorithm in our replication.

###  3.3 Results & Validation

For the daily replication, we compare our replicated values with the true EOD
VIX values. Our daily replication method leads to an MSE of $0.0025$, a MAE of
$0.0238$, and a mean percentage error of $0.184\%$.

Figure 1: Replication error of the VIX index for daily close values Figure 2:
Percentage replication error of the VIX for daily close values

To better evaluate the original VIX methodology, we also perform an intraday
replication. As for the replication performance, we validated our replicated
values with the real VIX values and obtain an MSE of $0.0325$, an MAE of
$0.0637$, and a mean percentage error of $0.336\%$.

Figure 3: Replication error of the VIX index for intraday minute-by-minute
values Figure 4: Percentage replication error of the VIX index for intraday
minute-by-minute values

Although the MSE and MAE of both the daily and the intraday replication are
acceptable, there are still some minor discrepancies between our replicated
values and the actual values. The causes of the discrepancies are the
following three. The first one are time-stamp differences. The intraday S&P
500 options dataset records all the information at every minute starting from
09:31:00 EST. On the other hand, the intraday VIX spot value dataset not only
does not record the information starting at 09:31:00 ET, but also at different
seconds on every day. For example, the information was recorded starting from
09:31:29 on January 2, 2018, while starting from 09:31:17 EST on January 10,
2018. Such varying time stamps on every day cause difficulties in adjusting
the intraday replication to have matching time stamps. Moreover, on a daily
level, the VIX spot price data normally does not have its EOD record at
exactly 3:15 p.m. EST, rather it fluctuates between 1 to 15 seconds earlier.
In summary, the time-stamp difference between the recording time of the
options data and the VIX spot value data explains the discrepancies in both
the daily and the intraday replication. The second and third sources of error
are the usage of a constant risk-free interest rate and the exclusion of the
VIX index filtering algorithm. As mentioned before, we have decided to make
those two changes in order to simplify the comparison and understanding of the
results, knowing that they will have small influences on the replication
performance.

Besides using the MSE and the MAE to analyse our replications, we have also
obtained the correlation between the replicated values and the real VIX values
as 0.999 for both replications. The high values indicate that the replicated
values are strongly correlated to the real values. In addition, one of the
findings is that the number of put options selected during the option
selection process is generally higher than the number of call options, an
expected results based on the well-known richness of out-of-the-money put
options. We have re-computed the number of call and put options selected
during the intraday replication minute by minute, and the average number of
put options selected for every minute is 258 while the average number of call
options is 109. This reflects the financial market dynamics that investors are
looking for downside protection with put options, causing more put options to
have nonzero bid price and thereby being selected by the VIX methodology. The
following graph shows the average number of put and call options selected on
every day, calculated by taking the average of all the minute-by-minute counts
on that day.

Figure 5: Count of call and put options selected in intraday replication

###  3.4 Selecting the most relevant options

In this section, we describe how we select a subset of options rather than
using all the options selected based on the original methodology to replicate
VIX. In other words, we only use a portion of options selected by the official
methodology, while simultaneously still using the official formulas for
computing the VIX index. There are two approaches we implemented using a
subset selection, and we discuss their methodologies and compare their
performance.

####  3.4.1 Selecting a subset of options every minute

This approach is to select a subset of options that are nearest to the at-the-
money implied underlying price $K_{0}$, the original criteria in the VIX White
Paper, for both near-term and next-term options in every minute. For
implementation, after selecting all the options using the original
methodology, we are only keeping a given number of near-term and next-term
options that are closest to the near-term and next-term $K_{0}$ respectively.
In our implementation, we have computed the VIX based on
$1,2,3,4,5,10,25,50,100,200$, and all available options.

Next, since we are using only a subset of options and the VIX index is a
weighted average of the option prices, we have to adjust the corresponding
result to match our reduced sample of options. We first scale the
corresponding result by a constant value, to have equal means of both time-
series, before examining the error. Despite very good replication results, a
clear benefit from using more options is still visible.

Figure 6: MSE of selecting a subset of options every minute with different
numbers of options (n) to replicate the VIX index (after scaling) Figure 7:
Error of selecting a subset of options every minute with different numbers of
options (n) to replicate the VIX index (after scaling)

We also compute the correlation between the replicated values using a subsets
of options and the real VIX values. Once again the high correlation proves
that a small subset of options will give a good performance in replicating the
VIX index.

Figure 8: Correlation of selecting a subset of options every minute with
different numbers of options (n) to replicate the VIX index

####  3.4.2 Selecting a subset of options at the beginning of every day

We proceed as in the previous section, with the exception of fixing the subset
of options at the beginning of the day at 09:31:00 EST. As expected,
replication results are not as good, but still sufficient for actual trading.

Figure 9: MSE of selecting a subset of options at the beginning of every day
with different numbers of options (n) to replicate the VIX index (after
scaling) Figure 10: Error of selecting a subset of options at the beginning of
every day with different numbers of options (n) to replicate the VIX (after
scaling)

As before, we compute the correlation between the replicated values and the
real VIX values.

Figure 11: Correlation of selecting a subset of options at the beginning of
every day with different numbers of options(n) to replicate VIX

Comparing both approaches, we conclude that selecting a new subset of options
every minute gives a better replication performance, leading to a lower MSE.
As a result, we are using this approach to generate the input of the neural
network models in the following section.

Figure 12: Comparison of the replication MSE of both approaches

##  4 Neural networks for the VIX and VIX futures

To better understand the VIX index, we use basic neural network models and
LSTM models to replicate the intraday VIX index.

###  4.1 A basic neural network

Before applying the more sophisticated model, we first build a basic neural
network model and study its performance and the relation between the VIX index
and its underlying options.

####  4.1.1 Data

The data used in the model is the intraday S&P500 data and VIX index from
January 2, 2018 to February 28, 2018. Our subset of options is based on the
selection as per the CBOE methodology. For every minute, we choose 52 options,
including both near-term and next term, as features, in particular every third
option based on strike price.

####  4.1.2 Methodology and Result

We first build a basic neural network model using the option prices as the
only feature (Model 1), so the inputs of the model are the subset of option
prices at every minute every day in the two months period and the per minute
VIX index in the same period. The model contains two dense layers with each
having 64 nodes, and one output layer (see Table 1). The activation function
is the rectified linear activation function, the loss function is mean squared
error and the optimizer is Adam. We randomly assign 70% of the data as the
training data and the remaining 30% as the test data. After training the model
and doing the prediction, we got the in-sample MSE of $0.064$ and out-of-
sample MSE of $0.095$. We also performed 5-fold cross- validation and obtained
an MSE of $0.162$.

In addition, we set up additional regressors including a decision tree
regressor, random forest regressor and a support vector regressor as
benchmarks. Both the decision tree regressor and the random forest regressor
were implemented with a maximum number of leaf nodes of eight and the random
forest regressor with ten trees. For the support vector regressor, we used a
radial basis function kernel. The MSE of the decision tree regressor, random
forest regressor, and support vector regressor, are $1.188,1.080,$ and $0.337$
respectively. Compared with these regressors, the neural network model has the
smallest MSE.

Besides option prices, we also trained another neural network model (Model 2)
and we used the option prices plus the option greeks including theta and vega
as the features. Since the number of the features increases, we built a
slightly more complex model than before to fit the data. We expanded the
layers to 4 and the number of nodes to 128 and as a result we obtained an in-
sample MSE of 0.062 and an out-of-sample MSE of 0.090. The 5-fold cross-
validation gives a MSE of 0.126. Thus, using the additional features as
inputs, the neural network model has an smaller MSE than all of the previously
tested approaches. We also set up additional regressors as benchmarks. The MSE
of the decision tree regressor, random forest regressor, and support vector
regressor are 1.051, 0.925 and 0.371, respectively. The MSE of the neural
network model with more features is still the smallest one compared to the
benchmarks.

(a) Basic neural network, Model 1 Layer (type) | Output Shape | Param #
---|---|---
flatten (Flatten) | (None, 52) | 0
dense_1 (Dense) | (None, 64) | 3392
dense_2 (Dense) | (None, 64) | 4160
dense_3 (Dense) | (None, 1) | 65
Total params: 7,617 |  |
Trainable params: 7,617 |  |
Non-trainable params: 0 |  |

(b) Basic neural network, Model 2 Layer (type) | Output Shape | Param #
---|---|---
flatten (Flatten) | (None, 156) | 0
dense_1 (Dense) | (None, 128) | 20096
dense_2 (Dense) | (None, 128) | 16512
dense_3 (Dense) | (None, 128) | 16512
dense_4 (Dense) | (None, 128) | 16512
dense_5 (Dense) | (None, 1) | 129
Total params: 69,761 |  |
Trainable params: 69,761 |  |
Non-trainable params: 0 |  |

Figure 13: Summary: Performance of both models

(a) Training Loss for the basic neural network, model 1

(b) Training Loss for the basic neural network, model 2

###  4.2 LSTM

Lastly, we use Long Short Term Memory networks (LSTM) to study the long-term
dependencies of S&P options data.

LSTM is a type of a recurrent neural network (RNN), a neural network that
persists information of previous events to inform later ones. However, one of
the drawbacks of RNNs is that they are unable to learn to connect the
information when there is a larger context to process. An LSTM is a
specialized RNN that is capable of learning long term dependencies.

####  4.2.1 Data and Methodology

The data used in the model is the same as described in Section 4.1.1 We use
the intraday option price data as our input, and the VIX index value as the
value to predict. After splitting the data into training and test set, we
scaled the data to the unit interval.

We analysed two different model architectures, a flat model and a multi-layer
model. For each model architecture, we also investigated two different input
data sets. One of them selects options by order, and the other selects every
third option as input. The latter ensures our input to be spread more evenly
across the price to eliminate duplicate information.

For each model, we used a learning rate of $0.000005$ and $300$ epochs. We
also introduced two callbacks, an early stopping callback with a patience of
1500 on the validation loss and a tensorboard callback for better
visualizations.

####  4.2.2 Model 1: The Flat Model

The flat model contains an LSTM layer with 1000 output units and a dense layer
with one neuron (see Table 2).

Table 2: LSTM Model: Flat Layer (type) | Output Shape | Param #
---|---|---
lstm_1 (LSTM) | (None, 1000) | 4,212,000
dense_1 (Dense) | (None, 1) | 1,001
Total params: 4,213,001 |  |
Trainable params: 4,213,001 |  |
Non-trainable params: 0 |  |

For the model using options in order, the MSE of the predicted value vs the
actual value is $5.632$. For the model using every third option, the MSE is
$7.797$. The model training loss and the predictions vs the actual VIX values
are shown below. Both models give us an acceptable performance.

(a) Options in Order: Model Training Loss

(b) Every Third Option: Model Training Loss

(a) Options in Order: Predictions vs Real VIX by Quote_time

(b) Every Third Option: Predictions vs Real VIX by Quote_time

(a) Options in Order: Predictions vs Real VIX by Date

(b) Every Third Option: Predictions vs Real VIX by Date

####  4.2.3 Model 2: The Multi-Layer Model

The multi-layer model contains two LSTM layers and one dropout layer. The LSTM
layers have 500 output units, and the $30\%$ dropout layers are for the
purpose of overfitting prevention. The model also contains another LSTM layer
with one output unit, and a dense layer to return our prediction value (see
Table 3).

Table 3: LSTM Model: Multi-Layer Layer (type) | Output Shape | Param #
---|---|---
lstm_1 (LSTM) | (None, 10, 500) | 1,106,000
dropout_1 (Dropout) | (None, 10, 500) | 0
lstm_2 (LSTM) | (None, 10, 500) | 2,002,000
dropout_2 (Dropout) | (None, 10, 500) | 0
lstm_3 (LSTM) | (None, 500) | 2,002,000
dense (Dense) | (None, 1) | 501
Total params: 5,110,501 |  |
Trainable params: 5,110,501 |  |
Non-trainable params: 0 |  |

The model using options in order is the only model that has not triggered
early stopping at epoch $1500$. Its MSE of the predicted value against the
actual value is $17.918$, the worst performance among all. For model using
every third option, the MSE of the predicted value against the actual value is
$2.963$. The model training loss and the predictions vs the actual VIX are
shown below. While the performance of model using options in order is not as
good, the model using Every Third Option give us the best performance among
the four.

(a) Options in Order: Model Training Loss

(b) Every Third Option: Model Training Loss

(a) Options in Order: Predictions vs Real VIX by Quote_time

(b) Every Third Option: Predictions vs Real VIX by Quote_time

(a) Options in Order: Predictions vs Real VIX by Date

(b) Every Third Option: Predictions vs Real VIX by Date

##  5 VIX futures

Monthly VIX futures are futures contracts with monthly expirations, being
cash-settled with the settlement amount being determined on the Wednesday that
is thirty days prior to the third Friday of the calendar month immediately
following the month in which the contract expires. The final settlement value
is determined in a Special Opening Auction with the VIX normally being
calculated based on actual traded option prices.

The fair value of a VIX futures contract is the square root of the implied
variance minus an adjustment representing the concavity of the forward
position (Hancock(2012)).

There has been strong evidence observed in the market that VIX and SPX option
prices are negatively correlated with each other, meaning that including VIX
in a portfolio will be a good hedging choice. The VIX exposure for downside
protection is illustrated by constructing two hypothetical portfolios, one
holding only the SPY and the other holding 90 percent of the wealth in SPY and
10 percent in the VIX. In the study by Tim Leung, when negative news stroke
the market, like credit rating downgrades by Standard and Poor’s in 2011, the
former portfolio would lose up to 10 percent with a much more volatile
trajectory while the latter one (VIX included) would have yield a positive
return, remaining stable throughout the period when markets collapsed. But the
fact that VIX is not a traded asset, but rather an anticipated benchmark for
the US stock market volatility leads to the existence of VIX derivatives like
VIX futures and VIX options. Most of the time volatility exposure is achieved
using these two products. These instruments give speculators to establish a
position on market volatility with a long position in the VIX futures contract
meaning a bet on the increase of market volatility and a decline in the market
index.

##  6 Relationship between VIX and VIX futures

Numerous research has been done on the VIX and VIX futures and it is obvious
that these two are highly correlated since the settlement price of VIX futures
is the future variance adjusted downward by the concavity. Regressing the
returns on VIX futures against returns on VIX yields a slope coefficient
statistically significant less than 1 and a negative intercept for all
contracts with different maturities, showing that futures returns are less
volatile than spot returns and that when the spot price does not move, futures
prices still tend to fall as futures prices will converge to the spot price as
the maturity date approaches.

##  7 LSTM for predicting VIX futures

###  7.1 Data and Methodology

The input data is the same as used earlier. For the output data, we substitute
the VIX index with VIX futures values starting from 2018-01-02 to 2018-02-28.
Specifically for VIX futures, we utilize minute-by-minute data from 9:30:00 am
to 16:14:00 p.m. every day in order to match the time-stamps of the S&P500
option prices data. Again, we choose 52 out-of-the-money options (every third
option as per strike price), including both near-term and next term, as
features.

###  7.2 Model 1: The Flat Model

The first model is the same as considered before, having an LSTM layer with
1000 output units, and a dense layer to output the predicted value (see Table
4).

Table 4: LSTM Model 1 Layer (type) | Output Shape | Param #
---|---|---
lstm_1 (LSTM) | (None, 1000) | 4,212,000
dense_1 (Dense) | (None, 1) | 1,001
Total params: 4,213,001 |  |
Trainable params: 4,213,001 |  |
Non-trainable params: 0 |  |

For model using every option in order, the MSE of the predicted value against
the actual value is $22.295$ and the MAE is 4.417. For model using every third
option, the MSE of the predicted value against the actual value is $19.051$
and MAE is $4.070$. We can see that the MSEs for both models are significantly
larger than previous performances when predicting VIX values and both
triggered early stopping at epoch $1757$ and epoch $1500$ respectively.

(a) Every Option in Order: Model Training Loss

(b) Every Third Option: Model Training Loss

(a) Every Option in Order: Predictions vs actual VIX

(b) Every Third Option: Predictions vs actual VIX

(a) Every Option in Order: Predictions vs actual VIX

(b) Every Third Option: Predictions vs actual VIX

###  7.3 Model 2: The Multi-Layer Model

We use the same architecture for the neural network as before (see Table 3).
The model using every option in order triggered early stopping at epoch
$1500$. Its MSE of the predicted value against the actual value is $4.437$ and
MAE being $1.724$, achieving the best performance among all. For model using
every third option, it also triggered early stopping at epoch $1500$. The MSE
of the predicted value against the actual value is $28.896$ and MAE being
$5.014$, achieving the worst performance among all models. The model using
every options gives us the best performance among the four considered
versions.

(a) Every Option in Order: Model Training Loss

(b) Every Third Option: Model Training Loss

(a) Every Option in Order: Predictions vs actual VIX

(b) Every Third Option: Predictions vs Real VIX by Quote_time

(a) Every Option in Order: Predictions vs actual VIX

(b) Every Third Option: Predictions vs actual VIX

##  8 Conclusion and Summary

###  8.1 VIX replication and prediction

In this study, we first performed a VIX replication using both daily and
intraday minute-by-minute data. After selecting options according to the
official method, we selected subsets of them through our subset selection
methods. We replicated the VIX using the official formulas and analysed the
results. We then selected the subset with the best performance as the input of
our feed-forward neural network and LSTM models.

The performance of the basic neural network model is acceptable. After adding
more features, the model gives an even better performance. For LSTM models, we
tried two different model architectures each with two different input options.
Among those, the flat model using the options in order as well as the
multiple-layer model using every-third options have better performance.

Thus, our work shows that instead of using all options selected by the CBOE’s
methodology, which are about 300 options per minute, we can apply machine
learning and deep learning techniques to replicate the VIX index by using a
smaller subset containing only 52 options per minute (including both near-term
and next-term options) with a small MSE.

For further studies, one can develop a selection methodology that is not
dependent on the CBOE’s methodology in the first place, but that uses a neural
network for choosing an optimal set of otpions.

###  8.2 VIX futures replication and prediction

Using the same network architecture as before, we are predicting VIX futures.
Instead of using all options selected by CBOE’s methodology and adjusting it
with a downward concavity term to obtain the VIX futures index, our work shows
that using deep learning models based on a LSTM architecture to predict the
future index with only a small subset of 52 option prices per minute is
feasible. The multi layer LSTM model using every option price in order yielded
rather satisfatory results (MSE $4.437$), but the other three models have
performances not quite satisfactory, so we conclude that a multi-layer LSTM
model works better in predicting the results than a flat LSTM model. Also, it
is of importance that we choose the proper input data, consecutive option
prices, as opposed to every third option prices, turns out better in the
prediction of VIX futures.

## References

  * [1] Osterrieder, Joerg and Vetter, Lars and Röschli, Kevin.  The VIX Volatility Index - A Very Thorough Look at It.  January 7, 2019. https://ssrn.com/abstract=3311727 or http://dx.doi.org/10.2139/ssrn.3311727.
  * [2] Kumar, P., and D. Seppi.  1992\. Futures manipulation with cash settlement.  The Journal of Finance 47:1485-1502.
  * [3] Joe Pimbley, Gene Phillips.  Fix the VIX: Reducing Manipulation in the Volatility Index, p. 2018 - 2022.  Posted: 2018.
  * [4] Saha, A., Malkiel, B.G. & Rinaudo.  Has the VIX index been manipulated?.  J Asset Manag 20, 1–14 (2019). https://doi.org/10.1057/s41260-018-00102-4
  * [5] Ran Wei, Edi Grgeta, Ging Cee Ng and Brendan Perry; Analysis Group, Inc.  Examining The Evidence On VIX Manipulation.  May 15, 2019. https://www.analysisgroup.com
  * [6] Osterrieder, J., Kucharczyk, D., Rudolf, S. et al.  Neural networks and arbitrage in the VIX.  Digit Finance (2020). https://doi.org/10.1007/s42521-020-00026-y
  * [7] Exchange, C.B.O. (2009)  VIX white paper.  https://www.cboe.com/micro/vix/vixwhite.pdf.
  * [8] Oleg Torben G Andersen, Maria T Bondarenko, Gonzalez-Perez  Exploring Return Dynamics via Corridor Implied Volatility.  The Review of Financial Studies, volume 28, issue 10, p. 2902 - 2945 Posted: 2015
  * [9] H Keith, Black  Improving hedge fund risk exposures by hedging equity market volatility, or how the VIX ate my kurtosis.  The Journal of Trading, volume 1, issue 2, p. 6 - 15 Posted: 2006
  * [10] Menachem Brenner, Dan Galai  New Financial Instruments for Hedge Changes in Volatility.  Financial Analysts Journal, volume 45, issue 4, p. 61 - 65 Posted: 1989
  * [11] Larry Connors  Timing your S&P trades with the VIX.  Futures, volume 31, issue 7, p. 46 - 46 Posted: 2002
  * [12] G.D’Anne Hancock.  VIX and VIX Futures Pricing Algorithms: Cultivating Understanding.  Modern Economy, 2012,3,284-294  May 2020. https://www.SciRP.org/journal/me
  * [13] T. Leung and B. Ward, Tracking VIX with VIX Futures: Portfolio Construction and Performance.  Handbook of Applied Investment Research, J. Guerard and W. Ziemba eds. World Scientific Publishing Co, 2020  World Scientific Publishing Co, in print, 2020
  * [14] CBOE Global Markets (2020)  VIX Weeklys Options and Futures

##  Appendix A Appendix

Figure 27: MSE of selecting a subset of options every minute with different
numbers of options (n) to replicate VIX (before scaling) Figure 28: Error of
selecting a subset of options every minute with different numbers of options
(n) to replicate VIX (before scaling) Figure 29: MSE of selecting a subset of
options at the beginning of every day with different numbers of options (n) to
replicate VIX (before scaling) Figure 30: Error of selecting a subset of
options at the beginning of every day with different numbers of options(n) to
replicate VIX (before scaling)


#  On minimal surfaces immersed in three dimensional Kropina Minkowski space

Ranadip Gangopadhyay111E-mail<EMAIL_ADDRESS>Ashok
Kumar222E-mail<EMAIL_ADDRESS>and Bankteshwar Tiwari333E-mail:
<EMAIL_ADDRESS>
DST-CIMS, Banaras Hindu University, Varanasi-221005, India

###### Abstract

In this paper we consider a three dimensional Kropina space and obtain the
partial differential equation that characterizes a minimal surfaces with the
induced metric. Using this characterization equation we study various
immersions of minimal surfaces. In particular, we obtain the partial
differential equation that characterizes the minimal translation surfaces and
show that the plane is the only such surface.

##  1 Introduction

The study of minimal surfaces in Riemannian manifolds has been extensively
developed [7, 12, 15]. Various well developed techniques have played key roles
in the differential geometry and partial differential equations. For instance,
the estimates for nonlinear equations based on the maximum principle arising
in Bernstein’s classical work, and even in the Lebesgue’s definition of the
integral, that he developed in his thesis, is based on the Plateau problem for
minimal surfaces [16]. However, minimal surfaces in Finsler spaces have not
been studied and developed at the same pace. The fundamental contribution to
the minimal surfaces of Finsler geometry was made by Shen [17]. Shen
introduced the notion of mean curvature for immersions into Finsler manifolds
and established some of its properties. As in the Riemannian case, if the mean
curvature is identically zero, then the immersion is said to be minimal.

The Randers metric is the simplest class of non-Riemannian Finsler metric,
defined as $F=\alpha+\beta$, where $\alpha$ is a Riemannian metric and $\beta$
is a one-form. M. Souza and K. Tenenblat studied the surfaces of revolution in
Minkowski space $\mathbb{R}^{3}$ with Randers metric [21] and found the
condition for such surfaces to be minimal. Further Souza et al. obtained a
Bernstein type theorem for minimal surfaces in a Randers space [22]. Minimal
surfaces in a Randers space have been studied by several authors [8, 9, 10,
18]. Recently, minimal surface with Matsumoto metric has been studied in [11].

It is well known that the classification of constant flag curvature Randers
space has been done by Bao et al using Zermello Navigation for a time-
independent vector field $W$ satisfying $h(W,W)<1$, in a Riemannian space
$(M,h)$. In fact the solution of a Zermello Navigation problem in a Riemannian
manifold $(M,h)$ with a time independent vector field $W$ satisfying
$h(W,W)<1$, gives a unique Randers metric on the manifold $M$ and conversely.
However, for vector field $W$, satisfying $h(W,W)=1$, recently R. Yoshikawa
and S.V. Sabau prove that the time minimizing path would be the geodesics of a
Kropina metric which is given by $F=\frac{\alpha^{2}}{\beta}$, where $\alpha$
is a Riemannian metric and $\beta$ is a one-form [23]. Kropina metric was
introduced by V.K. Kropina [13] and has numerous applications in physics and
biology [1]. V. Balan studied the constant mean curvature surfaces in
Minkowski space with Kropina metric and obtain some characteristic
differential equations for these surfaces [2].

In this paper we study the minimal surfaces immersed in three-dimensional
Kropina spaces. We first obtain the characterization partial differential
equation for a surface to be minimal in the Kropina space. Then we consider
three different immersions, the surfaces of revolution, the graph of a smooth
functions, the translation surfaces in the Kropina space. In Section $3$, we
study the minimal surfaces of revolution and find the generating curves which
gives the minimal surfaces explicitly. That is, we obtain the following
theorem:

###### Theorem 1.1

Let $(\mathbb{R}^{3},F=\alpha^{2}/\beta)$ be a Kropina space, where $\alpha$
is the Euclidean metric, and $\beta=bdx^{3}$ is a $1$-form with norm $b$,
satisfying $b>0$. Let $\varphi:M^{2}\to\mathbb{R}^{3}$ be an immersion given
by $\varphi(x^{1},x^{2})=(f(x^{1})\cos x^{2},f(x^{1})\sin x^{2},x^{1})$, $f$
is a smooth function and $f(x^{1})>0$. Then $\varphi$ is minimal, if and only
if, either it is an open cone with generating curve
$f(x^{1})=\dfrac{1}{\sqrt{2}}x^{1}+c$, or a surface with the generating curve
$x^{1}=(-f^{2/3}-2a)\sqrt{a-f^{2/3}}+d$, here $a,c,d$ being real constants.

In Section $4$, we study the minimal surfaces of graph of a smooth function
and obtain a Bernstein-type theorem as follows:

###### Theorem 1.2

If a minimal surface in the Kropina space $(\mathbb{R}^{3},F)$ is the graph of
a smooth function $f:U\subset\mathbb{R}^{2}\to\mathbb{R}$ then it is a plane
provided $f_{x_{1}}^{2}+f_{x_{2}}^{2}\geq 2$.

In Section $5$, we study the minimal translation surfaces and prove that plane
is the only such surface.

###### Theorem 1.3

If a minimal translation surface in the Kropina space $(\mathbb{R}^{3},F)$ is
given by the immersion $\varphi:U\subset\mathbb{R}^{2}\to(\mathbb{R}^{3},F)$,
$\varphi(x^{1},x^{2})=(x^{1},x^{2},f(x^{1})+g(x^{2}))$ then it is a plane.

##  2 Preliminaries

Let $M$ be an $n$-dimensional smooth manifold. $T_{x}M$ denotes the tangent
space of $M$ at $x$. The tangent bundle of $M$ is the disjoint union of
tangent spaces $TM:=\sqcup_{x\in M}T_{x}M$. We denote the elements of $TM$ by
$(x,y)$ where $y\in T_{x}M$ and $TM_{0}:=TM\setminus\left\\{0\right\\}$.

###### Definition 2.1

[6] A Finsler metric on $M$ is a function $F:TM\to[0,\infty)$ satisfying the
following condition:
(i) $F$ is smooth on $TM_{0}$,
(ii) $F$ is a positively 1-homogeneous on the fibers of tangent bundle $TM$,
(iii) The Hessian of $\frac{F^{2}}{2}$ with element
$g_{ij}=\frac{1}{2}\frac{\partial^{2}F^{2}}{\partial y^{i}\partial y^{j}}$ is
positive definite on $TM_{0}$.
The pair $(M,F)$ is called a Finsler space and $g_{ij}$ is called the
fundamental tensor.

In general, the explicit calculations of geometric objects in Finsler geometry
is very tedious and complicated. This is perhaps one of the reason why the
study of special Finsler spaces has attracted its attention and specailly the
geometry of Finsler metrics, namely, $(\alpha,\beta)$-metric, introduced by M.
Matsumoto [14] has taken much attention in recent years. The
$(\alpha,\beta)$-metric is defined as, $F=\alpha\phi(\frac{\beta}{\alpha})$
where $\alpha$ is a Riemannian metric, $\beta$ is a one form and $\phi$ is a
smooth function. This class of Finsler metrics contains many interesting
subclass of Finsler metrics such as Randers metrics, Matsumoto mountain
metric, Kropina metric etc.

For an $n$-dimensional Finsler manifold $(M^{n},F)$, the Busemann-Hausdorff
volume form is defined as $dV_{BH}=\sigma_{BH}(x)dx$, where

$\sigma_{BH}(x)=\frac{vol(B^{n}(1))}{vol\left\\{(y^{i})\in
T_{x}M:F(x,y)<1\right\\}},$ (1)

$B^{n}(1)$ is the Euclidean unit ball in $\mathbb{R}^{n}$ and $vol$ is the
Euclidean volume.

###### Proposition 2.1

[5] Let $F=\alpha\phi(s)$, $s=\beta/\alpha$, be an $(\alpha,\beta)$-metric on
an $n$-dimensional manifold $M$. Then the Busemann-Hausdorff volume form
$dV_{F}$ of the $(\alpha,\beta)$-metric $F$ is given by

$dV_{F}=\frac{\int\limits_{0}^{\pi}\sin^{n-2}(t)dt}{\int\limits_{0}^{\pi}\frac{\sin^{n-2}(t)}{\phi(b\cos(t))^{n}}dt}dV_{\alpha}$

where, $dV_{\alpha}=\sqrt{det(a_{ij})}dx$ denotes the volume form of
Riemannian metric $\alpha$.

A Kropina metric on $M$ is a Finsler structure $F$ on $TM$ is given by
$F=\frac{\alpha^{2}}{\beta}$, where $\alpha=\sqrt{a_{ij}y^{i}y^{j}}$ is a
Riemannian metric and $\beta=b_{i}y^{i}$ is a one-form with $b>0$.

Let $(\tilde{M}^{m},\tilde{F})$ be a Finsler manifold, with local coordinates
$(\tilde{x}^{1},\dots,\tilde{x}^{m})$ and
$\varphi:M^{n}\to(\tilde{M}^{m},\tilde{F})$ be an immersion. Then $\tilde{F}$
induces a Finsler metric on $M$, defined by

$F(x,y)=\left(\varphi^{*}\tilde{F}\right)(x,y)=\tilde{F}\left(\varphi(x),\varphi_{*}(y)\right),\quad\forall(x,y)\in
TM.$ (2)

The following convention is in order: the greek letters
$\epsilon,\eta,\gamma,\tau,\dots$ are the indices ranging from $1$ to $n$ and
the latin letters $i,j,k,l,\dots$ are the indices ranging from $1$ to $n+1$.

A Minkowski space is the vector space $\mathbb{R}^{n}$ equipped with a
Minkowski norm $F$ whose indicatrix is strongly convex. Equivalently, we can
say that $F(x,y)$ depends only on $y\in T_{x}(\mathbb{R}^{n})$. In this paper
we consider the hypersurface $M^{n}$ in the Minkowski Kropina space
$\mathbb{R}^{n+1}$ given by the immersion
$\varphi:M^{n}\to(\mathbb{R}^{n+1},F)$, where $F=\frac{\alpha^{2}}{\beta}$,
$\alpha$ is the Euclidean metric, and $\beta$ is a one-form with Euclidean
norm $b>0$. Without loss of generality we consider $\beta=bdx^{n+1}$. If
$M^{n}$ has local coordinates $x=(x^{\epsilon}),\epsilon=1,...,n$, and
$\varphi(x)=\left(\varphi^{i}(x^{\epsilon})\right)\in\mathbb{R}^{n+1}$,
$i=1,\dots,n+1$, we define

$\mathcal{F}(x,z)=\frac{vol(B^{n})}{vol(D^{n}_{x})},$ (3)

where,

$D^{n}_{x}=\left\\{(y^{1},y^{2},...,y^{n})\in\mathbb{R}^{n}:F(x,y)<1\right\\},\quad
y=y^{\epsilon}z^{i}_{\epsilon}\quad\textnormal{and}\quad
z=\left(z^{i}_{\epsilon}\right)=\left(\frac{\partial\varphi^{i}}{\partial
x^{\epsilon}}\right)$ (4)

The mean curvature $\mathcal{H}_{\varphi}$, for the immersion $\varphi$, along
the vector $v$ introduced by Z. Shen [17] and is given by

$\mathcal{H}_{\varphi}(v)=\frac{1}{\mathcal{F}}\left\\{\frac{\partial^{2}\mathcal{F}}{\partial
z^{i}_{\epsilon}\partial
z^{j}_{\eta}}\frac{\partial^{2}\mathcal{\varphi}^{j}}{\partial
x^{\epsilon}\partial x^{\eta}}+\frac{\partial^{2}\mathcal{F}}{\partial
z^{i}_{\epsilon}\partial\tilde{x}^{j}}\frac{\partial\mathcal{\varphi}^{j}}{\partial
x^{\epsilon}}-\frac{\partial\mathcal{F}}{\partial\tilde{x}^{i}}\right\\}v^{i}.$

Here $v=(v^{i})$ is a vector field over $\mathbb{R}^{n+1}$.
$\mathcal{H}_{\varphi}(v)$ depends linearly on $v$ and the mean curvature
vanishes on $\varphi_{*}(TM)$. Since, $(\mathbb{R}^{n+1},F)$ is a Minkowski
space, $F=F(y)$. Hence, the expression of the mean curvature reduces to

$\mathcal{H}_{\varphi}(v)=\frac{1}{\mathcal{F}}\left\\{\frac{\partial^{2}\mathcal{F}}{\partial
z^{i}_{\epsilon}\partial
z^{j}_{\eta}}\frac{\partial^{2}\mathcal{\varphi}}{\partial
x^{\epsilon}\partial x^{\eta}}\right\\}v^{i}.$ (5)

The immersion $\varphi$ is said to be minimal when $\mathcal{H}_{\varphi}=0$.
In this paper we consider an immersed surface in three dimensional Minkowski
space. Using the definition of pullback metric given in (2), we show that if
$\tilde{F}$ is a Kropina metric, then the induced pullback metric on the
surface is again a Kropina metric.

###### Proposition 2.2

Let $\varphi:M^{2}\to(\mathbb{R}^{3},\tilde{F}=\frac{\alpha^{2}}{\beta})$,
where $\alpha$ is the Euclidean metric and $\beta=bdx^{3}$, $(b>0)$ be an
immersion in a Kropina space with local coordinates $(\varphi^{i}(x^{a}))$.
Then the pull back metric defined in (2) is a Kropina metric.

### Proof:

Let
$\varphi\left(x^{1},x^{2}\right)=\left(\varphi^{1}(x^{1},x^{2}),\varphi^{2}(x^{1},x^{2}),\varphi^{3}(x^{1},x^{2})\right)$
be an immersion. Then, for any tangent vector $v\in TM$,
$(\phi^{*}(\tilde{F}))(v)=\tilde{F}(\phi_{*}v)=\frac{\delta_{ij}\frac{\partial\phi^{i}}{\partial
x^{\epsilon}}\frac{\partial\phi^{j}}{\partial
x^{\delta}}v^{\epsilon}v^{\delta}}{b\frac{\partial\phi^{j}}{\partial
x^{\eta}}v^{\eta}}=\frac{A_{\epsilon\delta}v^{\epsilon}v^{\delta}}{bz^{3}_{\eta}v^{\eta}}$,
where

$A=\left(A_{\tau\gamma}\right)=\left(\sum\limits_{i=1}^{3}z^{i}_{\tau}z^{i}_{\gamma}\right)$
(6)

Hence, $F=\phi^{*}(\tilde{F})$ is again a Kropina metric where,
$\alpha^{2}=A_{\epsilon\delta}v^{\epsilon}v^{\delta}$ and
$\beta=bz^{3}_{\eta}v^{\eta}$. $\square$

###### Theorem 2.1

Let $\varphi:M^{2}\to(\mathbb{R}^{3},F)$ be an immersion in a Kropina space
with local coordinates $(\varphi^{i}(x^{a}))$. Then $\varphi$ is minimal if
and only if

$\begin{split}\frac{\partial^{2}\varphi^{j}}{\partial x^{\epsilon}\partial
x^{\eta}}v^{i}\\{-2C^{2}E\frac{\partial^{2}E}{\partial
z^{i}_{\epsilon}\partial z^{j}_{\eta}}+4C^{2}\frac{\partial E}{\partial
z^{i}_{\epsilon}}\frac{\partial E}{\partial
z^{j}_{\eta}}-6CE\left(\frac{\partial C}{\partial
z^{i}_{\epsilon}}\frac{\partial E}{\partial z^{j}_{\eta}}+\frac{\partial
E}{\partial z^{i}_{\epsilon}}\frac{\partial C}{\partial
z^{j}_{\eta}}\right)\\\ -6CE^{2}\frac{\partial C}{\partial
z^{i}_{\epsilon}}\frac{\partial C}{\partial
z^{j}_{\eta}}+3E^{2}\frac{\partial^{2}C^{2}}{\partial z^{i}_{\epsilon}\partial
z^{j}_{\eta}}&\\}=0,\end{split}$ (7)

where

$E=b^{2}\sum\limits_{k=1}^{3}(-1)^{\gamma+\tau}z^{k}_{\tilde{\gamma}}z^{k}_{\tilde{\tau}}z^{3}_{\gamma}z^{3}_{\tau},\qquad\tilde{\tau}=\delta_{\tau
2}+2\delta_{\tau 1},\quad C=\sqrt{detA}$ (8)

### Proof:

For Kropina surface we have $\phi(s)=\frac{1}{s}$ and $n=2$. Therefore, we
have

$\begin{split}dV_{BH}=\frac{\int\limits_{0}^{\pi}dt}{\int\limits_{0}^{\pi}(b^{\prime}\cos
t)^{2}dt}\sqrt{det(A)}dx=\frac{4}{b^{\prime 2}}\sqrt{det(A)}dx\end{split}$ (9)

Here, $b^{\prime 2}=b^{2}A^{\epsilon\delta}z^{3}_{\epsilon}z^{3}_{\delta}$ is
the norm of $\beta$ with respect to the pullback Kropina metric $F$. Hence,
the Euclidean volume of $D^{n}_{x}$ is given by

$vol(D^{n}_{x}):=4\frac{vol(B^{n})\sqrt{detA}}{b^{2}A^{\epsilon\eta}z^{3}_{\epsilon}z^{3}_{\eta}},$
(10)
$A=\left(A_{\tau\gamma}\right)=\left(\sum\limits_{i=1}^{3}z^{i}_{\tau}z^{i}_{\gamma}\right)$
(11)

Therefore, using (3) and (8) in (10) we have

$\mathcal{F}(x,z)=\frac{4C^{3}}{E}$ (12)

It should be noted that

$\frac{\partial^{2}C^{2}}{\partial z^{i}_{\epsilon}\partial
z^{j}_{\eta}}=\frac{\partial}{\partial z^{j}_{\eta}}\left(2C\frac{\partial
C}{\partial z^{i}_{\epsilon}}\right)=2\frac{\partial C}{\partial
z^{i}_{\epsilon}}\frac{\partial C}{\partial
z^{j}_{\eta}}+2C\frac{\partial^{2}C}{\partial z^{i}_{\epsilon}\partial
z^{j}_{\eta}}$ (13)

Now differentiating (12) twice first with respect to $z^{i}_{\epsilon}$ and
then with respect to $z^{j}_{\eta}$ and using (13) we get

$\begin{split}\frac{\partial^{2}\mathcal{F}}{\partial z^{i}_{\epsilon}\partial
z^{j}_{\eta}}=\frac{8C^{3}}{E^{3}}\frac{\partial E}{\partial
z^{i}_{\epsilon}}\frac{\partial E}{\partial
z^{j}_{\eta}}-\frac{12C^{2}}{E^{2}}\left(\frac{\partial C}{\partial
z^{j}_{\eta}}\frac{\partial E}{\partial z^{i}_{\epsilon}}+\frac{\partial
E}{\partial z^{j}_{\eta}}\frac{\partial C}{\partial
z^{i}_{\epsilon}}\right)-\frac{4C^{3}}{E^{2}}\frac{\partial^{2}E}{\partial
z^{i}_{\epsilon}\partial z^{j}_{\eta}}\\\ +\frac{12C}{E}\frac{\partial
C}{\partial z^{j}_{\eta}}\frac{\partial C}{\partial
z^{i}_{\epsilon}}+\frac{6C}{E}\frac{\partial^{2}C^{2}}{\partial
z^{i}_{\epsilon}\partial z^{j}_{\eta}}\end{split}$ (14)

Now using (14) in (5) we obtain the proof of the theorem. $\square$

##  3 Minimal surfaces of revolution

In this section we consider the surface of revolution given by the immersion
$\varphi:M^{2}\to\mathbb{R}^{3}$ such that
$\varphi(x^{1},x^{2}):=\left(\varphi^{i}(x^{\epsilon})\right)=(f(x^{1})\cos
x^{2},f(x^{1})\sin x^{2},x^{1})$, with $f(x^{1})>0$. The curve
$(x^{1},f(x^{1}))$ is the generating curve for the surface of revolution. Here
we note that, the mean curvature vanishes on tangent vectors of the immersion
$\varphi$. Therefore, we only need to consider a vector field $v$ such that
the set $\left\\{\varphi_{x^{1}},\varphi_{x^{2}},v\right\\}$ is linearly
independent. Hence, we consider $v=(-\cos x^{2},-\sin
x^{2},f^{\prime}(x^{1}))$. Therefore, the coordinates of the vector $v$ can be
written as

$v^{i}=-\delta_{i1}\cos x^{2}-\delta_{i2}\sin
x^{2}+\delta_{i3}f^{\prime}(x^{1}).$ (15)

We also have

$\begin{split}z^{i}_{\epsilon}:=\frac{\partial\varphi^{i}}{\partial
x^{\epsilon}}=\delta_{\epsilon 1}[\delta_{i1}f^{\prime}(x^{1})\cos
x^{2}+\delta_{i2}f^{\prime}(x^{1})\sin x^{2}+\delta_{i3}]\\\ +\delta_{\epsilon
2}[-\delta_{i1}f(x^{1})\sin x^{2}+\delta_{i2}f(x^{1})\cos x^{2}]\end{split}$
(16)

Therefore, in view of (11) we obtain the followings:

$A=\begin{pmatrix}1+(f^{\prime}(x^{1}))^{2}&0\\\ 0&(f(x^{1})^{2}\\\
\end{pmatrix}$ (17) $C^{2}=f^{2}(x^{1})(1+f^{\prime 2}(x^{1})),\quad
E=b^{2}f^{2}(x^{1}).$ (18)

From the above values of $A$, $C$ and $E$ we further obtain

$\frac{\partial C}{\partial z^{i}_{\epsilon}}v^{i}=0$ (19) $\frac{\partial
E}{\partial z^{i}_{\epsilon}}v^{i}=2b^{2}\delta_{\epsilon
1}f^{2}(x^{1})f^{\prime}(x^{1})$ (20) $\frac{\partial C}{\partial
z^{j}_{\eta}}\frac{\partial^{2}\varphi^{j}}{\partial x_{\epsilon}\partial
x_{\eta}}=\frac{f^{\prime}(x^{1})}{\sqrt{1+f^{\prime
2}(x^{1})}}\delta_{\epsilon
1}\left[f(x^{1})f^{\prime\prime}(x^{1})+1+f^{\prime 2}(x^{1})\right]$ (21)
$\frac{\partial E}{\partial
z^{j}_{\eta}}\frac{\partial^{2}\varphi^{j}}{\partial x_{\epsilon}\partial
x_{\eta}}=2b^{2}\delta_{\epsilon 1}f(x^{1})f^{\prime}(x^{1})$ (22)
$\frac{\partial^{2}E}{\partial z^{i}_{\epsilon}\partial
z^{j}_{\eta}}\frac{\partial^{2}\varphi^{j}}{\partial x_{\epsilon}\partial
x_{\eta}}v^{i}=2b^{2}f(x^{1})\left[1+2f^{\prime 2}(x^{1})\right]$ (23)
$\frac{1}{2}\frac{\partial^{2}C^{2}}{\partial z^{i}_{\epsilon}\partial
z^{j}_{\eta}}\frac{\partial^{2}\varphi^{j}}{\partial x_{\epsilon}\partial
x_{\eta}}v^{i}=f(x^{1})\left[-f(x^{1})f^{\prime\prime}(x^{1})+1+f^{\prime
2}(x^{1})\right]$ (24)

Then putting all these values together in (7) and after some simplifications
we have

$\left(1-2f^{\prime 2}\right)\left(1+3ff^{\prime\prime}+f^{\prime 2}\right)=0$
(25)

Therefore, either $\left(1-2f^{\prime 2}\right)=0$, or,
$1+3ff^{\prime\prime}+f^{\prime 2}=0$

If $\left(1-2f^{\prime 2}\right)=0$ then
$f^{\prime}(x^{1})=\frac{1}{\sqrt{2}}$. Therefore, the minimal surface is an
open cone with the generating line $f(x^{1})=\frac{1}{\sqrt{2}}x^{1}+c$. Here
$c$ is a positive real number.
Now suppose $1+3ff^{\prime\prime}+f^{\prime 2}=0$. Let us consider
$p=f^{\prime}$. Then $f^{\prime\prime}=p\frac{dp}{df}$. Substituting these
values in the above ODE it reduces to

$1+3fp\frac{dp}{df}+p^{2}=0$ (26) $i.e.\frac{3p}{1+p^{2}}dp+\frac{1}{f}df=0$
(27)

Integrating we have,

$\frac{3}{2}\ln(1+p^{2})+\ln f=\frac{3}{2}\ln a,\quad\textnormal{where $a$ is
an integrating constant}.$

Simplifying we have,

$p=\sqrt{af^{-2/3}-1}$ (28) $i.e.\frac{df}{dx^{1}}=\sqrt{af^{-2/3}-1}$ (29)

Integrating, we have

$x^{1}=(-f^{2/3}-2a)\sqrt{a-f^{2/3}}+d,\quad\textnormal{where $d$ is an
integrating constant}.$ (30)

This is the precise generating curve for which the surface becomes a Finsler
minimal surface and therefore we obtain the proof of the Theorem 1.1.

##  4 Graph of a smooth function

In this section we study the graph of a smooth function
$f:U\subset\mathbb{R}^{2}\to\mathbb{R}$ in the Kropina space
$(\mathbb{R}^{3},F)$, where $\tilde{\alpha}$ is the Euclidean metric and
$\tilde{\beta}=bdx^{3}$ is a one-form with $b>0$. Here we consider the
immersion $\varphi:U\subset\mathbb{R}^{2}\to(\mathbb{R}^{3},F)$ given by
$\varphi(x^{1},x^{2})=(x^{1},x^{2},f(x^{1},x^{2}))$. Before proving Theorem
1.2. we need the following proposition:

###### Proposition 4.1

An immersion $\varphi:U\subset\mathbb{R}^{2}\to(\mathbb{R}^{3},F)$ given by
$\varphi(x^{1},x^{2})=(x^{1},x^{2},f(x^{1},x^{2}))$, where $f$ is a real
valued smooth function on $U\subset\mathbb{R}^{2}$ is minimal, if and only if,
$f$ satisfies

$\sum\limits_{\epsilon,\eta=1,2}\left[(W^{2}-1)(W^{2}-3)\left(\delta_{\epsilon\eta}-\frac{f_{x^{\epsilon}}f_{x^{\eta}}}{W^{2}}\right)+2(W^{2}+3)\frac{f_{x^{\epsilon}}f_{x^{\eta}}}{W^{2}}\right]f_{x^{\epsilon}x^{\eta}}=0,$
(31)

where, $W^{2}=1+f^{2}_{x^{1}}+f^{2}_{x^{2}}$.

### Proof:

As we already know that the mean curvature vanishes along the tangent vectors
of the immersed surface, we need to consider a vector field $v$ such that the
set $\left\\{v,\varphi_{x^{1}},\varphi_{x^{2}}\right\\}$ is linearly
independent. Therefore, we consider $v=\varphi_{x^{1}}\times\varphi_{x^{2}}$.
Then $v=\left(v^{1},v^{2},v^{3}\right)=\left(-f_{x^{1}},-f_{x^{2}},1\right)$.
From (11) we obtain the following:

$A=\begin{pmatrix}1+f^{2}_{x^{1}}&f_{x^{1}}f_{x^{2}}\\\
f_{x^{1}}f_{x^{2}}&1+f^{2}_{x^{2}}\\\ \end{pmatrix},\quad
C=\sqrt{detA}=W,\quad E=b^{2}\left(W^{2}-1\right),$ (32)

By some simple calculations we can have

$\frac{\partial C}{\partial z^{i}_{\epsilon}}v^{i}=0,\qquad\frac{\partial
E}{\partial z^{i}_{\epsilon}}v^{i}=2b^{2}(\delta_{\epsilon
1}f_{x^{1}}+\delta_{\epsilon 2}f_{x^{2}}),$ (33) $\frac{\partial C}{\partial
z^{j}_{\eta}}\frac{\partial^{2}\varphi^{j}}{\partial x^{\epsilon}\partial
x^{\eta}}v^{i}=\frac{f_{x^{1}}f_{x^{\epsilon}x^{1}}+f_{x^{2}}f_{x^{\epsilon}x^{2}}}{W},$
(34) $\frac{\partial E}{\partial
z^{j}_{\eta}}\frac{\partial^{2}\varphi^{j}}{\partial x^{\epsilon}\partial
x^{\eta}}=2b^{2}(f_{x^{1}}f_{x^{\epsilon}x^{1}}+f_{x^{2}}f_{x^{\epsilon}x^{2}}),$
(35) $\frac{\partial^{2}E}{\partial z^{i}_{\epsilon}\partial
z^{j}_{\eta}}\frac{\partial^{2}\varphi^{j}}{\partial x^{\epsilon}\partial
x^{\eta}}v^{i}=2b^{2}\left[(1+f^{2}_{x^{2}})f_{x^{1}x^{1}}-2f_{x^{1}}f_{x^{2}}f_{x^{1}x^{2}}+(1+f^{2}_{x^{1}})f_{x^{2}x^{2}}\right],$
(36) $\frac{1}{2}\frac{\partial^{2}C^{2}}{\partial z^{i}_{\epsilon}\partial
z^{j}_{\eta}}\frac{\partial^{2}\varphi^{j}}{\partial x^{\epsilon}\partial
x^{\eta}}v^{i}=\left[(1+f^{2}_{x^{2}})f_{x^{1}x^{1}}-2f_{x^{1}}f_{x^{2}}f_{x^{1}x^{2}}+(1+f^{2}_{x^{1}})f_{x^{2}x^{2}}\right].$
(37)

Then putting all these values in (7) we get (31). Hence, we complete the
proof. $\square$

###### Remark 4.1

If $W^{2}$ is constant then we have $\left(\frac{\partial f}{\partial
x^{1}}\right)^{2}+\left(\frac{\partial f}{\partial x^{2}}\right)^{2}=k$, where
$k$ is some real number. Then solving this partial differential equation we
get $f(x^{1},x^{2})=ax^{1}+\sqrt{k-a^{2}}x^{2}+c$, where $a$ and $c$ are
arbitrary constants. Therefore, the surface becomes a plane and clearly it
satisfies (31). Hence, it is minimal.

###### Definition 4.1

[19] A differential equation is said to be a elliptic equation of mean
curvature type on a domain $\Omega\subset\mathbb{R}^{2}$ if

$\sum\limits_{\epsilon,\eta=1,2}a_{\epsilon\eta}(x,f,\nabla
f)f_{x^{\epsilon}x^{\eta}}=0$ (38)

where $a_{\epsilon\eta},\epsilon,\eta=1,2$ are given real-valued functions on
$\Omega\times\mathbb{R}\times\mathbb{R}^{2}$, $x\in\Omega$,
$f:\Omega\to\mathbb{R}$ with

$|\xi|^{2}-\frac{(p\cdot\xi)^{2}}{1+|p|^{2}}\sum\limits_{\epsilon,\eta=1,2}a_{\epsilon\eta}(x,u,p)\xi_{\epsilon}\xi_{\eta}\leq\left(1+\mathcal{C}\right)\left[|\xi|^{2}-\frac{(p\cdot\xi)^{2}}{1+|p|^{2}}\right]$
(39)

for all $u\in\mathbb{R}$, $p\in\mathbb{R}^{2}$ and
$\xi\in\mathbb{R}^{2}\setminus\left\\{0\right\\}$.

Therefore, we can write equation (4.1) as

$\sum\limits_{\epsilon,\eta=1,2}a_{\epsilon\eta}(x,f,\nabla
f)f_{x^{\epsilon}x^{\eta}}=0$ (40)

where,

$a_{\epsilon\eta}:=\begin{cases}\delta_{\epsilon\eta}-\frac{f_{x^{\epsilon}}f_{x^{\eta}}}{W^{2}}+2\frac{W^{2}+3}{(W^{2}-1)(W^{2}-3)}\left(\frac{f_{x^{\epsilon}}f_{x^{\eta}}}{W^{2}}\right)&\mbox{if
}W^{2}>3\\\
(W^{2}+3)\left(\frac{f_{x^{\epsilon}}f_{x^{\eta}}}{W^{2}}\right)&\mbox{if
}W^{2}=3\end{cases}$ (41)

Case-1: Now let us assume $W^{2}=3$. Then clearly, (31) satisfies (39).
Case-2: Now assume $W^{2}>3$. Let us consider
$\xi\in\mathbb{R}^{2}\setminus\left\\{0\right\\}$, $x,t\in\mathbb{R}^{2}$ and
$u\in\mathbb{R}$ and we define

$h_{\epsilon\eta}(u)=\delta_{\epsilon\eta}-\frac{t_{\epsilon}t_{\eta}}{W^{2}(u)}.$
(42)

Hence, we have,

$\sum\limits_{\epsilon,\eta=1}^{2}h_{\epsilon\eta}(t)\xi_{\epsilon}\xi_{\eta}=\frac{|\xi|^{2}}{W^{2}}(1+|t|^{2}\sin^{2}\theta),$
(43)

where, $\theta$ is the angle function between $t$ and $\xi$. We also have from

$\sum\limits_{\epsilon,\eta=1}^{2}a_{\epsilon\eta}(x,u,t)\xi_{\epsilon}\xi_{\eta}=\sum\limits_{\epsilon\eta=1}^{2}h_{\epsilon\eta}(t)\xi_{\epsilon}\xi_{\eta}+2\frac{W^{2}+3}{(W^{2}-1)(W^{2}-3)}\left[\frac{t\cdot\xi}{W}\right]^{2},$
(44)

where $\cdot$ represents the Euclidean inner product.
Hence, it is evident from (43) and (44) that for all
$\xi\in\mathbb{R}^{2}\setminus\left\\{0\right\\}$,

$\sum\limits_{\epsilon,\eta=1}^{2}a_{\epsilon\eta}(x,u,t)\xi_{\epsilon}\xi_{\eta}>0.$
(45)

Hence, (40) is an elliptic equation.
Now we prove that it is a differential equation of mean curvature type for
which we need to show that there exists a constant $\mathcal{C}$ such that,
for all

$\sum\limits_{\epsilon,\eta=1}^{2}h_{\epsilon\eta}(x,u,t)\xi_{\epsilon}\xi_{\eta}\leq\sum\limits_{\epsilon,\eta=1}^{2}a_{\epsilon\eta}(x,u,t)\xi_{\epsilon}\xi_{\eta}\leq(1+\mathcal{C})\sum\limits_{\epsilon,\eta=1}^{2}h_{\epsilon\eta}(x,u,t)\xi_{\epsilon}\xi_{\eta}.$
(46)

The first inequality is immediate from (45). To prove the second inequality we
need to show that

$2\frac{W^{2}+3}{(W^{2}-1)(W^{2}-3)}\left[\frac{t\cdot\xi}{W}\right]^{2}\leq\mathcal{C}\sum\limits_{\epsilon,\eta=1}^{2}h_{\epsilon\eta}(x,u,t)\xi_{\epsilon}\xi_{\eta},$
(47)

That is, we need to show that

$\frac{2(W^{2}+3)|t|^{2}}{(W^{2}-1)(W^{2}-3)(1+|t|^{2}\sin^{2}\theta)}\leq\mathcal{C}$
(48)

The left hand side of (48) is a rational function of $|t|$. Here the numerator
is of degree less than or equal to $4$, and denominator is of degree $6$.
Therefore, it is a bounded function of $|t|$ as $|t|$ goes to infinity.
Now if $W^{2}>m$, for some real number $m>3$, then again the left hand side of
(48) is bounded by the similar reason given above.
Therefore, from the above two cases and by the continuity of $W^{2}$ it can be
said that (31) satisfies (39) whenever $W^{2}\geq 3$.

Now the theorem proved by L. Simon (Theorem 4.1 of [20]) and from the above
discussion, we conclude Theorem 1.2.

##  5 The characterization of minimal surfaces of translation surfaces

In this section we study the minimal translation surface $M^{2}$ in Kropina
space $(\mathbb{R}^{3},F)$, where $\mathbb{R}^{3}$ is a real Minkowski space
and $F=\frac{{\tilde{\alpha}}^{2}}{\tilde{\beta}}$ is a Kropina metric, where
$\tilde{\alpha}$ is the Euclidean metric and $\tilde{\beta}=bdx^{3}$ is a one-
form with $b>0$. Here we consider the immersion
$\varphi:U\subset\mathbb{R}^{2}\to(\mathbb{R}^{3},F)$ given by
$\varphi(x^{1},x^{2})=(x^{1},x^{2},f(x^{1})+g(x^{2}))$.

Let us consider the following immersion:

$\varphi(x^{1},x^{2})=(\varphi^{1},\varphi^{2},\varphi^{3})=\left(x^{1},x^{2},f(x^{1})+g(x^{2})\right)$
(49)

Then we can write

$\varphi^{j}=\delta_{j1}x_{1}+\delta_{j2}x_{2}+(f+g)\delta_{j3},\quad
1\leq\epsilon\leq 3.$ (50)

Therefore, from (11) and (49) we get

$A=\begin{pmatrix}1+f^{2}_{x^{1}}&f_{x^{1}}g_{x^{2}}\\\
f_{x^{1}}g_{x^{2}}&1+g^{2}_{x^{2}}\\\ \end{pmatrix},\quad
C=\sqrt{detA}=\sqrt{1+f^{2}_{x^{1}}+g^{2}_{x^{2}}}\quad\textnormal{and}\quad
E=b^{2}(f^{2}_{x^{1}}+g^{2}_{x^{2}})$ (51)

Here we choose $v=\varphi_{x^{1}}\times\varphi_{x^{2}}$. Then
$v=(v^{1},v^{2},v^{3})=(-f_{x^{1}},-g_{x^{2}},1)$. Hence,
$v^{i}=-\delta_{i3}f_{x^{1}}-\delta_{i3}g_{x^{2}}+\delta_{i3},\quad 1\leq
i\leq 3$.
By some simple calculations we can have

$\frac{\partial C}{\partial z^{i}_{\epsilon}}v^{i}=0,\quad\frac{\partial
E}{\partial z^{i}_{\epsilon}}v^{i}=2b^{2}(\delta_{\epsilon
1}f_{x^{1}}+\delta_{\epsilon 2}g_{x^{2}}),$ (52) $\frac{\partial C}{\partial
z^{j}_{\eta}}\frac{\partial^{2}\varphi^{j}}{\partial x^{\epsilon}\partial
x^{\eta}}=\frac{\delta_{\epsilon 1}f_{x^{1}}f_{x^{1}x^{1}}+\delta_{\epsilon
2}g_{x^{2}}g_{x^{2}x^{2}}}{C},$ (53) $\frac{\partial E}{\partial
z^{j}_{\eta}}\frac{\partial^{2}\varphi^{j}}{\partial x^{\epsilon}\partial
x^{\eta}}=2b^{2}(\delta_{\epsilon 1}f_{x^{1}}f_{x^{1}x^{1}}+\delta_{\epsilon
2}g_{x^{2}}g_{x^{2}x^{2}}),$ (54) $\frac{\partial^{2}E}{\partial
z^{i}_{\epsilon}\partial z^{j}_{\eta}}\frac{\partial^{2}\varphi^{j}}{\partial
x^{\epsilon}\partial
x^{\eta}}v^{i}=2b^{2}\left[(1+g^{2}_{x^{2}})f_{x^{1}x^{1}}+(1+f^{2}_{x^{1}})g_{x^{2}x^{2}}\right],$
(55) $\frac{\partial^{2}C^{2}}{\partial z^{i}_{\epsilon}\partial
z^{j}_{\eta}}\frac{\partial^{2}\varphi^{j}}{\partial x_{\epsilon}\partial
x_{\eta}}v^{i}=2\left[(1+g^{2}_{x^{2}})f_{x^{1}x^{1}}+(1+f^{2}_{x^{1}})g_{x^{2}x^{2}}\right],$
(56)

Then substituting all these above values in (7) we obtain

$\lambda f_{x^{1}x^{1}}+\mu g_{x^{2}x^{2}}=0$ (57)

where,

$\lambda=8f^{2}_{x^{1}}+2f^{2}_{x^{1}}(f^{2}_{x^{1}}+g^{2}_{x^{2}})+\left(1+g^{2}_{x^{2}}\right)(f^{2}_{x^{1}}+g^{2}_{x^{2}})(f^{2}_{x^{1}}+g^{2}_{x^{2}}-2)$
(58)

and

$\mu=8g^{2}_{x^{2}}+2g^{2}_{x^{2}}(f^{2}_{x^{1}}+g^{2}_{x^{2}})+\left(1+f^{2}_{x^{1}}\right)(f^{2}_{x^{1}}+g^{2}_{x^{2}})(f^{2}_{x^{1}}+g^{2}_{x^{2}}-2)$
(59)

Let $r=\left(f_{x^{1}}\right)^{2}$ and $s=\left(g_{x^{2}}\right)^{2}$. Then

$f_{x^{1}x^{1}}=\frac{r_{f}}{2},\qquad g_{x^{2}x^{2}}=\frac{s_{g}}{2}.$ (60)

Then (58) and (59) becomes

$\lambda=8r+2r(r+s)+(1+s)(r+s)(r+s-2)$ (61)

and

$\mu=8s+2s(r+s)+(1+r)(r+s)(r+s-2)$ (62)

And (57) becomes

$r_{f}\lambda+s_{g}\mu=0$ (63)

Therefore, we have two cases:
Case 1: If $r_{f}=0$ or, $s_{g}=0$, then $r$ and $s$ are constant functions.
And hence $f$ and $g$ are linear functions. Therefore, $M^{2}$ is a plane in
$(V^{3},F)$ locally.
Case 2: Let $r_{f}\neq 0$ and $s_{g}\neq 0$. Then we have, $\lambda\neq 0$ and
$\mu\neq 0$. Suppose

$\kappa=\frac{r_{f}}{\mu}=-\frac{s_{g}}{\lambda}.$ (64)

Which implies that

$(r_{f})_{g}=\mu_{g}\kappa+\mu\kappa_{g}=0\quad\textnormal{and}\quad(s_{g})_{f}=\lambda_{f}\kappa+\lambda\kappa_{f}=0$

Hence, we have,

$\left(\log{\kappa}\right)_{g}=\frac{\kappa_{g}}{\kappa}=-\frac{\mu_{g}}{\mu}\quad\textnormal{and}\quad(\log{\kappa})_{f}=\frac{\kappa_{f}}{\kappa}=-\frac{\lambda_{f}}{\lambda}.$
(65)

Since, $(\log{\kappa})_{fg}=(\log{\kappa})_{gf}$, we have,

$\left(\frac{\lambda_{f}}{\lambda}\right)_{g}=\left(\frac{\mu_{g}}{\mu}\right)_{f}.$
(66)

We can easily observe that, $r_{g}=(r_{f})_{g}=0$ and $s_{f}=(s_{g})_{f}=0$.
Therefore, we have,

$\left(\frac{\lambda_{f}}{\lambda}\right)_{g}=\left(\frac{\lambda_{r}r_{f}}{\lambda}\right)_{g}=\left(\frac{\lambda_{r}}{\lambda}\right)_{g}r_{f}=\left(\frac{\lambda_{r}}{\lambda}\right)_{s}r_{f}s_{g}$
(67)

and

$\left(\frac{\mu_{g}}{\mu}\right)_{f}=\left(\frac{\mu_{s}s_{g}}{\mu}\right)_{f}=\left(\frac{\mu_{s}}{\mu}\right)_{f}s_{g}=\left(\frac{\mu_{s}}{\mu}\right)_{r}r_{f}s_{g}$
(68)

Using (67) and (68) in (66) we get,

$\left(\frac{\lambda_{r}}{\lambda}\right)_{s}=\left(\frac{\mu_{s}}{\mu}\right)_{r}.$
(69)

That is,

$\left(\log\frac{\lambda}{\mu}\right)_{rs}=0$ (70)

Let $p=r+s$ and $q=r-s$. Then we have

$\lambda=K(p)-L(p)q,\qquad\mu=K(p)+L(p)q$ (71)

where,

$K(p)=4p+p^{2}+p(p-2)+\frac{1}{2}p^{2}(p-2)$ (72) $L(p)=\frac{1}{2}p(p-2)-p-4$
(73)

Now from (70) it follows that

$\left(\log\frac{\lambda}{\mu}\right)_{rs}=\left(\log\frac{\lambda}{\mu}\right)_{pp}-\left(\log\frac{\lambda}{\mu}\right)_{qq}=0$
(74)

Now substitute the values of $\lambda$ and $\mu$ in (74) we get

$\begin{split}q^{3}\left(K_{pp}L^{3}-KL^{2}L_{pp}-2K_{p}L_{p}L^{2}+2KLL^{2}_{p}\right)\\\
+q\left(-K_{pp}K^{2}L+K^{3}L_{pp}-2K_{p}K^{2}L_{p}+2K^{2}_{p}KL-2KL^{3}\right)=0.\end{split}$
(75)

Since, $q$ is an arbitrary function we get,

$K_{pp}L^{3}-KL^{2}L_{pp}-2K_{p}L_{p}L^{2}+2KLL^{2}_{p}=0$ (76)
$-K_{pp}K^{2}L+K^{3}L_{pp}-2K_{p}K^{2}L_{p}+2K^{2}_{p}KL-2KL^{3}=0$ (77)

Multiplying (76) by $\frac{N}{M}$, and (77) by $\frac{M}{N}$ and then adding
them together, we obtain

$\left[\left(\frac{K}{L}\right)_{p}\right]^{2}=1.$ (78)

Again from (72) and (73) we have

$\frac{K}{L}=p+\frac{4p^{2}+12p}{p^{2}-4p-8}$ (79)

Now differentiating (79) with respect to $p$ we get

$\left(\frac{K}{L}\right)_{p}=1-4\frac{p^{2}+34p+12}{(p^{2}-4p-8)^{2}}$ (80)

Squaring both sides of (80) gives

$\left[\left(\frac{K}{L}\right)_{p}\right]^{2}=1-8\frac{p^{2}+34p+12}{(p^{2}-4p-8)^{2}}+16\left(\frac{p^{2}+34p+12}{(p^{2}-4p-8)^{2}}\right)^{2}$
(81)

Hence, the right hand side of (81) is a rational function of $p$. Therefore,
it can not be equal to $1$ identically. This argument contradicts (78). Hence
it leads to the fact that case 1 is true and hence we obtain Theorem 1.3.

## References

  * [1] Antonelli, P.L., Ingarden, R.S., Matsumoto, M.: The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology. Kluwer, Dordrecht (1993)
  * [2] Balan, V.: Constant mean curvature submanifolds in $(\alpha,\beta)$-Finsler spaces. Differential Geom. Appl. 45–55 (2007)
  * [3] Bao, D., Robbles, C., Shen Z.: Zermelo navigation on Riemannian manifolds. J. Differential Geom. 66(3), 377–435 (2004).
  * [4] Caratheodory, C.: Calculus of Variations and Partial Differential Equations of the First Order, (Translated by Robert B. Dean). AMS Chelsea Publishing, Berlin (2006). (Originally published 1935, Berlin)
  * [5] Cheng, X., Shen, Z., A class of Finsler metrics with isotropic S-curvature. Israel. J. Math. 169, 317–340 (2009)
  * [6] Chern, S.S., Shen, Z.: Riemannian-Finsler geometry. World Scientific Publisher, Singapore (2005).
  * [7] Colding, T., Minicozzi, W.P.: A Course in Minimal Surfaces. American Mathematical Society (2011).
  * [8] Cui, N.: On minimal surfaces in a class of Finsler 3-spheres. Geom. Dedicata. 168, 87–100 (2014)
  * [9] Cui, N.: Nontrivial minimal surfaces in a class of Finsler 3-spheres. Int. J. Math. 25(4), 1450034(17 pages) (2014).
  * [10] Cui, N., Shen, Y.B.: Nontrivial minimal surfaces in a hyperbolic Randers space. Math. Nachr. 290(4), 570–582 (2017).
  * [11] Gangopadhyay, R., Tiwari, B.: Minimal surfaces in three-dimensional Matsumoto space. arXiv:2007.09439 [math.DG]. (2020).
  * [12] Itoh, T., Minimal surfaces in a Riemannian manifold of constant curvature. Kodai Math. Sem. Rep. 25(2), 202-214 (1973)
  * [13] Kropina, V.K.: On projective two-dimensional finsler spaces with special metric. Trudy Sem. Vector. Tensor. Anal. 11, 277 (1961).
  * [14] Matsumoto, M., Theory of Finsler spaces with $(\alpha,\beta)$-metric. Rep. Math. Phys. 31(1), 43-83 (1992).
  * [15] Osserman, R.: A Survey of minimal surfaces. Dover Publication (2002).
  * [16] Rado, T.: On Plateau’s proble. Ann. of Math. 31, 457-469 (1930).
  * [17] Shen, Z.: On Finsler geometry of submanifolds. Math. Ann., 311, 549-576 (1998).
  * [18] da Silva, R.M., Tenenblat, K.: Helicoidal Minimal Surfaces in a Finsler Space of Randers Type, Canad. Math. Bull. 57(4), 765–779 (2014)
  * [19] Simon, L.: Equations of mean curvature type in 2 independent variables. Pac. J. Math. 69, 245–268 (1977)
  * [20] Simon, L.: A Holder Estimate for Quasiconformal Maps Between Surfaces in Euclidean Space. Acta Math. 139, 19–51 (1977).
  * [21] Souza, M., Tenenblat, K.: Minimal surfaces of rotation in Finsler space with a Randers metric. Math. Ann. 325, 625–642 (2003).
  * [22] Souza, M., Spruck, J., Tenenblat, K.: A Berstein type theorem on a Randers space. Math. Ann. 329, 291–305 (2004).
  * [23] Yoshikawa, R., Sabau, S.V.: Kropina metrics and Zermelo navigation on Riemannian manifolds. Geom. Dedicata. 171, 119–148 (2014).