# Explaining deep learning of galaxy morphology with saliency mapping

Prabh Bhambra,1 Benjamin Joachimi,1 Ofer Lahav1
1Department of Physics and Astronomy, University College London, Gower Street,
London WC1E 6BT, UK E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We successfully demonstrate the use of explainable artificial intelligence
(XAI) techniques on astronomical datasets in the context of measuring galactic
bar lengths. The method consists of training convolutional neural networks on
human classified data from Galaxy Zoo in order to predict general galaxy
morphologies, and then using SmoothGrad (a saliency mapping technique) to
extract the bar for measurement by a bespoke algorithm. We contrast this to
another method of using a convolutional neural network to directly predict
galaxy bar lengths. These methods achieved correlation coefficients of 0.76
and 0.59, and root mean squared errors of 1.69 and 2.10 respective to human
measurements. We conclude that XAI methods outperform conventional deep
learning in this case, which could be reasonably explained by the larger
datasets available when training the models. We suggest that our XAI method
can be used to extract other galactic features (such as the bulge-to-disk
ratio) without needing to collect new datasets or train new models. We also
suggest that these techniques can be used to refine deep learning models as
well as identify and eliminate bias within training datasets.

###### keywords:

methods: data analysis – techniques: image processing – galaxies: general –
galaxies: bar

††pubyear: 2021††pagerange: Explaining deep learning of galaxy morphology with
saliency mapping–B

##  1 Introduction

From autonomous vehicles to targeted advertising, machine learning (ML)
algorithms have become more and more ubiquitous in society and a greater
emphasis is being put on understanding the decisions they make. We call an
algorithm interpretable if it can be understood by humans on its own without
much analysis. Algorithms themselves vary in their interpretability, with the
likes of decisions trees tending towards being more interpretable, while
neural networks tend towards being some of the least interpretable (Došilović
et al., 2018). Despite their opacity, neural networks are widely used due to
their relatively good performance compared to other ML algorithms. The field
of explainable artificial intelligence (XAI) seeks to help explain the
behaviour of difficult to interpret algorithms (Adadi & Berrada, 2018).

ML has been widely adopted in the the field of astronomy, although to date
relatively little thought has been given to the field of XAI. This paper aims
to use XAI methods to generalise ML models trained on astronomical datasets.
Due to astronomy’s reliance on telescopic imaging, we considered datasets
primarily consisting of images. Due to the richness of the data, the Galaxy
Zoo project was chosen (Willett et al., 2013b). To our knowledge this is the
first attempt at quantitatively using saliency mapping for image processing in
astronomy, although there are recent cases of them being used qualitatively
for spectral analysis (Peruzzi et al., 2021) and image processing (Villanueva-
Domingo & Villaescusa-Navarro, 2021). This paper intends to act as a proof of
concept that these techniques are effective when applied to this field, and
that they can be used to infer quantitative measurements.

Galaxy Zoo is a citizen science project in which volunteer participants are
shown images of galaxies, and then asked a series of questions regarding their
morphologies. Human classification of galaxy morphologies is extremely time
consuming and requires expert level domain knowledge. However the use of
citizen science platforms helps to reduce the reliance on these limited
resources. Showing the same image to multiple participants can be used reach a
crowd-sourced consensus of the galaxy morphology without requiring extreme
time commitment from any single participant (Lintott et al., 2008).

There have been early applications of ML methods to aid in the classifications
of galaxies (e.g. Lahav et al., 1995; Lahav et al., 1996; Banerji et al.,
2010). Recently, the Galaxy Zoo project has provided a rich data source for
the training of these ML models. Many implementations can be seen on the
Kaggle competition “Galaxy Zoo - The Galaxy Challenge” in which participants
were tasked to train an ML model on a subset of Galaxy Zoo data in order to
predict galaxy morphologies. Dieleman et al. (2015) provided the winning
submission to the contest by using convolutional neural networks (CNNs), a
model architecture which is widely used in image processing due to their
excellent performance (Aloysius & Geetha, 2017). More recently, Domínguez
Sánchez et al. (2019) used Galaxy Zoo data to train ML models which were then
used to classify galaxies from the Dark Energy Survey. Walmsley et al. (2020)
used Bayesian CNNs, which can more effectively learn from examples with
uncertain labels, to predict posteriors for the morphologies of galaxies.

XAI methods can be used to produce saliency maps to show which areas in an
image are most important to a CNN’s classification of an image. This paper
aims to use SmoothGrad, an XAI method developed to give visual explanations of
CNNs, to produce saliency maps explaining which parts of an image are
important in the image’s classification as a barred galaxy. From these
saliency maps the length of the bars can then be measured and compared to
other methods of measurement. Galactic bars are of particular interest due to
their ability to contribute to the redistribution of matter in a galaxy
through transferring angular momentum between the disc, bulge, and any spiral
arms or rings present (Kormendy & Kennicutt, 2004; Athanassoula et al., 2009).
Bar lengths are an especially good application of XAI due to the simplicity of
their geometric structure. Galaxies are seen in 2D projection, and the highly
linear structure of bars makes it straightforward to quantify the output of
XAI methods and compare this quantification to other methods of measurement.
This comparison can be used to demonstrate the predictive power of CNN
classifications. We aim to show that XAI methods can be used to extract new
information, which has never been presented to the CNN, from its
classifications.

This paper is structured as follows: Section 2 describes the data used
throughout this paper, while Sections 3 and 4 describe the XAI methods used
and the modelling process respectively. Section 5 describes the results of
comparing our methods to human measurements. Finally, Section 6 contains our
conclusion.

##  2 Data

The data used to train the models was taken from Kaggle’s “Galaxy Zoo - The
Galaxy Challenge” dataset111https://www.kaggle.com/c/galaxy-zoo-the-galaxy-
challenge/. This dataset consists of galaxy images each with dimensions
$424\times 424\times 3$ (i.e. a height and width of 424 pixels, and 3 (red,
green, and blue) colour channels). The images were partitioned into a training
set and a test set. The training set consisted of 61,578 images with
associated galaxy classification probability distributions. The probability
distributions represent the proportion of Galaxy Zoo volunteers who associated
the galaxy into each of the 37 classes. The test set consisted of 79,975
galaxy images without labels (as the testing of the models is handled solely
by Kaggle’s submission system). The galaxy images are composites of images
taken by the Sloan Digital Sky Survey (SDSS) camera through its u, g, r, i,
and z filters (Gunn et al., 1998).

The Hoyle bar length catalogue (Hoyle et al., 2011) was collected as an
extension to the Galaxy Zoo project. Galaxy Zoo participants were asked to
measure the length of the bars of 3,150 barred galaxies by drawing a line on
the images using a web interface. With multiple participants measuring the
bars on each galaxy, participants were able to reproduce each other’s
measurements with a scatter of 17 per cent (Hoyle et al., 2011). The bar
length measurements in the Hoyle catalogue as well as the Galaxy Zoo images
were scaled by the petroR90_r value (the radius containing 90 per cent of the
r-band Petrosian aperture flux) for the respective galaxy (Willett et al.,
2013b; Hoyle et al., 2011), hence this data was required to undo this scaling.
These values were taken from the Galaxy Zoo metadata file hosted on the
project’s data page222https://data.galaxyzoo.org/. The use of the bar length
catalogue also required a full set of Galaxy Zoo 2 images (Willett et al.,
2013a).

The Hoyle bar length catalogue also includes an error that introduces a
proportional offset. The values in the catalogue are quoted in units of
$h^{-1}~{}\rm{kpc}$, but are actually in units of $\rm{kpc}$ with $h$ assumed
to be equal to $0.7$. Therefore, in order to use the catalogue in its intended
units of $h^{-1}~{}\rm{kpc}$, we multiplied through by $h=0.7$ (Walmsley &
Geron, personal communication, 15 November 2021).

##  3 Saliency mapping

Saliency mapping is a term used to refer to a family of techniques seeking to
explain the behaviour of CNNs. Saliency maps are heatmaps used to demonstrate
the relative importance of parts of an image to the classification of that
image into a certain class by the CNN in question (Smilkov et al., 2017).
These heatmaps are often computed by taking gradients from the output layer of
the CNN to various other layers, in this way the resulting saliency map can be
seen as visualising the sensitivity of an output to a given input (Simonyan et
al., 2013).

Various methods were considered for use. Grad-CAM (Selvaraju et al., 2016) and
Grad-CAM++ (Chattopadhyay et al., 2017) take gradients from the output of the
CNN to the penultimate convolution layer in order to assign importance to
neurons within the CNN. This approach produces saliency maps which are able to
locate features well but are unable to pull out finer detail within the image.
Conversely, Guided Backpropagation (Springenberg et al., 2014) manages to
highlight features well, but is unable to strongly isolate important areas of
the image. We ultimately decided to use SmoothGrad (Smilkov et al., 2017),
which provides explanations up to the pixel space of the input image. This
allows SmoothGrad to pick out the finer detail in the image when explaining
the behaviour of the CNN.

SmoothGrad builds on top of the idea of traditional saliency mapping (Simonyan
et al., 2013), in which the importance ($L$) of a pixel’s intensity ($x$) to
the classification of class ($c$) is given by the gradient of the score of the
relevant class ($y$) with respect to the pixel intensity:

$L^{c}(x)=\frac{\partial y^{c}}{\partial x}.$ (1)

$L^{c}$ is calculated for each pixel within the image and then a two-
dimensional saliency map is constructed by stacking the individual values of
$L^{c}$.

However, due to modern models combining many linear and non-linear (e.g.
Rectified Linear Unit) operations, these gradients are locally unstable. Small
fluctuations in the value of a pixel create visually similar images, however
these can result in wildly different saliency maps (Smilkov et al., 2017).
SmoothGrad aims to address this issue by creating $N$ visually similar samples
of the same image by adding Gaussian noise to each, and then averaging the
saliency maps for each sample:

$\hat{L}^{c}(x)=\frac{1}{N}\sum^{N}_{i=1}L^{c}[x+\mathcal{N}(0,\sigma^{2})],$
(2)

where $\mathcal{N}(0,\sigma^{2})$ is the probability density function for a
Gaussian distribution with a mean of $0$ and a standard deviation of
$\sigma^{2}$. Both $N$ and $\sigma^{2}$ are free parameters in the SmoothGrad
algorithm.

Examples of SmoothGrad saliency maps can be found in Figure 1. The galaxy
image was used to generate saliency maps explaining why models classified the
galaxy as a barred galaxy with a prominent bulge and spiral arms. The saliency
maps were compiled by averaging the individual saliency maps generated from
each of the models described in Section 4.2, using SmoothGrad parameters of
$N=256$ and $S_{\rm{noise}}=0.1$ (where $S_{\rm{noise}}$ is a transformation
of $\sigma^{2}$, see Section 4.5). This example demonstrates SmoothGrad’s
ability to pick out features of the galaxy when explaining the CNN’s decisions
to classify the image into various classes. It can be seen that SmoothGrad is
successfully able to highlight the bar, bulge, and spiral arms in each
successive saliency map.

Figure 1: Example SmoothGrad saliency maps explaining the classification of a
galaxy image (left) as a barred (centre-left) galaxy with a prominent bulge
(centre-right) and spiral arms (right). Red pixels represent pixels with the
highest importance $\hat{L}^{c}$, while blue represents those with the least
importance. The saliency maps were generated using the models described in
Section 4.2. All images were cropped to the centre $100\times 100$ pixels for
this figure. The galaxy shown is identified by the SDSS Objid of
587722983363248237, which was chosen due to its clear demonstration of
multiple features. Similar saliency maps for a random selection of galaxies
can be found in Appendix A.

##  4 Machine learning model

###  4.1 Data augmentation

Initial attempts at training models directly on unaugmented images resulted in
poor convergence, therefore image augmentation was used. The images were down-
sampled to dimensions of $224\times 224\times 3$ in order to reduce the impact
of noise, and were also subjected to random augmentations during each epoch in
order to reduce overfitting:

  1. 1.

random rotations in the range $0^{\circ}-360^{\circ}$,

  2. 2.

random horizontal and vertical flips,

  3. 3.

random horizontal and vertical shifts in the range of 0-5 per cent of the
image dimensions,

  4. 4.

random brightness scaling in the range of 70-130 per cent.

###  4.2 Model architecture

In total, three different models were trained to predict galaxy morphologies,
each based on a common image classification model (the VGG16 (Simonyan &
Zisserman, 2014), ResNet50v2 (He et al., 2016), and Xception (Chollet, 2016)
architectures). Each model was created with a number of input nodes to match
the size of the down-sampled images, and 37 (i.e. the number of Galaxy Zoo
classes) nodes in the output layer. All models were implemented with the
“imagenet” pre-trained weights from the Keras API for TensorFlow (Abadi et
al., 2016).

###  4.3 Training

The models were trained with the augmented images as the input and the Galaxy
Zoo class probability distributions as the targeted output. 5 per cent of the
training dataset was used for validation to monitor overfitting.

The models were trained in order to minimise the root mean squared error
(RMSE) between the model’s predictions and the Galaxy Zoo class probability
distributions of galaxy morphologies. This was done by using an Adam optimiser
(Kingma & Ba, 2014). The learning rate was also set to reduce by a factor of
0.5 if the loss on the validation dataset did not decrease over the course of
five epochs. The models were set to be trained for up to 100 epochs; however
training was halted in the case that the RMSE of the validation set did not
decrease over the course of 10 epochs.

The learning curves for the models can be found in Figure 2. The final metrics
for each model can be found in Table 1 which shows the number of epochs each
model trained for before stopping, as well as the RMSE on the training,
validation, and test sets. The models were also ensembled by averaging the
predictions of each individual model. The evaluation of the models on the test
set was handled by Kaggle’s submission system. Table 1 shows little variation
in the performance of the models, which demonstrates the robustness of our
training methods to model architecture choices. These models performed
slightly worse than those trained by Dieleman et al. (2015) due to not using
as intensive data augmentation practices.

Figure 2: The learning curves for the three CNNs architectures trained. The losses for the VGG16 (red), ResNet50v2 (green), and Xception (blue) based models are shown for both the training (solid) and validation (dashed) sets. Table 1: RMSE on the training, validation, and test sets for each model architecture, including an ensembled model created by averaging the predictions of the other three models. Model | Epochs | Train RMSE | Val RMSE | Test RMSE
---|---|---|---|---
VGG16 | 58 | 0.07315 | 0.07818 | 0.07939
ResNet50v2 | 53 | 0.07584 | 0.07874 | 0.08110
Xception | 29 | 0.07206 | 0.07774 | 0.07933
Ensemble | N/A | N/A | N/A | 0.07747

###  4.4 Bar length measurement

We wanted to find a way to quantify how well SmoothGrad was able to explain
the behaviour of the CNNs and how much information could be extracted from the
CNN’s classifications. Attempting to measure bar lengths from saliency maps
was decided upon due to the availability of human measured bar lengths which
could act as “ground truth” measurements. SmoothGrad was used to generate
saliency maps highlighting the pixels within each galaxy in the Hoyle Bar
Length catalogue that had the most importance (i.e. highest $\hat{L}^{c}$) in
classifying the images as barred galaxies by the models. SmoothGrad has two
hyperparameters (Smilkov et al., 2017):

  1. 1.

$N$ \- the number of samples to consider,

  2. 2.

$\sigma^{2}$ \- the standard distribution of the Gaussian noise to be added to
each sample.

$N$ was set to 256. A greater number of samples would lead to less noisy
saliency maps, but this would increase the required computing time
significantly. The optimal value of $\sigma^{2}$ is dependent on the dataset
being considered, therefore this was considered as a hyperparameter to be
optimised.

In addition, elementwise multiplication of the saliency map with the input
image is considered. This hyperparameter will be denoted as $X_{\rm{input}}$
and acts as a boolean flag. As we expect the bar of the galaxy to be brighter
than the average brightness of the image, this should bias the saliency maps
towards highlighting features of the galaxy rather than mistakenly
highlighting any other artefacts or parts of the background.

Algorithm 4 was used to measure the bar length of a galaxy from its respective
saliency map. The algorithm isolates the bar of the galaxy by zeroing out
pixels in the saliency map below a specific threshold. This threshold is
incremented until either the non-zero pixels are linearly distributed over the
image with a Pearson correlation coefficient greater than a user-specified
value, or the number of non-zero pixels is less than a user-specified required
number of pixels. This algorithm requires two hyperparameters to be optimised:

  1. 1.

$T_{\rm{c}}$ the Pearson correlation threshold over which the non-zero pixels
are considered linearly distributed;

  2. 2.

$T_{\rm{l}}$ \- the lower bound on the number of non-zero pixels required.

The bar lengths returned by Algorithm 4 are measured in units of pixels.
However, Galaxy Zoo images are scaled (Willett et al., 2013b) by
$(\texttt{petroR90\\_r}\times 0.02\times\texttt{pixel\\_scale})$ for each
galaxy, where pixel_scale is equal to $0.396~{}\rm{arcsec~{}pixel}^{-1}$ (Gunn
et al., 1998). Therefore our measured bar lengths were later multiplied by
this value to convert the measurements to units of $h^{-1}~{}\rm{kpc}$. The
measured lengths also had to be multiplied by a factor of $(424/224)$ in order
to undo the image down-sampling described in Section 4.1.

Input: saliencyMap, $T_{\rm{c}}$, $T_{\rm{l}}$

Output: barLength

for  _$T_{h}$ = 0.5 to 1.0 by 0.0001_ do

x, y = where(saliencyMap < $T_{h}$)

corr = corr(x, y)

size = len(x)

if  _corr > $T_{\rm{c}}$ or size < $T_{\rm{l}}$_ then

break

end if

end for

x, y = where(saliencyMap > $T_{h}$)

barLength = 0

for  _i = 0 to len(x)_ do

for  _j = 0 to len(x)_ do

length = $\sqrt{(x[i]-x[j])^{2}+(y[i]-y[j])^{2}}$

if  _length > barLength_ then

barLength = length

end if

end for

end for

return barLength

Algorithm 1 Measuring the bar length from a SmoothGrad saliency map. 444The
corr() function calculates the Pearson correlation coefficient between two
variables, and the where() function returns the indices of an array at which
the condition is evaluated as True. Figure 3: Examples of isolating the bar
from galaxy images for three different galaxies. The leftmost column shows the
initial image cropped to the centre $100\times 100$ pixels. The middle column
shows a saliency map for each galaxy highlighting the pixels most important in
classifying the images as barred galaxies. The rightmost column shows the use
of Algorithm 4 in isolating the bar from the relevant saliency maps. The first
two examples were chosen at random, however the bottom example was chosen to
present a case where this method fails (discussed in Section 5.1). The
galaxies shown are identified by their SDSS Objids 588848901007933476,
588017977830342894, and 588016878291124448 from top to bottom. Similar
saliency maps for a random selection of galaxies can be found in Appendix B.

###  4.5 Hyperparameter optimisation

Overall, optimising the method of measuring bar lengths requires searching the
space of four hyperparameters: $X_{\rm{input}}$, $\sigma^{2}$, $T_{\rm{c}}$,
and $T_{\rm{l}}$. The values of hyperparameters were chosen as follows:

  1. 1.

$X_{\rm{input}}\in\\{\rm{True},\rm{False}\\}$;

  2. 2.

$S_{\rm{noise}}=\sigma/(x_{\rm{max}}-x_{\rm{min}})\in\\{0.1,0.15,0.2,0.25\\}$;

  3. 3.

$T_{\rm{c}}\in\\{0.7,0.75,0.8,0.85,0.9\\}$;

  4. 4.

$T_{\rm{l}}\in\\{10,20,30,40,50\\}$,

where $S_{\rm{noise}}$ is a transformation of $\sigma^{2}$, and $x_{\rm{max}}$
and $x_{\rm{min}}$ are the values of the most and least intense pixels in the
image being evaluated.

The Hoyle bar length catalogue was separated into two sets, a validation set
of 1,050 galaxies used for hyperparameter selection, and a test set of 2,100
galaxies used for final evaluation. The measured bar lengths were then
compared to the length_scaled values from the Hoyle bar length catalogue. The
catalogue values had been scaled by the petroR90_r value for each galaxy
(Hoyle et al., 2011), therefore this scaling was undone to ensure the values
being used were in units of $h^{-1}~{}\rm{kpc}$.

Bar lengths were measured for each combination of hyperparameters using the
following methods:

  1. 1.

saliency maps generated from the VGG16 based model;

  2. 2.

saliency maps generated from the ResNet50v2 based model;

  3. 3.

saliency maps generated from the Xception based model;

  4. 4.

saliency maps generated by elementwise averaging of the saliency maps from
(i), (ii), and (iii);

  5. 5.

averaging the bar lengths found by methods (i), (ii), and (iii).

Table 2 shows the Pearson correlation coefficients ($\rho$) between the
measured bar lengths and the catalogue values for the optimal set of
hyperparameters of each method. The hyperparameters were optimised in order to
maximise $\rho$, which is used as a measure of predictiveness.

Table 2: Best Pearson correlation coefficients ($\rho$) found for the validation set, and required hyperparameters for each method (for all, the hyperparameters $X_{\rm{input}}$ and $S_{\rm{noise}}$ were found to be False and 0.1 respectively). Method | $T_{\rm{c}}$ | $T_{\rm{l}}$ | $\rho$
---|---|---|---
VGG16 | 0.9 | 40 | 0.68
ResNet50v2 | 0.85 | 30 | 0.71
Xception | 0.8 | 40 | 0.67
Average Saliency Maps | 0.85 | 30 | 0.75
Average Measurements | 0.85 | 40 | 0.76

###  4.6 Control model

We also trained a CNN in order to directly predict the bar lengths of galaxies
in the Hoyle catalogue from their respective galaxy images. Section 4.3 showed
the the Xception architecture performed best at classifying galaxies, hence
this architecture was chosen again to directly predict bar lengths. The same
image augmentation methods were used as in Section 4.1, however the number of
output nodes in the model was changed to one. The model was trained to
minimise the RMSE of its predictions with respect to the length_scaled value
from the catalogue. The model was trained for 100 epochs with early stopping
implemented such that training was ceased if there was no improvement over the
course of 10 epochs.

For the purpose of training this model, the 3,150 galaxies in the catalogue
were randomly split into training, validation, and test sets containing 64,
16, and 20 per cent of the images, respectively. Training was halted after 36
epochs at which the RMSE on the validation set was 0.10203.

##  5 Results

###  5.1 Bar length measurements

Table 2 shows that the highest $\rho$ on the validation set were gained from
the “Average Measurements” method. Therefore, this method was chosen to be
evaluated on the test set of the catalogue (and will hereafter referred to as
the “Explainable Deep Learning” method). Evaluation of this method on the test
set was performed twice to reduce and quantify any uncertainty due to the
Gaussian noise added as part of the SmoothGrad algorithm. The measured bar
lengths are computed as the mean of the bar lengths found during each run.
Performance was shown to be similar between the validation and test sets, with
measured bar lengths having $\rho=0.76$, and an RMSE of 1.69 respective to
human measured values (using hyperparameters: $X_{\rm{input}}=\rm{False}$,
$S_{\rm{noise}}=0.1$, $T_{\rm{c}}=0.85$, and $T_{\rm{l}}=40$).

In comparison, the CNN described in Section 4.6 (hereafter referred to as the
“Direct Deep Learning” method) only managed to achieve $\rho=0.59$, and an
RMSE of 2.10 between the measured and catalogue bar length values on the
model’s test set. This relatively poor performance could reasonably be
explained by the much smaller size of the bar length catalogue used to train
the model, in comparison to the size of the full Galaxy Zoo data used to train
the other models. Although we could decouple the size of the dataset from the
method, we feel like this would be intentionally handicapping XAI methods as
one of their distinct advantages is that they do not require specialised data
which is often harder to collect.

In most cases XAI methods are able to correctly highlight the bar of a galaxy
image and measure it accordingly, as can be seen in the middle row of Figure
3. XAI methods can result in a few extreme outliers as seen in the bottom row
of Figure 3. In this case the CNN mistakenly sees the entire galaxy as a bar
due to the galaxy’s inclination in the image, resulting in a vastly
overestimated bar length measurement. The “double blob” structure in the
saliency maps for the first example in Figure 3 can be explained by the model
not requiring the use of the prominent bulge in classifying these galaxies as
barred, the bulge is not unique to barred galaxies and hence holds little
importance in the model’s classification.

We did not expect machine learning methods to be able to reproduce absolute
values of human bar length measurements. Defining the start and end points of
the bar is a difficult task for both humans and machines, so we expect
differences between the two methods. However, a strong correlation between the
two methods would be considered a success. Linear regression fits were
performed by Scikit-learn (Pedregosa et al., 2011) between the Hoyle bar
length catalogue and the measured lengths for both the “Explainable Deep
Learning” and “Direct Deep Learning” methods. The regression fits were forced
to go through the origin, and were weighted by the uncertainty associated with
each data point. For both fits the error on the Hoyle bar lengths were set to
17 per cent as specified in Hoyle et al. (2011). Errors for the bar lengths
measured by the “Explainable Deep Learning” method were estimated using the
spread of the measured bar lengths from each run over the test set. The spread
for each galaxy ($\delta^{g}$) was computed as $\delta^{g}=\lvert
l_{1}-l_{2}\rvert$, where $l_{1}$ and $l_{2}$ are the first and second
measurements taken. The galaxies were then binned into deciles by their
measured bar lengths. $\delta^{g}$ was averaged over all galaxies in each
decile, and this average was used as the error for all the galaxies in the
decile. Errors for the predictions of the “Direct Deep Learning” method were
not quantified.

Figure 4 shows that the “Explainable Deep Learning” method not only correlates
strongly with the catalogue values but also matches the values extremely well,
as demonstrated by the close relation between the regression fit and the one-
to-one relation. Although the regression fit for the “Direct Deep Learning”
method is also close to unity, it is important to remember that the most
important metric for evaluating the performance of these methods is $\rho$.
The distribution of the data points on the “Direct Deep Learning” plot has
more scatter than for “Explainable Deep Learning” plot, and is thus it is a
less predictive method.

Figure 4: Bar lengths measured by the “Explainable Deep Learning” (top) and
“Direct Deep Learning” (bottom) methods vs the Hoyle bar length catalogue for
each method’s test dataset. Weighted regression fits are also shown (red
lines), as well as the one-to-one relation (black lines). A kernel density
estimate overlay (dark blue contours) is provided to help illustrate the
distribution of data points in each panel and error bars are not displayed for
the sake of clarity. The regression fits for both are close to unity, however
the “Explainable Deep Learning” has a smaller scatter than that of the “Direct
Deep Learning” method indicating stronger predictive power.

###  5.2 Misclassification

Throughout this study, we came across many instances where the XAI model
predictions differed from the Galaxy Zoo consensus (i.e. the model’s predicted
answer to a Galaxy Zoo question was different to the answer with the highest
share of votes amongst Galaxy Zoo participants). For the entire Kaggle dataset
we observed that the XAI model predictions on whether a galaxy has any
discernible features differed from the Galaxy Zoo consensus 8 per cent of the
time. We present two examples in Figure 5 in which our ensembled models
predict different morphologies to the consensus. These examples were chosen in
order to present cases both where the model predicted features where Galaxy
Zoo participants did not, and vice versa.

The top row shows a galaxy which our XAI model classifies as smooth (i.e. no
bar, spiral arms, or other structure present) with 72 per cent confidence.
Contrarily, only 39 per cent of Galaxy Zoo participants classified it as a
smooth galaxy, with most classifying it as a having features (and some even
classifying it as a galaxy with spiral arms). We can only speculate that these
classifications are due to the neighbouring objects in the image (although
Galaxy Zoo participants are instructed to only focus on the object in the
centre of the image). The related saliency map shows the models being unable
to pick out structure in this galaxy apart from a slight overdensity in the
centre of the image. The bottom row shows a galaxy where 40 per cent of Galaxy
Zoo participants have classified it as a smooth galaxy. Again, our model
disagrees with this with a confidence of 53 per cent, and we can see from the
saliency map that the model confidently picks out what looks like either a
ring structure or dust lane in the galaxy image.

Looking at these examples, we agree more with the XAI model predictions than
the Galaxy Zoo consensus. The SmoothGrad saliency maps shown in Figure 5 can
help us understand why the models make their decisions, however we can only
speculate as to why the Galaxy Zoo participants classified these examples in
the way they did. It is possible to ask participants for their reasoning via
the Galaxy Zoo Forum555https://www.galaxyzooforum.org/, although this would be
extremely time consuming to do for a large number of galaxies and does not
guarantee responses from the participants who provided the responses in the
dataset. This brings up an interesting scenario in which the citizen science
method of classifying galaxies is less easily explained than deep learning
methods.

Figure 5: Two examples where the machine learning prediction disagrees with
the Galaxy Zoo citizen science consensus. The image (left column) is shown
along with SmoothGrad saliency maps explaining the model’s classification of
this image (right column). The top row shows a galaxy where the Galaxy Zoo
consensus is that of a galaxy with features and structure, while the bottom
row shows an example where the consensus is that of a smooth galaxy. Our model
disagrees with both of these. The Galaxy Zoo consensus as well as the model
predictions probabilities are given for both examples. The galaxies shown are
identified by the filenames 102271.jpg and 120250.jpg from the Kaggle dataset.

##  6 Conclusions

Although machine learning (ML) is widely used in astronomy, we believe that we
are the first to quantitatively demonstrate the effectiveness of saliency
mapping within the field. We used SmoothGrad to extract bar length
measurements from convolutional neural networks (CNN) trained on Galaxy Zoo
data. This method of measuring bar lengths correlated with human measurements
with a Pearson correlation coefficient of 0.76 and achieved a root mean
squared error of 1.69. This vastly outperformed a CNN trained specifically to
predict bar lengths, which was only able to achieve a correlation of 0.59 and
root mean squared error of 2.10, with respect to human measurements.

The superiority of explainable artificial intelligence (XAI) methods could
reasonably be explained due to the larger dataset that was used to train the
ML models. The Galaxy Zoo dataset used contained 61,578 galaxy images for the
training of models which can predict galaxy morphologies. In comparison, only
2,520 galaxy images were used in the training of the model tasked with
directly predicting galactic bar lengths. The larger size of the Galaxy Zoo
dataset allowed more general learning of galactic features and helped increase
the predictiveness of the XAI based methods. A smaller dataset would reduce
the performance of the initial model (Henghes et al., 2021) and thus would
reduce the ability of XAI methods to extract galactic features.

The major advantage of utilising XAI methods is that information which the
model was never presented with can be extracted from the model predictions.
This greatly reduces the reliance of collecting data in order to learn
features of galaxies. We focused on bar lengths, however we could also extract
other galactic features (such as the bulge-to-disk ratio) from the same
models. Traditional deep learning methods would require a new and extensive
dataset in order to be able to predict these features.

We also briefly discussed the difficulties in reconciling differences between
the ML model predictions and the Galaxy Zoo consensus. One possible path for
future work is to use models trained only on galaxies where there is an
extremely strong consensus between Galaxy Zoo participants (see e.g. Domínguez
Sánchez et al., 2018), models designed to handle galaxies with uncertain
labels (see e.g. Walmsley et al., 2020), or to use pretrained models (see e.g.
Walmsley et al., 2022). This could improve the ML model predictions, and lead
to cleaner saliency maps and more accurate bar length measurements. Another
possibility would be to use the saliency maps in order to identify and
eliminate bias from the training dataset, which would lead to a higher quality
dataset for training models.

We conclude that SmoothGrad was able to explain the decisions made by the
models in a way that could be quantified and compared favourably to human
action. This demonstrates that saliency mapping was successful in explaining
the ML models used in this paper within the context of bar lengths, and could
see a wide range of applications within the field of astronomy.

## Acknowledgements

The authors would like to thank A. Bevan for drawing our attention to saliency
mapping techniques, and M. Walmsley for their insightful and helpful feedback.
PB is supported by the STFC UCL Centre for Doctoral Training in Data Intensive
Science. BJ acknowledges support by an STFC Consolidated Grant, grant no.
ST/V000780/1. OL acknowledges STFC Consolidated Grant ST/R000476/1 and a
Visiting Fellowship at All Souls College, Oxford.

## Data Availability

All code required to reproduce this work will be made publicly available on
acceptance. The code includes a readme detailing the steps required to
reproduce the study, including downloading all data and setting the seeds used
when randomly splitting the datasets and augmenting training data. Please note
that the version of the saliency mapping package used does not support the use
of random seeds, however the high number of samples used in the SmoothGrad
algorithm should minimise the impact of this. GitHub .

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##  Appendix A Randomised Saliency Mapping Examples

Here we present further examples similar to Figure 1, of SmoothGrad saliency
maps for three chosen classes on five randomly selected galaxies. The galaxies
selected are from the Kaggle dataset used to train the CNNs in Section 4.2,
with the condition that the selected galaxy must have at least a 50 per cent
of voters classifying it as having a bar, a bulge, and spiral arms.

Figure 6: Similar to Figure 1, but with examples randomly selected from the
Kaggle dataset, with the conditions that the selections must have at least 50
per cent of voters classifying it as having a bar, a bulge, and spiral arms.

##  Appendix B Randomised Bar Length Measurement Examples

Here we present further examples similar to Figure 3, of SmoothGrad saliency
maps isolating the bar of five randomly selected galaxies. The galaxies
selected are from the test set of the Hoyle catalogue used to evaluate the
performance of the XAI based methods in Section 5.1.

Figure 7: Similar to Figure 3, but with examples randomly selected from the
Hoyle catalogue.


[title=Index of Notation]

# A strong min-max property for level sets of Phase Transitions

Érico Melo Silva  Department of Mathematics, Princeton University, Washington
Road, Princeton, NJ, 08540, USA<EMAIL_ADDRESS>
###### Abstract.

We show that certain functions whose nodal sets lie near a fixed nondegenerate
minimal hypersurface satisfy a strong min-max principle for the Allen–Cahn
energy which is analogous to the strong min-max principle for non-degenerate
minimal hypersurfaces first proved by Brian White [Whi94].

##  1\. Introduction

Given a Riemannian manifold $(M^{n},g)$ a minimal hypersurface
$\Sigma^{n-1}\subset M$ is one which arises as a critical point for the area
functional. The variational properties of minimal hypersurfaces have been a
rich source of study for geometric analysis. In particular, the properties of
the stability operator, $L_{\Sigma}$, of a minimal hypersurface often
determine important aspects of geometric variational problems.

In [Whi94], B. White proved that any minimal submanifold which is
nondegenerate (in the sense that it admits no non-zero Jacobi fields, i.e. the
equation $L_{\Sigma}f=0$ has no nontrivial solutions), and finite Morse index
is the solution of a strong min-max problem.

The Allen–Cahn energy for functions $u\in W^{1,2}(M)$ is given by

(1.1) $E_{\varepsilon}(u)\doteq\int_{M}\frac{\varepsilon|\nabla
u|^{2}}{2}+\frac{W(u)}{\varepsilon},$

where $W(t)=\frac{(1-t^{2})^{2}}{4}$ is the standard double–well potential.
The Euler–Lagrange equation of 1.1 is known as the Allen–Cahn equation and is
given by

(1.2) $\varepsilon^{2}\Delta_{g}u=W^{\prime}(u).$

The solutions to 1.2 have a well known correspondence with minimal
hypersurfaces, beginning with Modica–Mortola’s work (see e.g., [Mod85],
[Mor77]) on the $\Gamma$–convergence of the energies 1.1 to the perimeter
functional acting on sets of finite perimeter. A rich body of work has emerged
in recent years utilizing min–max properties of the Allen–Cahn equation 1.2 to
provide variational constructions of minimal hypersurface. Of particular note
in this direction is the work of Guaraco [Gua18], Guaraco–Gaspar [GG18],
Chodosh–Mantoulidis [CM20], and many others.

In the other direction, constructing solutions to the Allen–Cahn equation
given a fixed minimal hypersurface, we have the following result of
Pacard–Ritoré [PR03].

######  Theorem 1.1 (Theorem 1.1 [PR03]).

Assume that $(M,g)$ is an $n$-dimensional closed Riemannian manifold and
$\Sigma^{n-1}\subseteq M^{n}$ is a $2$-sided, nondegenerate minimal
hypersurface. Then there exists $\varepsilon_{0}>0$ such that
$\forall\varepsilon<\varepsilon_{0}$, there exist solutions,
$u_{\varepsilon}$, to equation (1.2) such that $u_{\varepsilon}$ converges to
$+1$ (resp. -1) on compact subsets of $(M^{+})^{o}$ (resp. $(M^{-})^{o}$) and

$E_{\varepsilon}(u_{\varepsilon})\xrightarrow{\varepsilon\to
0}2\sigma_{0}\mathcal{A}(\Sigma)$

where $\mathcal{A}(\Sigma)$ the $(n-1)$–dimensional area of $\Sigma$ and
$\Sigma$ is a constant which depends on the choice of potential in 1.1.

In [MS23] the author, together with J. Marx–Kuo, gave a variant of the
Allen–Cahn energy 1.1 suited to the study of the geometry of level sets of
solutions. In this paper, we prove that our variant of the Allen–Cahn energy
satisfies a strong min–max property analogous to that proved in White [Whi94],
provided that we restrict to surfaces which lie over the minimal hypersurface
as small normal graphs. As a geometric application of our result, we give a
new, purely variational proof of 1.1.

######  Remark 1.2.

The first variational proof of 1.1 was given recently by Pigati–De Philippis
[PP22]. In their work, the candidate solutions to the Allen–Cahn equation are
constructed using gradient flow methods. In the present work, our construction
relies on the variational properties of the associated surface energy first
defined in [MS23], as well as a variant of the classical mountain pass theorem
for solutions to elliptic semilinear equations.

###  1.1. Acknowledgements.

The author is indebted to his advisor, Fernando Codá Marques, for his patience
and essential conversations regarding the completion of the present work. The
author would also like to thank Jared Marx–Kuo for innumerable mathematical
conversations.

##  2\. Preliminaries and Main Results

###  2.1. Notation and conventions.

  * •

$(M^{n},g)$ always denotes an orientable, closed Riemannian manifold.

  * •

Unless specified otherwise $\Sigma\subset M$ denotes an embedded, closed
hypersurface, i.e. a submanifold of $M$ of codimension $1$.

  * •

$B^{k}$ is the unit ball in $\bf{R}^{k}$.

  * •

If $E\subset M$ is a Borel set, then $\mathcal{C}(E)$ denotes the space of
sets of finite perimeter (Caccioppoli sets) $\Omega\subset E$.

  * •

We say a sequence of Caccioppoli sets $\Omega_{n}\to\Omega$ if the measure of
the symmetric difference tends to $0$, (i.e. $|\Omega_{n}\Delta\Omega|\to 0$.

  * •

We denote by $C^{k,\alpha}_{\varepsilon}(M)$ the Banach space of
$C^{k,\alpha}$ functions on $M$ equipped with the $\varepsilon$-weighted
Hölder norms

(2.1) $\displaystyle\|f\|_{C^{k,\alpha}_{\varepsilon}}\doteq\sum_{|\beta|\leq
k}\varepsilon^{|\beta|\|D^{\beta}}f\|_{C^{0}(M)}+\varepsilon^{k+\alpha}[D^{k}f]_{\alpha},$

where $\beta$ are multiindices and $[\cdot]_{\alpha}$ is the usual
$\alpha$–Hölder seminorm.

  * •

We denote by $W^{1,2}_{\varepsilon}(M)$ a Banach space we call the
$\varepsilon$-weighted Sobolev space, with norm

(2.2)
$\displaystyle\|f\|_{W^{1,2}_{\varepsilon}(M)}^{2}\doteq\varepsilon\|f\|_{L^{2}(M)}+\varepsilon^{3}\|\nabla
f\|_{L^{2}(M)}.$

Similarly we denote by $W^{1,2}_{0,\varepsilon}(M)$ those functions in
$W^{1,2}_{\varepsilon}(M)$ which have trace $0$.

  * •

The properties of the one-dimensional solution
$\operatorname{\bf{H}}(z)=\tanh(t/\sqrt{2})$ to the Allen–Cahn equation
$\partial_{z}^{2}u-W^{\prime}(u)=0$. We define four functions as the unique
solutions to auxiliary ODEs: $\omega,\rho,\tau,\kappa$,

(2.3)
$\displaystyle\omega^{\prime\prime}-W^{\prime\prime}(\operatorname{\bf{H}})\omega$
$\displaystyle=\partial_{z}\operatorname{\bf{H}}^{\prime},$ (2.4)
$\displaystyle\rho^{\prime\prime}-W^{\prime\prime}(\operatorname{\bf{H}})\rho$
$\displaystyle=\omega^{\prime},$ (2.5)
$\displaystyle\tau^{\prime\prime}-W^{\prime\prime}(\operatorname{\bf{H}})\tau$
$\displaystyle=z\operatorname{\bf{H}}^{\prime},$ (2.6)
$\displaystyle\kappa^{\prime\prime}-W^{\prime\prime}(\operatorname{\bf{H}})\kappa$
$\displaystyle=\operatorname{\bf{H}}\omega,$

all with the boundary conditions $f(0)=0$, $\lim_{z\to\infty}f(z)=0$.

  * •

We define constants

$\displaystyle\sigma_{0}\doteq\int_{0}^{\infty}\partial_{z}\operatorname{\bf{H}}(z)dz=\frac{\sqrt{2}}{{3}},$
$\displaystyle\sigma\doteq\operatorname{\bf{H}}^{\prime}(0)=\sqrt{2}^{-1}.$

  * •

We write $\ell_{\varepsilon,u}\doteq\Delta_{g}-W^{\prime\prime}(u)$ for the
linearized Allen–Cahn operator at a solution $u$.

  * •

For any function $f:\bf{R}\to\bf{R}$ we define $f_{\varepsilon}(z)\doteq
f(z/\varepsilon)$.

  * •

We define a cut-off of the heteroclinic, $\overline{\operatorname{\bf{H}}}$,
as follows. Let $\ell>5$ $\chi_{\varepsilon}:\bf{R}\to\bf{R}$ be a cutoff
function.

$\displaystyle\begin{cases}\chi_{\varepsilon}(z)=1,&|z|\in[0,-\ell\log(\varepsilon)],\\\
\chi_{\varepsilon}(z)=0,&|z|\not\in[0,-2\ell\log(\varepsilon)].\end{cases}$

Then

$\displaystyle\overline{\operatorname{\bf{H}}}(z)\doteq\chi_{\varepsilon}(z)\operatorname{\bf{H}}(z)+(1-\chi_{\varepsilon}(z)).$

Similarly, we can define the cut-off functions $\overline{\omega}$,
$\overline{\rho}$, $\overline{\tau}$ and $\overline{\kappa}$, where these each
cut-off to $0$ instead of to $1$, as the corresponding functions decay to $0$
at infinity. We note the cut-offs satisfy

$\displaystyle\left(\frac{d^{2}}{dz^{2}}-W^{\prime\prime}(\overline{\operatorname{\bf{H}}})\right)\overline{\operatorname{\bf{H}}}^{\prime}=E,$
$\displaystyle\left(\frac{d^{2}}{dz^{2}}-W^{\prime\prime}(\overline{\operatorname{\bf{H}}})\right)\overline{\omega}^{\prime}=E+\overline{\operatorname{\bf{H}}}^{\prime},$

with $\|E\|_{W^{1,k}}\leq C\varepsilon^{\ell}$.

  * •

If $\Sigma$ is a sufficiently smooth minimal hypersurface, we denote by
$H_{\Sigma}$ its mean curvature, and denote by $A_{\Sigma}$ its second
fundamental form.

  * •

We denote by $\nu$ the outward pointing unit normal to $\Sigma$.

  * •

We write by $N(\eta)$ a tubular neighborhood of $\Sigma$ of height $\eta$.
Provided $\eta$ is sufficiently small, we may find a parametrization
$\Sigma\times(-\eta,\eta)\to N(\eta)$, by

$(s,z)\mapsto\exp_{s}(z\nu(s)).$

We call this parametrization Fermi coordinates in the tubular neighborhood
$N(\eta)$.

  * •

If $f\in C^{k,\alpha}(\Sigma)$ and $f$ is sufficently small, we denote by
$\Gamma(f)$ the normal graph of $f$ over $\Sigma$. That is, in Fermi
coordinates,

$\Gamma(f)\doteq\\{x\in M|x=\exp_{s}(f(s)\nu(s)\\}.$

In [BO86] Brezis-Oswald considered the solvability of the Dirichlet problem
for the equation 1.2.

######  Theorem 2.1 ([BO86] Theorem $1$).

Suppose $\Omega\subset M$ is an open domain with smooth boundary. If
$\varepsilon<\lambda_{1}(\Omega)^{-\frac{1}{2}}$, then

(2.7) $\begin{cases}\Delta_{g}u=\frac{W^{\prime}(u)}{\varepsilon^{2}}&\text{
in }\Omega,\\\ u>0&\text{ in }\Omega,\\\ u=0&\text{ on
}\partial\Omega,\end{cases}$

has a unique solution. Moreover, the solution minimizes the energy 1.1 over
all functions in $W^{1,2}_{0}(\Omega)$.

It follows from 2.1 that for a separating hypersurface $\Sigma\subset M$ along
which $M$ splits into a disjoint union $M=M^{+}\sqcup_{\Sigma}M^{-}$, there
exists an $\varepsilon_{0}$ sufficiently small so that for all
$\varepsilon<\varepsilon_{0}$ the problem 2.7 is uniquely solvable in both
$M^{+}$ and $M^{-}$. This observation led the author, together with J.
Marx–Kuo in [MS23] to define the balanced energy of $\Sigma$.

######  Definition 2.2.

Let $\Sigma\subset M$ be a $2$–sided, separating hypersurface splitting $M$
into two disjoint open sets $M^{+}$ and $M^{-}$. Suppose $\varepsilon_{0}$ is
sufficiently small that the problem 2.7 admits a solution for both $M^{+}$ and
$M^{-}$. Call these solutions $u^{+}_{\Sigma,\varepsilon}$ and
$u^{-}_{\Sigma,\varepsilon}$, respectively. Then we define

(2.8)
$\displaystyle\operatorname{\mathcal{B}_{\varepsilon}}(\Sigma)\doteq\int_{M^{+}}\frac{\varepsilon|\nabla
u^{+}_{\Sigma,\varepsilon}|^{2}}{2}+\frac{W(u^{+}_{\Sigma,\varepsilon})}{\varepsilon}+\int_{M^{-}}\frac{\varepsilon|\nabla
u^{-}_{\Sigma,\varepsilon}|^{2}}{2}+\frac{W(u^{-}_{\Sigma,\varepsilon})}{\varepsilon}$

When $u^{+}_{\Sigma,\varepsilon}$ and $u^{-}_{\Sigma,\varepsilon}$ are
simultaneously defined, we write

$u_{\Sigma,\varepsilon}\doteq\begin{cases}u^{+}_{\Sigma,\varepsilon}&\text{in
}M^{+},\\\ u^{-}_{\Sigma,\varepsilon}&\text{in }M^{-},\\\ 0&\text{on
}\Sigma,\end{cases}$

and we call $u_{\Sigma,\varepsilon}$ the broken $\varepsilon$–phase transition
associated to $\Sigma$. If $\Sigma$ and $\varepsilon$ are understood, we omit
writing them.

###  2.2. Preliminary results

Suppose $\Sigma\subset M$ is a smooth, closed, embedded minimal hypersurface.
Suppose further that $\Sigma$ is nondegenerate and has Morse index $k$. In
[Whi94] B. White proved the following theorem.

######  Theorem 2.3 ([Whi94], Theorem $4$).

With $\Sigma$ as above, there exists a tubular neighborhood $U$ of $\Sigma$,
and a smooth $k$–parameter family of hypersurfaces homologous to $\Sigma$,
$\Sigma_{v},v\in B^{k}$ so that the following hold.

  1. (1)

$\Sigma_{0}=\Sigma$.

  2. (2)

$\sup_{v\in B^{k}}\mathcal{A}(\Sigma_{v})=\mathcal{A}(\Sigma)$.

  3. (3)

If $\tilde{\Sigma}_{v},v\in B^{k}$ is any other smooth $k$–parameter family of
smooth, closed, embedded hypersurfaces in $U$, homologous to $\Sigma$, and
$\tilde{\Sigma}_{v}=\Sigma_{v}$ for all $v\in\partial B^{k}$, then

(2.9) $\sup_{v\in
B^{k}}\mathcal{A}(\tilde{\Sigma}_{v})\geq\mathcal{A}(\Sigma),$

with equality if and only if $\tilde{\Sigma}_{v}=\Sigma$ for some $v\in
B^{k}$.

In [Mar23], J. Marx–Kuo computed an expansion of the solutions to 2.7 in
$\varepsilon$ to derive an asymptotic formula for solutions to the Dirichlet
problem which makes clear the geometric dependency of solutions on the
boundary of the domain.

######  Theorem 2.4 ([Mar23], Theorem 1.5).

Let $u:\Omega\to\bf{R}$ be the solution to 2.7 afforded by 2.1, where
$\Sigma=\partial\Omega$. Then

(2.10) $\displaystyle u(s,z)$
$\displaystyle=\overline{\operatorname{\bf{H}}}_{\varepsilon}(z)+\varepsilon
H_{\Sigma}(s)\overline{\omega}_{\varepsilon}(z)$
$\displaystyle\quad\quad+\varepsilon^{2}\bigg{(}(\textnormal{Ric}(\nu(s),\nu(s))+|A_{\Sigma}(s)|^{2})\overline{\tau}_{\varepsilon}(z)$
$\displaystyle\quad\quad+H_{\Sigma}^{2}(s)\bigg{(}\overline{\rho}_{\varepsilon}(z)+\frac{1}{2}\overline{\kappa}_{\varepsilon}(z)\bigg{)}\bigg{)}+\phi_{0,\varepsilon},$

where the remainder term $\phi_{0,\varepsilon}$ satisfies the estimate,

(2.11)
$\displaystyle\|\phi_{0,\varepsilon}\|_{C^{2,\alpha}_{\varepsilon}(\Omega)}\leq
C(\Sigma)\varepsilon^{3}.$

In [MS23], it was shown that the solutions $u_{\Sigma}$ depend in a smooth way
on the hypersurface $\Sigma$. This allows one to take the first and second
variations of the energies $\operatorname{\mathcal{B}_{\varepsilon}}$. We
collect the following results.

######  Theorem 2.5 ([MS23], Corollary 2.1).

If $\Sigma_{t}$ is a smooth $1$-parameter family of closed, $2$–sided,
separating hypersurfaces for which $\operatorname{\mathcal{B}_{\varepsilon}}$
is defined, and the variational vector field $X$ is given by $f\nu$ for some
$f\in C^{\infty}(\Sigma_{0})$, then

(2.12)
$\displaystyle\frac{d}{dt}\bigg{|}_{t=0}\operatorname{\mathcal{B}_{\varepsilon}}(\Sigma_{t})=\frac{\varepsilon}{2}\int_{\Sigma_{0}}f((u^{+}_{\nu})^{2}-(u^{-}_{\nu})^{2})=-2\sigma_{0}\int_{\Sigma_{0}}H_{\Sigma_{0}}fd\mathcal{H}^{n-1}+E(f),$

where $E(f)\leq C(\Sigma_{0})\varepsilon$.

Since $u_{t}\doteq u_{\Sigma_{t},\varepsilon}$ is a well defined family of
functions in $W^{1,2}(M)$ smooth in $t$, we denote by $\dot{u}_{t}$ its time
derivative, which can be considered as a function in $W^{1,2}(M)$ which,
following [MS23], satisfies the boundary value problem

(2.13)
$\displaystyle\begin{cases}\ell_{\varepsilon,u_{t}}\dot{u}_{t}=0&\text{in
}M^{+}_{t},\\\ \dot{u}_{t}=-f\frac{\partial u_{t}}{\partial\nu_{t}}&\text{on
}\Sigma_{t}.\end{cases}$

We have the following decomposition for the function $\dot{u}_{t}$.

######  Proposition 2.6.

[MS23], Lemma 9.1 In fermi coordinates adapted to $\Sigma_{t}$, we may write

(2.14) $\dot{u}_{t}^{\pm}(s,z)=-f(s)\frac{\partial
u_{t}^{\pm}}{\partial\nu_{t}}\dot{\overline{\operatorname{\bf{H}}}}_{\varepsilon}(z)+\phi_{1,\varepsilon}^{\pm},$

where $\|\phi_{1,\varepsilon}^{\pm}\|_{W^{1,2}_{0,\varepsilon}(M^{\pm})}\leq
C(\Sigma)\varepsilon\|f\|_{W^{1,2}(\Sigma)}$.

For the second variation, the following was shown.

######  Theorem 2.7 ([MS23], Theorem 2.6).

Let $\Sigma$ be a critical point for
$\operatorname{\mathcal{B}_{\varepsilon}}$. Then if $\Sigma_{t}$ is any normal
variation with variational vector field given by $f\nu$ as above, then

(2.15)
$\displaystyle\frac{d^{2}}{dt^{2}}\bigg{|}_{t=0}\operatorname{\mathcal{B}_{\varepsilon}}(\Sigma_{t})$
$\displaystyle=\varepsilon\int_{\Sigma}fu_{\nu}(\dot{u}^{+}_{\nu}-\dot{u}^{-}_{\nu}),$
$\displaystyle=2\sigma_{0}\int|\nabla
f|^{2}-(\textnormal{Ric}(\nu,\nu)+|A_{\Sigma}|^{2})f^{2}d\mathcal{H}^{n-1}+\tilde{E}(f),$

where $E(f)\leq C(\Sigma)\varepsilon^{\frac{1}{2}}\|f\|_{H^{1}(\Sigma)}.$

###  2.3. Main results.

We now state the main results of this paper.

######  Theorem 2.8.

Let $\Sigma\subset(M,g)$ be a smooth, closed, embedded, $2$-sided, separating
minimal hypersurface with Morse index equal to $k$ and nullity equal to $0$.
Let $\Sigma_{v},$ $v\in B^{k}$ be the family afforded by 2.3, and write
$\mathcal{F}\doteq\\{\tilde{\Sigma}_{v}|\text{ for every }v\in\partial
B^{k},\tilde{\Sigma}_{v}=\Sigma_{v}\\}$ for the set of all families of normal
graphs over $\Sigma$ which agree with the canonical family $\Sigma_{v}$ on
$\partial B^{k}$. Then, for any tubular neighborhood of $\Sigma$ contained in
the one afforded by Theorem 2.3, and for any $\delta>0$ there is
$\varepsilon_{0}>0$ so that for any $\varepsilon<\varepsilon_{0}$,

$\inf_{\tilde{\Sigma}_{v}\in\mathcal{F}}\sup_{v\in
B^{k}}\operatorname{\mathcal{B}_{\varepsilon}}(\tilde{\Sigma}_{v})\geq\operatorname{\mathcal{B}_{\varepsilon}}(\Sigma)-\delta.$

######  Remark 2.9.

Effectively, Theorem 2.8 says that any Allen–Cahn solution produced as a limit
of broken phase transitions over surfaces which are small normal graphs over
$\Sigma$ must have Allen–Cahn energy which is controlled by the area of
$\Sigma$ from below.

Arguing as in the proof of the classical mountain pass principle for solutions
to semilinear PDEs, we can show a mountain pass principle for the
$\operatorname{\mathcal{B}_{\varepsilon}}$ energies. Let $\mathcal{U}$ be a
sufficently small neighborhood of $0$ in $C^{k,\alpha}(\Sigma)$ (here we take
$k>5$) so that the normal graph of any member of $\mathcal{U}$ is contained in
the tubular neighborhood $U$ given by 2.8.

######  Theorem 2.10.

Consider a family of normal graphs parametrized by the boundary of the unit
$k$-ball $\\{\Sigma_{v}\\}_{v\in\partial B^{k}}$. Define $\mathcal{P}^{L},c$
and $d$ as follows.

(2.16) $\displaystyle\mathcal{P}^{L}$ $\displaystyle\doteq\\{p\in
C(B^{k},\mathcal{U})|\text{ for every }v\in\partial
B^{k},\Gamma(p(v))=\Sigma_{v},\mathcal{A}(\Gamma(p(v)))\leq L\\},$ (2.17)
$\displaystyle c$ $\displaystyle\doteq\max_{v\in\partial
B^{k}}\operatorname{\mathcal{B}_{\varepsilon}}(\Sigma_{v}),$ (2.18)
$\displaystyle d$ $\displaystyle\doteq\inf_{p\in\mathcal{P}^{L}}\sup_{v\in
B^{k}}\operatorname{\mathcal{B}_{\varepsilon}}(\Gamma(p(v))).$

If $d>c$, then there exists a smooth phase transition $u$ with
$E_{\varepsilon}(u)=d$, and moreover the level set $u^{-1}(0)\subset U$.

By combining 2.3 and 2.10, we can then show

######  Corollary 2.11.

If $\Sigma$ is a smooth, closed, embedded, separating, nondegenerate minimal
hypersurface in a closed Riemannian manifold $M$, then there exists a sequence
of phase transitions $u_{i}$ solving 1.2 for a sequence $\varepsilon_{i}\to 0$
as $i\to\infty$ with

$E_{\varepsilon_{i}}(u_{i})\to 2\sigma_{0}\mathcal{A}(\Sigma).$

Moreover, $u_{i}^{-1}(0)$ Hausdorff-converges to $\Sigma$ as $i\to\infty$.

##  3\. First and second variations of energy

The goal of this section is to compute the first and second variations of
$\operatorname{\mathcal{B}_{\varepsilon}}$ at an arbitrary point, and to prove
a result analogous to 2.15 for sufficiently small normal graphs over a minimal
hypersurface.

Let $F_{t}:M\to M$ be a $1$-parameter family of diffeomorphisms and set,

  * •

$F_{0}=\text{id}_{M}$,

  * •

$\partial_{t}F_{t}\doteq X_{t}$. We write $X\doteq X_{0}$ and call $X$ the
variational vector field of $F_{t}$.

  * •

$Z_{t}\doteq\nabla_{\partial_{t}}X_{t}$. We write $Z\doteq Z_{0}$ and call
$Z_{t}$ the acceleration vector field for $F_{t}$.

######  Remark 3.1.

We can take $X=f\nu$ for $f\in C^{\infty}(\Sigma)$ and we can take
$Z_{t}\equiv 0$ for all sufficiently small $t$ by assuming $F_{t}$ is of the
form

(3.1) $F_{t}(s,z)=\exp_{s}(tf(s)\chi(z)\nu(s)),$

where $\chi(z)$ is a suitably chosen cutoff function which is identically $1$
in a neighborhood of $z=0$.

Write $M^{\pm}_{t}\doteq F_{t}(M^{\pm})$ and $\Sigma_{t}\doteq F_{t}(\Sigma)$.
For small $t$, we may compute the first and second variations of
$\operatorname{\mathcal{B}_{\varepsilon}}(\Sigma_{t})$. For the sake of
concision we omit the subscript $t$.

######  Proposition 3.2.

Let $u^{\pm},\dot{u}^{\pm}$ be as above. Then,

(3.2)
$\displaystyle\frac{d}{dt}\operatorname{\mathcal{B}_{\varepsilon}}(\Sigma_{t})=-\frac{\varepsilon}{2}\int_{\Sigma_{t}}\left(\left(\frac{\partial
u^{+}}{\partial\nu}\right)^{2}-\left(\frac{\partial
u^{-}}{\partial\nu}\right)^{2}\right)\langle X,\nu\rangle d\mathcal{H}^{n-1},$

and

(3.3)
$\displaystyle\frac{d^{2}}{dt^{2}}\operatorname{\mathcal{B}_{\varepsilon}}(\Sigma_{t})$
$\displaystyle=-\varepsilon\int_{\Sigma}\left(\frac{\partial
u^{+}}{\partial\nu}\frac{\partial\dot{u}^{+}}{\partial\nu}-\frac{\partial
u^{-}}{\partial\nu}\frac{\partial\dot{u}^{-}}{\partial\nu}\right)\langle
X,\nu\rangle$ $\displaystyle\quad\quad+\left(\frac{\partial
u^{+}}{\partial\nu}\frac{\partial^{2}u^{+}}{\partial\nu^{2}}-\frac{\partial
u^{-}}{\partial\nu}\frac{\partial^{2}u^{-}}{\partial\nu^{2}}\right)\langle
X,\nu\rangle^{2}$
$\displaystyle\quad\quad+\frac{1}{2}\left(\left(\frac{\partial
u^{+}}{\partial\nu}\right)^{2}-\left(\frac{\partial
u^{-}}{\partial\nu}\right)^{2}\right)(\langle Z,\nu\rangle+\langle
X,\dot{\nu}\rangle-H_{\Sigma}\langle X,\nu\rangle^{2})d\mathcal{H}^{n-1}$

###### Proof.

By symmetry, it suffices to compute the variation on each side and then add
together, so suppressing $+$ and $-$ signs and computing gives,

$\displaystyle\frac{d}{dt}E_{\varepsilon}(u_{t};M_{t})$
$\displaystyle=\frac{d}{dt}\int_{M_{t}}\frac{\varepsilon}{2}|\nabla
u_{t}|^{2}+\frac{W(u)}{\varepsilon}d\mathcal{H}^{n}$
$\displaystyle=\int_{M_{t}}\varepsilon\langle\nabla
u_{t},\nabla\dot{u}_{t}\rangle+\frac{W^{\prime}(u_{t})\dot{u}_{t}}{\varepsilon}d\mathcal{H}^{n}$
$\displaystyle\quad\quad+\int_{\Sigma_{t}}\left(\frac{\varepsilon}{2}|\nabla
u_{t}|^{2}+\frac{W(u_{t})}{\varepsilon}\right)\langle X_{t},\nu_{t}\rangle
d\mathcal{H}^{n-1}$ $\displaystyle=I_{1}+I_{2}.$

Integrating $I$ by parts and noting that $u_{t}$ vanishes along $\Sigma_{t}$
gives

$\displaystyle I_{1}$
$\displaystyle=\int_{M_{t}}\dot{u}_{t}\left(\frac{W^{\prime}(u_{t})}{\varepsilon}-\varepsilon\Delta_{g}u_{t}\right)d\mathcal{H}^{n}+\int_{\Sigma_{t}}\varepsilon\frac{\partial
u_{t}}{\partial\nu_{t}}\dot{u}_{t}d\mathcal{H}^{n-1}$
$\displaystyle=-\int_{\Sigma_{t}}\varepsilon\left(\frac{\partial
u_{t}}{\partial\nu_{t}}\right)^{2}\langle X_{t},\nu_{t}\rangle
d\mathcal{H}^{n-1}.$

Now note for $I_{2}$ that since $u_{t}$ vanishes identically along
$\Sigma_{t}$, it follows that $\nabla u_{t}$ is parallel to $\nu_{t}$. By the
choice of the standard double well potential it similarly follows that
$W(u_{t})\equiv\frac{1}{4}$ along $\Sigma_{t}$. Putting that together gives

$\displaystyle
I_{2}=\int_{\Sigma_{t}}\left(\frac{\varepsilon}{2}\left(\frac{\partial
u_{t}}{\nu_{t}}\right)^{2}+\frac{1}{4\varepsilon}\right)\langle
X_{t},\partial\nu_{t}\rangle d\mathcal{H}^{n-1}.$

From which it follows that

$\displaystyle\frac{d}{dt}E_{\varepsilon}(u_{t})=\int_{\Sigma_{t}}\left(-\frac{\varepsilon}{2}\left(\frac{\partial
u_{t}}{\nu_{t}}\right)^{2}+\frac{1}{4\varepsilon}\right)\langle
X_{t},\partial\nu_{t}\rangle d\mathcal{H}^{n-1}.$

And so,

$\displaystyle\frac{d}{dt}\operatorname{\mathcal{B}_{\varepsilon}}(\Sigma_{t})$
$\displaystyle=\frac{d}{dt}E_{\varepsilon}(u_{t}^{+};M^{+}_{t})+\frac{d}{dt}E_{\varepsilon}(u^{-}_{t};M^{-}_{t})$
$\displaystyle=-\frac{\varepsilon}{2}\int_{\Sigma_{t}}\left(\left(\frac{\partial
u^{+}_{t}}{\partial\nu_{t}}\right)^{2}-\left(\frac{\partial
u^{-}_{t}}{\partial\nu_{t}}\right)^{2}\right)\langle X_{t},\nu_{t}\rangle
d\mathcal{H}^{n-1}.$

To compute the second variation it suffices to compute

$\displaystyle I_{1}=\frac{d}{dt}\int_{\Sigma_{t}}\left(\frac{\partial
u^{+}_{t}}{\partial\nu_{t}}\right)^{2}\langle X_{t},\nu_{t}\rangle
d\mathcal{H}^{n-1}.$

For the purposes of simplifying notation we write $\Phi(x,t)=(\partial
u^{+}_{t}/\partial\nu_{t})^{2}(x,t)$. We pull back by the variation $F_{t}$
and compute.

$\displaystyle I_{1}$
$\displaystyle=\int_{\Sigma}\left(\frac{d}{dt}\Phi(F_{t}(x),t)\right)\langle
X_{t}(F_{t}(x)),\nu_{t}(F_{t}(x))\rangle F_{t}^{\ast}d\mathcal{H}^{n-1}(x)$
$\displaystyle\quad\quad+\int_{\Sigma}\Phi(F_{t}(x),t)\left(\frac{d}{dt}\langle
X_{t}(F_{t}(x)),\nu_{t}(F_{t}(x))\rangle\right)F_{t}^{\ast}d\mathcal{H}^{n-1}(x).$
$\displaystyle\quad\quad+\int_{\Sigma}\Phi(F_{t}(x),t)\langle
X_{t}(F_{t}(x)),\nu_{t}(F_{t}(x))\rangle\left(\frac{d}{dt}F_{t}^{\ast}d\mathcal{H}^{n-1}(x)\right)$
$\displaystyle=I_{1}+I_{2}+I_{3}.$

The term in $I_{3}$ is the same as in the usual first variation of the area
functional.

$\displaystyle I_{3}=-\int_{\Sigma_{t}}\Phi H_{\Sigma_{t}}\langle
X_{t},\nu_{t}\rangle^{2}d\mathcal{H}^{n-1}.$

Dealing now with $I_{1}$, we compute

$\displaystyle\int_{\Sigma_{t}}\left(\frac{\partial\Phi}{\partial
t}+\langle\nabla^{M}\Phi,X_{t}\rangle\right)\langle X_{t},\nu_{t}\rangle
d\mathcal{H}^{n-1}.$

Computing $I_{2}$ gives

$\displaystyle I_{2}$
$\displaystyle=\int_{\Sigma_{t}}\Phi\left(\langle\nabla_{\partial_{t}}X_{t},\nu_{t}\rangle+\langle
X_{t},\nabla_{\partial_{t}}\nu_{t}\rangle\right)d\mathcal{H}^{n-1}$
$\displaystyle=\int_{\Sigma_{t}}\Phi\langle Z_{t},\nu_{t}\rangle
d\mathcal{H}^{n-1},$

Putting all three parts together gives us

$\displaystyle I_{1}+I_{2}+I_{3}$
$\displaystyle=\int_{\Sigma_{t}}\dot{\Phi}\langle
X_{t},\nu_{t}\rangle+(\Phi_{\nu_{t}}-\Phi H_{\Sigma_{t}})\langle
X_{t},\nu_{t}\rangle^{2}+$ $\displaystyle\quad\quad\Phi(\langle
Z_{t},\nu_{t}\rangle+\langle
X_{t},\nabla_{\partial_{t}}\nu_{t}\rangle)d\mathcal{H}^{n-1},$

which is precisely 3.3. ∎

######  Remark 3.3.

By selecting our variation $F_{t}$ as in 3.1, we see that the terms in the
second variation involving $Z$ and $\dot{\nu}$ vanish for all small $t$, and
in this case we have simplified expressions for the first and second
variation.

######  Corollary 3.4.

Let $F_{t}$ be a variation such that $X_{t}=f\nu_{t}$ and $Z_{t}\equiv 0$ for
$t$ sufficiently small. Then for such $t$ we have,

(3.4)
$\displaystyle\frac{d}{dt}\operatorname{\mathcal{B}_{\varepsilon}}(\Sigma_{t})=-\frac{\varepsilon}{2}\int_{\Sigma_{t}}f\left(\left(\frac{\partial
u^{+}}{\partial\nu}\right)^{2}-\left(\frac{\partial
u^{-}}{\partial\nu}\right)^{2}\right)d\mathcal{H}^{n-1},$

and

(3.5)
$\displaystyle\frac{d^{2}}{dt^{2}}\operatorname{\mathcal{B}_{\varepsilon}}(\Sigma_{t})$
$\displaystyle=-\varepsilon\int_{\Sigma}f\left(\frac{\partial
u^{+}}{\partial\nu}\frac{\partial\dot{u}^{+}}{\partial\nu}-\frac{\partial
u^{-}}{\partial\nu}\frac{\partial\dot{u}^{-}}{\partial\nu}\right)$
$\displaystyle\quad\quad+f^{2}\left(\frac{\partial
u^{+}}{\partial\nu}\frac{\partial^{2}u^{+}}{\partial\nu^{2}}-\frac{\partial
u^{-}}{\partial\nu}\frac{\partial^{2}u^{-}}{\partial\nu^{2}}\right)$
$\displaystyle\quad\quad-H_{\Sigma}f^{2}\frac{1}{2}\left(\left(\frac{\partial
u^{+}}{\partial\nu}\right)^{2}-\left(\frac{\partial
u^{-}}{\partial\nu}\right)^{2}\right)d\mathcal{H}^{n-1}$

###  3.1. Estimate for the second variation.

In this section, we use the asymptotic formulae 2.10, 2.14 for $u^{\pm}$ and
$\dot{u}^{\pm}$ to expand the second variation formula 3.3. More precisely we
prove the following asymptotic formula for the second variation.

######  Theorem 3.5.

Let $\Sigma_{t}$ be a $1$-parameter family of hypersurfaces as in Corollary
3.4. Then

(3.6)
$\displaystyle\frac{d^{2}}{dt^{2}}\operatorname{\mathcal{B}_{\varepsilon}}(\Sigma_{t})$
$\displaystyle=2\sigma_{0}\int_{\Sigma_{t}}|\nabla^{\Sigma_{t}}f|^{2}+(\textnormal{Ric}(\nu,\nu)+|A_{\Sigma_{t}}|^{2}-H_{\Sigma_{t}}^{2})f^{2}d\mathcal{H}^{n-1}+E(f)$

Where $E(f)\leq C(\Sigma)\varepsilon\|f\|_{W^{1,2}(\Sigma)}$.

###### Proof.

The proof is identical to the proof of Theorem 2.6 in [MS23] except in a few
respects. The terms contributed by the variation of the volume form of
$\Sigma_{t}$ and the terms containing two normal derivatives of the Dirichlet
phase transitions do not appear in the calculation of the second variation at
a critical point, so we deal with them here. The term containing a normal
derivative of $\dot{u}$ is treated in exactly the same way, so we refer to the
appendix of [MS23] for the proof. We work in Fermi coordinates $(s,z)$ adapted
to $\Sigma_{t}$. Recall,

(3.7) $\displaystyle u_{\nu}^{+}(s,0)$
$\displaystyle=\frac{1}{\varepsilon\sqrt{2}}-\frac{2}{3}H_{\Sigma}+O(\varepsilon),$
(3.8) $\displaystyle u_{\nu\nu}^{+}(s,0)$
$\displaystyle=\frac{H_{\Sigma}}{\varepsilon\sqrt{2}}-\frac{2}{3}H_{\Sigma}^{2}+O(\varepsilon),$
(3.9) $\displaystyle\dot{u}_{\nu}^{+}(s,0)$ $\displaystyle=\phi^{+}_{z}(s,0).$

substituting in 3.7, 3.8 and 3.9 into the first two terms in 3.5 gives the
following integral terms.

(3.10) $\displaystyle
I_{1}\doteq-\varepsilon\int_{\Sigma_{t}}f\left(\frac{1}{\varepsilon\sqrt{2}}+O(1)\right)(\partial_{z}\phi^{+}-\partial_{z}\phi^{-})d\mathcal{H}^{n-1},$

(3.11) $\displaystyle I_{2}$
$\displaystyle\doteq-\varepsilon\int_{\Sigma_{t}}f^{2}\left(\frac{1}{\varepsilon\sqrt{2}}-\frac{2}{3}H_{\Sigma_{t}}+O(\varepsilon)\right)\left(\frac{H_{\Sigma_{t}}}{\varepsilon\sqrt{2}}-\frac{2}{3}H_{\Sigma_{t}}^{2}+O(\varepsilon)\right)$
$\displaystyle\quad\quad+f^{2}\left(-\frac{1}{\varepsilon\sqrt{2}}+\frac{2}{3}H_{\Sigma_{t}}+O(\varepsilon)\right)\left(\frac{H_{\Sigma_{t}}}{\varepsilon\sqrt{2}}-\frac{2}{3}H_{\Sigma_{t}}^{2}+O(\varepsilon)\right)d\mathcal{H}^{n-1}$

The third term contributes,

$\displaystyle\frac{\varepsilon}{2}\int_{\Sigma_{t}}H_{\Sigma_{t}}f^{2}\frac{1}{2}\left(\left(\frac{\partial
u^{+}}{\partial\nu}\right)^{2}-\left(\frac{\partial
u^{-}}{\partial\nu}\right)^{2}\right)d\mathcal{H}^{n-1}=2\sigma_{0}\int_{\Sigma_{t}}H_{\Sigma_{t}}^{2}f^{2}+C(\Sigma)\varepsilon\|f\|_{L^{\infty}}^{2}$

by 2.12.

The second summand in the integral $I_{2}$ appears with a reversed sign owing
to the fact that the expansion of $u^{-}$ occurs with respect to the opposite
normal vector as in Theorem 2.10. It follows that this term contributes only
an error term which is $O(\varepsilon)$ as $\varepsilon\to 0$. Finally, to
deal with the term $I_{1}$, we follow the proof of Theorem $(2.6)$ in [MS23],
only stating what the difference is in the current calculation.

In the Fermi coordinates $(s,z)$, the Laplacian
$\Delta_{g}=\partial_{z}^{2}-H_{z}\partial_{z}+\Delta_{z}$, where $H_{z}$ and
$\Delta_{z}$ represent the mean curvature of the parallel hypersurface
$\\{p\in M|\text{d}(p,\Sigma_{t})=z$, d the signed distance function, and the
Laplacian along the parallel hypersurface of distance $z$ respectively. Using
the decomposition 2.14, it follows that
$\ell_{\varepsilon,u\pm}(\phi^{\pm})=-\ell_{\varepsilon,u^{\pm}}(h\overline{\operatorname{\bf{H}}}_{\varepsilon}^{\prime})$,
where $h=-f(s)u_{\nu}^{\pm}(s,0)$. Expanding this using Fermi coordinates
gives

$\displaystyle\ell_{\varepsilon,u}\phi=-\varepsilon^{2}\Delta_{z}(h)\overline{\operatorname{\bf{H}}}_{\varepsilon}^{\prime}+h\varepsilon
H_{z}+(W^{\prime\prime}(u)-W^{\prime\prime}(\overline{\operatorname{\bf{H}}}_{\varepsilon})+E)h\overline{\operatorname{\bf{H}}}_{\varepsilon}^{\prime}$

Substituting this into the expression for each term of $I_{1}$,

(3.12) $\displaystyle\int_{\Sigma_{t}}f\phi^{+}_{z}$
$\displaystyle=-\frac{\sqrt{2}}{\varepsilon^{2}}\int_{\Sigma_{t}}f\int_{0}^{-2\ell\log\varepsilon}\bigg{(}-\varepsilon^{2}\Delta_{z}(h)\overline{\operatorname{\bf{H}}}_{\varepsilon}^{{}^{\prime}2}+\varepsilon
H_{z}h\overline{\operatorname{\bf{H}}}_{\varepsilon}^{{}^{\prime\prime}}$
$\displaystyle\quad\quad+(W^{\prime\prime}(u)-W^{\prime\prime}(\overline{\operatorname{\bf{H}}}_{\varepsilon})+E)h\overline{\operatorname{\bf{H}}}^{{}^{\prime}2}_{\varepsilon}$
$\displaystyle\quad\quad-\sqrt{2}(\varepsilon^{2}\Delta_{z}(\phi^{+})-H_{z}\varepsilon^{2}\phi^{+}_{z}$
$\displaystyle\quad\quad-(W^{\prime\prime}(u)-W^{\prime\prime}(\overline{\operatorname{\bf{H}}}_{\varepsilon})+E)\phi^{+})\overline{\operatorname{\bf{H}}}_{\varepsilon}^{\prime}\bigg{)}\sqrt{\det
g(s,z)}dzds$

That this term is equal to

$\int_{\Sigma_{t}}|\nabla
f|^{2}-(\textnormal{Ric}(\nu,\nu)+|A_{\Sigma_{t}}|^{2})f^{2}d\mathcal{H}^{n-1}+E(f)$

where $E(f)\leq C\varepsilon^{\frac{1}{2}}\|f\|_{W^{1,2}}$ is the subject of
[MS23], section 11.4, the only difference in our situation, is that $H_{z}$ is
not necessarily small in $\varepsilon$, and so cannot be absorbed into the
remainder, instead we note that the terms which appear with $H_{z}$ in the
integral 3.12 appear with opposite contributions from $\phi^{+}_{z}$ and
$\phi^{-}_{z}$ respectively. ∎

##  4\. The Strong min-max property

We are ready to prove 2.8.

###### Proof of Theorem 2.8.

We proceed by contradiction. Suppose there exists a $\delta>0$ so that we can
find for each $i$ a $k$-parameter family of surfaces $\\{\Sigma_{p}^{i}|p\in
B^{k}\\}$ in $\mathcal{U}$ with

$\sup_{p\in
B_{k}}\operatorname{\mathcal{B}_{\varepsilon_{i}}}(\Sigma_{p}^{i})\leq\operatorname{\mathcal{B}_{\varepsilon_{i}}}(\Sigma)-\delta.$

In particular, for each $i$ we can find a sequence of functions $f_{i}$ so
that each normal graph $\Gamma(f_{i})$ satisfies

$\operatorname{\mathcal{B}_{\varepsilon_{i}}}(\Gamma(f_{i}))\leq\operatorname{\mathcal{B}_{\varepsilon_{i}}}(\Sigma)-\frac{\delta}{2}.$

Taylor expanding $\operatorname{\mathcal{B}_{\varepsilon_{i}}}$ around $f_{i}$
and applying Corollary 2.1 in [MS23] gives

$\operatorname{\mathcal{B}_{\varepsilon_{i}}}(\Gamma(f_{i}))=\operatorname{\mathcal{B}_{\varepsilon_{i}}}(\Sigma)+R_{\varepsilon_{i}}+T_{\varepsilon_{i}}(f_{i}),$

where $T_{\varepsilon_{i}}$ is the remainder term in the first order Taylor
expansion and,

$R_{\varepsilon_{i}}=-\int_{\Sigma}H_{\Sigma}f+E_{i}(f),$

and $|E_{i}(f)|\leq C\varepsilon_{i}$ where $C$ is a constant depending only
on $\Sigma$. Since $\Sigma$ is a minimal surface it follows that
$R_{\varepsilon_{i}}=O(\varepsilon_{i})$ as $i\to\infty$. It follows from
Taylor’s theorem that

$T_{i}(f_{i})=\frac{1}{2}\frac{d^{2}}{dt^{2}}\bigg{|}_{t=\xi_{i}}\operatorname{\mathcal{B}_{\varepsilon_{i}}}(\Gamma(\xi_{i}f_{i})),$

where $\xi_{i}\in(0,1)$. However, from Theorem 3.5, this is

$\sigma_{0}\frac{d^{2}}{dt^{2}}\bigg{|}_{t=\xi_{i}}\mathcal{A}(\Gamma(\xi_{i}f_{i}))+\tilde{R}_{\varepsilon_{i}}$

where $R_{\varepsilon_{i}}\to 0$ as $i\to\infty$. In particular, it follows
that $T_{\varepsilon_{i}}(f_{i})-2\sigma_{0}T(f_{i})\to 0$ as $i\to\infty$,
where $T(f_{i})$ is the Taylor remainder term when expanding the area
functional about $f_{i}$. Therefore it follows that, for $i$ sufficiently
large

$|\operatorname{\mathcal{B}_{\varepsilon_{i}}}(\Gamma(f_{i}))-\mathcal{A}(\Gamma(f_{i}))|\leq\frac{\delta}{4}.$

So for any such $i$ we have

$\sigma_{0}\sup_{p\in
B^{k}}\mathcal{A}(\Sigma_{p}^{i})\leq\frac{\delta}{4}+\sup_{p\in
B^{k}}\operatorname{\mathcal{B}_{\varepsilon_{i}}}(\Sigma_{p}^{i})\leq\operatorname{\mathcal{B}_{\varepsilon_{i}}}(\Sigma)-\frac{\delta}{4}.$

However,
$\operatorname{\mathcal{B}_{\varepsilon_{i}}}(\Sigma)\to\mathcal{A}(\Sigma)$,
so this implies that

$\sup_{p\in B^{k}}\mathcal{A}(\Sigma_{p}^{i})<\mathcal{A}(\Sigma),$

which is impossible.

∎

##  5\. A weak Palais-Smale condition

Let $\mathcal{S}^{k,\alpha}$ denote the space of all $2$-sided, closed,
$C^{k,\alpha}$-hypersurfaces of $M$ which are homologous to $\Sigma$ (and
moreover which are therefore boundaries). We wish to show the following
Palais-Smale type compactness condition holds.

######  Proposition 5.1.

Let $\\{\Sigma_{n}\\}$ be a sequence of hypersurfaces in
$\mathcal{S}^{k,\alpha}$ and suppose $d>0$. Then if

  1. (1)

$\delta\operatorname{\mathcal{B}_{\varepsilon}}(\Sigma_{n})\to 0$,

  2. (2)

$\operatorname{\mathcal{B}_{\varepsilon}}(\Sigma_{n})\to d$,

  3. (3)

$\sup_{n}\textnormal{Area}(\Sigma_{n})<\infty$

then, up to taking a subsequence, there exists a Caccioppoli set $M^{+}$ for
which $M^{+}_{n}\to M^{+}$ and $\Sigma_{\infty}\doteq\partial M^{+}$, and
moreover there is a smooth phase transition $u:M\to\bf{R}$ which vanishes on
$\Sigma_{\infty}$

######  Remark 5.2.

The boundary $\Sigma_{\infty}$ need not be smooth, as there can be points
along $\Sigma_{\infty}$ where $\nabla u=0$. However, the phase transition $u$
will be smooth everywhere by standard elliptic regularity.

###### Proof of Proposition 5.1.

Suppose $u_{n}$ denotes the broken phase transition associated to the
hypersurface $\Sigma_{n}$. Then it is easy to see that $u_{n}$ satisfies

(5.1) $\displaystyle E_{\varepsilon}^{\prime}(u_{n})\to 0$ (5.2)
$\displaystyle E_{\varepsilon}(u_{n})\to d$

Therefore, by the classical Palais-Smale condition, there is a function
$u_{\infty}$ with $u_{n}\xrightharpoonup[W^{1,2}]{}u_{\infty}$. It follows
that $u_{\infty}$ is a weak solution to the Allen-Cahn equation in
$W^{1,2}(M)$, and therefore by elliptic regularity it is a smooth phase
transition. By 5.2, $M^{+}\doteq\\{u_{\infty}>0\\}$ and
$M^{-}\doteq\\{u_{\infty}<0\\}$ are nonempty. We note that on $M^{+}$,
$u_{\infty}$ is a positive Dirichlet minimizer for

$\begin{cases}\Delta_{g}u_{\infty}=\frac{W^{\prime}(u)}{\varepsilon^{2}},&\text{in
}M^{+},\\\ u_{\infty}\equiv 0,&\text{on }\partial M^{+}.\end{cases}$

By the weak convergence of $u_{n}$ to $u_{\infty}$ in $W^{1,2}$, it follows
from Rellich’s theorem that $u_{n}\to u_{\infty}$ strongly in $L^{2}$.
Moreover, by compactness of $M$, we may pass to a further subsequence with
$u_{n}\to u_{\infty}$ in $L^{1}$.

######  Claim 5.3.

$M_{n}^{+}\to M^{+}$.

###### Proof of Claim 5.3.

Suppose it is not the case. Then, possibly up to taking a subsequence, we may
find a $\delta>0$ so that $|M^{+}_{n}\Delta M^{+}|>\delta$. However, the area
bound on $\Sigma_{n}$ implies that $\\{M_{n}^{+}\\}$ is of uniformly bounded
variation, so BV-compactness ([Mag12], Theorem 12.26) then implies that there
is a measurable set $M_{\infty}^{+}$ so that

$M_{n}^{+}\to M_{\infty}^{+}.$

Therefore it follows that

$|M_{\infty}^{+}\Delta M^{+}|\geq\delta.$

So moreover there is, for some $k_{0}<0$, a set $A\subset M_{\infty}^{+}$ of
positive measure so that

$u_{\infty}\big{|}_{A}\leq k_{0}.$

However, $u_{n}\geq 0$ along $A$ by construction, and
$u_{n}\big{|}_{A}\xrightarrow[L^{1}]{}u_{\infty}\big{|}_{A}$, which is a
contradiction. ∎

The proof is now complete. ∎

##  6\. A mountain pass theorem

###  6.1. A Deformation Lemma

In order to prove an appropriate mountain-pass type theorem, we need to prove
some variant of the usual deformation lemma [Rab11]. The classical deformation
lemma does not quite work because our Palais-Smale type condition requires us
to use a weaker notion of convergence than convergence as boundaries of
Caccioppoli sets.

For $c\in\mathbf{R}$ we write

(6.1) $\displaystyle
A_{c}^{L}\doteq\\{\Sigma\in\mathcal{S}^{k,\alpha}|\operatorname{\mathcal{B}_{\varepsilon}}(\Sigma)\leq
c,\mathcal{H}^{n}\leq L\\},$ (6.2) $\displaystyle
K_{c}\doteq\\{\Omega\in\mathcal{C}(M)|\partial\Omega=u^{-1}(0),E_{\varepsilon}(u)=c,\delta
E_{\varepsilon}(u)=0\\}.$

######  Remark 6.1.

Some care must be taken in the definition of the sets 6.1. As stated,
$\operatorname{\mathcal{B}_{\varepsilon}}$ is defined only for those
hypersurfaces which split $M$ into two components for which the first
eigenvalue satisfies the condition $\lambda_{1}<\varepsilon^{2}$. We may still
uniquely define $\operatorname{\mathcal{B}_{\varepsilon}}$ on all of
$\mathcal{S}^{k,\alpha}$ by taking the Dirichlet phase transition associated
to a side of $\tilde{\Sigma}\in\mathcal{S}^{k,\alpha}$ to be $0$ if 2.1 admits
no solution. This extended definition of
$\operatorname{\mathcal{B}_{\varepsilon}}$ remains smooth on
$\mathcal{S}^{k,\alpha}$, but 3.5 is only true in the regime where the
condition $\lambda_{1}<\varepsilon^{2}$ on both sides of $\tilde{\Sigma}$.

We wish to show that if $K_{c}$ is empty, we can always deform $A_{c+\eta}$
into $A_{c-\eta}$ for some appropriate $\eta>0$ without running into a
critical point.

######  Lemma 6.2 (Deformation).

Suppose $c>0$ is such that $K_{c}=\emptyset$. For each $\eta>0$ taken
sufficiently small, there are constants $0<\bar{\eta}<\eta$ and $\theta>0$ and
a function $F\in C([0,1]\times\mathcal{S}^{k,\alpha};\mathcal{S}^{k,\alpha})$
so that

  1. (1)

$F(0,\Sigma)=\Sigma,$

  2. (2)

$F(1,\Sigma)=\Sigma$ for all $\Sigma\not\in
BE^{-1}_{\varepsilon}[c-\eta,c+\eta]$,

  3. (3)

$\operatorname{\mathcal{B}_{\varepsilon}}(F(t,\Sigma))\leq F(\Sigma)$,

  4. (4)

$F(1,A_{c+\bar{\eta}}^{L})\subset A_{c-\bar{\eta}}^{\theta L}$.

###### Proof.

First we claim the following.

######  Claim 6.3.

There are constants $0<\mu,\eta<1$ so that

$\|\delta\operatorname{\mathcal{B}_{\varepsilon}}(\Sigma)\|\geq\mu$

whenever $\Sigma\in A^{L}_{c+\eta}\backslash A^{L}_{c-\eta}$.

###### Proof of claim 6.3.

Suppose this is not the case. Then there are sequences $\mu_{k},\eta_{k}\to 0$
as $k\to\infty$ and $\Sigma_{k}\in A^{L}_{c+\eta_{k}}\backslash
A^{L}_{c-\eta_{k}}$ with

(6.3)
$\displaystyle\|\delta\operatorname{\mathcal{B}_{\varepsilon}}(\Sigma_{k})\|\leq\mu_{k},$
(6.4) $\displaystyle\operatorname{\mathcal{B}_{\varepsilon}}(\Sigma_{k})\to
c.$

The Palais-Smale condition 5.1 then implies the existence of
$\Sigma_{\infty}\in K_{c}$, which contradicts our assumption that
$K_{c}=\emptyset$. ∎

Now set $\bar{\eta}$ so that

$0<\bar{\eta}<\eta,0<\bar{\eta}<\frac{\mu^{2}}{2}.$

We also set

$\displaystyle
A\doteq\\{\Sigma\in\mathcal{S}^{k,\alpha}|\operatorname{\mathcal{B}_{\varepsilon}}\leq
c-\eta\text{ or }\operatorname{\mathcal{B}_{\varepsilon}}\geq
c+\eta,\mathcal{H}^{n}(\Sigma)\leq L\\},$ $\displaystyle
B\doteq\\{\Sigma\in\mathcal{S}^{k,\alpha}|c-\bar{\eta}\leq\operatorname{\mathcal{B}_{\varepsilon}}(\Sigma)\leq
c+\bar{\eta},\mathcal{H}^{n}(\Sigma)\leq\theta L\\}.$

We define the functions $d:\mathcal{S}^{k,\alpha}\to\mathbf{R}$,
$h:\mathbf{R}^{+}\to\mathbf{R}$ by

$\displaystyle
d(\Sigma)=\text{dist}(\Sigma,A)(\text{dist}(\Sigma,A)+\text{dist}(\Sigma,B))^{-1},$
$\displaystyle h(t)=\begin{cases}1,&0\leq t\leq 1,\\\ 1/t,&t\geq
1.\end{cases}$

The function $d$ can be easily seen to satisfy $0\leq d\leq 1$, $d|_{A}\equiv
0,d|_{B}\equiv 1$.It can readily be seen that $d$ is Lipschitz continuous.

Now we may define a potential $G:\mathcal{S}^{k,\alpha}\to
T\mathcal{S}^{k,\alpha}$ by

$G(\Sigma)\doteq-d(\Sigma)h(\|\operatorname{\mathcal{B}_{\varepsilon}}^{\prime}(\Sigma)\|)\operatorname{\mathcal{B}_{\varepsilon}}^{\prime}(\Sigma).$

We observe that $G$ is bounded and globally Lipschitz on
$\mathcal{S}^{k,\alpha}$. Therefore by the general existence and uniqueness
theory for ODEs on Banach spaces (see [Lan85], Chapter IV), a solution exists
for some time $t\geq 0$ to the equation

(6.5) $\displaystyle\begin{cases}\dot{F}(t)=G\cdot F(t),\\\
F(0)=\Sigma.\end{cases}$

Moreover, we may observe that, by the area formula, for any bounded subset of
$\mathcal{S}^{k,\alpha}$ there is a $\theta>0$ so that if $\Sigma$ satisfies
$\mathcal{H}^{n-1}(\Sigma)\leq L$ then
$\mathcal{H}^{n-1}(F(t,\Sigma))\leq\theta L$.

We may now compute

$\displaystyle\frac{d}{dt}\operatorname{\mathcal{B}_{\varepsilon}}(F(t,\Sigma))$
$\displaystyle=\operatorname{\mathcal{B}_{\varepsilon}}^{\prime}(\Sigma)\dot{F}(t,\Sigma)$
$\displaystyle=-d(F(t,\Sigma))h(\|\operatorname{\mathcal{B}_{\varepsilon}}^{\prime}(F(t,\Sigma))\|)\|\operatorname{\mathcal{B}_{\varepsilon}}^{\prime}(F(t,\Sigma))\|^{2}$
$\displaystyle\leq 0.$

Finally, suppose that $\Sigma\in A_{c+\bar{\eta}}^{L}$. We wish to show that
$F(1,\Sigma)\in A^{\theta L}_{c-\bar{\eta}}.$ Since we have just shown that
deforming by $F$ causes the $\operatorname{\mathcal{B}_{\varepsilon}}$ to
decrease, then if $\Sigma\not\in B$ for some time $t$ we are done, so we may
assume that $F(t,\Sigma)\in B$ for all $t\in[0,1]$. This implies that
$g(F(t,\Sigma))\equiv 1$ for $0\leq t\leq 1$. As a result our previous
calculation gives

$\frac{d}{dt}\operatorname{\mathcal{B}_{\varepsilon}}(F(t,\Sigma))=-h(\|\operatorname{\mathcal{B}_{\varepsilon}}^{\prime}(F(t,\Sigma))\|)\|\operatorname{\mathcal{B}_{\varepsilon}}(F(t,\Sigma))\|^{2}.$

If $\|\operatorname{\mathcal{B}_{\varepsilon}}^{\prime}(F(t,\Sigma))\|\geq 1$
we consequently get

$\frac{d}{dt}\operatorname{\mathcal{B}_{\varepsilon}}(F(t,\Sigma))=-\|\operatorname{\mathcal{B}_{\varepsilon}}(F(t,\Sigma))\|\leq-\mu\leq-\mu^{2}.$

Otherwise if
$\|\operatorname{\mathcal{B}_{\varepsilon}}^{\prime}(F(t,\Sigma))\|\leq 1,$
then by definition of $h$ we get

$\frac{d}{dt}\operatorname{\mathcal{B}_{\varepsilon}}(F(t,\Sigma))\leq-\mu^{2}.$

Either way we find

$\operatorname{\mathcal{B}_{\varepsilon}}(F(1,\Sigma))\leq\operatorname{\mathcal{B}_{\varepsilon}}(\Sigma)-\mu^{2}\leq
c-\bar{\eta},$

which completes the proof. ∎

###  6.2. $1$-parameter mountain pass

With the deformation lemma in hand, we are now ready to prove a mountain pass
type result. For simplicity, we present the $1$-parameter version.

######  Theorem 6.4 (Mountain Pass Theorem).

Let $\Sigma_{0},\Sigma_{1}\in\mathcal{S}^{k,\alpha}$ be distinct. Define
$\mathcal{P}^{L},c,d$ by

(6.6) $\displaystyle\mathcal{P}^{L}\doteq\\{p\in
C([0,1],\mathcal{U})|p(0)=\Sigma_{0},p(1)=\Sigma_{1},\mathcal{H}^{n}(p(t))\leq
L\\},$ (6.7) $\displaystyle
c\doteq\max\\{\operatorname{\mathcal{B}_{\varepsilon}}(\Sigma_{0}),\operatorname{\mathcal{B}_{\varepsilon}}(\Sigma_{1})\\},$
(6.8) $\displaystyle
d\doteq\inf_{p\in\mathcal{P}^{L}}\max_{t\in[0,1]}\operatorname{\mathcal{B}_{\varepsilon}}(p(t)).$

Then, if $d>c$, $K_{d}\neq\emptyset$.

###### Proof.

Suppose that $K_{d}$ is empty. Choosing a sufficiently small number $\eta<d$
we can apply our deformation lemma to find positive constants
$0<\bar{\eta}<\eta$, $0<\theta<1$ and a homeomorphism
$F:\mathcal{U}\to\mathcal{U}$ satisfying

(6.9) $\displaystyle F(A_{d+\bar{\eta}}^{L})\subset A^{\theta
L}_{d-\bar{\eta}},$ (6.10) $\displaystyle F(\Sigma)=\Sigma$
$\displaystyle\text{ if
}\Sigma\not\in\operatorname{\mathcal{B}_{\varepsilon}}^{-1}[d-\eta,d+\eta].$

Take some path $a\in\mathcal{P}^{L}$ satisfying

$\max_{t\in[0,1]}\operatorname{\mathcal{B}_{\varepsilon}}(a(t))\leq
d+\bar{\eta}.$

Then $\bar{a}\doteq F(a)\in\mathcal{P}^{\theta L}\subset\mathcal{P}^{L}$. The
deformation lemma now implies that

$\max_{t\in[0,1]}\operatorname{\mathcal{B}_{\varepsilon}}(\bar{a}(t))\leq
d-\bar{\eta},$

a contradiction. ∎

The proof of the $k$-parameter version, 2.10, is identical. We now prove the
corollary 2.11 as a consequence of the mountain–pass construction.

###### Proof of Corollary 2.11.

.

Given a nondegenerate minimal hypersurface $\Sigma$ with Morse index $k$, we
may write

$L_{\Sigma}=\Delta_{\Sigma}+(\text{Ric}(\nu,\nu)+|A_{\Sigma}|^{2})$

for its stability operator. Since $\Sigma$ is nondegenerate, the eigenvalues
of $L_{\Sigma}$ satisfy

$\lambda_{1}<\lambda_{2}\leq\dots\leq\lambda_{k}<0<\lambda_{k+1}\leq\dots.$

Let $\varphi_{1},\dots,\varphi_{k}$ be the $k$ eigenfunctions which correspond
to the negative eigenvalues of $L_{\Sigma}$. Let $\eta>0$ be a constant chosen
sufficently small that each $N(\eta\phi_{k})$ is contained in the tubular
neighborhood $N(\frac{1}{n})$ of height $\frac{1}{n}$ about $\Sigma$. The
smooth $k$-parameter family $F:B^{k}\to S^{k,\alpha}$ given by
$F:(v_{1},\dots,v_{k})\mapsto\Gamma(\eta(v_{1}\varphi_{1}+\dots+v_{k}\varphi_{k}))$
is precisely the family of hypersurfaces satisfying $(2)$ in 2.3.

As in the proof of Theorem 2.8, we have
$\operatorname{\mathcal{B}_{\varepsilon}}(F(v))\to\mathcal{A}(F(v))$ uniformly
in $\varepsilon$. Let $\delta<\frac{1}{n}$. Then applying 2.8 to
$N(\frac{1}{n})$ we find an $\tilde{\varepsilon}_{n}$ so that for all
$\varepsilon<\varepsilon_{n}$

$\inf_{\tilde{\Sigma}_{v}}\sup_{v}\operatorname{\mathcal{B}_{\varepsilon}}(\tilde{\Sigma}_{v})\geq\operatorname{\mathcal{B}_{\varepsilon}}(\Sigma)-\frac{1}{n}.$

where $\tilde{\Sigma}_{v}$ ranges over all families which agree with the
canonical family $F(v)$ on $\partial B^{k}$. Now by uniformity of the
convergence as $\varepsilon\to 0$ we may simply select
$\varepsilon_{n}<\tilde{\varepsilon}_{n}$ sufficiently small that

$\max_{v\in\partial
B^{k}}\operatorname{\mathcal{B}_{\varepsilon}}(F(v))<\operatorname{\mathcal{B}_{\varepsilon}}(\Sigma)-\frac{1}{n}$

for every $\varepsilon\leq\varepsilon_{n}$. Therefore by 2.10, there is a
smooth phase transition $u_{\varepsilon}$, $\varepsilon<\varepsilon_{n}$, with

$E_{\varepsilon}(u_{\varepsilon})=\inf_{\tilde{\Sigma}_{v}}\sup_{v}\operatorname{\mathcal{B}_{\varepsilon}}(\tilde{\Sigma}_{v}),$

and moreover $u_{\varepsilon}^{-1}(0)\subset\overline{N(\frac{1}{n})}$.

Now the result follows, since $E_{\varepsilon}(u_{\varepsilon})\to
2\sigma_{0}\mathcal{A}(\Sigma)$ by construction.

∎

##  Appendix A Smooth dependence on the domain.

Let $k>2$ be a given integer, and suppose that $\Omega\subset(M,g)$ is a
domain with $C^{k,\alpha}$ boundary (i.e., the inclusion map $i_{\Omega}\in
C^{k,\alpha}(\Omega;M)$) which supports a positive Dirichlet phase transition
vanishing on the boundary. Take a neighborhood $U\subset
C^{k,\alpha}(\Omega;M)$ consisting only of $C^{k,\alpha}$ embeddings of
$\Omega$. Suppose that $F\in U$ and let $u_{F}$ be the unique positive
solution, afforded by [BO86], to the problem

$\begin{cases}\varepsilon^{2}\Delta_{g}u=W^{\prime}(u)&\text{in }F(\Omega),\\\
u\equiv 0&\text{on }\partial F(\Omega).\end{cases}$

In [MS23] it was shown that the dependence $F\mapsto u_{F}$ is smooth,
provided the neihgborhood $U$ is taken sufficiently small. Let $F_{t}$ be a
smooth one parameter family of diffeomorphisms with $F_{0}=\text{id}_{M},$
$\partial_{t}F_{t}\doteq X_{t}$ and $\nabla_{\partial_{t}}X_{t}=0$. Moreover,
set $X\doteq X_{0}$ for the variation vector field. It follows that $t\mapsto
u_{t}\doteq u_{F_{t}}$ is a smooth one parameter family of maps. It is a
straightforward computation to show that the time-derivative $\dot{u}_{t}$
satisfies the boundary value problem:

(A.1)
$\displaystyle\begin{cases}\Delta_{g}\dot{u}_{t}=W^{\prime\prime}(u_{t})\dot{u_{t}}\text{
in }\Omega_{t},\\\ \dot{u}_{t}=-\langle X_{t},\nabla^{M}u_{t}\rangle\text{ on
}\partial\Omega_{t},\end{cases}$

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