# On the (Limited) Generalization of MasterFace Attacks and
Its Relation to the Capacity of Face Representations

Philipp Terhörst12, Florian Bierbaum3, Marco Huber23, Naser Damer2,
Florian Kirchbuchner23, Kiran Raja1, Arjan Kuijper23
1Norwegian University of Science and Technology, Gjøvik, Norway
2Fraunhofer Institute for Computer Graphics Research IGD, Darmstadt, Germany
3Technical University of Darmstadt, Darmstadt, Germany

###### Abstract

A MasterFace is a face image that can successfully match against a large
portion of the population. Since their generation does not require access to
the information of the enrolled subjects, MasterFace attacks represent a
potential security risk for widely-used face recognition systems. Previous
works proposed methods for generating such images and demonstrated that these
attacks can strongly compromise face recognition. However, previous works
followed evaluation settings consisting of older recognition models, limited
cross-dataset and cross-model evaluations, and the use of low-scale testing
data. This makes it hard to state the generalizability of these attacks. In
this work, we comprehensively analyse the generalizability of MasterFace
attacks in empirical and theoretical investigations. The empirical
investigations include the use of six state-of-the-art FR models, cross-
dataset and cross-model evaluation protocols, and utilizing testing datasets
of significantly higher size and variance. The results indicate a low
generalizability when MasterFaces are training on a different face recognition
model than the one used for testing. In these cases, the attack performance is
similar to zero-effort imposter attacks. In the theoretical investigations, we
define and estimate the face capacity and the maximum MasterFace coverage
under the assumption that identities in the face space are well separated. The
current trend of increasing the fairness and generalizability in face
recognition indicates that the vulnerability of future systems might further
decrease. Future works might analyse the utility of MasterFaces for
understanding and enhancing the robustness of face recognition models.

##  1 Introduction

In face recognition (FR), a MasterFace is a face image that can be
successfully matched against a large portion of the population [23]. Such a
face can be used to impersonate any user without having access to information
about the enrolled subject [17]. Since face recognition systems (FRS) are
spreading worldwide and are increasingly involved in our daily life activities
[26], successful MasterFace attacks pose a great threat to these systems.

Figure 1: Visualization of theoretical concepts - Assuming an embedding space
(on a unit-sphere) with perfect identity-separation, the three concepts are
visualized for two and three-dimensional embeddings for a fixed decision
threshold $r=0.4$. In 2D, an optimal MasterFace can cover 12.5% of the
identities. In 3D, this drops to 7.5% indicating that for higher-dimensional
representations the effectiveness might further decrease.

Consequently, MasterFaces attracted attention in both media and academia. For
instance, the work from Shmelkin et al. [23] won the Google Best Paper Award
at IEEE International Conference on Automatic Face and Gesture Recognition
2021 for their work on generating MasterFaces. In that work, the authors
report that a single face can cover more than 20% of the identities in the
used database (for a given FR model at a threshold for a false match rate of
$10^{-3}$). Also, Nguyen et al. [17, 16] proposed a similar method for
generating MasterFaces and demonstrated that these attacks can successfully
compromise FRS. However, these works (a) make use of older FRS, (b) consist of
only limited cross-dataset and cross-model evaluations, and (c) conduct their
experiments on testing data of small size and variance. This does not allow to
make significant statements on the generalizability of these attacks.

In this work, we empirically and theoretically analysed the generalizability
of MasterFace attacks. Contrarily to previous works, our experiments (a) make
use of six state-of-the-art FR models, (b) are based on cross-dataset and
cross-model evaluations, and (c) utilize testing datasets that are of a
significantly higher size and variance. To ensure comparability with previous
works [23, 17], we use the original MasterFace images reported in the Arxiv
versions of these papers (CC-BY-4.0) for the experiments. The empirical
investigations showed that only an insignificant proportion of the testing
identities are successfully matched against the MasterFaces and that the
comparison scores resulting from comparisons with MasterFaces closely
resembles the score distribution of zero-effort imposter. Consequently, in our
experiments, no utilized FRS showed any significant vulnerability to
MasterFaces attacks. The observations lead to the assumption that successful
MasterFace attacks exploit regions in the embedding space of a specific FRS
characterized by poor identity-separability.

To assess the threat of MasterFace attacks regarding the rapid development of
more generalizable FRS, we conduct a theoretical analysis on the maximum
effectiveness of MasterFace attacks assuming an FRS that can perfectly
differentiate between identities. For this, we define and estimate the face
capacity as the maximum number of identities that can be differentiated and
the maximum MasterFace Coverage as the maximum number of perfectly-separable
identity representations that are successfully matched with a MasterFace
template. We define the ratio of the coverage and the capacity as the maximum
MasterFace effectiveness. The theoretical investigations showed that the face
capacity is several magnitudes higher than the maximum MasterFace coverage
demonstrating that the maximum MasterFace effectiveness becomes
insignificantly low when dealing when a perfect identity-separation is
achieved. In Figure 1, this is visualized for lower-dimensional
representations. With the current trend of increasing the generalizability of
FRS, this indicates that the vulnerability of MasterFace attacks might further
decrease in the future. In summary, the empirically and theoretically
investigations showed that MasterFace attacks do not generalize well to unseen
data and FR models.

##  2 Background

###  2.1 The Dodding Zoo and MasterFaces

Generally, biometric systems perform differently on different groups of users
[25, 3, 8]. In [27, 5], several user groups have been characterized depending
on their effect on the biometric system. This concept, formalized by
Doddington et al. [5], is known as biometric menagerie or Doddington Zoo.
Originally, this consists of four members: Sheep, Goats, Lambs, and Wolves.
Sheep match well against themselves and poorly against others. Goats are
generally difficult to match. Lambs are vulnerable to impersonation attacks
and Wolves are highly successful in impersonating lambs. MasterFace attacks
aim at creating synthetic and highly-effective wolve-samples [17]. They aim
for generating face images that pass biometric identity verification for a
large portion of the population and thus, can be used to impersonate, with a
high probability of success, any user, without having access to any user-
information [23]. This is the reason why in [17], MasterFace attacks are also
named Wolf attacks.

###  2.2 Generation of MasterFaces

Previous works [23, 17, 16] share a similar idea on the MasterFace generation
process. The works make use of a pre-trained StyleGAN [12] model $\mathcal{G}$
for the generation of a high-quality face image $\mathcal{I}$ from a given
latent vector $x$ and search in the latent space for such a MasterFace vector

$\displaystyle
x_{MF}=\arg\max_{x}\sum_{\mathcal{I}\in\mathcal{D}}f(\mathcal{M}(\mathcal{G}(x)),\mathcal{M}(\mathcal{I}),\theta)$
(1)

that maximizes its coverage with the images $\\{I\\}$ from a given dataset
$\mathcal{D}$. Here, $\mathcal{M}(\mathcal{I})$ is a face recognition model
that takes an image $\mathcal{I}$ as an input and output a face embedding $z$.
The function $f(x,y,\theta)$ is a matching function that compares the
similarity between the embeddings $x$ and $y$ and returns 1 or 0 depending on
a decision threshold $\theta$. To solve this optimization problem, a
Covariance Matrix Adaptation Evolution Strategy (CMA-ES) [19] is used. In
[23], multiple evolutionary strategies are compared and a novel approach is
proposed that uses a neural network to guide the search without a fitness
evaluation. The final MasterFace image $\mathcal{I}_{MF}$ is given by
$\mathcal{G}(x_{MF})$.

###  2.3 Limitations

The idea of creating a synthetic biometric sample to match a larger portion of
the population was first introduced for fingerprint images known as
MasterPrints [20, 21, 1]. MasterPrints build on the fact that in some
applications, such as smartphones, the fingerprint sensor is small and thus,
only partial fingerprint images are used for matching [1]. In contrast to
MasterPrints, works on MasterFaces report high performances even when non-
partial information is available. However, this might be explained by the
limitation of their experimental setups.

  * •

Weak assumption \- Previous works [23, 17, 16] build on the assumption that
many identities might cluster at specific points in the embedding space to
motivate the existence of MasterFace images. However, this means that the
identities of such a cluster are already similar to each other and thus, might
already match each other. This is supported by the observation that generated
MasterFaces mainly show very old or young faces [23]. Since, current FRS aim
to maximize the margin between identities in the embedding space, and thus to
distribute in the whole space, in this work, we will theoretically analyse how
well these attacks work if this assumption is not given. Moreover, building on
the assumption of clustered identities in the embedding space makes the
attacks strongly dependent on the face recognition model and thus, shows the
need for investigating the inter-model generalizability.

  * •

Older face recognition models \- Previous works limited their analysis to the
usage of rather old face recognition models that have less advanced
generalization mechanisms. For instance, in [23], Dlib [13] (2009), SphereFace
[14] (2017), and FaceNet [22] (2015) were investigated. In this work, we will
analyse the effectiveness of MasterFace attacks on more recent and advanced FR
models building on stronger generalization techniques.

  * •

Limited cross-dataset/model evaluations \- In previous works [23, 17, 16], the
MasterFace generation model was mainly (a) optimized for the effectiveness of
a specific FR model used for testing and (b) evaluated with data coming from
the same distribution as the training data111Please note that no cross-model
generalizability was claimed in [23].. Both points might easily result in
overfitted MasterFaces and demonstrate the need for cross-dataset and cross-
model evaluations. In this work, we focus on these two evaluation settings to
elaborate on the generalizability of these attacks.

  * •

Small testing data \- Previous works conducted their experiments with low-
scale testing data. This includes the use of the LFW dev set (1711 images)
[23, 17], and the dev set of the MOBIO dataset (58 identities) [17, 16]. Since
the amount and variance of testing data might strongly affect the performance
of MasterFace attacks, we will analyse this for both aspects on a larger scale
than previous research.

##  3 Methodology

Besides the comprehensive empirical evaluation, we propose a theoretical
analysis of the effectiveness of MasterFace attacks. Previous attacks [23, 17,
16] build on the assumption that many identities might cluster at specific
points in the embedding space. In the following, we will analyse the
effectiveness of optimal MasterFace attacks when such an assumption is not
given. This is achieved in three steps. First, the capacity of a face space is
calculated stating the optimal number of perfectly separable identities in
this space. Second, the maximum coverage of a MasterFace is computed in the
same context. Third, we will extrapolate the face capacity and MasterFace
coverage to analyse which portion of the face space can maximally be covered
by a MasterFace in high dimensions.

###  3.1 The Capacity of the Face Space

In the following, we assume a face space on a $d$-dimensional sphere (unit-
length face embeddings) in which every identity is perfectly separated and
matching function $r$ stating a non-match if two face embeddings $x^{(i)}$ and
$x^{(j)}$ have an Euclidean distance of $D(x^{(i)},x^{(j)})>r$. The capacity
$N$ of a face space (defined over $d$ and $r$) describes the maximum number of
identities that can be encoded in such a space with an optimal separation. For
a given $d$ and $r$, the following optimization problem

$\displaystyle\min_{x}\sum_{i=1}^{n}\sum_{j=i+1}^{n}\frac{1}{D(x^{(i)},x^{(j)})}$
(2)
$\displaystyle\text{with}\qquad||x^{(i)}||=1\qquad\forall\,\,\,i\in[1,\dots,n]$
(3)

equally distributes $n$ identities in the face space. The optimization
constraint $||x^{(i)}||=1$ ensures that all embeddings lie on a sphere and
that the conclusions also generalize to cosine similarity (since for unit-
length embeddings both measures only differ by a constant factor). By
iteratively solving this problem with increasing $n$ and determining the
nearest neighbour distance $r^{\prime}$, the capacity of the face space is
determined by the greatest $n$ that fulfils $r^{\prime}<r$.

###  3.2 Maximum MasterFace Coverage

The maximum MasterFace coverage $\overline{N}$ describes the maximum number of
perfectly-separable identity embeddings $x^{(i)}$ that are matched with a
MasterFace embedding $x^{(MF)}$. To achieve this, for a given $d$ and $r$, the
optimization problem

$\displaystyle\min_{x}\frac{1}{n}\sum_{i=1}^{n}\left(D(x^{(i)},x^{(MF)})-r\right)^{2}$
(4)
$\displaystyle\text{with}\,\,\,D(x^{(i)},x^{(j)})>r\,\,\,\text{and}\,\,\,||x^{(i)}||=1\,\,\,\forall\,i,j$
(5)

can check if $n$ identity embeddings $x^{(i)}$ can be placed around the
MasterFace embedding $x^{(MF)}$ such that these can be successfully matched
against the MasterFace embedding without being matched against each other.
Minimizing Eq. 4 ensures that the MasterFace can successfully match the $n$
identities while the optimization constraint $D(x^{(i)},x^{(j)})>r$ ensures
that these identities are still separable from each other. The highest $n$
that fulfils the described criteria determines the maximum MasterFace coverage
$\overline{N}$.

###  3.3 Maximum MasterFace Effectiveness

The maximum MasterFace effectiveness describes the maximal proportion of all
perfectly-separable identities in the embedding space that a MasterFace can
successfully match with. Consequently, we can define the maximum MasterFace
effectiveness $\eta(r,d)=\frac{\overline{N}(r,d)}{N(r,d)}$ over the ratio of
the MasterFace coverage $\overline{N}$ and the face capacity $N$. Since the
calculations of these values require computationally-expensive efforts that
are not feasible for higher dimensions, we solve these explicitly for
$d\in[3,10]$ and $r\in[0.65,1.35]$ with a Sequential Least Squares Programming
(SLSQP) algorithm and extrapolate these to get an estimate about the magnitude
of $\eta(r,d)$ for larger $d$. We found that the face capacity can be
accurately described through

$\displaystyle N(r,d)=\exp\left[d\,(\alpha+\gamma\,r)+\beta+\delta\,r\right]$
(6)

while the extrapolation of the MasterFace coverage follows

$\displaystyle\overline{N}(r,d)=(\overline{\alpha}\,d^{3}+\overline{\beta})\cdot\sigma\left(\phi\,(r^{\epsilon}-1)\right)+\overline{\gamma}\,d^{3}+\overline{\delta}.$
(7)

Here, $\exp(\cdot)$ and $\sigma(\cdot)$ are the exponential and the sigmoid
functions. In Eq. 7, we manually set $\phi=10000$ and $\epsilon=0.0005$ to
determine the shape of the sigmoid function $\sigma(\cdot)$ regarding $r$. The
learned parameters for both functions are shown in Table 1. In Section 5.2, we
will first discuss the suitability of the capacity and coverage approximations
before discussing its implications on the MasterFace effectiveness under the
assumption of face space with perfect identity-separation.

Table 1: Learned parameters for the capacity and coverage. | $\alpha$ | $\beta$ | $\gamma$ | $\delta$
---|---|---|---|---
Capacity | 0.993 | 3.701 | -0.436 | -3.706
Coverage | -0.172 | 8.258 | 0.153 | 0.315
|  |  |  |

##  4 Experimental setup

###  4.1 Databases and images

The experiments were conducted on three publicly-available datasets. The
Adience dataset [6] consists of 26k images from over 2k different subjects.
The images of the Adience dataset possess a wide range in terms of image
quality and additionally consists of many young faces. Contrarily to previous
works that focus only on around 10% of the images and identities of LFW [9]
for testing, we analyse the performance on the whole database containing 13k
images of over 5k identities. The ColorFeret database [18] consists of 14k
high-resolution face images from over 1k different individuals. The data
possess a variety of face poses (from frontal to profile) and facial
expressions under well-controlled conditions. For the experiments, the
reported MasterFace images from both groups that worked on this problem [23,
17] were used as included in their respective Arxiv papers (CC-BY-4.0). For
the sake of simplicity, we refer to the images from [17] as WolfAttacks and
from [23] as MasterFace in the experiments.

###  4.2 Evaluation metrics

Following the international standard for biometric evaluation [11], we report
the face verification error in terms of false non-match rate (FNMR) at a
threshold for a fixed false match rate (FMR). Extending the mean set coverage
score from [23], we introduce the concept of sample- and identity-coverage to
analyse the effectiveness of MasterFace attacks on the image- and identity-
level. Comparing the MasterFaces against every image of a given database, the
sample-coverage states the mean portion of images that are successfully
matched per MasterFace image. On the other hand, the identity-coverage states
the mean portion of identities that are matched successfully requiring at
least one image per identity to be matched with a MasterFace.

###  4.3 Utilized models

In the experiments, six state-of-the-art FR approaches are utilized for
evaluating the MasterFace attacks. This include ArcFace [4] (2019),
CurricularFace [10] (2020), Elastic-Arc and Elastic-Cos [2] (2021), MagFace
[15] (2021), and QMagFace [24] (2021). For the experiments, we used the
(ResNet-100/iResNet-101) models from the official repositories of the authors
to extract the face templates. Comparing two templates was done based on the
standard cosine similarity.

###  4.4 Investigations

This work analyses the generalizability of MasterFace attacks through
empirical and theoretical investigations. The empirical investigations aim to
analyse the vulnerability of six state-of-the-art FRS to MasterFace attacks.
This includes analysing sample- and identity-coverage rates as well as
analysing the results on the score-level.

To approximate the vulnerability of future FRS with increasing identity-
separation, the theoretical investigations analyses the face capacity in
relation to the maximum MasterFace coverage under the assumption of a face
space with perfect identity-separation. This allows studying the maximum
MasterFace effectiveness.

##  5 Results

###  5.1 Empirical Analysis

#### Investigating Coverage-Rates:

To analyse the vulnerability of six state-of-the-art FRS on MasterFace
attacks, Table 2 shows sample- and identity-coverage rates on three datasets
for WolfAttack and MasterFace images on the mentioned FRS. To cover a wide
range of applications, the decision thresholds range between FMR of $10^{-1}$
to $10^{-4}$.

Generally, the sample- and identity coverage drops drastically for lower FMRs
and the investigated FRS show no vulnerability to the MasterFace attacks. This
is also true for LFW. Despite that both methods use large parts of LFW as
training data, the coverage rates are close to zero. Consequently, we assume
that the main factor for the reported success of MasterFace attacks rely on
the fact that these attacks are optimized for very specific FR models.

In [17], FMRs of 6-35% (at a threshold for FMRs between 1-3%) were reported
for the WolfAttacks depending on the testing data and the FRS. This relates to
our sample-coverage at an FMR around $10^{-1}$ which varies between 2.96-8.94%
in our experiments. In [23], over 40% of the testing identities were matched
with 10 MasterFaces on three FRS. However, this was achieved by optimizing the
MasterFace generation on the target model and the testing data. For the
identity-separated evaluation, [23] reported an identity-coverage between
17-22% at an FMR threshold of $10^{-3}$. This is significantly lower than the
observed performance of 0.01-0.31% at the same FMR threshold in our
experiments using larger datasets and FR models that the attacks are not
trained on.

Overall, the results show a very limited generalizability of the MasterFace
attacks on recent FRS. The experiments demonstrate that the utilized FRS show
no significant vulnerability to any MasterFace attacks if these attacks are
not directly optimized on the deployed FRS. This holds especially true for FMR
of $10^{-3}$ and smaller as recommended by the European Boarder Guard Agency
Frontex [7]. In this range, the identity-coverage lies below 0.31% in all
investigated cases.

#### Investigating Score Distributions:

To get an understanding of how much the MasterFace attacks differ from zero-
effort imposter comparisons, Figure 2 show score distributions for the six FR
models on the three databases. In each plot, the score distributions for
genuine (green) and imposter (blue) comparisons are shown together with the
score distributions that result by comparing the MasterFace (red) and
WolfAttack (orange) images with all images of the database. The well-
separation of the genuine and imposter distributions demonstrates that the
utilized FR models perform well. However, in all cases, the MasterFace attack
distributions closely resemble the imposter distributions. This demonstrates
that the utilized FR solutions show no vulnerability to the MasterFace
attacks. This holds also true for the LFW dataset. Despite that both methods
use large parts of LFW for training, this supports our claim that the success
of current MasterFace attacks is based on optimizing the attacks on a specific
FR model.

Table 2: Analysis of coverage - The effectiveness of the MasterFace attacks from [23] (MasterFace) and [17] (WolfAttack) are analysed. Sample- and identity-coverage rates are reported on three datasets on six state-of-the-art FR models for several decision thresholds ranging from $10^{-1}$ to $10^{-4}$ FMRs. The results show no significant vulnerabilities to MasterFace attacks. |  |  |  | Sample-Coverage (%) at FMR |  | Identity-Coverage (%) at FMR
---|---|---|---|---|---|---
Dataset | Attack | FRS |  | $10^{-1}$ | $10^{-2}$ | $10^{-3}$ | $10^{-4}$ |  | $10^{-1}$ | $10^{-2}$ | $10^{-3}$ | $10^{-4}$
Adience | WolfAttack | ArcFace |  | 7.13 | 0.76 | 0.07 | 0.01 |  | 6.74 | 0.50 | 0.05 | 0.01
|  | CurricularFace |  | 5.93 | 0.72 | 0.14 | 0.03 |  | 6.83 | 0.56 | 0.06 | 0.01
|  | Elastic-Arc |  | 6.95 | 0.76 | 0.15 | 0.04 |  | 6.60 | 0.56 | 0.08 | 0.03
|  | Elastic-Cos |  | 6.89 | 0.59 | 0.09 | 0.03 |  | 6.96 | 0.47 | 0.05 | 0.01
|  | MagFace |  | 6.17 | 0.61 | 0.08 | 0.01 |  | 5.87 | 0.43 | 0.04 | 0.01
|  | QMagFace |  | 6.14 | 0.59 | 0.08 | 0.01 |  | 5.85 | 0.43 | 0.04 | 0.01
| MasterFace | ArcFace |  | 4.46 | 0.23 | 0.17 | 0.00 |  | 1.09 | 0.19 | 0.23 | 0.02
|  | CurricularFace |  | 5.31 | 0.21 | 0.21 | 0.01 |  | 1.11 | 0.21 | 0.25 | 0.04
|  | Elastic-Arc |  | 4.83 | 0.20 | 0.21 | 0.01 |  | 1.01 | 0.19 | 0.31 | 0.02
|  | Elastic-Cos |  | 4.91 | 0.20 | 0.12 | 0.00 |  | 1.09 | 0.19 | 0.21 | 0.03
|  | MagFace |  | 4.38 | 0.19 | 0.18 | 0.00 |  | 0.96 | 0.18 | 0.22 | 0.01
|  | QMagFace |  | 4.35 | 0.19 | 0.18 | 0.00 |  | 0.96 | 0.18 | 0.22 | 0.02
LFW | WolfAttack | ArcFace |  | 2.96 | 0.10 | 0.00 | 0.00 |  | 3.59 | 0.20 | 0.01 | 0.00
|  | CurricularFace |  | 5.59 | 0.34 | 0.02 | 0.00 |  | 5.73 | 0.60 | 0.05 | 0.01
|  | Elastic-Arc |  | 5.59 | 0.36 | 0.02 | 0.00 |  | 5.40 | 0.53 | 0.04 | 0.01
|  | Elastic-Cos |  | 6.11 | 0.39 | 0.02 | 0.00 |  | 5.78 | 0.59 | 0.05 | 0.01
|  | MagFace |  | 3.07 | 0.13 | 0.01 | 0.00 |  | 3.65 | 0.22 | 0.02 | 0.00
|  | QMagFace |  | 3.08 | 0.13 | 0.01 | 0.00 |  | 3.65 | 0.22 | 0.02 | 0.00
| MasterFace | ArcFace |  | 4.20 | 0.22 | 0.02 | 0.01 |  | 0.87 | 0.21 | 0.03 | 0.02
|  | CurricularFace |  | 8.73 | 0.73 | 0.06 | 0.02 |  | 1.05 | 0.48 | 0.07 | 0.02
|  | Elastic-Arc |  | 8.53 | 0.71 | 0.06 | 0.02 |  | 1.04 | 0.47 | 0.07 | 0.02
|  | Elastic-Cos |  | 8.94 | 0.73 | 0.06 | 0.01 |  | 1.03 | 0.46 | 0.07 | 0.02
|  | MagFace |  | 5.53 | 0.33 | 0.03 | 0.01 |  | 0.89 | 0.28 | 0.05 | 0.02
|  | QMagFace |  | 5.53 | 0.34 | 0.03 | 0.02 |  | 0.89 | 0.29 | 0.04 | 0.02
ColorFeret | WolfAttack | ArcFace |  | 4.76 | 0.32 | 0.01 | 0.00 |  | 10.21 | 1.58 | 0.13 | 0.00
|  | CurricularFace |  | 4.40 | 0.29 | 0.01 | 0.00 |  | 10.80 | 1.49 | 0.13 | 0.00
|  | Elastic-Arc |  | 5.06 | 0.33 | 0.02 | 0.00 |  | 10.18 | 1.49 | 0.10 | 0.00
|  | Elastic-Cos |  | 4.81 | 0.29 | 0.02 | 0.00 |  | 10.65 | 1.51 | 0.12 | 0.00
|  | MagFace |  | 4.72 | 0.26 | 0.00 | 0.00 |  | 10.24 | 1.35 | 0.02 | 0.00
|  | QMagFace |  | 4.74 | 0.26 | 0.00 | 0.00 |  | 10.26 | 1.37 | 0.02 | 0.00
| MasterFace | ArcFace |  | 6.50 | 0.40 | 0.03 | 0.01 |  | 3.45 | 0.75 | 0.18 | 0.08
|  | CurricularFace |  | 7.02 | 0.43 | 0.04 | 0.01 |  | 3.33 | 0.76 | 0.19 | 0.07
|  | Elastic-Arc |  | 6.96 | 0.45 | 0.04 | 0.01 |  | 3.85 | 0.77 | 0.20 | 0.05
|  | Elastic-Cos |  | 6.80 | 0.38 | 0.03 | 0.01 |  | 2.94 | 0.73 | 0.16 | 0.05
|  | MagFace |  | 7.09 | 0.46 | 0.04 | 0.01 |  | 3.33 | 0.73 | 0.19 | 0.12
|  | QMagFace |  | 7.11 | 0.46 | 0.04 | 0.01 |  | 3.33 | 0.73 | 0.19 | 0.09
|  |  |  |  |  |  |  |  |  |  |  |

(a) Adience
ArcFace

(b) Adience
CurricularFace

(c) Adience
Elastic-Arc

(d) Adience
Elastic-Cos

(e) Adience
MagFace

(f) Adience
QMagFace

(g) LFW
ArcFace

(h) LFW
CurricularFace

(i) LFW
Elastic-Arc

(j) LFW
Elastic-Cos

(k) LFW
MagFace

(l) LFW
QMagFace

(m) ColorFeret
ArcFace

(n) ColorFeret
CurricularFace

(o) ColorFeret
Elastic-Arc

(p) ColorFeret
Elastic-Cos

(q) ColorFeret
MagFace

(r) ColorFeret
QMagFace

Figure 2: Analysis of the score distributions - The score distributions for
the genuine (green) and imposter (blue) comparisons are shown together with
the scores distributions that result when comparing the WolfAttack (orange)
and MasterFace (red) images against the database. The results are shown for
six state-of-the-art FRS on three databases. In all cases the MasterFace
attacks closely resemble the imposter distributions showing that the utilized
FR models are not vulnerable to these attacks.

###  5.2 Theoretical Analysis

The results from the empirical analysis suggests that the MasterFace
generation exploits regions in the embedding space of a specific FR model with
a weak identity separation. Since the generalization capabilities of FR models
increases drastically with every year, regions that MasterFace attacks might
exploit continuously decrease. In this section, we analyse the border case of
an FR model with perfect identity-separation as described in Section 3.

Figure 3 analyses the face capacity and MasterFace coverage fits defined in
Equations 6 and 7. In Figures 3a and 3b, the solutions of the optimization
problem are shown as dots, while the solid lines represent the fits. In Figure
3a, an exponential behaviour concerning $r$ is observed, while Figure 3b
follows a sigmoid function with respect to $r$. Both figures demonstrate that
the functions are well suitable for the fit and result in reasonable
approximations. Figure 3c extrapolates222Please note that these functions
result from an extrapolation process and thus, might not show the optimal
values. We recommend to use these only to get a general idea of the magnitude
of the face capacity and the maximum MasterFace coverage. the face capacity
and the maximum MasterFace coverage to higher dimensions that are more
frequently used in FR models, such as $d=128$ and $d=512$. The coverage and
capacity functions are shown for three different $r$ values that refer to
different FMR thresholds of the utilized models ($r=1.12$ for $10^{-4}$ FMR,
$r=1.18$ for $10^{-3}$ FMR, $r=1.25$ for $10^{-2}$ FMR). Independent of the
choice of $r$, the face capacity shows a stronger (exponential) growth than
the (polynomial growth) of the maximum MasterFace coverage for higher
dimensions.

This has a significant impact on the maximum MasterFace effectiveness as shown
in Figure 4. With higher dimensions $d$ (and lower decision thresholds $r$)
the maximum MasterFace effectiveness $\eta$ drops significantly. Consequently,
effective MasterFace attacks are only possible in very low-dimensional face
representations (e.g. $d<10$) assuming a perfect identity-separation in the
face space.

(a) Face Capacity

(b) Maximum MasterFace Coverage

(c) High-Dimensional Behavior

Figure 3: Analysis of the face capacity and MasterFace coverage fits - In (a),
the maximum number of identities $N$ with a distance of greater than $r$ to
each other are shown (capacity). In (b), the same is represented for the
maximum number of identities that have at distance of greater $r$ to each
other while the respective distance to a potential MasterFace is $r$ (maximum
coverage). The dots represent the solutions of their optimization problems,
while the solid lines represent the fits. In (c), the high-dimensional
behaviour of both functions are shown. Figure 4: Analysis of the maximum
MasterFace effectiveness $\eta$ \- The maximal proportion of all perfectly-
separable identities in the embeddings space that can successfully match with
a MasterFace are shown for different $d$ and $r$. For embedding dimensions
usually used in FRS ($d\geq 128)$), the effectiveness becomes very small. This
demonstrates that FRS with a perfect identity separation are not significantly
vulnerable to MasterFace attacks.

##  6 Summary of Observations

In contrast to previous works, the experiments of this work make use of six
state-of-the-art FR model, multiple testing datasets of significantly higher
size and variance, and cross-dataset and cross-model evaluations. This leads
to some key observations. Based on the empirical and theoretical
investigations of this work, we observe

  1. 1.

Low sample- and identity coverage rates of MasterFace attacks in cross-model
evaluations \- While in [23] identity-coverage between 17-22% at an FMR
threshold of $10^{-3}$ were reported, our experiments show significant lower
identity-coverage between 0.01-0.31% at the same threshold when the attacks
are not trained on the specific FR model.

  2. 2.

Score distributions of MasterFace attacks very similar to imposter
distributions \- For all FR model and dataset combination, we observe that the
score distribution of MasterFace attacks closely resemble the distribution of
imposter comparisons in cross-dataset and cross-model evaluation settings.

  3. 3.

No performance improvement of MasterFace attacks trained on testing data \-
Despite that both MasterFace attack methods were trained on large parts of the
LFW dataset, we observe no improvement of the MasterFace attack performance on
this dataset compared to the others.

  4. 4.

Insignificant maximum MasterFace effectiveness in face spaces with perfect
identity-separation \- The theoretical analysis showed that the maximum number
of perfectly-separable identities in a face space (capacity) is several
magnitudes higher than the number of perfectly-separable identities that a
MasterFaster can maximally cover.

Observation 1 and 2 demonstrates a low generalization of MasterFace attacks to
cross-modal and cross-dataset evaluation settings and shows that these
MasterFace images perform similar to zero-effort imposter comparisons.
Observation 3 eliminates the cross-dataset factor since training MasterFace
attacks on part of the training data does not increase the success of these
attacks. Consequently, we assume that an FR model might be only vulnerable to
MasterFace attacks when these are specifically trained on the individual
model. Observation 4 states that for a face space with perfect identity-
separation, even an optimal MasterFace will not cover a significant portion of
possible identities. With the current trend of increasing generalizability in
FRS, this indicates that the vulnerability of future FRS to MasterFace attacks
might further decrease.

##  7 Conclusion

In this work, we comprehensively analysed the generalizability of MasterFace
attacks empirically and theoretically. Contrarily to previous works, our
empirical investigations included the use of six state-of-the-art FR model,
larger testing datasets, and cross-dataset and cross-model evaluation
settings. The results show a MasterFace attack performance, similar to zero-
effort imposter, in our evaluation setting. This indicates a low cross-model
generalizability, meaning that MasterFace attacks might only be successful
when the same FRS is used for training and testing. In the theoretical
investigation, we defined and investigated the face capacity and the maximum
MasterFace coverage to analyse the maximum MasterFace effectiveness under the
assumption of an FRS with perfect identity-separation. The investigations
showed an insignificantly low maximum MasterFace effectiveness for such FRS.
Since research on FR still aims for higher generalizability, this indicates
that the vulnerability of future FRS for MasterFaces might further decrease.
We emphasize that future works might investigate the use of MasterFaces for
analysing and enhancing the robustness of FR models.

#### Acknowledgement

This work was carried out during the tenure of an ERCIM ’Alain Bensoussan‘
Fellowship Programme. Parts of this research work has been funded by the
German Federal Ministry of Education and Research and the Hessen State
Ministry for Higher Education, Research and the Arts within their joint
support of the National Research Center for Applied Cybersecurity ATHENE.
Portions of the research in this paper use the FERET database of facial images
collected under the FERET program, sponsored by the DOD Counterdrug Technology
Development Program Office.

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## Supplementary

Due to the strict page limit, the paper focuses on only briefly describing the
functions for the face capacity (Eq. 6) and the maximum MasterFace coverage
(Eq. 7) in Section 3.3 and discussing its suitability in Section 3.3. In this
section, we will explain the choice of these functions and their suitability
in more details.

(a) Face Capacity Para. $A$

(b) Face Capacity Para. $B$

(c) MasterFace Coverage Para. $\overline{A}$

(d) MasterFace Coverage Para. $\overline{B}$

Figure 5: Parameter values for the face capacity and MasterFace coverage for
single dimensions $d$.

### Deriving the Functions

For both, the face capacity and the maximum MasterFace coverage, we computed
the optimal solutions for $d=[3,\dots,10]$ with a Sequential Least Squares
Programming (SLSQP) algorithm as described in Section 3.3. These solutions are
shown as dots in Figures 3a and 3b. For finding the best functions that
describe these solutions, we tried a wide range of functions such as different
polynomial, exponential, and sigmoid functions and evaluated these on single
fixed dimensions $d$. For the final function, we choose the simplest function
that described the behaviour of the data well. This resulted in a face
capacity function

$\displaystyle N(r,d)=\exp[A(d)+r\,B(d)]$ (8)

with two parameters $A$ and $B$ describing a linear function regarding $r$ in
the argument of the exponential function. Each $d$ resulted in different
parameter values for $A$ and $B$. These values are shown in Figure 5a and 5b
for the different dimensions $d$. Based on the linearity of these, we fit the
following linear functions

$\displaystyle A(d)$ $\displaystyle=\alpha\,d+\beta$ (9) $\displaystyle B(d)$
$\displaystyle=\gamma\,d+\delta$ (10)

on the parameter points, resulting in the parameter values
$\alpha,\beta,\gamma,\delta$ as shown in Table 1. Inserting $A(d)$ and $B(d)$
into Eq. 8 leads to the used face capacity function in Eq. 6.

The same procedure was applied for the maximum MasterFace coverage. Similarly,
to the face capacity function, the function for the maximum MasterFace
coverage

$\displaystyle\overline{N}(r,d)=\overline{A}(d)\cdot\sigma\left(\phi\,(r^{\epsilon}-1)\right)+\overline{B}(d),$
(11)

consists of two parameters $\overline{A}$ and $\overline{B}$ that are
determined for each $d$ separately. However, in contrast the face capacity,
the MasterFace coverage follows a sigmoid shape with respect to $r$ (see
Figure 3b). Consequently, we used a modified sigmoid function that is squeezed
and shifted to the desired shape. The values of $\overline{A}$ and
$\overline{B}$ for different dimensions $d$ are shown in Figures 5c and 5d.
These follow a behaviour that can be well described through polynomial
functions of third order

$\displaystyle\overline{A}(d)=\overline{\alpha}\,d^{3}+\overline{\beta}$ (12)
$\displaystyle\overline{B}(d)=\overline{\gamma}\,d^{3}+\overline{\delta}.$
(13)

As before, the resulting parameters are shown in Table 1 and inserting
$\overline{A}(d)$ and $\overline{B}(d)$ into Eq. 11 results in the used
maximum MasterFace coverage function 7.

### Function Suitability

The advantage of dividing the fitting process into two steps is that we can
directly observe the quality of the fits with respect to the dimension
parameter $d$ as shown in Figure 5. For the face capacity function, we can
observe an outlier for $d=8$ that slightly degrade the quality of the fit.
However, assuming a linear function for $A(d)$ seems fitting (see Figure 5a)
and will probably result in reasonable parameter values for larger $d$. For
the function $B(d)$, assuming a linear function was more of safe choice since
the stability of the solver and the outlier for $d=8$ results in a less clear
function behaviour. Combing the observations for $A(d)$ and $B(d)$, the face
capacity function $N(d,r)$ might significantly differ from the optimal values.
Consequently, we recommend to use this function only to get a general idea of
the magnitude of the face capacity as we mentioned and did in the main paper.
For the maximum MasterFace coverage, both functions $\overline{A}$ and
$\overline{B}$ well represents the parameter values as observable in Figures
5c and 5d. This indicates that the maximum MasterFace coverage function
$\overline{N}$ might result in highly suitable estimates with respect to $d$.


# A uniform Sobolev inequality for ancient Ricci flows with bounded Nash
entropy

Pak-Yeung Chan, Zilu Ma, and Yongjia Zhang

###### Abstract

This note is a continuation of [CMZ21]. We shall show that an ancient Ricci
flow with uniformly bounded Nash entropy must also have uniformly bounded
$\nu$-functional. Consequently, on such an ancient solution there are uniform
logarithmic Sobolev and Sobolev inequalities. We emphasize that the main
theorem in this paper is true so long as the theory in [Bam20c] is valid, and
in particular, when the underlying manifold is closed.

##  1 Introduction

Qi S Zhang [Zhq07] proved a uniform Sobolev inequality for Ricci flows on
closed manifolds. His method depends on Perelman’s monotonicity formula.
Indeed, Perelman’s $\mu$-functional is known to be increasing in time, and by
this reason, is bounded from below by the geometric data of the initial
manifold. On the other hand, the $\mu$-functional is the logarithmic Sobolev
constant. The Sobolev inequality then follows from these facts.

If the underlining Ricci flow is an ancient solution, then there is no such
“initial manifold” to make use of when estimating the $\mu$-functional.
Nevertheless, Perelman’s asymptotic soliton [Per02, Proposition 11.2] provides
a substitute. This is exactly how [CMZ21, Theorem1.4] is proved. In that
theorem, the authors show that, on an ancient solution admitting an asymptotic
soliton, the $\mu$-functional at any time and at any scale is bounded from
below by the entropy of the asymptotic shrinker. This also means that the
$\nu$-functional is uniformly bounded.

However, Perelman’s asymptotic soliton is not known to exist in general cases.
This fact is a great restriction to the application of the logarithmic Sobolev
and Sobolev inequalities proved in [CMZ21, Theorem 1.4, Corollary 1.5]. In
this article, we shall remove this restriction by showing that Bamler’s
tangent flow at infinity [Bam20c, Theorem 1.41], which is known to be a metric
soliton, can also be used to estimate the $\mu$-functional for the ancient
solution. For most of the definitions which appear in this article, the reader
will find a brief introduction in [CMZ21, Section 2]; the notations in [CMZ21]
are completely adopted in this article.

Let $(M^{n},g(t))_{t\in(-\infty,0]}$ be a complete ancient Ricci flow.
Throughout this article, we shall make a technical assumption that $g(t)$ has
bounded curvature within each compact time interval, namely,

$\displaystyle\sup_{M\times[t_{1},t_{2}]}\big{|}{\operatorname{Rm}}_{g_{t}}\big{|}<\infty\quad\text{
for all }\quad-\infty<t_{1}\leq t_{2}\leq 0.$ (1.1)

Note that the curvature bound may depend on the interval $[t_{1},t_{2}]$, and
is not uniform on $M\times(-\infty,0]$. We further assume the existences of a
point $p_{0}$ $\in M$ and a constant $Y\in(0,\infty)$, such that for all
$\tau>0$, the Nash entropy based at $(p_{0},0)$ is bounded from below by $-Y$,
i.e.,

$\mathcal{N}_{p_{0},0}(\tau)\geq-Y\quad\text{ for all }\quad\tau>0.$ (1.2)

To apply the results in [Bam20c], we make an additional assumption: if the
sequence in [Bam20c, (1.2)] is taken to be

$\left\\{\big{(}(M,g_{i}(t))_{t\in(-\infty,0]},(\nu^{i}_{t})_{t\in(-\infty,0]}\big{)}\right\\}_{i=1}^{\infty},$

where $\tau_{i}\to\infty$, $g_{i}(t):=\tau_{i}^{-1}g(\tau_{i}t)$, and
$\nu^{i}_{t}=\nu_{p_{0},0\,|\,\tau_{i}t}$, then the theorems in [Bam20c]
(especially [Bam20c, Theorem 1.6] and all the statements contained in [Bam20c,
Theorem 1.15]) are still valid. In particular, this is the case if $M^{n}$ is
closed. Furthermore, we strongly believe that all the results in [Bam20c] can
be verified with only minor modifications under the assumptions (1.1) and
(1.2) alone. In other words, the assumption that the manifolds in [Bam20c,
(1.2)] are closed can be replaced by (1.1).

The main theorem of this article is:

######  Theorem 1.1.

Under the assumptions stated above, we have

$\displaystyle\inf_{t\leq 0}\nu(g(t))=\mu_{\infty}>-\infty,$ (1.3)

where

$\mu_{\infty}:=\inf_{\tau>0}\mathcal{N}_{p_{0},0}(\tau)=\lim_{\tau\rightarrow\infty}\mathcal{N}_{p_{0},0}(\tau)\geq-Y.$

In particular, this is true for closed ancient Ricci flows with bounded Nash
entropy.

Once the $\nu$-functional is known to be bounded, the following logarithmic
Sobolev inequalities and Sobolev inequalities are simply the consequences of
some straightforward computations (c.f. [Zhq07, LW20]).

######  Corollary 1.2 (The logarithmic Sobolev and Sobolev inequalities).

Under the assumptions as stated above, we have

  1. (1)

Logarithmic Sobolev inequality: for any compactly supported locally Lipschitz
function $u$ on $(M,g(t))$, where $t\leq 0$, and positive scale $\tau>0$, we
have

$\displaystyle\int_{M}u^{2}\log
u^{2}dg_{t}-\left(\int_{M}u^{2}dg_{t}\right)\int_{M}u^{2}dg_{t}+\left(\mu_{\infty}+n+\frac{n}{2}\log(4\pi\tau)\right)\int_{M}u^{2}dg_{t}$
$\displaystyle\leq\tau\int_{M}(4|\nabla u|^{2}+Ru^{2})dg_{t}$ .

  2. (2)

Sobolev inequality: for any compactly supported locally Lipschitz function $u$
on $(M,g(t))$, where $t\leq 0$, we have

$\left(\int_{M}|u|^{\frac{2n}{n-2}}dg_{t}\right)^{\frac{n-2}{n}}\leq
C(n)e^{-\frac{2\mu_{\infty}}{n}}\int_{M}(4|\nabla u|^{2}+Ru^{2})dg_{t}.$

Here $\mu_{\infty}$ is the lower bound of the $\nu$-functional in (1.3). In
particular, this is true for closed ancient Ricci flows with bounded Nash
entropy.

As an application of the Sobolev inequality, we shall also prove the following
volume growth lower bound for steady gradient Ricci solitons. Note that we do
not make any curvature assumptions in the following theorem.

######  Theorem 1.3.

Let $(M^{n},g,f)$ be a complete steady gradient Ricci soliton normalized in
the way that

$\operatorname{Ric}=\nabla^{2}f,\quad R+|\nabla f|^{2}=1.$

Suppose that the induced ancient Ricci flow satisfies the same assumptions as
in Theorem 1.1. Fix a point $o\in M.$ Then

$|B_{r}(o)|\geq c(n)e^{\mu_{\infty}}r^{n/2}\quad\text{ for all }\quad r>10,$

where $c(n)$ is a dimensional constant and $\mu_{\infty}$ is as defined in
Theorem 1.1. Here, we denote by $|\Omega|$ the volume of a measurable subset
$\Omega\subset M$ and by $B_{r}(o)$ the geodesic ball centered at $o$ with
radius $r.$

This result generalizes the previous results on volume growth obtained in
[MS13] without too much extra restrictions. And the volume lower bound here is
in a similar form as that for shrinking Ricci solitons proved in [LW20,
Proposition 6].

In dimension three, it is a consequence of the Hamilton-Ivey pinching estimate
that any complete steady gradient Ricci soliton has nonnegative sectional
curvature (see [CBl09] and the references therein). When the curvature of the
soliton is strictly positive, i.e., the manifold does not split locally, a
result of Catino-Mastrolia-Monticelli [CMM16] gives the following lower bound
of the volume growth:

$|B_{r}(o)|\geq cr^{2}\quad\text{ for all $r$ large enough}.$ (1.4)

It will be interesting to see if the quadratic growth in (1.4) is sharp or
not. Under the additional condition that the scalar curvature attains its
maximum, we show that the steady gradient Ricci soliton must have quadratic
volume growth. We do not impose any non-collapsed condition. In particular,
the result also applies to the flying-wing examples constructed by Lai
[Lai20].

######  Theorem 1.4.

Suppose $(M^{3},g,f)$ is a three dimensional complete steady gradient Ricci
soliton with positive sectional curvature. Assume that the scalar curvature
attains its maximum somewhere on $M$. Then there exists a positive constant
$C$ such that for all large $r$, it holds that

$C^{-1}r^{2}\leq|B_{r}(o)|\leq Cr^{2}.$ (1.5)

Acknowledgement. The first-named author was partially supported by an
AMS–Simons Travel Grant and would like to thank Professor Jiaping Wang for
fruitful discussions on the volume of steady solitons. The second-named author
would like to thank Yuxing Deng for very enlightening discussions about steady
solitons during his stay in La Jolla.

##  2 Proof of Theorem 1.1

For any sequence $\tau_{i}\to\infty,$ there is a subsequence of
$g_{i}(t):=\tau_{i}^{-1}g(\tau_{i}t)$ which has an $\mathbb{F}$-limit in sense
of [Bam20b], i.e.,

$\displaystyle\left((M,g_{i}(t))_{t\in(-\infty,0]},(\nu^{i}_{t})_{t\in(-\infty,0]}\right)\xrightarrow{\makebox[28.45274pt]{$\mathbb{F},\mathfrak{C}$}}\left(\mathcal{X}^{\infty},(\nu^{\infty}_{t})_{t\in(-\infty,0)}\right)\quad\text{
as }\quad i\to\infty,$ (2.1)

where $\nu^{i}_{t}=\nu_{p_{0},0\,|\,\tau_{i}t}$ is the conjugate heat kernel,
$\left(\mathcal{X}^{\infty},(\nu^{\infty}_{t})_{t\in(-\infty,0)}\right)$ is a
metric flow pair over $(-\infty,0]$, and $\mathfrak{C}$ is a correspondence
(c.f. [Bam20b, Definition 5.4]). Note that this fact is true assuming only
(1.1) and (1.2), because it follows from [Bam20b, Theorem 7.8], and does not
depend on [Bam20c]. According to our assumptions at the beginning of section
1, we shall then fix a sequence $\\{\tau_{i}\\}_{i=1}^{\infty}$, such that
[Bam20c] is valid for the sequence in (2.1).

######  Theorem 2.1.

([Bam20c]) Under the assumptions indicated at the begining of section 1,
$\left(\mathcal{X}^{\infty},(\nu^{\infty}_{t})_{t\in(-\infty,0)}\right)$ is a
metric soliton and $\mathcal{X}^{\infty}_{0}=\\{x^{\infty}\\}$ consists of a
singular point. Furthermore, we have
$d\nu^{\infty}_{t}=d\nu_{x^{\infty}\,|\,t}$ and

$\displaystyle\lim_{i\rightarrow\infty}\mathcal{N}_{p_{0},0}^{g_{i}}(\tau)=\mathcal{N}_{x^{\infty}}(\tau):=\int_{\mathcal{R}^{\infty}}f^{\infty}(\cdot,-\tau)d\nu^{\infty}_{-\tau}-\frac{n}{2}\quad\text{
for all }\quad\tau>0,$ (2.2)

where $\mathcal{N}_{p_{0},0}^{g_{i}}(\tau)$ is the Nash entrpy of the Ricci
flow $g_{i}$, $\mathcal{R}^{\infty}$ is the regular part of
$\mathcal{X}^{\infty}$, $f^{\infty}$ is a smooth function on
$\mathcal{R}^{\infty}$ defined as
$d\nu^{\infty}_{t}:=(4\pi|t|)^{-\frac{n}{2}}e^{-f^{\infty}(\cdot,t)}dg_{\infty,t}$
on $\mathcal{R}_{\infty}$, and
$\nu^{\infty}_{t}(\mathcal{X}^{\infty}\setminus\mathcal{R}^{\infty})=0$ for
all $t<0$. The convergence in (2.1) is smooth on $\mathcal{R}^{\infty}$ in the
sense of [Bam20b, Theorem 9.31].

###### Proof.

By [Bam20c, Theorem 1.4], the singular part
$\mathcal{X}^{\infty}\setminus\mathcal{R}^{\infty}$ always has zero measure.
By [Bam20c, Theorem 1.6], the convergence in (2.1) is smooth on
$\mathcal{R}^{\infty}$. The convergence of the Nash entropy (2.2) follows from
[Bam20c, Theorem 1.15]. Since the Nash entropy is monotonically decreasing in
$\tau$, we have that

$\mathcal{N}_{x^{\infty}}(\tau)=\lim_{i\rightarrow\infty}\mathcal{N}_{p_{0},0}^{g_{i}}(\tau)=\lim_{i\rightarrow\infty}\mathcal{N}^{g}_{p_{0},0}(\tau_{i}\tau)$

is a constant independent of $\tau$. It then follows from [Bam20c, Theorem
1.19] that
$\left(\mathcal{X}^{\infty},(\nu^{\infty}_{t})_{t\in(-\infty,0)}\right)$ is a
metric soliton. ∎

Next, to show that the $\nu$-functional is uniformly bounded on
$(M,g(t))_{t\in(-\infty,0]}$, we shall use the same method as we have applied
in [CMZ21, Section 9]. Let us arbitrarily fix $t_{0}\in(-\infty,0]$,
$\tau_{0}>0$, and $u_{0}$ satisfying $u_{0}\geq 0$, $\sqrt{u_{0}}\in
C_{0}^{\infty}(M)$, and $\int_{M}u_{0}dg_{t_{0}}=1$. We shall estimate

$\overline{\mathcal{W}}(g(t_{0}),u_{0},\tau_{0}):=\int_{M}\left(\tau_{0}\left(\frac{|\nabla
u_{0}|^{2}}{u_{0}^{2}}+R_{g_{t_{0}}}\right)-\log
u_{0}-\frac{n}{2}\log(4\pi\tau_{0})-n\right)u_{0}\,dg_{t_{0}}.$

As in [CMZ21, Section 9], we shall, without loss of generality, assume
$t_{0}<0$.

Let us solve the conjugate heat equation coupled with
$(M,g(t))_{t\in(-\infty,t_{0}]}$, with $u_{0}$ being its initial value. Then
the solution is

$\displaystyle
u(x,t):=\int_{M}K(\cdot,t_{0}\,|\,x,t)u_{0}\,dg_{t_{0}}\quad\text{ for all
}\quad(x,t)\in M\times(-\infty,t_{0}].$

By the fact that the conjugate heat equation preserves the integral, we also
have that

$d\mu_{t}:=u(\cdot,t)\,dg_{t}\quad\text{ for all }\quad t\leq t_{0}$

is a conjugate heat flow. Our first observation is the following.

######  Lemma 2.2.

We have

$\displaystyle\sup_{t\in(-\infty,\tau_{i}^{-1}t_{0}]}{\rm
dist}_{W_{1}}^{g^{i}_{t}}(\mu^{i}_{t},\nu^{i}_{t})\leq\frac{C}{\sqrt{\tau_{i}}},$
(2.3)

where $\mu^{i}_{t}:=\mu_{\tau_{i}t}$,
$\nu^{i}_{t}=\nu_{p_{0},0\,|\,\tau_{i}t}$, ${\rm dist}_{W_{1}}$ is the
$1$-Wassernstein distance, and $C$ is a constant depending on $u_{0}$ but
independent of $i$.

###### Proof.

Let $\nu_{t}=\nu_{p_{0},0\,|\,t}$. By [Bam20b, Proposition 3.24(b)], for all
$t\leq t_{0}$, we have

${\rm dist}_{W_{1}}^{g_{t}}(\mu_{t},\nu_{t})\leq{\rm
dist}_{W_{1}}^{g_{t_{0}}}(\mu_{t_{0}},\nu_{t_{0}}).$ (2.4)

Hence, we need only to estimate the right-hand-side of (2.4). Let us fix a
large positive constant $D_{0}$ such that $\operatorname{spt}u_{0}\subseteq
B_{t_{0}}(p_{0},D_{0})$. Let $\phi$ be an arbitrary bounded $1$-Lipschitz
function with respect to the metric ${\rm dist}_{g_{t_{0}}}$, then we may
compute

$\displaystyle\int_{M}\phi\,d\mu_{t_{0}}-\int_{M}\phi\,d\nu_{t_{0}}$
$\displaystyle=$
$\displaystyle\int_{M}(\phi-\phi(p_{0}))\,d\mu_{t_{0}}-\int_{M}(\phi-\phi(p_{0}))\,d\nu_{t_{0}}$
$\displaystyle\leq$ $\displaystyle
D_{0}\int_{M}u_{t_{0}}dg_{t_{0}}-\int_{M}(\phi-\phi(p_{0}))K(p_{0},0|\cdot,t_{0})\,dg_{t_{0}}$
$\displaystyle\leq$ $\displaystyle D_{0}+\int_{M}{\rm
dist}_{t_{0}}(\cdot,p_{0})K(p_{0},0|\cdot,t_{0})\,dg_{t_{0}}$
$\displaystyle\leq$ $\displaystyle C,$

where we have used the Gaussian upper bound in [CCGGIIKLLN10, Theorem 26.25]
and (1.1) in the last inequality. Here $C$ is a constant depending only on the
curvature bound on $M\times[t_{0},0]$ and the lower bound of
$\operatorname{Vol}_{g_{0}}\big{(}B_{g_{0}}(p_{0},1)\big{)}$. It follows from
the Kantorovich-Rubinstein Theorem that

$\begin{split}{\rm
dist}_{W_{1}}^{g_{t_{0}}}(\mu_{t_{0}},\nu_{t_{0}})&=\sup_{\phi}\left(\int_{M}\phi\,d\mu_{t_{0}}-\int_{M}\phi\,d\nu_{t_{0}}\right)\\\
&\leq C,\end{split}$ (2.5)

where the supremum is taken over all bounded $1$-Lipschitz functions $\phi$
with respect to the metric ${\rm dist}_{g_{t_{0}}}$.

Finally, let us arbitrarily fix $i\in\mathbb{N}$ and
$t\in(-\infty,\tau_{i}^{-1}t_{0}]$. Recall that the $W_{1}$-Wassernstein
distance between two probability measures $\mu,\nu\in\mathcal{P}(X)$, where
$X$ is a metric space, is defined as

${\rm dist}_{W_{1}}(\mu,\nu)=\inf_{q}\int_{X\times X}{\rm
dist}(x,y)\,dq(x,y),$

where the infimum is taken over all couplings $q\in\mathcal{P}(X\times X)$ of
$\mu$ and $\nu$. Hence, using (2.4) and (2.5), we can pick a coupling $q$ of
$\mu^{i}_{t}(=\mu_{\tau_{i}t})$ and $\nu^{i}_{t}(=\nu_{\tau_{i}t})$ such that

$\int_{M\times M}{\rm dist}_{g_{\tau_{i}t}}(x,y)\,dq(x,y)\leq 2C.$

Therefore, we have

$\displaystyle{\rm dist}_{W_{1}}^{g_{i,t}}(\mu^{i}_{t},\nu^{i}_{t})$
$\displaystyle\leq$ $\displaystyle\int_{M\times M}{\rm
dist}_{g_{i,t}}(x,y)\,dq(x,y)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\tau_{i}}}\int_{M\times M}{\rm
dist}_{g_{\tau_{i}t}}(x,y)\,dq(x,y)$ $\displaystyle\leq$
$\displaystyle\frac{2C}{\sqrt{\tau_{i}}}.$

∎

The next observation follows from the exactly same reasoning as [CMZ21,
Proposition 9.5]

######  Lemma 2.3.

There is a constant $C_{0}$, depending on $p_{0}$ and the function $u_{0}$,
such that

$\displaystyle u(x,t)\leq C_{0}K(p_{0},0\,|\,x,t)\quad\text{ for all
}\quad(x,t)\in M\times(-\infty,t_{0}).$

For the convenience of the proof, we shall next fix some notations. Let

$\displaystyle d\mu_{t}$ $\displaystyle:=$
$\displaystyle(4\pi|t|)^{-\frac{n}{2}}e^{-\bar{f}(\cdot,t)}dg_{t}\quad\text{
for }\quad t<t_{0},$ $\displaystyle d\mu^{i}_{t}$ $\displaystyle:=$
$\displaystyle(4\pi|t|)^{-\frac{n}{2}}e^{-\bar{f}^{i}(\cdot,t)}dg_{t}\quad\text{
for }\quad t<\tau_{i}^{-1}t_{0},$ $\displaystyle d\nu_{t}$ $\displaystyle:=$
$\displaystyle(4\pi|t|)^{-\frac{n}{2}}e^{-f_{p_{0},0}(\cdot,t)}dg_{t}\quad\text{
for }\quad t<0,$ $\displaystyle d\nu^{i}_{t}$ $\displaystyle:=$
$\displaystyle(4\pi|t|)^{-\frac{n}{2}}e^{-f^{i}_{p_{0},0}(\cdot,t)}dg_{t}\quad\text{
for }\quad t<0,$

and we define

$\displaystyle\mathcal{N}_{p_{0},0}^{i}(\tau)$ $\displaystyle:=$
$\displaystyle\mathcal{N}_{p_{0},0}^{g_{i}}(\tau)=\int_{M}f^{i}_{p_{0},0}(\cdot,-\tau)d\nu^{i}_{-\tau}-\frac{n}{2}\quad\text{
for }\quad\tau>0,$ $\displaystyle\overline{\mathcal{N}}^{i}(\tau)$
$\displaystyle:=$
$\displaystyle\int_{M}\bar{f}^{i}_{p_{0},0}(\cdot,-\tau)d\mu^{i}_{-\tau}-\frac{n}{2}\quad\text{
for }\quad\tau>\tau_{i}^{-1}|t_{0}|.$

######  Lemma 2.4.

We have

$\displaystyle\left((M,g_{i}(t))_{t\in(-\infty,\tau_{i}^{-1}t_{0}]},(\mu^{i}_{t})_{t\in(-\infty,\tau_{i}^{-1}t_{0}]}\right)\xrightarrow{\makebox[28.45274pt]{$\mathbb{F},\mathfrak{C}$}}\left(\mathcal{X}^{\infty},(\nu^{\infty}_{t})_{t\in(-\infty,0)}\right)\quad\text{
as }\quad i\to\infty,$ (2.6)

where $\mathfrak{C}$ is the same correspondence as in (2.1). The convergence
is smooth on $\mathcal{R}^{\infty}$ in the sense of [Bam20b, Theorem 9.31],
and in particular, we have

$\displaystyle f^{i}_{p_{0},0}\rightarrow
f^{\infty},\quad\bar{f}^{i}\rightarrow f^{\infty}$ (2.7)

locally smoothly on $\mathcal{R}^{\infty}$. Here both $f^{\infty}$ and
$\mathcal{R}^{\infty}$ are defined in the statement of Theorem 2.1, and, in
the convergence, $f^{i}_{p_{0},0}$ and $\bar{f}^{i}$ should be understood as
being pulled back by the diffeomorphisms provided by [Bam20b, Theorem 9.31].

###### Proof.

Combining Lemma 2.2 and [Bam20b, Lemma 5.19], we obtain (2.6). By [Bam20b,
Lemma 6.17], we then have

$\displaystyle(\mu^{i}_{t})_{t\in(-\infty,\tau_{i}^{-1}t_{0}]}\xrightarrow{\makebox[28.45274pt]{$\mathfrak{C}$}}(\nu^{\infty}_{t})_{t\in(-\infty,0)}\quad\text{
as }\quad i\to\infty,$ (2.8)

where the convergence is in the sense of [Bam20b, Definition 6.14]. By
[Bam20c, Theorem 1.6], we have that the convergence in (2.1) is smooth on
$\mathcal{R}^{\infty}$. Combining this fact with (2.8) and applying [Bam20c,
Theorem 9.31(f)], we have that the convergence of $\mu_{t}^{i}$ is also
locally smooth on $\mathcal{R}^{\infty}$. Hence, the convergence in (2.6) is
smooth on $\mathcal{R}^{\infty}$. This finishes the proof of the lemma.

∎

######  Lemma 2.5.

We have

$\displaystyle\lim_{i\rightarrow\infty}\mathcal{N}^{i}_{p_{0},0}(\tau)=\lim_{i\rightarrow\infty}\overline{\mathcal{N}}^{i}(\tau)\equiv\mathcal{N}_{x^{\infty}}(\tau)\quad\text{
for all }\quad\tau>0.$ (2.9)

All the terms above are constants independent of $\tau>0$.

###### Proof.

Since the convergences in (2.1), (2.6), and (2.7) are all smooth on
$\mathcal{R}^{\infty}$, and since
$\nu^{\infty}_{t}(\mathcal{X}^{\infty}\setminus\mathcal{R}^{\infty})\equiv 0$,
one may argue in the same way as the proof of [Bam20c, Theorem 14.45(a)]. The
only statement we need to check is: fixing any $\tau>0$, there is a constant
$C$ independent of $i$, such that

$\displaystyle\int_{M}e^{-\tfrac{1}{2}\bar{f}^{i}(\cdot,-\tau)}dg_{i,-\tau_{i}}\leq
C\quad\text{ for all $i$ large enough}.$ (2.10)

We shall now prove this estimate. By Lemma 2.3, we have

$e^{-\tfrac{1}{2}\bar{f}^{i}(\cdot,-\tau)}\leq
C_{0}^{\frac{1}{2}}e^{-\tfrac{1}{2}f^{i}_{p_{0},0}(\cdot,-\tau)}.$

Hence, [Bam20c, Proposition 5.5] implies that

$\displaystyle\int e^{-\tfrac{1}{2}\bar{f}^{i}(\cdot,-\tau)}dg^{i}_{-\tau}$
$\displaystyle\leq\int
C_{0}^{\frac{1}{2}}e^{-\tfrac{1}{2}f^{i}_{p_{0},0}(\cdot,-\tau)}dg^{i}_{-\tau}=C_{0}^{\frac{1}{2}}(4\pi\tau)^{\frac{n}{2}}\int
e^{\tfrac{1}{2}f^{i}_{p_{0},0}(\cdot,-\tau)}d\nu^{i}_{-\tau}$
$\displaystyle\leq
C_{0}^{\frac{1}{2}}(4\pi\tau)^{\frac{n}{2}}e^{\frac{n}{2}},$

where we have also used the fact that $R\geq 0$ on an ancient solution (c.f.
[CBl09]); this is exactly (2.10). The rest of the proof is the same as the
proof of [Bam20c, Theorem 14.45(a)]. ∎

In order to estimate $\overline{\mathcal{W}}(g_{t_{0}},u_{0},\tau_{0})$, we
shall now defined another set of notations. Let

$\displaystyle\tau_{t}$ $\displaystyle:=$ $\displaystyle
t_{0}+\tau_{0}-t\quad\text{for all }\quad t\leq t_{0}$ (2.11) $\displaystyle
u(\cdot,t)$ $\displaystyle:=$
$\displaystyle(4\pi\tau_{t})^{-\frac{n}{2}}e^{-f(\cdot,t)}\quad\text{for all
}\quad t\leq t_{0},$ $\displaystyle\overline{\mathcal{W}}(t)$
$\displaystyle:=$
$\displaystyle\overline{\mathcal{W}}\big{(}g_{t},u(\cdot,t),\tau_{t}\big{)}\quad\text{for
all }\quad t\leq t_{0},$ $\displaystyle\mathcal{N}(t)$ $\displaystyle:=$
$\displaystyle\int_{M}f(\cdot,t)u(\cdot,t)dg_{t}-\frac{n}{2}\quad\text{for all
}\quad t\leq t_{0}.$

The following lemma is the same as [CMZ21, Lemma 9.1, Theorem 9.3].

######  Lemma 2.6.

We have

$\frac{d}{dt}\overline{\mathcal{W}}(t)\geq 0\quad\text{ for all }\quad
t<t_{0}$

and

$\lim_{t\rightarrow
t_{0}-}\overline{\mathcal{W}}(t)=\overline{\mathcal{W}}(g_{t_{0}},u_{0},\tau_{0}).$

######  Lemma 2.7.

We have

$\lim_{t\rightarrow\infty}\mathcal{N}(\tau_{i}t)=\lim_{i\rightarrow\infty}\overline{\mathcal{N}}^{i}(\tau)\equiv\mathcal{N}_{x^{\infty}}(\tau).$

All the terms above are constants independent of $t<0$ or $\tau>0$.

###### Proof.

Let us fix an arbitrary $t<0$. Since

$\displaystyle f(\cdot,\tau_{i}t)$ $\displaystyle=$
$\displaystyle\bar{f}^{i}(\cdot,t)-\frac{n}{2}\log\left(\frac{\tau_{\tau_{i}t}}{\tau_{i}|t|}\right)$
$\displaystyle=$
$\displaystyle\bar{f}^{i}(\cdot,t)-\frac{n}{2}\log\left(1+\frac{\tau_{0}+t_{0}}{\tau_{i}|t|}\right),$

we have, by the definition of $\mathcal{N}$ and $\overline{\mathcal{N}}^{i}$,

$\mathcal{N}(\tau_{i}t)=\overline{\mathcal{N}}^{i}(|t|)-\frac{n}{2}\log\left(1+\frac{\tau_{0}+t_{0}}{\tau_{i}|t|}\right)\quad\text{
for all $i$ large enough}.$

Taking $i\rightarrow\infty$, the lemma then follows immediately. ∎

With the preparations above, we are ready to show that the limit in Lemma 2.7
is a lower bound of $\overline{\mathcal{W}}(g_{t_{0}},u_{0},\tau_{0})$ and
thereby prove our main theorem.

###### Proof of Theorem 1.1.

In the proof, we shall retain the notations in (2.11). First of all, we
compute

$\displaystyle-\frac{d}{dt}\mathcal{N}(t)$ $\displaystyle=$
$\displaystyle-\frac{d}{dt}\int_{M}fu\,dg_{t}=-\int_{M}u\Box
fdg_{t}+\int_{M}f\Box^{*}u\,dg_{t}$ $\displaystyle=$
$\displaystyle-\int_{M}u\Box f\,dg_{t}=\int_{M}\left(-\frac{\partial
f}{\partial t}+\Delta f\right)u\,dg_{t}$ $\displaystyle=$
$\displaystyle\int_{M}\left(2\Delta f-|\nabla
f|^{2}+R-\frac{n}{2\tau_{t}}\right)u\,dg_{t}$ $\displaystyle=$
$\displaystyle\int_{M}\left(|\nabla
f|^{2}+R\right)u\,dg_{t}-\frac{n}{2\tau_{t}}.$

Hence

$\displaystyle-\tau_{t}\frac{d}{dt}\mathcal{N}(t)$ $\displaystyle=$
$\displaystyle\int_{M}\left(\tau_{t}\left(|\nabla
f|^{2}+R\right)+f-n\right)u\,dg_{t}-\mathcal{N}(t)$ $\displaystyle=$
$\displaystyle\overline{\mathcal{W}}(t)-\mathcal{N}(t),$

and

$-\frac{d}{dt}\left(\tau_{t}\mathcal{N}(t)\right)=\overline{\mathcal{W}}(t).$

By the monotonicity of $\overline{\mathcal{W}}(t)$, we have

$\displaystyle\frac{1}{\tau_{i}}\big{(}\tau_{-2\tau_{i}}\mathcal{N}(-2\tau_{i})-\tau_{-\tau_{i}}\mathcal{N}(-\tau_{i})\big{)}$
$\displaystyle=$
$\displaystyle-\frac{1}{\tau_{i}}\int_{-2\tau_{i}}^{-\tau_{i}}\frac{d}{dt}\big{(}\tau_{t}\mathcal{N}(t)\big{)}dt=\frac{1}{\tau_{i}}\int_{-2\tau_{i}}^{-\tau_{i}}\overline{\mathcal{W}}(t)dt\leq\overline{\mathcal{W}}(g_{t_{0}},u_{0},\tau_{0}).$

Letting $i\to\infty$, and using (2.7), we have

$\mathcal{N}_{x^{\infty}}(\tau)=\lim_{i\rightarrow\infty}\left(\frac{\tau_{-2\tau_{i}}}{\tau_{i}}\mathcal{N}(-2\tau_{i})-\frac{\tau_{-\tau_{i}}}{\tau_{i}}\mathcal{N}(-\tau_{i})\right)\leq\overline{\mathcal{W}}(g_{t_{0}},u_{0},\tau_{0}).$
(2.12)

Since $t_{0}$, $u_{0}$, and $\tau_{0}$ are arbitrarily fixed, this finishes
the proof of the theorem. ∎

##  3 Applications to Steady Ricci Solitons

Let $(M^{n},g,f)$ be a complete steady gradient Ricci soliton satisfying

$\operatorname{Ric}=\nabla^{2}f,\quad R+|\nabla f|^{2}=1.$

By [CBl09], $R\geq 0$ everywhere on $M$, and as a consequence

$|\nabla f|\leq 1.$

A complete steady gradient Ricci soliton generates a canonical solution $g(t)$
to the Ricci flow. If $\phi_{t}$ is the flow of the vector field $-\nabla f$
with $\phi_{0}=\text{ id }$, then $\phi_{t}$ exists for all time because of
$|\nabla f|\leq 1$, and $(g(t):=\phi_{t}^{*}g)_{t\in\mathbb{R}}$ solves the
Ricci flow equation with $g(0)=g$. We prove a volume lower bound for steady
gradient solitons whose canonical form $g(t)$ satisfy all the assumptions in
Theorem 1.1. The argument only requires a Sobolev inequality on $M$, so we
shall show a slightly more general statement:

######  Proposition 3.1.

Suppose that we have a Sobolev inequality on $(M^{n},g,f):$

$\left(\int_{M}|u|^{\frac{2n}{n-2}}\,dg\right)^{\frac{n-2}{n}}\leq C_{\rm
Sob}\int_{M}4|\nabla u|^{2}+Ru^{2}\,dg,$

for any compactly supported and locally Lipschitz function $u$, where $C_{\rm
Sob}<\infty$ is the Sobolev constant. Fix a point $o\in M.$ Then there is a
constant $c$ depending on $n$ and $C_{\rm Sob}$ such that

$|B_{r}(o)|\geq cr^{n/2}\quad\text{ for all }\quad r>10.$

###### Proof.

The proof follows verbatim as Theorem 3.1.5 in [SC] with minor changes. For
simplicity, we write

$V(r):=|B_{r}(o)|.$

We choose a Lipschitz test function

$u(x):=(r-d(x,o))_{+}.$

Then

$\displaystyle\left(\int_{M}|u|^{\frac{2n}{n-2}}\,dg\right)^{\frac{n-2}{n}}$
$\displaystyle\geq\frac{r^{2}}{4}V(r/2)^{\frac{n-2}{n}},$
$\displaystyle|\nabla u|^{2}$ $\displaystyle\leq 1,\ \text{a.e.},$
$\displaystyle\int_{M}Ru^{2}=\int_{B_{r}(o)}u^{2}\Delta f$
$\displaystyle=\int_{\partial
B_{r}(o)}u^{2}\partial_{r}f-\int_{B_{r}(o)}\nabla f\cdot\nabla u^{2}$
$\displaystyle\leq 2\int_{B_{r}(o)}u\leq 2rV(r),$

where we integrated by parts and used the fact that $|\nabla f|\leq 1.$ Hence,

$\frac{r^{2}}{4}V(r/2)^{a}\leq C_{\rm Sob}(4+2r)V(r)\leq 4C_{\rm
Sob}(1+r)V(r),$

where $a=\frac{n-2}{n}.$ So

$\displaystyle V(r)\geq c_{1}\frac{r}{2}V(r/2)^{a},$ $\displaystyle\quad\text{
for }r\geq 1;$ $\displaystyle V(r)\geq c_{1}\frac{r^{2}}{4}V(r/2)^{a},$
$\displaystyle\quad\text{ for }r\in(0,1),$

where $c_{1}=\min\left\\{\frac{1}{16C_{\rm Sob}},1\right\\}.$ Fixing $r>10$
and let $k=k(r)\geq 1$ be the integer such that

$1\leq 2^{-k}r<2.$

Iterating the inequality above, we have

$\displaystyle
V(r)\geq(c_{1}r)^{\sum_{j=0}^{k}a^{j}}2^{-\sum_{j=0}^{k}(j+1)a^{j}}V\left(2^{-1-k}r\right)^{a^{1+k}}.$

Write $s=2^{-1-k}r\in[1/2,1).$ For any $m\geq 1$, we have

$\displaystyle V(s)$ $\displaystyle\geq c_{1}(s/2)^{2}V(s/2)^{a}$
$\displaystyle\geq\cdots$
$\displaystyle\geq(c_{1}s^{2})^{\sum_{i=0}^{m}a^{i}}2^{-\sum_{i=0}^{m}2(i+1)a^{i}}V(2^{-1-m}s)^{a^{1+m}}.$

Since $a\in(0,1),$ we have

$\lim_{m\to\infty}V\left(2^{-m}s\right)^{a^{m}}=1.$

Therefore, taking $m\to\infty,$ we have

$V(s)\geq c(n)c_{1}^{n/2}s^{n}\geq c(n)c_{1}^{n/2},$

where we have used the fact $s\in[1/2,1).$ So

$V(r)\geq c(n)c_{1}^{n/2}r^{\frac{n}{2}(1-a^{1+k})}.$

By the definition of $k$, we have

$r^{-\frac{n}{2}a^{1+k(r)}}\geq c(n)\quad\text{ if }\quad r>10.$

Hence, for $r>10,$ we have

$V(r)\geq c(n)c_{1}^{n/2}r^{n/2}.$

∎

Remark. The estimate on the integral involving $R$ are similar to Theorem 5.1
in [MS13] (c.f. Lemma 4.3 in [D16]).

###### Proof of Theorem 1.3.

From the proof of Proposition 3.1, we can see that if the Sobolev constant is
$C_{\rm Sob}=C(n)e^{-\frac{2\mu_{\infty}}{n}}$, then the constant $c$ in the
previous proposition can be taken as

$c=c(n)C_{\rm Sob}^{-n/2}=c(n)e^{\mu_{\infty}}.$

This finishes the proof of Theorem 1.3. ∎

We end this section by establishing the quadratic volume estimate of
positively curved three dimensional steady gradient Ricci soliton.

###### Proof of Theorem 1.4.

Catino-Mastrolia-Monticelli [CMM16, Corollary 1.7] showed that a 3 dimensional
steady gradient soliton with $\liminf_{r\to\infty}r^{-2}|B_{r}(o)|=0$ must
either be flat or split isometrically as a quotient of
$\mathbb{R}\times\Sigma$, where $\Sigma$ is the cigar soliton. The lower
estimate in (1.5) then follows from the positively curved condition. To get
the upper bound on the volume, we look at the area growth of the level sets.
Since $\nabla^{2}f=\operatorname{Ric}>0$ and $R$ attains its maximum, by a
result of Cao-Chen [CC12, Proposition 2.3], there exist positive constants
$\alpha$ $\in(0,1)$ and $c_{0}$ such that for any $x$ $\in M$, it holds that

$\alpha\,{\rm dist}(x,o)-c_{0}\leq f(x)\leq{\rm dist}(x,o)+c_{0}.$ (3.1)

By the convexity of $f$ and the identity $\nabla R=-2\operatorname{Ric}(\nabla
f)$, $f$ and $R$ have the same unique critical point, say $p_{0}$, then
$R(p_{0})=1-|\nabla f|^{2}(p_{0})=1$ and $\max_{M}R=1$. Moreover, by the Morse
Lemma, the level sets $\Gamma_{s}:=\\{x:\,f(x)=s\\}$ are all diffeomorphic to
$\mathbb{S}^{2}$ for all $s>\min_{M}f$. Using $\operatorname{Ric}>0$ and
$\nabla R=-2\operatorname{Ric}(\nabla f)$, we can find $\varepsilon_{0}>0$ and
a large positive constant $s_{0}$ such that on $\\{x:\,f(x)\geq s_{0}\\}$, it
holds that

$R\leq\max_{M}R-\varepsilon_{0}=1-\varepsilon_{0}\,;$ $|\nabla
f|^{2}=1-R\geq\varepsilon_{0}.$ (3.2)

As $\Gamma_{s}$ is a level set of $f$, its second fundamental form $A_{s}$
(w.r.t. the normal $\frac{\nabla f}{|\nabla f|}$) is given by
$\frac{\nabla^{2}f}{|\nabla f|}=\frac{\operatorname{Ric}}{|\nabla f|}\geq 0$.
For any $q$ $\in\Gamma_{s}$, we can find an orthonormal frame
$\\{e_{i}\\}_{i=1}^{2}$ which is an eigenbasis of $A_{s}$ with eigenvalues
$\sigma_{i}\geq 0$, $i=1,2$. Let $\tilde{h}_{s}$ be the induced metric on
$\Gamma_{s}$. Then by the Gauss equation

$\begin{split}K_{s}&=R_{1221}+\sigma_{1}\sigma_{2}>0\,;\\\
2K_{s}&=R-2\operatorname{Ric}(\mathbf{n},\mathbf{n})+2\sigma_{1}\sigma_{2}\\\
&\geq R-2\operatorname{Ric}(\mathbf{n},\mathbf{n}),\end{split}$ (3.3)

where $K_{s}$ and $\mathbf{n}$ denote the Gauss curvature of
$(\Gamma_{s},\tilde{h}_{s})$ and the normal vector $\frac{\nabla f}{|\nabla
f|}$, respectively. We then consider the flow $\psi_{s}$ of the vector field
$\frac{\nabla f}{|\nabla f|^{2}}$ with $\psi_{s_{0}}=\text{ id }$. When
restricted on $\Gamma_{s_{0}}$,
$\psi_{s}:\Gamma_{s_{0}}\longrightarrow\Gamma_{s}$ are diffeomorphisms for all
$s\geq s_{0}$. Let $h_{s}$ be the pull back metric $\psi_{s}^{*}\tilde{h}_{s}$
on $\Gamma_{s_{0}}$. Then we may compute

$\displaystyle\frac{\partial}{\partial s}h_{s}$ $\displaystyle=$
$\displaystyle\psi_{s}^{*}\mathcal{L}_{\frac{\nabla f}{|\nabla
f|^{2}}}g=\psi_{s}^{*}\left(\frac{2\nabla^{2}f}{|\nabla
f|^{2}}\right)=\psi_{s}^{*}\left(\frac{2\operatorname{Ric}}{|\nabla
f|^{2}}\right),$

where $\mathcal{L}_{\frac{\nabla f}{|\nabla f|^{2}}}$ is the Lie derivative
with respect to $\frac{\nabla f}{|\nabla f|^{2}}$. We denote by $dh_{s}$ the
volume form induced by the metric $h_{s}$, then, applying (3.2) and (3.3), we
have

$\begin{split}\frac{\partial}{\partial
s}dh_{s}&=\psi_{s}^{*}\left(\frac{R-\operatorname{Ric}(\mathbf{n},\mathbf{n})}{|\nabla
f|^{2}}\right)dh_{s}\\\
&\leq\psi_{s}^{*}\left(\frac{2K_{s}+\operatorname{Ric}(\mathbf{n},\mathbf{n})}{|\nabla
f|^{2}}\right)dh_{s}\\\
&\leq\varepsilon_{0}^{-1}\psi_{s}^{*}(2K_{s}+\operatorname{Ric}(\mathbf{n},\mathbf{n}))\,dh_{s}.\end{split}$
(3.4)

By virtue of $\nabla R=-2\operatorname{Ric}(\nabla f)$, we have
$\psi_{s}^{*}(\operatorname{Ric}(\mathbf{n},\mathbf{n}))=-\frac{1}{2}\frac{\partial}{\partial
s}R\circ\psi_{s}$. It follows from (3.4) and $R\leq 1$ that

$\begin{split}\frac{\partial}{\partial
s}\left(e^{\frac{R\circ\psi_{s}}{2\varepsilon_{0}}}\,dh_{s}\right)&\leq\varepsilon_{0}^{-1}\psi_{s}^{*}(2e^{\frac{R}{2\varepsilon_{0}}}K_{s})\,dh_{s}\\\
&\leq\varepsilon_{0}^{-1}e^{\frac{1}{2\varepsilon_{0}}}\psi_{s}^{*}(2K_{s})\,dh_{s}.\end{split}$
(3.5)

Hence by the Gauss-Bonnet Theorem, we have

$\begin{split}\frac{d}{ds}\int_{\Gamma_{s_{0}}}e^{\frac{R\circ\psi_{s}}{2\varepsilon_{0}}}\,dh_{s}&=\int_{\Gamma_{s_{0}}}\frac{\partial}{\partial
s}\left(e^{\frac{R\circ\psi_{s}}{2\varepsilon_{0}}}\,dh_{s}\right)\\\
&\leq\frac{8\pi e^{\frac{1}{2\varepsilon_{0}}}}{\varepsilon_{0}}.\end{split}$
(3.6)

Integrating the above differential inequality with respect to $s$, we can
choose a $s_{1}>s_{0}$ such that for all $s\geq s_{1}$, it holds that

$\displaystyle\text{ Area }(\Gamma_{s})$ $\displaystyle\leq$
$\displaystyle\int_{\Gamma_{s_{0}}}e^{\frac{R\circ\psi_{s}}{2\varepsilon_{0}}}\,dh_{s}$
$\displaystyle\leq$ $\displaystyle\frac{8\pi
e^{\frac{1}{2\varepsilon_{0}}}}{\varepsilon_{0}}(s-s_{0})+e^{\frac{1}{2\varepsilon_{0}}}\text{
Area }(\Gamma_{s_{0}})$ $\displaystyle\leq$ $\displaystyle Cs.$

We used $R\geq 0$ in the first inequality. The Coarea formula then implies
that for all $s\gg s_{1}$, it holds that

$\displaystyle|\\{x:\,f(x)\leq s\\}|$ $\displaystyle=$
$\displaystyle|\\{x:\,f(x)<s_{1}\\}|+|\\{x:\,s_{1}\leq f(x)\leq s\\}|$
$\displaystyle\leq$
$\displaystyle|\\{x:\,f(x)<s_{1}\\}|+\frac{Cs^{2}}{2\sqrt{\varepsilon_{0}}}$
$\displaystyle\leq$ $\displaystyle C^{\prime}s^{2}.$

By (3.1), for all large $r$, we have

$|B_{r}(o)|\leq|\\{x:\,f(x)\leq r+c_{0}\\}|\leq 4C^{\prime}r^{2}.$

∎

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Department of Mathematics, University of California, San Diego, CA, 92093
E-mail address<EMAIL_ADDRESS>`

Department of Mathematics, University of California, San Diego, CA, 92093
E-mail address<EMAIL_ADDRESS>

School of Mathematics, University of Minnesota, Twin Cities, MN, 55414
E-mail address<EMAIL_ADDRESS>