11institutetext: École Polytechnique Fédérale de Lausanne (EPFL), Lausanne,
Switzerland 22institutetext: Inception Institute of AI (IIAI), Abu Dhabi, UAE
33institutetext: Lausanne University Hospital (CHUV), Lausanne, Switzerland

# AMAE: Adaptation of Pre-Trained Masked Autoencoder for Dual-Distribution
Anomaly Detection in Chest X-Rays

Behzad Bozorgtabar CONTACT Behzad Bozorgtabar. Email:
<EMAIL_ADDRESS>Dwarikanath Mahapatra 22 Jean-Philippe Thiran
1133

###### Abstract

Unsupervised anomaly detection in medical images such as chest radiographs is
stepping into the spotlight as it mitigates the scarcity of the labor-
intensive and costly expert annotation of anomaly data. However, nearly all
existing methods are formulated as a one-class classification trained only on
representations from the normal class and discard a potentially significant
portion of the unlabeled data. This paper focuses on a more practical setting,
dual distribution anomaly detection for chest X-rays, using the entire
training data, including both normal and unlabeled images. Inspired by a
modern self-supervised vision transformer model trained using partial image
inputs to reconstruct missing image regions– we propose AMAE, a two-stage
algorithm for adaptation of the pre-trained masked autoencoder (MAE). Starting
from MAE initialization, AMAE first creates synthetic anomalies from only
normal training images and trains a lightweight classifier on frozen
transformer features. Subsequently, we propose an adaptation strategy to
leverage unlabeled images containing anomalies. The adaptation scheme is
accomplished by assigning pseudo-labels to unlabeled images and using two
separate MAE based modules to model the normative and anomalous distributions
of pseudo-labeled images. The effectiveness of the proposed adaptation
strategy is evaluated with different anomaly ratios in an unlabeled training
set. AMAE leads to consistent performance gains over competing self-supervised
and dual distribution anomaly detection methods, setting the new state-of-the-
art on three public chest X-ray benchmarks - RSNA, NIH-CXR, and VinDr-CXR.

###### Keywords:

Anomaly detection Chest X-ray Masked autoencoder.

##  1 Introduction

To reduce radiologists’ reading burden and make the diagnostic process more
manageable, especially when the number of experts is scanty, computer-aided
diagnosis (CAD) systems, particularly deep learning-based anomaly detection
[2, 19, 26, 1, 3, 4], have witnessed the flourish due to their capability to
detect rare anomalies for different imaging modalities including chest X-ray
(CXR). Nonetheless, unsupervised anomaly detection methods [30, 24] are
strongly preferred due to the difficulties of highly class-imbalanced learning
and the tedious annotation of anomaly data for developing such systems. Most
current anomaly detection methods are formulated as a one-class classification
(OCC) problem [22], where the goal is to model the distribution of normal
images used for training and thus detect abnormal cases that deviate from
normal class at test time. On this basis, image reconstruction based, e.g.,
autoencoder [11] or GANs [24, 12], self-supervised learning (SSL) based, e.g.,
contrastive learning [30], and embedding-similarity-based methods [9] have
been proposed for anomaly detection. Some recent self-supervised methods
proposed synthetic anomalies using cut-and-paste data augmentation [15, 23] to
approximate real sub-image anomalies. Nonetheless, their performances lag due
to the lack of real anomaly data. More importantly, these methods have often
ignored readily available unlabeled images. More recently, similar to our
method, DDAD [5] leverages readily available unlabeled images for anomaly
detection, but it requires training an ensemble of several reconstruction-
based networks. Self-supervised model adaptation on unlabeled data has been
widely investigated using convolutional neural networks (CNNs) in many vision
tasks via self-training [21], contrastive learning [30, 26], and anatomical
visual words [13]. Nonetheless, the adaptation of vision transformer (ViT)
[10] architectures largely remains unexplored, particularly for anomaly
detection. Recently, masked autoencoder (MAE) [14] based models demonstrated
great scalability and substantially improved several self-supervised learning
benchmarks [31].
In this paper, inspired by the success of the MAE approach, we propose a two-
stage algorithm for “Adaptation of pre-trained Masked AutoEncoder” (AMAE) to
leverage simultaneously normal and unlabeled images for anomaly detection in
chest X-rays. As for Stage 1 of our method, (i) AMAE creates synthetic
anomalies from only normal training images, and the usefulness of pre-trained
MAE [14] is evaluated by training a lightweight classifier using a proxy task
to detect synthetic anomalies. (ii) For the Stage 2, AMAE customizes the
recipe of MAE adaptation based on an unlabeled training set. In particular, we
propose an adaptation strategy based on reconstructing the masked-out input
images. The rationale behind the proposed adaptation strategy is to assign
pseudo-labels to unlabeled images and train two separate modules to measure
the distribution discrepancy between normal and pseudo-labeled abnormal
images. (iii) We conduct extensive experiments across three chest X-ray
datasets and verify the effectiveness of our adaptation strategy in
apprehending anomalous features from unlabeled images. In addition, we
evaluate the model with different anomaly ratios (ARs) in an unlabeled
training set and show consistent performance improvement with increasing AR.

Figure 1: Schematic overview of AMAE training (Stage 1). Top. Illustration of
Anatpaste augmentation [23] generated from normal training images. Bottom.
Starting from MAE initialization, only the MLP-based projection head (Proj.)
is trained to classify synthetic anomalies.

##  2 Method

Notation. We first formally define the problem setting for the proposed dual-
distribution anomaly detection. Contrary to previous unsupervised anomaly
detection methods, AMAE fully uses unlabeled images, yielding a training data
$\mathcal{T}_{train}=\mathcal{T}_{n}\cup\mathcal{T}_{u}$ consisting of both
normal $\mathcal{T}_{n}$ and unlabeled $\mathcal{T}_{u}$ training sets. We
denote the normal training set as
$\mathcal{T}_{n}=\left\\{\boldsymbol{x}_{ni}\right\\}_{i=1}^{N}$, with $N$
normal images, and the unlabeled training set as
$\mathcal{T}_{u}=\left\\{\boldsymbol{x}_{ui}\right\\}_{i=1}^{M}$, with $M$
unlabeled images to be composed of both normal and abnormal images. At test
time, given a test set
$\mathcal{T}_{test}=\left\\{\left(\boldsymbol{x}_{ti},y_{i}\right)\right\\}_{i=1}^{S}$
with $S$ normal or abnormal images, where $y_{i}\in\left\\{0,1\right\\}$ is
the corresponding label to $\boldsymbol{x}_{ti}$ (0 for normal (negative) and
1 for abnormal (positive) image), the trained anomaly detection model should
identify whether the test image is abnormal or not.

Architecture. Our architecture is
$\blacktriangleright\mathrel{\mkern-4.0mu}<$-shaped: the ViT-small (ViT-S/16)
[10] encoder $f$ followed by a self-supervised ViT head $g$ and lightweight
(3-layer) multilayer perception (MLP) anomaly classifier (projection head)
$h$, simultaneously. Starting from the pre-trained MAE on 0.3M unlabeled chest
X-rays and officially released checkpoints, we use exactly the same ViT
encoder $f$ and decoder $g$ as MAE [31].

###  2.1 Stage 1- Proxy Task to Detect Synthetic Anomalies

AMAE starts the first training stage using only normal training images by
defining a proxy task to detect synthetic anomalies shorn of real known
abnormal images. For this purpose, we utilize the state-of-the-art (SOTA)
anatomy-aware cut-and-paste augmentation, AnatPaste [23], to create synthetic
anomalies from only a set of normal training images $\mathcal{T}_{n}$.
AnatPaste integrates an anatomical mask $\boldsymbol{x}_{mask}$ created from
unsupervised lung region segmentation, which guides generating anomalous
images via cutting a patch from a normal chest radiograph $\boldsymbol{x}_{n}$
and randomly pasting it at another image location $\boldsymbol{x}_{paste}$ as:

$\displaystyle\mathbf{Aug}\left(\boldsymbol{x}_{n}\right)=\boldsymbol{x}_{n}\ast\left(1-\boldsymbol{x}_{mask}\right)+\boldsymbol{x}_{paste}\ast\boldsymbol{x}_{mask}$
(1)

where $\mathbf{Aug\left(\cdot\right)}$ is the AnatPaste augmentation (see [23]
for more details).

Given a normal training set $\mathcal{T}_{n}$, for each normal image
$\boldsymbol{x}_{n}\sim\mathcal{T}_{n}$, we create a synthetic anomaly,
denoted as $\mathbf{Aug}\left(\boldsymbol{x}_{i}\right)$. In preparation for
input to the frozen ViT encoder, $f_{0}$ (obtained by MAE pre-training), each
input image with the $h\times w$ spatial resolution is split into
$T=\left(h/p\right)\times\left(w/p\right)$ patches of size $p\times p$. Then,
for every input patch, a token is created by linear projection with an added
positional embedding. The sequence of tokens is then fed to the frozen ViT
encoder $f_{0}$ consisting stack of transformer blocks, yielding the
embeddings of tokens
$\boldsymbol{z}_{i}^{1},\boldsymbol{z}_{i}^{2}\in\mathbb{R}^{T\times d}$
corresponding to $i^{th}$ normal and synthetic anomaly images. The returned
embeddings
$\boldsymbol{z}_{i}^{1},\boldsymbol{z}_{i}^{2}\in\mathbb{R}^{T\times d}$ are
pooled via average pooling to form $d$-dimensional embeddings, which are fed
to an MLP anomaly classifier projection head $h$ (see Fig. 1 for schematic
overview). Subsequently, we only train an anomaly classifier projection head
$h$ on top of the frozen embeddings to detect synthetic anomalies using the
cross-entropy loss $l_{ce}$ as follows:

$\displaystyle
h_{0}=\arg\min_{h}\operatorname{\mathbb{E}}_{\boldsymbol{x}_{n}\sim\mathcal{T}_{n}}\left[l_{ce}\left(h\circ
f_{0}\left(\boldsymbol{x}_{n}\right),0\right)+l_{ce}\left(h\circ
f_{0}\left(\mathbf{Aug}\left(\boldsymbol{x}_{n}\right)\right),1\right)\right]$
(2)

We set the label for the normal image to 0 and 1 otherwise (synthetic
anomaly). The above gradient-based optimization produces a trained classifier
projection head $h_{0}$. Thus, the whole architecture can be trained with much
fewer parameters while making only the classifier projection head specialized
at recognizing anomalies without influencing the ViT encoder.

Figure 2: Schematic overview of AMAE training (Stage 2) and test stage. Top.
Starting from MAE initialization, our adaptation strategy first assigns pseudo
labels to unlabeled images using a pre-trained classifier from Stage 1 and
uses two separate MAE modules to model the normative and anomalous
distributions of pseudo-labeled images. Bottom. At test time, the discrepancy
between the average of multiple reconstructions from two modules is used to
compute the anomaly score.

###  2.2 Stage 2- MAE Inter-Discrepancy Adaptation

The proposed MAE adaptation scheme is inspired by [5] to model the dual
distribution of training data. Unlike [5], which treats all unlabeled images
similarly, we propose assigning pseudo-labels to unlabeled images and
formulating anomaly detection by measuring the distribution discrepancy
between normal and pseudo-labeled abnormal images from unlabeled sets. We use
a pre-trained anomaly classifier (Stage 1) to assign pseudo labels to
unlabeled images. To begin with, for each unlabeled image
$\boldsymbol{x}_{u}\sim\mathcal{T}_{u}$, we consider the anomaly detection
model’s confidence from Stage 1 ($h_{0}\circ
f_{0}\left(\boldsymbol{x}_{u}\right)$). Those images on which the model is
highly confident (normal or abnormal) are treated as reliable images used for
adaptation. For this purpose, we collect all output probabilities and opt for
a threshold per class $t_{c}$ corresponding to each class’s top $K$-th
percentile ($K$=50) of all given confidence values. Those unlabeled images
deemed reliable yield two subsets: a subset of pseudo-labeled normal images
$\mathcal{T}_{un}$ and a subset of pseudo-labeled abnormal images
$\mathcal{T}_{ua}$. We then utilize two MAE-based modules, Module A and Module
B (see Fig. 2, Top), using the same MAE architecture and pixel-wise mean
squared error (MSE) optimization in [14]. Within each module, the input
patches for each image are masked out using a set of $L$ random masks
$\left\\{\boldsymbol{m}^{\left(j\right)}\in\left\\{0,1\right\\}^{T}\right\\}_{j=1}^{L}$
in which a different small subset of the patches (ratio of 25%) is retained
each time to be fed to the ViT encoder $f$. The lightweight ViT decoder $g$
receives unmasked patches’ embeddings and adds learnable masked tokens to
replace the masked-out patches. Subsequently, the full set of embeddings of
visible patches and masked tokens with added positional embeddings to all
tokens is processed by the ViT decoder $g$ to reconstruct the missing patches
of each image in pixels. This yields the reconstructed image
$\hat{\boldsymbol{x}}^{\left(j\right)}=g\circ
f\left(\boldsymbol{m}^{\left(j\right)}\left(\boldsymbol{x}\right)\right)$,
which is then compared against the input image $\boldsymbol{x}$ to optimize
both ViT encoder $f$ and decoder $g$:

$\displaystyle
f^{\ast},g^{\ast}=\arg\min_{f,g}\mathbb{E}_{\boldsymbol{x}\sim\mathcal{T}_{train}}\left[\frac{1}{L}\sum_{j=1}^{L}l_{mse}\left(\boldsymbol{m}^{\left(j\right)}\left(\hat{\boldsymbol{x}}^{\left(j\right)}\right),\boldsymbol{m}^{\left(j\right)}\left(\boldsymbol{x}\right)\right)\right]$
(3)

All pixels in the $t^{th}$ patch of both input image and reconstructed images
are multiplied by
$\left(\boldsymbol{m}^{\left(j\right)}\right)_{t}\in\left\\{0,1\right\\}$. The
above self-supervised loss term averages $L$ pixel-wise mean squared errors
for each image. Module A is trained on a combination of the normal training
set $\mathcal{T}_{n}$ and pseudo-labeled normal images from the unlabeled set
$\mathcal{T}_{un}$. In contrast, Module B is trained using only pseudo-labeled
abnormal images from an unlabeled set $\mathcal{T}_{ua}$. Optimization for Eq.
3 always starts from pre-trained $f_{0}$ and $g_{0}$, and we reset the MAE
weights to $f_{0}$ and $g_{0}$ before training each module. A high discrepancy
between the reconstruction outputs of the two modules can indicate potential
abnormal regions. Similar to the training stage, we apply $L$ random masks to
the test image $\boldsymbol{x}_{t}\sim\mathcal{T}_{test}$ to obtain $L$
reconstructions (see Fig. 2, Bottom). Thus, the anomaly score based on the
inter-discrepancy of the two MAE modules is computed as follows:

$\displaystyle\mathcal{A}_{inter}^{p}=\left|\hat{\mu}_{A}^{p}-\hat{\mu}_{B}^{p}\right|$
(4)

where $p$ is the index of pixels, $\hat{\mu}_{A}$ and $\hat{\mu}_{B}$ are the
mean maps of $L$ reconstructed images from Module A and Module B,
respectively. The pixel-level anomaly scores for each image are averaged,
yielding the image-level anomaly score.

##  3 Experiments

Table 2: Performance comparison with SOTA methods (Avg. over four replicate).
The two best results for each protocol are highlighted in bold and underlined.
Note that “IN” refers to “ImageNet-Pretrained,” and “e2e” refers to end-to-end
training. The experimental results of competing methods† are taken from [6].

Method |  Protocol |  Taxonomy | CXR Datasest & Metrics
---|---|---|---
RSNA | VinDr-CXR | NIH-CXR
AUC % | AP % | AUC % | AP % | AUC % | AP %
AE† | N | Rec. | 66.9 | 66.1 | 55.9 | 60.3 | 70.0 | 65.4
AE-U† [17] | N | Rec. | 86.7 | 84.7 | 73.8 | 72.8 | 91.0 | 83.2
f-AnoGAN† [24] | N | Rec. | 79.8 | 75.6 | 76.3 | 74.8 | 84.1 | 79.2
IGD† [8] | N | Rec. | 81.2 | 78.0 | 59.2 | 58.7 | 85.2 | 80.3
DRAEM† [32] | N | Rec.+SSL | 62.3 | 61.6 | 63.0 | 68.3 | 65.5 | 62.4
CutPaste${}^{\text{e2e}}$† [25] | N | SSL | 55.0 | 58.0 | 54.6 | 55.5 | 58.0 | -
AnatPaste [23] | N | SSL | 83.1 | 83.7 | 66.0 | 66.2 | 94.0 | 93.5
FPI† [27] | N | SSL | 47.6 | 55.7 | 48.2 | 49.9 | 70.5 | -
PII† [28] | N | SSL | 82.9 | 83.6 | 65.9 | 65.8 | 92.2 | -
NSA† [25] | N | SSL | 82.2 | 82.6 | 64.4 | 65.8 | 94.1 | -
DDAD-AE [5] | N | Rec. | 69.4 | - | 60.1 | - | 73.0 | 71.5
AMAE${}^{\text{IN}}$ \- Stage 1 | N | SSL | 83.3 | 83.8 | 65.9 | 66.0 | 94.1 | 93.7
AMAE \- Stage 1 | N | SSL | 86.8 | 84.9 | 74.2 | 72.9 | 95.0 | 94.9
CutPaste${}^{\text{e2e}}$† [25] | Y | SSL | 59.8 | 61.7 | 59.2 | 60.0 | -
AnatPaste [23] | Y | SSL | 84.4 | 85.5 | 67.1 | 67.5 | -
FPI† [27] | Y | SSL | 46.6 | 53.8 | 47.4 | 49.4 | -
PII† [28] | Y | SSL | 84.3 | 85.4 | 66.8 | 67.2 | -
NSA† [25] | Y | SSL | 84.2 | 84.3 | 64.4 | 64.8 | -
DDAD-AE [5] ($\mathcal{A}_{iner}$) | Y | Rec. | 81.5 | 81.0 | 71.0 | - | -
AMAE \- Stage 2 ($\mathcal{A}_{inter}$ ) | Y | Rec. | 91.4 | 91.7 | 86.1 | 84.5 | -

Datasets. We evaluated our method on three public CXR datasets: 1) the RSNA
Pneumonia Detection Challenge dataset111https://www.kaggle.com/c/rsna-
pneumonia-detection-challenge, 2) the VinBigData Chest X-ray Abnormalities
Detection Challenge dataset (VinDrCXR)222https://www.kaggle.com/c/vinbigdata-
chest-xray-abnormalities-detection [18], and a subset of 3) the curated NIH
dataset (NIH-CXR)333https://nihcc.app.box.com/v/ChestXray-
NIHCC/file/371647823217 [29, 25], by including only posteroanterior view
images of both male and female patients aged over 18. We show a summary of
each dataset’s repartitions in Table 1. Except for NIH, where we use only the
normal set $\mathcal{T}_{n}$ (OCC setting), for the other two datasets, we
utilize both $\mathcal{T}_{n}$ and $\mathcal{T}_{u}$ and exact repartition
files from [5] for model training.

Implementation details. We adopt AdamW [16] optimizer and set the learning
rate ($lr$) and batch size to $2.5e\text{-}4$ and 16, where we linearly warm
up the $lr$ for the first 20 epochs and decay it following a cosine schedule
thereafter till 150 epochs. We follow the exact recipe as [31] for other
hyperparameters. The number of generated masks $L$ per image is set to 2 and 4
for the adaptation and test stages. We use PyTorch 1.9 [20] and train each
model on a single GeForce RTX 2080 Ti GPU. We use the area under the ROC curve
(AUC) and average precision (AP) for the evaluation metrics.

Figure 3: Ablations on the RSNA dataset. (a) Performance comparison with a
varying AR of $\mathcal{T}_{u}$. (b) The $\chi^{2-\text{distance}}$ of AS
histograms with and without pseudo labeling. (c) Ablation for different
backbones and pre-training schemes. Test AUC performances are presented as
$mean\pm 1.96std$ averaged over four replicates. Samples are statistically
tested for _H 0: means are similar_, using a bilateral Welch t-test. _ns_ non-
significant, ** $pvalue\in[0.05,0.01]$, *** $pvalue\in[0.01,0.001]$.

Comparison with SOTA Methods. Table 2 compares AMAE with a comprehensive
collection of SOTA methods, including self-supervised synthetic anomaly and
reconstruction (Rec.) based methods using their official codes and under two
experimental protocols. We use the Y protocol to indicate if access to the
unlabeled images is possible in which an AR of 60% of $\mathcal{T}_{u}$ is
assumed in the experiments; otherwise, we use N. Under protocol N (OCC
setting), except for VinDr-CXR, AMAE-Stage 1 achieves SOTA results on two CXR
benchmarks, demonstrating the effectiveness of pre-trained ViT using MAE and
synthetic anomalies. In particular, AMAE-Stage 1 surpasses the best-performing
synthetic anomaly-based method, AnatPaste [23], with the same synthesis
approach as ours but using ResNet18 as a feature extractor. Furthermore,
outperforming MAE pre-training on ImageNet (e.g., improved AUC from 65.9% to
74.2% on the VinDr-CXR) indicates the importance of in-domain adaptation.
Under the Y protocol, our adaptation strategy, AMAE\- Stage 2
($\mathcal{A}_{inter}$), outperforms the current SSL methods by a larger
margin, underlining the importance of modeling the dual distribution and
leveraging unlabeled images more effectively.

Ablation Studies. To understand the effectiveness of AMAE in capturing
abnormal features from unlabeled images, we conduct ablation experiments on
the RSNA with the AR of $\mathcal{T}_{u}$ varying from 0 to 100% (Fig. 3 (a)).
Concerning reconstruction-based methods, the baseline aggregating multiple
reconstructions via MAE achieves consistent improvement compared with the AE
baseline (+14.6% AUC), implying better capturing of fine-grained texture
information. With an increasing AR of $\mathcal{T}_{u}$, our MAE adaptation
strategy (AMAE \- Stage 2 ($\mathcal{A}_{inter}$ )) performs favorably against
the SOTA method (DDAD [5]), and AMAE without adaptation (Stage 1).
Furthermore, We consider an additional baseline for model adaptation based on
the patch reconstruction objective in [14] on pooled normal and unlabeled
images, denoted as (AMAE \- Stage 2 (Mask Rec. )). AMAE \- Stage 2
($\mathcal{A}_{inter}$ ) rises with a steeper slope than AMAE \- Stage 2 (Mask
Rec. ), e.g., improved AUC from 89% to 92% on AR=80%, suggesting high-quality
pseudo-labeled images. We also analyze the discriminative capability of our
adaptation with and without pseudo labeling by levering all unlabeled images
in Module B. We utilize the $\chi^{2-\text{distance}}$ between the histograms
of anomaly scores (AS) of normal and abnormal images in the RSNA test set
(Fig. 3 (b)), showing a more substantial discriminative capability of
incorporating pseudo labeling (improved $\chi^{2-\text{distance}}$ from 38.53
to 58.37). Finally, the ViT encoder obtained by MAE pre-training (Stage 1)
surpasses DenseNet-121 (DN121), either pre-trained by MAE [31] or an advanced
contrastive learning method (MoCo v2 [7]) on the RSNA dataset (Fig. 3 (c)).

##  4 Conclusion

We present AMAE, an adaptation strategy of the pre-trained MAE for dual
distribution anomaly detection in CXRs, which makes our method capable of more
effectively apprehending anomalous features from unlabeled images. Experiments
on the three CXR benchmarks demonstrate that AMAE is generalizable to
different model architectures, achieving SOTA performance. As for the
limitation, an adequate number of normal training images is still required,
and we will extend our pseudo-labeling scheme in our future work for robust
anomaly detection bypassing any training annotations.

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# Remarks on the size of apparent horizons

Gregory J. Galloway  Department of Mathematics
University of Miami

###### Abstract

Marginally outer trapped surfaces (also referred to as apparent horizons) that
are stable in $3$-dimensional initial data sets obeying the dominant energy
condition strictly are known to satisfy an area bound. The main purpose of
this note is to show (in several ways) that such surfaces also satisfy a
diameter bound.

##  1 Introduction

As reviewed in Section 2, marginally outer trapped surfaces (a.k.a. apparent
horizons) admit an important notion of stability, analogous, at some level, to
the stability of minimal surfaces in Riemannian manifolds. Heuristically, a
stable marginally outer trapped surface is infinitesimally outermost. It is
known that stable marginally outer trapped surfaces in $3$-dimensional initial
data sets that obey the dominant energy condition strictly satisfy an area
bound, as we now briefly recall.

An initial data set $(M,g,K)$ consists of a smooth orientable $n$-dimensional
manifold $M$ equipped with a Riemannian metric $g$ and a symmetric
$(0,2)$-tensor $K$. The main physical example is when $(M,g,K)$ is an initial
data set in a spacetime (time oriented Lorentzian manifold)
$(\overline{M},\overline{g})$, i.e. $M$ is a spacelike hypersurface in
$\overline{M}$, with induced metric $g$ and second fundamental form $K$.

The local energy density $\mu$ and the local current density $J$ of an initial
data set $(M,g,K)$ are given by

$\displaystyle\mu=\frac{1}{2}(S-|K|^{2}+(\mathrm{tr}\,K)^{2})\quad\mbox{and}\quad
J=\operatorname{div}(K-(\mathrm{tr}\,K)g),$

where $S$ is the scalar curvature of $(M,g)$. When $(M,g,K)$ is a spacetime
initial data set, these quantities are given by $\mu=G(u,u)$, $J=G(u,\cdot)$,
where $G$ is the Einstein tensor $G={\rm
Ric}_{\overline{M}}-\frac{1}{2}R_{\overline{M}}\overline{g}$.

The following was obtained with A. Mendes; see [8, Proposition 3.1].
Henceforth we use the shorthand ‘MOTS’ for ‘marginally outer trapped surface’.

######  Proposition 1.1.

Let $\Sigma$ be a stable MOTS in a $3$-dimensional initial data $(M,g,K)$.
Suppose there exists $c>0$, such that $\mu+J(\nu)\geq c$ on $\Sigma$, where
$\nu$ is the outward unit normal to $\Sigma$. Then $\Sigma$ is topologically a
$2$-sphere and its area satisfies,

$A(\Sigma)\leq\frac{4\pi}{c}\,.$ (1.1)

Moreover, if equality holds, $\Sigma$ is a round $2$-sphere, with Gaussian
curvature $\kappa_{\Sigma}=c$, the outward null second fundamental form $\chi$
of $\Sigma$ vanishes, and $\mu+J(\nu)=c$ on $\Sigma$.

In the statement of Proposition 3.1 in [8], for convenience it was assumed
that $\Sigma$ is topologically a $2$-sphere, but it is immediate from the
proof that this is necessarily the case.

As discussed in [8], the equality case holds in certain well known models. The
rigidity part of Proposition 1.1 was an important element in proving local and
global initial data rigidity results for outer area minimizing marginally
outer trapped $2$-spheres; see [8, Theorem 3.2] and [9, Theorem 3.1]. These
results extend to general initial data sets certain results of Bray, Brendle
and Neves [5] concerning area minimizing $2$-spheres in Riemannian
$3$-manifolds with positive scalar curvature.

We note that a positive lower bound on the quantity $\mu+J(\nu)$ arises
naturally due to the presence of matter fields and/or a positive cosmological
constant. Suppose $(M,g,K)$ is an initial data set in a spacetime
$(\overline{M},\overline{g})$ that satisfies the Einstein equation,

$G+\Lambda\bar{g}=\cal{T}\,,$ (1.2)

where $\cal T$ is the energy-momentum tensor. Setting $\ell=u+\nu$, where
$\nu$ is any unit vector tangent to $M$ and where $u$ is the future directed
unit normal to $M$, we have,

$\mu+J(\nu)=G(u,\ell)={\mathcal{T}}(u,\ell)+\Lambda\geq\Lambda\,.$

provided $\cal T$ obeys the dominant energy condition (which includes the
matter vacuum case $\mathcal{T}=0$). Hence, for positive $\Lambda$, the
inequality (1.1) becomes,

$A(\Sigma)\leq\frac{4\pi}{\Lambda}\,.$

Such an inequality was obtained by Hayward, Shiromizu, and Nakao [12] for
spacetimes with positive cosmological constant, using a slightly different
notion of stability in the spacetime setting. Finally, note that the condition
$\mu+J(\nu)\geq c$ is implied by the strict dominant energy condition
$\mu-|J|\geq c$.

The main purpose of this note is to point out that stable MOTS in a
$3$-dimensional initial data set obey a diameter bound, as well.

######  Theorem 1.2.

Let $\Sigma$ be a stable MOTS in a $3$-dimensional initial data set $(M,g,K)$.
Suppose there exists $c>0$, such that $\mu+J(\nu)\geq c$ on $\Sigma$, where
$\nu$ is the outward unit normal to $\Sigma$. Then the diameter of $\Sigma$
satisfies,

${\rm diam}(\Sigma)\leq\frac{2}{\sqrt{3}}\cdot\frac{\pi}{\sqrt{c}}\,.$ (1.3)

Here ${\rm diam}(\Sigma)$ has the usual meaning,

${\rm diam}(\Sigma)=\sup_{x,y\in\Sigma}d(x,y)\,,$

where $d(x,y)$ is the distance between $x$ and $y$ in $\Sigma$ with respect to
the induced metric on $\Sigma$. The obvious advantage of a diameter estimate,
is that when $\mu+J(\nu)$ is large, $\Sigma$ must, roughly speaking, be small
in all directions. In general, surfaces with small area can still have very
large diameter. We note, however, whereas the bound in (1.1) is sharp, the
bound in (1.3) is not.

As we shall discuss in Section 3, Theorem 1.2 can be obtained from one of the
recent spectral torical band inequalities for $n$-dimensional manifolds,
$2\leq n\leq 7$, by Hirsch et al., [13, Theorem 1.3], whose proof is based on
the theory of $\mu$-bubbles, together with a key property of stable MOTS [11]
(see Section 2). In Section 2, we also present a direct proof of Theorem 1.2,
using only elementary techniques, in addition to this property of MOTS. Some
comments about higher dimensions are presented at the end of Section 3. In
what follows all manifolds are assumed to be smooth, connected and orientable,
unless otherwise stated.

##  2 Preliminaries and a proof of Theorem 1.2

###  2.1 Preliminaries

Let $(M,g,K)$ be an $n$-dimensional, $n\geq 3$, initial data set. One can
always arrange for $(M,g,K)$ to be an initial data set in a spacetime
$(\overline{M},\overline{g})$ (see e.g. [3, Section 3.2]), and we find it
convenient to do so. However, all essential quantities depend only on the
initial data. Thus we may assume $M$ is a spacelike hypersurface, with induced
metric $g$ and second fundamental form $K$, in an $(n+1)$-dimensional
spacetime $(\overline{M},\overline{g})$. Our sign convention is such that, for
vectors $X,Y\in T_{p}M$, $K$ is defined as,
$K(X,Y)=\overline{g}(\bar{\nabla}_{X}u,Y)$, where $u$ is the future directed
timelike unit normal field to $M$.

Let $\Sigma$ be a closed two-sided surface in $M$. Then $\Sigma$ admits a
smooth unit normal field $\nu$ in $M$, unique up to sign. By convention, we
refer to such a choice as outward pointing. Then $l_{+}=u+\nu$ and
$l_{-}=u-\nu$ are future directed outward pointing and inward pointing,
respectively, null normal vector fields along $\Sigma$. The null second
fundamental forms $\chi_{+}$ and $\chi_{-}$ of $\Sigma$ in $\bar{M}$ are
defined by

$\chi_{\pm}(X,Y)=\overline{g}(\bar{\nabla}_{X}l_{\pm},Y),\quad X,Y\in
T_{p}\Sigma\,.$ (2.1)

The null expansion scalars $\theta_{\pm}$ of $\Sigma$ are obtained by tracing
$\chi_{\pm}$,

$\theta_{\pm}=\mathrm{tr}\,_{\Sigma}\chi_{\pm}=\operatorname{div}_{\Sigma}\ell_{\pm}\,.$
(2.2)

Physically, $\theta_{+}$ measures the divergence of the outgoing light rays
from $\Sigma$. In terms of the initial data $(M,g,K)$,

$\theta_{\pm}={\rm tr}_{\Sigma}K\pm H\,,$

where $H=\operatorname{div}_{\Sigma}\nu$ is the mean curvature of $\Sigma$
within $M$.

Penrose famously defined $\Sigma$ to be a trapped surface if both $\theta_{-}$
and $\theta_{+}$ are negative. Focusing attention on the outward null normal,
we say that $\Sigma$ is an outer trapped surface if $\theta_{+}<0$. If
$\theta_{+}$ vanishes identically we say that $\Sigma$ is a marginally outer
trapped surface (or MOTS). Note that in the ‘time-symmetric case’ ($K=0$) a
MOTS is simply a minimal surface.

We now recall the notion of stability of MOTS as introduced by Andersson, Mars
and Simon ([1, 2]; see also [14]).

Let $\Sigma$ be a MOTS in the initial data set $(M,g,K)$ with outward unit
normal $\nu$. We consider a normal variation of $\Sigma$ in $M$, i.e., a
variation $t\to\Sigma_{t}$ of $\Sigma=\Sigma_{0}$ with variation vector field
$V=\frac{\partial}{\partial t}|_{t=0}=\phi\nu$, $\phi\in C^{\infty}(\Sigma)$.
Let $\theta_{+}(t)$ be the null expansion of $\Sigma_{t}$ with respect to
$l_{t}=u+\nu_{t}$, where $\nu_{t}$ is the outer unit normal to $\Sigma_{t}$ in
$M$. A computation gives,

$\left.\frac{\partial\theta_{+}}{\partial t}\right|_{t=0}=L(\phi)\;,$ (2.3)

where $L:C^{\infty}(\Sigma)\to C^{\infty}(\Sigma)$ is the operator,

$\displaystyle L(\phi)$
$\displaystyle=-\triangle\phi+2g(X,\nabla\phi)+\left(\mathcal{Q}+{\rm
div}\,X-|X|^{2}\right)\phi\,,$ (2.4) $\displaystyle\mathcal{Q}$
$\displaystyle=\frac{1}{2}S_{\Sigma}-(\mu+J(\nu))-\frac{1}{2}|\chi_{+}|^{2}\,,$
(2.5)

where $\triangle$, $\nabla$ and ${\rm div}$ are the Laplacian, gradient and
divergence operators, respectively, on $\Sigma$, $S_{\Sigma}$ is the scalar
curvature of $\Sigma$ and $X$ is the vector field on $\Sigma$ dual to the one
form $K(\nu,\cdot)|_{T\Sigma}$.

The operator $L$ is not self-adjoint in general, but nevertheless admits a
principal eigenvalue.

######  Proposition 2.1.

There is a real eigenvalue $\lambda_{1}=\lambda_{1}(L)$, called the principal
eigenvalue, which is an eigenvalue with smallest real part. The associated
eigenfunction $\phi$, $L(\phi)=\lambda_{1}\phi$, is unique up to a
multiplicative constant, and can be chosen to be strictly positive.

In the time-symmetric case ($K=0$), $L$ reduces to the classical stability (or
Jacobi) operator for minimal surfaces in Riemannian manifolds. As such, we
refer to $L$ as the MOTS stability operator. In analogy with the minimal
surface case, we say that a MOTS is stable provided $\lambda_{1}(L)\geq 0$.
(In the minimal surface case this is equivalent to the second variation of
area being nonnegative.) Note that, by taking $\phi>0$ to be the principal
eigenfunction in (2.3), a stable MOTS $\Sigma$ admits an outward variation
$t\to\Sigma_{t}$ with $\frac{\partial\theta_{+}}{\partial t}|_{t=0}\geq 0$. It
can be shown, in a certain sense, that the converse also holds.

Stable MOTSs arise in various situations. For example, weakly outermost MOTSs
are stable. A separating MOTS $\Sigma$ is weakly outermost in $(M,g,K)$ if
there are no outer trapped surfaces outside of and homologous to $\Sigma$.
Indeed, if $\lambda_{1}(L)<0$, (2.3) implies that $\Sigma$ can be deformed
outward to an outer trapped surface. Weakly outermost MOTSs include, in
particular, compact cross sections of the event horizon in stationary black
hole spacetimes obeying the null energy condition, as well as the MOTS
foliating a dynamical horizon; cf. [4]. More generally, the boundary of the
trapped region is a weakly outermost MOTS (see e.g. [3, 6]).

The proof of Theorem 1.2 in Section 2 relies on the following key fact.
Consider the “symmetrized” operator $\hat{L}:C^{\infty}(\Sigma)\to
C^{\infty}(\Sigma)$,

$\hat{L}(\phi)=-\triangle\phi+\mathcal{Q}\phi\,,$ (2.6)

obtained formally from (2.4) by setting $X=0$.

######  Proposition 2.2 ([11]).

If $\Sigma$ is a stable MOTS in an $n$-dimensional, $n\geq 3$, initial data
set, then $\Sigma$ is ‘symmetric stable’, i.e. $\lambda_{1}(\hat{L})\geq 0$,
where $\lambda_{1}(\hat{L})$ is the principal eigenvalue of $\hat{L}$.

In fact, the arguments in [11] show that
$\lambda_{1}(\hat{L})\geq\lambda_{1}(L)$. These arguments were motivated in
part by computations made by Schoen and Yau [16] on graphical solutions of
Jang’s equation in the presence of a translational symmetry, in their proof of
the positive mass theorem in the general non-time-symmetric case.

Remark. The arguments in [11] apply equally well to the following. Let
$\Sigma^{n-1}$ be a stable MOTS in an $n$-dimensional, $n\geq 3$, initial data
set, and let $\Omega$ be a smooth connected compact $(n-1$)-dimensional
submanifold-with-boundary in $\Sigma$. Then $\lambda_{1}(\Omega)\geq 0$, where
$\lambda_{1}(\Omega)$ is the principal Dirichlet eigenvalue of the symmetrized
operator $\hat{L}$ restricted to $\Omega$. This mild variation of Proposition
2.2 will be used in Section 3.

###  2.2 Proof of Theorem 1.2

We now present a proof of the diameter estimate (1.3). The proof is based
largely on the proof of Theorem 3.2 in [10]. The latter result was motivated
by, and extends to the fully non-time-symmetric case, Theorem 1 in [17], but
uses a different measure of size, one which is amenable to the present
situation.

###### Proof.

Fix points $p,q\in\Sigma$ such that ${\rm diam}(\Sigma)=d(p,q)$. $\Sigma$ is a
stable MOTS, hence by Proposition 2.2, it is symmetric stable,
$\lambda_{1}=\lambda_{1}(\hat{L})\geq 0$. Let $\psi>0$ be an associated
eigenfunction. Substituting $\phi=\psi$ into Equation (2.6), we obtain,

$\triangle\psi=-(\mu+J(\nu)+\frac{1}{2}|\chi_{+}|^{2}+\lambda_{1}-\kappa)\psi$
(2.7)

where $\kappa=\frac{1}{2}S_{\Sigma}$ is the Gaussian curvature of $\Sigma$ in
the induced metric $h$.

Now consider $\Sigma$ in the conformally related metric $\tilde{h}=\psi^{2}h$.
The Gaussian curvature of $(\Sigma,\tilde{h})$ is given by,

$\tilde{\kappa}=\psi^{-2}\kappa-\psi^{-3}\triangle\psi+\psi^{-4}|\nabla\psi|^{2}\,.$
(2.8)

Substituting (2.7) into (2.8) gives,

$\tilde{\kappa}=\psi^{-2}(P+\psi^{-2}|\nabla\psi|^{2})\,,$ (2.9)

where,

$P=\mu+J(\nu)+\frac{1}{2}|\chi_{+}|^{2}+\lambda_{1}\,.$ (2.10)

Let $\gamma$ be a minimal geodesic from $p$ to $q$ in the metric $\tilde{h}$.
Then by Synge’s formula [15] for the second variation of arc length, we have
along $\gamma$,

$\int_{0}^{\tilde{\ell}}\left(\frac{df}{d\tilde{s}}\right)^{2}-\tilde{\kappa}f^{2}\,d\tilde{s}\geq
0\,,$ (2.11)

for all smooth functions $f$ defined on $[0,\tilde{\ell}]$ that vanish at the
end points, where $\tilde{\ell}$ is the $\tilde{h}$-length of $\gamma$ and
$\tilde{s}$ is $\tilde{h}$-arc length along $\gamma$. By making the change of
variable $s=s(\tilde{s})$, where $s$ is $h$-arc length along $\gamma$, and
using Equation (2.9), (2.11) becomes,

$\int_{0}^{\ell}\psi^{-1}(f^{\prime})^{2}-(P+\psi^{-2}|\nabla\psi|^{2})\psi^{-1}f^{2}\,ds\geq
0\,,$ (2.12)

for all smooth functions $f$ defined on $[0,\ell]$ that vanish at the
endpoints, where $\ell$ is the $h$-length of $\gamma$, and
${}^{\prime}=\frac{d}{ds}$.

Setting $k=\psi^{-1/2}f$ in (2.12), we obtain by a small computation,

$\int_{0}^{\ell}(k^{\prime})^{2}-P\,k^{2}-\big{(}\frac{3}{4}\psi^{-2}(\psi^{\prime})^{2}k^{2}-\psi^{-1}\psi^{\prime}kk^{\prime}\big{)}\,ds\geq
0\,,$ (2.13)

where $\psi^{\prime}$ is shorthand for $(\psi\circ\gamma)^{\prime}$, etc.
Completing the square on the bracketed term of the integrand shows that,

$\frac{3}{4}\psi^{-2}(\psi^{\prime})^{2}k^{2}-\psi^{-1}\psi^{\prime}kk^{\prime}\geq-\frac{1}{3}(k^{\prime})^{2}\,,$

which then implies,

$\int_{0}^{\ell}\frac{4}{3}(k^{\prime})^{2}-P\,k^{2}\,ds\geq 0\,.$ (2.14)

Since, from (2.10), $P\geq\mu+J(\nu)\geq c$, (2.14) gives,

$\frac{4}{3}\int_{0}^{\ell}(k^{\prime})^{2}\,ds\geq
c\int_{0}^{\ell}k^{2}\,ds\,.$ (2.15)

Setting $k=\sin\frac{\pi s}{\ell}$ in (2.15) then gives,

$\ell\leq\frac{2}{\sqrt{3}}\cdot\frac{\pi}{\sqrt{c}}\,.$

Since ${\rm diam}(\Sigma)\leq\ell$, the diameter bound (1.3) is established.∎

Remark. From the inequality $\mu+J(\nu)\geq c$, and the fact that
$\lambda_{1}(\hat{L})\geq 0$, one can also derive from (2.7) that the
principal eigenvalue of the operator $-\triangle\phi+\kappa\,\phi$ is greater
than or equal to $c$, i.e., $\lambda_{1}(-\triangle\phi+\kappa\,\phi)\geq c$.
In the case of $2$-dimensional compact (or, more generally, complete)
Riemannian manifolds, Kai Xu has shown for $\beta>\frac{1}{4}$, that the
eigenvalue condition $\lambda_{1}(-\triangle\phi+\beta\kappa\,\phi)\geq c$
leads to a diameter bound depending on $\beta$; see [18, Theorem 1.4]. In the
case $\beta=1$, one obtains the diameter bound (1.3). Xu gives two proofs, one
which uses a method similar to that used here, and one which uses the theory
of $\mu$-bubbles.

##  3 A spectral bound inequality of Hirsch et al. [13]

Given a smooth $n$-dimensional Riemannian manifold $(M^{n},g)$ and real number
$\xi$, in [13] the authors define what they call the $\xi$-spectral constant
$\Lambda_{\xi}$. As they point out, when $(M^{n},g)$ is a compact manifold
with boundary, $\Lambda_{\xi}$ is the principal Dirichlet eigenvalue of the
operator

$L_{\xi}(\phi)=-\triangle\phi+\xi R\phi\,,$ (3.1)

where $R$ is the scalar curvature of $(M^{n},g)$.

A compact Riemannian manifold-with-boundary $(N^{n},g)$ whose boundary
components are separated into two disjoint and non-empty collections,
$\partial N^{n}=\partial_{-}N^{n}\sqcup\partial_{+}N^{n}$, is referred to as a
Riemannian band. Following [13], a Riemannian band is called overtorical if
there exists a smooth map111In some places in the literature, the band map $F$
is only required to be continuous. In fact, while not needed here, Theorem 1.3
in [13] still holds in this case. There are situations where this may be
useful. We thank Marcus Khuri for a communication regarding this point.
$F:N^{n}\to T^{n-1}\times[-1,1]$ of nonzero degree, with
$F(\partial_{\pm}N^{n})\subset T^{n-1}\times\\{\pm 1\\}$.

The following result, which assumes $\xi=\frac{1}{2}$, is proved in [13].

######  Theorem 3.1 ([13], Theorem 1.3).

Let $(N^{n},\partial_{\pm}N^{n},g)$ be an overtorical band with $2\leq n\leq
7$. If $\Lambda_{\frac{1}{2}}>0$ then

$d(\partial_{-}N^{n},\partial_{+}N^{n})\leq\pi\sqrt{\frac{2n}{(n+1)\Lambda_{\frac{1}{2}}}}\,,$
(3.2)

where $d(\partial_{-}N^{n},\partial_{+}N^{n})$ is the distance between
$\partial_{-}N^{n}$ and $\partial_{+}N^{n}$.

We refer to $d(\partial_{-}N^{n},\partial_{+}N^{n})$ as the width of the
overtorical band.

As noted in the introduction, the proof of Theorem 3.1 is based on the theory
of $\mu$-bubbles. The dimension restriction, $n\leq 7$, comes from the
regularity theory of $\mu$-bubbles, which is tied to the regularity theory of
surfaces of prescribed mean curvature. We now use Theorem 3.1 in the case
$n=2$ to obtain Theorem 1.2.

Assume $(M,g,K)$ and $\Sigma$ satisfy the hypotheses of Theorem 1.2. Choose
points $p_{-},p_{+}\in\Sigma$ such that ${\rm diam}(\Sigma)=d(p_{-},p_{+})$.
Let $\gamma$ be a minimizing geodesic from $p_{-}$ to $p_{+}$ of length
$\ell$. Let $\Sigma_{\epsilon}$ be the compact surface-with-boundary obtained
from $\Sigma$ by removing from $\Sigma$ the interiors of two small geodesic
disks $D^{\pm}_{\epsilon}$ of radius $\epsilon$, centered at $p_{\pm}$.
$\Sigma_{\epsilon}$ has boundary consisting of two circles
$C^{\pm}_{\epsilon}=\partial D^{\pm}_{\epsilon}$.

Since, by Proposition 1.1, $\Sigma$ is diffeomorphic to a $2$-sphere,
$\Sigma_{\epsilon}$ is diffeomorphic to $S^{1}\times[-1,1]$, and hence is a
$2$-dimensional overtorical Riemannian band. By applying Theorem 3.1 in the
case $n=2$, we obtain,

$d(C^{-}_{\epsilon},C^{+}_{\epsilon})\leq\frac{2}{\sqrt{3}}\cdot\frac{\pi}{\sqrt{\Lambda_{\frac{1}{2}}}}\,.$
(3.3)

Using the fact that $\gamma$ is a minimizing geodesic from $p_{-}$ to $p_{+}$,
it follows that

$d(C^{-}_{\epsilon},C^{+}_{\epsilon})=\ell-2\epsilon={\rm
diam}(\Sigma)-2\epsilon\,,$

and hence,

${\rm
diam}(\Sigma)\leq\frac{2}{\sqrt{3}}\cdot\frac{\pi}{\sqrt{\Lambda_{\frac{1}{2}}}}+2\epsilon$
(3.4)

for all $\epsilon$ sufficiently small. Thus to complete this proof of Theorem
1.2, it suffices to observe that $\Lambda_{\frac{1}{2}}\geq c$.

By Proposition 2.2 and the remark shortly after its statement,
$\lambda_{1}(\Sigma_{\epsilon})\geq 0$, where $\lambda_{1}(\Sigma_{\epsilon})$
is the Dirichlet eigenvalue of the symmetrized operator

$\hat{L}(\phi)=-\triangle\phi+\frac{1}{2}S_{\Sigma}-\big{(}\mu+J(\nu)+\frac{1}{2}|\chi_{+}|^{2}\big{)}\phi\,.$
(3.5)

By Rayleigh’s formula this implies

$\int_{\Sigma_{\epsilon}}|\nabla\psi|^{2}+\Big{[}\frac{1}{2}S_{\Sigma}-\big{(}\mu+J(\nu)+\frac{1}{2}|\chi_{+}|^{2}\big{)}\Big{]}\psi^{2}\,dA\geq
0$ (3.6)

for all $\psi\in C_{0}^{\infty}(\Sigma_{\epsilon})$. Hence,

$\displaystyle\int_{\Sigma_{\epsilon}}|\nabla\psi|^{2}+\frac{1}{2}S_{\Sigma}\,dA$
$\displaystyle\geq\int_{\Sigma_{\epsilon}}\big{(}\mu+J(\nu)+\frac{1}{2}|\chi_{+}|^{2}\big{)}\psi^{2}\,dA$
(3.7) $\displaystyle\geq c\int_{\Sigma_{\epsilon}}\psi^{2}dA\,,$ (3.8)

for all $\psi\in C_{0}^{\infty}(\Sigma_{\epsilon})$. It now follows from
Rayleigh’s formula again that $\Lambda_{\frac{1}{2}}\geq c$. ∎

Comments on higher dimensions. The argument above to show that
$\Lambda_{\frac{1}{2}}\geq c$ holds for any stable MOTS $\Sigma$ in any
$n$-dimensional, $n\geq 3$, initial data set $(M,g,K)$ that satisfies
$\mu-J(\nu)\geq c$.

In [13], a Riemannian band $(N^{n},\partial_{\pm}N^{n},g)$ is referred to as a
nonPSC-band if $\partial_{-}N^{n}$ and $\partial_{+}N^{n}$ are not separable
by a smooth embedded hypersurface $V^{n-1}\subset N^{n}$ which admits a metric
of positive scalar curvature. As observed in [13], it follows from a classical
result of Schoen and Yau that overtorical bands in dimensions $3\leq n\leq 7$
are nonPSC-bands. Moreover, as is also pointed out, it follows from their
proof of Theorem 1.3, that the theorem holds more generally for nonPSC bands.
Then given an $n$-dimensional Riemannian manifold $\Omega^{n}$, the authors
define the torical-radius $\mathrm{Rad}_{t}(\Omega^{n})$ to be the supremum of
widths of all nonPSC-bands $(N^{n},\partial_{\pm}N^{n},g)$ that are
isometrically immersed into $\Omega$. Putting these facts together, one easily
obtains the following.

######  Theorem 3.2.

Let $\Sigma^{n-1}$ be a stable MOTS in an $n$-dimensional, $3\leq n\leq 8$,
initial data set $(M^{n},g,K)$. Suppose there exists $c>0$, such that
$\mu+J(\nu)\geq c$ on $\Sigma^{n-1}$, where $\nu$ is the outward unit normal
to $\Sigma^{n-1}$. Then the torical-radius satisfies,

${\rm Rad}_{t}(\Sigma^{n-1})\leq\pi\sqrt{\frac{2(n-1)}{n\,c}}\,.$ (3.9)

In [13] this inequality is shown to hold, with $3\leq n\leq 7$, for MOTS that
arise from the cylindrical blow-up of Jang’s equation, in connection with the
authors’s generalization of a result of Schoen and Yau [17] on the existence
of black holes. As the analysis of Schoen and Yau in [16] showed for
$3$-dimensional initial data sets (which was extended to $n$-dimensions,
$3\leq n\leq 7$, by Eichmair [7]), the Jang graph satisfies a stability
property that is inherited by MOTS arising from the blow-up of Jang’s
equation, and implies that such MOTS are symmetric stable. The inequality
(3.9) for such MOTS is a result of this analysis and Theorem 1.3 in [13] (and
its extension to nonPSC bands). The point we wish to make here is that (3.9)
holds for all stable MOTS up to dimension 7, regardless of how they arise.
Finally, we note that for $n\geq 3$ the torical-radius
$\mathrm{Rad}_{t}(\Omega^{n})$ does not directly relate to the diameter.

Acknowledgements. We are grateful to Marcus Khuri and Abraão Mendes for
helpful comments on earlier versions of this paper. We also acknowledge
research support from the Simons Foundation, under Award No. 850541.

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