# Sampling and Filtering of Neural Machine Translation Distillation Data

Vilém Zouhar
Institute of Formal and Applied Linguistics, Charles University
<EMAIL_ADDRESS>
###### Abstract

In most of neural machine translation distillation or stealing scenarios, the
goal is to preserve the performance of the target model (teacher). The
highest-scoring hypothesis of the teacher model is commonly used to train a
new model (student). If reference translations are also available, then better
hypotheses (with respect to the references) can be upsampled and poor
hypotheses either removed or undersampled.

This paper explores the importance sampling method landscape (pruning,
hypothesis upsampling and undersampling, deduplication and their combination)
with English to Czech and English to German MT models using standard MT
evaluation metrics. We show that careful upsampling and combination with the
original data leads to better performance when compared to training only on
the original or synthesized data or their direct combination.

##  1 Introduction

Model distillation is a process of transferring the knowledge of one or more,
usually larger, model(s) into another, usually smaller, model Buciluǎ et al.
(2006). A variation of this is training a new model in a way that its
performance is similar to that of the already trained one. This is achieved by
making use of either teacher predictions (black-box) or other products of the
workings of the teacher, such as attention-score or decoder score (grey/glass-
box). Assuming we have access to a parallel corpus, we focus on sampling the
translation hypotheses and making use not only of the teacher scores but also
of their comparison to the reference.

There are various possible motivations for model distillation. The student
model can be much smaller than the teacher model, which has the benefit of
faster inference speed Germann et al. (2020). It can also be used for model
stealing, where an adversary tries to copy the teacher functionality. This is
a practical concern for production-level MT systems Wallace et al. (2020).

One of the approaches for knowledge distillation is to use the teacher model
to generate a new dataset for the student model to train on. Having access to
a trained teacher model, this approach does not require parallel data and can
leverage large monolingual corpora. Reference translations, however, help with
determining which of the teacher’s translations are good and which are of low
quality.

We focus on this approach and propose and compare several importance sampling
approaches to prepare training data for student models that leverage access to
reference translations. These include pruning, upsampling and undersampling,
deduplication and their combination. We show that a combination of these
methods improves the student performance over just using the reference or the
best hypothesis (by the decoder score), which is a common distillation
practice.

The experiment code is available open-source.111github.com/zouharvi/reference-
mt-distill

###  1.1 Related work

The general methodology for knowledge distillation in the form of teacher-
student has been proposed by Hinton et al. (2015). For the MT task
specifically, Tan et al. (2019) focus on vastly reducing the number of
parameters, while retaining the performance of a multi-lingual teacher. Wei et
al. (2019) and Gordon and Duh (2020) use distillation during training to
further improve the model performance.

The work of Kim and Rush (2016) shows that taking either the top sentence with
respect to the teacher decoder score or BLEU Papineni et al. (2002) improves
the performance. Germann et al. (2020) presented student models that distil
knowledge from a larger teacher model with a negligible loss in performance.
They manipulate the queried data based on target sentence quality, such as by
removing sentences that are not correctly recognized by a language identifier.
For the parallel part of the data, they extract the best BLEU scoring sentence
out of 8 hypotheses. Freitag et al. (2017) experiment with pruning sentences
that are below some TER Snover et al. (2006) threshold (lower is better). They
further document the effect of using an ensemble of teachers and also reducing
the student model size.

##  2 Methods

The evaluation of every sampling method follows the following three-step
process. First, the specific parallel corpus (Section 2.1) is translated by
the teacher model (Section 2.2) for the considered translation direction. New
datasets based on metrics are then created. The reference is taken into
consideration during the hypothesis selection. We train new models (students)
on these datasets and measure their performance. There are 12 hypotheses
(default in Marian NMT) provided by the teacher using beam search for every
source sentence which we consider when composing a new dataset.

Figure 1: Scheme of an example of hypothesis sampling with BLEU metric.

Figure 1 shows an example of the sampling process with BLEU. Twelve
translations are made of Source and each receives a score against the provided
reference. The new data contain Translation 2 three times, because of its high
score. Translation 12 is omitted because of its low score. This upsampling is
explained in detail in Section 2.3.

###  2.1 Data

We make use of the Europarl v10 parallel corpus Koehn (2005) for English-Czech
(0.6M sentences) and English-German (1.8M sentences). The sentences are longer
(23 target words per sentence on average) than in the WMT News Task domain
Barrault et al. (2020). To modern standards, this dataset is relatively small
and very domain restricted. This was chosen deliberately because of
computational limitations.222$\sim$4500 GPU hours in total for the whole
experiment Despite that it demonstrates the results of the different sampling
methods with respect to each other. These results may not be transferable to
large parallel corpora in which training data is abundant.

For every language pair, we randomly sample 15k sentences as development
dataset (used only for determining the best epoch and early stopping) and 15k
sentences for final test evaluation which is reported. The WMT News test
dataset is not used for student evaluation, because the students are trained
on a limited amount of data and on a different domain. Out of the WMT20 News
tokens, $0.18\%$ are not present in the Europarl training set. This would
introduce a higher variance into the WMT News test evaluation, which would be
largely dependent on the diversity of the teacher vocabulary.

###  2.2 Models

The teachers333Version student.base at github.com/browsermt/students in this
experiment are transformer-based Vaswani et al. (2017), speed optimized and
were themselves created by knowledge distillation from state-of-the-art models
Popel et al. (2020); Junczys-Dowmunt (2019), as proposed by Germann et al.
(2020). The Czech$\leftrightarrow$English model is described by Germann et al.
(2020) and the English$\rightarrow$German model by Bogoychev et al. (2020).
Our student models follow the teacher’s architecture with half the size of the
embedding vector (256 instead of 512) and half of the attention heads (4
instead of 8). Student models were trained with an early stopping of 20
evaluations on validation data with evaluation performed every 10k sentences.
Vocabularies were not shared from the teacher because they did not affect the
results, and not using them makes fewer assumptions regarding the level of
access to the teacher model. Marian NMT Junczys-Dowmunt et al. (2018) is used
for teacher decoding and student training.

Table 1 shows the teacher performance measured on WMT20 News and the test
subset of Europarl. Czech models performed better on the Europarl than on the
News task, while for the German model the trend was the opposite. This may be
caused by the fact that the models were distilled from a system that had
Europarl as part of the training data, CzEng 2.0 Kocmi et al. (2020).

Dataset: | CS$\rightarrow$EN | EN$\rightarrow$CS | EN$\rightarrow$DE
---|---|---|---
BLEU: |  |  |
WMT20 News | $28.2$ | $35.8$ | $42.7$
Europarl | $46.1$ | $38.2$ | $32.1$
ChrF: |  |  |
WMT20 News | $0.57$ | $0.55$ | $0.66$
Europarl | $0.69$ | $0.64$ | $0.61$
TER: |  |  |
WMT20 News | $0.57$ | $0.71$ | $0.51$
Europarl | $0.41$ | $0.50$ | $0.61$
Table 1: Teacher models BLEU, ChrF and TER scores on WMT20 News Task dataset
and Europarl domain.

###  2.3 Sampling

Concerning the sampling metrics (always between the considered hypothesis and
the reference), we make use of BLEU, ChrF Popović (2015), TER (negative), the
difference (negative of absolute value) in subword unit counts by
SentencePiece Kudo and Richardson (2018) (SP) and decoder probability divided
by the number of output tokens (score). TER and SP are negative in Section 3
so that higher is always better. The motivation for SP is to capture the
difference in length of the hypotheses with respect to the reference. This is
a very naive metric, but we can use it to see the performance and the
behaviour of all the other metrics. Although BLEU is a document-level metric,
it can also be used to determine sentence similarity. Standard machine
translation metrics444 Sacrebleu metrics version strings:
BLEU+case.mixed+numrefs.1+smooth.exp+tok.13a+v1.4.14
ChrF2+numchars.6+space.false+v1.4.14
TER+tok.tercom-nonorm-punct-noasian-uncased+v1.4.14  are computed using
Sacrebleu Post (2018). Different sampling methods are used even though the
goal is to maximize the BLEU scores of the student models. There is no reason
to assume that sampling only based on BLEU will lead to the best results.

The number of training sentences differs for every method. We define the
following notation.

  * •

$T$ \- top; $T_{\text{metric}}^{n}$ takes $n$ top translation hypotheses
according to metric; equal to $S_{\text{metric}}^{1,1,\ldots 1(n)}$. The
student model may benefit from seeing e.g. the second best hypothesis, even
though it’s not the best available. This results in $n$ times the number of
original sentences which are all different.

  * •

$S$ \- skewed; $S_{\text{metric}}^{k_{1},k_{2},\ldots k_{n}}$ takes
$k_{1}\times$ the top translation hypotheses according to metric,
$k_{2}\times$ the second top translation, etc. As opposed to
$T_{\text{metric}}^{n}$, this method tries to preserve the information of the
ordering by setting $k_{1}\geq k_{2}\geq\ldots k_{n}$. This results in $(\sum
k_{i})$ times the number of original sentences but only $n$ times of which are
different sentences.

  * •

$Dedup[X]$ deduplicates sentence pairs of $X$. It is used after joining the
results of other methods. This method is useful for emulating the or
operation: $Dedup[A+B]$ then means “all sentences in either $A$ or $B$.” The
output size is strictly dependent on their overlap.

  * •

$G$ \- greater than; $G_{\text{metric}}^{m}$ takes all sentence translations
with metric at least $m$. This results in sentences that are close to the
reference according to the metric. The number of output sentences highly
dependent on the threshold and is discussed in the corresponding section.

Sampling methods can be combined: $T_{\text{bleu}}^{2}+G_{\text{score}}^{-10}$
joins the top 2 sentences measured by BLEU and adds them to the hypotheses
with decoder score of at least $-10$. Duplicates are intentionally not
removed; thus, hypotheses in both sampling methods are upsampled.

##  3 Results

#### Baseline.

Table 2 shows results for baseline sampling methods. Original corresponds to
training only to the provided parallel corpus (references).
$T^{1}_{\text{score}}$ takes only the highest-scoring hypothesis from the
decoder, which is related to the scenario where the reference is not
available, and the decoder score is the best measure for hypothesis
quality.555MT quality estimation tools could be used to approximate the
sentence translation quality or language models to use sentence fluency in
lieu of translation quality. The sampling method $T^{12}_{-}$ takes all
available hypotheses (metric does not matter).

Dataset | CS$\rightarrow$EN | EN$\rightarrow$CS | EN$\rightarrow$DE
---|---|---|---
Original | $41.6$ | $31.8$ | $25.1$
$T^{1}_{\text{score}}$ | $40.0$ | $31.2$ | $28.5$
$T^{12}_{-}$ | $41.1$ | $31.6$ | $28.4$
Table 2: BLEU scores for students trained on baseline datasets

Training on the original data leads to better results than training on the
best scoring hypotheses. Training on all hypotheses results in slightly lower
BLEU performance. This may be caused by the small amount of training data
available in which case taking all hypotheses just improves the vocabulary and
language modelling capacity.

#### Best hypotheses.

The results of datasets created by taking either the best one or the four best
hypotheses for every source sentence is shown in Table 3. In the case of
multiple hypotheses having the same score, the one with the highest decoder
score is chosen. The top one and top four hypotheses were chosen to show that
the optimum is neither the top one nor the top twelve (all) hypotheses.

On average, the hypothesis overlap666Overlap computed as
$\text{average}_{m1\neq m2}{|T^{1}_{m1}\cap T^{1}_{m2}|}/{n}$ and
$\text{average}_{m1\neq m2}{|T^{4}_{m1}\cap T^{4}_{m2}|}/{(4n)}$. Original
data size is $n$. in sampling between metrics is $29\%$ for $T^{1}$ and $51\%$
for $T^{4}$. This is expected and shows that when more top hypotheses are
taken into the new dataset, the individual metrics tend to matter less.

Dataset | CS$\rightarrow$EN | EN$\rightarrow$CS | EN$\rightarrow$DE
---|---|---|---
$T^{1}_{\text{BLEU}}$ | $42.6$ | $34.4$ | $29.5$
$T^{1}_{\text{ChrF}}$ | $43.8$ | $33.9$ | $30.5$
$T^{1}_{\text{TER}}$ | $43.0$ | $36.1$ | $28.5$
$T^{1}_{\text{SP}}$ | $39.9$ | $29.5$ | $28.2$
$T^{4}_{\text{BLEU}}$ | $44.0$ | $33.3$ | $29.3$
$T^{4}_{\text{ChrF}}$ | $44.3$ | $34.9$ | $29.6$
$T^{4}_{\text{TER}}$ | $44.2$ | $32.0$ | $28.8$
$T^{4}_{\text{SP}}$ | $41.8$ | $32.3$ | $27.9$
$T^{4}_{\text{score}}$ | $44.2$ | $32.0$ | $28.8$
Table 3: BLEU scores for students trained on best-one and best-four hypotheses
datasets

Taking only the top-scoring hypothesis of reference-based metrics, $T^{1}$
showed better results than the baseline (training on the original data, taking
the highest decoder scoring hypothesis or taking all hypotheses). In all cases
the $T^{4}$ outperformed $T^{1}$. The main gains were on CS$\rightarrow$EN and
EN$\rightarrow$CS. Although the results on EN$\rightarrow$DE are only slightly
better than the baseline, they are systematic across all metrics except for
SP. The effect of choosing the metric for the top four hypotheses seems
marginal, even compared to sampling based on the decoder score. The only
exception is the SP difference, which leads to lower results.

#### Thresholding.

Determining a single threshold for all datasets leads to a vastly different
number of hypotheses being selected (the use of $G^{65}_{\text{BLEU}}$ results
in $1.3\times$ the original dataset for CS$\rightarrow$EN, but $0.6$ for
EN$\rightarrow$DE). Therefore, we establish different metric thresholds for
every dataset so that the new datasets are $1\times$ to $1.5\times$ the
original size for consistent results across language pairs.

Some of the source sentences were easier to translate, and more of their
hypotheses were put into the new dataset. Others had no hypothesis above a
given threshold and were not included in the new data at all. On average only
$25\%$ of original sentences were preserved for BLEU, ChrF, TER and SP. For
the decoder score metric, it is $46\%$. The high loss of source sentences is
expected since most of the hypotheses share large portions of the target
sentence and only differ in a few words. All of them will then behave
similarly with respect to the metric.

Dataset | CS$\rightarrow$EN | EN$\rightarrow$CS | EN$\rightarrow$DE
---|---|---|---
$G_{\text{BLEU}}$ | $39.0_{\,65}$ | $30.2_{\,60}$ | $27.2_{\,55}$
$G_{\text{ChrF}}$ | $37.4_{\,0.82}$ | $29.2_{\,0.81}$ | $26.5_{\,0.80}$
$G_{\text{TER}}$ | $37.8_{\,-0.2}$ | $30.2_{\,-0.25}$ | $25.2_{\,-0.24}$
$G_{\text{SP}}$ | $32.5_{\,\text{--}1}$ | $19.6_{\,\text{--}2}$ | $23.0_{\,\text{--}1}$
$G_{\text{score}}$ | $39.0_{\,\text{--}0.08}$ | $32.0_{\,\text{--}0.09}$ | $27.6_{\,\text{--}0.11}$
Table 4: BLEU scores for students trained on datasets made of hypotheses above
threshold of different metrics. Metrics thresholds are in subscript.

The highest performance is achieved using $G_{\text{score}}$ which can be
explained by how much of the original sentences were preserved.
$G_{\text{score}}$ shows that it is possible to achieve a performance
comparable to $T^{1}_{\text{score}}$ with less than half of the source
sentences by only taking all hypotheses with a decoder score above a
threshold. $G_{\text{BLEU}}$ gets worse results (on average $-1.1$ BLEU), but
with only $27\%$ source sentences preserved.

Better performance could be achieved by lowering the threshold to allow more
source sentences and by intersecting the result with some of the other
sampling methods, thus eliminating only the very low-quality sentence pairs.
This is the approach (done with 5 hypotheses) done by Freitag et al. (2017):
$T^{1}_{score}\cap G^{-0.8}_{TER}$.

#### Upsampling.

In the first upsampling case, $S^{4,3,2,1}$, the best hypothesis is present
four times, the second-best three times, the third-best two times and the
fourth-best once. The reason for upsampling better hypotheses is that we want
to force the optimizer to make bigger steps for sentence pairs that are of
high quality, but at the same time, we want to present other hypotheses to
enlarge the vocabulary and improve the student’s language model. The most
straightforward approach is to put multiple copies of the high-quality example
into the dataset. We also experiment with $S^{2,2,1,1}$, because the
upsampling intensity for every hypothesis rank is an independent variable as
well. Both of these schemes are relatively conservative so that they can be
compared to each other and to $T^{4}$. Results for upsampling within a single
metric are shown in Table 5.

Dataset | CS$\rightarrow$EN | EN$\rightarrow$CS | EN$\rightarrow$DE
---|---|---|---
$S^{4,3,2,1}_{\text{BLEU}}$ | $\boldsymbol{45.2}$ | $\boldsymbol{37.1}$ | $29.7$
$S^{4,3,2,1}_{\text{ChrF}}$ | $42.9$ | $36.6$ | $\boldsymbol{30.1}$
$S^{4,3,2,1}_{\text{TER}}$ | $44.4$ | $36.9$ | $29.8$
$S^{4,3,2,1}_{\text{SP}}$ | $41.8$ | $30.7$ | $28.5$
$S^{4,3,2,1}_{\text{score}}$ | $41.4$ | $33.7$ | $27.9$
$S^{2,2,1,1}_{\text{BLEU}}$ | $44.3$ | $36.5$ | $29.6$
$S^{2,2,1,1}_{\text{ChrF}}$ | $45.2$ | $36.1$ | $29.8$
$S^{2,2,1,1}_{\text{TER}}$ | $43.5$ | $33.4$ | $29.6$
$S^{2,2,1,1}_{\text{SP}}$ | $41.8$ | $33.3$ | $28.9$
$S^{2,2,1,1}_{\text{score}}$ | $43.5$ | $33.4$ | $29.6$
Table 5: BLEU scores for students trained on datasets made by upsampling top
hypotheses within a single metric using $S^{4,3,2,1}$ and $S^{2,2,1,1}$

Both versions of upsampling ($S^{4,3,2,1}$ and $S^{2,2,1,1}$) outperformed all
of the previous results. There seems to be no systematic difference between
$S^{4,3,2,1}$ and $S^{2,2,1,1}$. With the exception of SP and decoder score,
the metrics are comparable. A direct comparison can be made to
$T^{4}=S^{1,1,1,1}$ because both $T^{4}$ and the upsampling methods contain
all source sentences and even the same hypotheses. The only difference is that
in the upsampling case, the better hypothesis is upsampled. In this case
$S^{2,2,1,1}$ had higher results over $T^{4}$ with $p<0.005$ by Student’s
t-test.777Average was subtracted from the three directions so that $T^{4}$ and
$S^{2,2,1,1}$ could be treated as only two distributions.

#### Combination.

For the combination scenarios, the newly sampled datasets are joined together.
This is shown in Table 6. In the first four cases, the sampling methods were
joined with the original data. A baseline to this is
$T^{1}_{\text{score}}+\text{Original}$, which is commonly used for
distillation.

Deduplicating the top four hypotheses according to BLEU or decoder score and
adding them to the original data did not improve over the baseline. Combining
the upsampling according to the decoder score with the original data also did
not help. Replacing the decoder score with BLEU resulted in a significant
improvement. The original data is upsampled so that the ratio of synthetic to
original data is 4:1 in the first case and 2:1 in the second one.

For the rest of the cases, the methods are combined without the original data.
Baselines are shown in Table 2. The combination of the top four hypotheses
($T^{4}_{\text{BLEU}}$ or $T^{4}_{\text{score}}$) with all of the hypotheses,
$T^{12}_{-}$, improved over the baseline, including $T^{12}_{-}$, but
performed poorly with respect to the other methods. Taking hypotheses that are
in the top four according to either BLEU or decoder score leads to the best
results in this section. The top one hypothesis, according to BLEU, is
upsampled at least two and at most four times. This seems to work best for
EN$\rightarrow$DE where the training data were three times larger.

Dataset | CS$\rightarrow$EN | EN$\rightarrow$CS | EN$\rightarrow$DE
---|---|---|---
$T^{1}_{\text{score}}+\text{Original}$ | $44.4$ | $36.4$ | $28.3$
$Dedup[T^{4}_{\text{BLEU}}+T^{4}_{\text{score}}]+\text{Original}$ | $43.7$ | $35.3$ | $29.1$
$S^{4,3,2,1}_{\text{score}}+2\times\text{Original}$ | $43.9$ | $36.1$ | $28.3$
$S^{4,3,2,1}_{\text{BLEU}}+2\times\text{Original}$ | $\boldsymbol{45.5}$ | $\boldsymbol{37.3}$ | $28.8$
$S^{4,3,2,1}_{\text{BLEU}}+4\times\text{Original}$ |  $\boldsymbol{45.5}$ $\star$ |  $\boldsymbol{37.4}$ $\star$ | $28.9$
$T^{4}_{\text{score}}+T^{12}_{-}$ | $41.6$ | $33.2$ | $28.3$
$T^{4}_{\text{BLEU}}+T^{12}_{-}$ | $42.6$ | $33.9$ | $28.7$
$T^{4}_{\text{BLEU}}+T^{4}_{\text{score}}$ | $43.3$ | $33.2$ | $28.9$
$Dedup[\sum T^{2}_{\text{metric}}]$ | $43.6$ | $34.7$ | $29.1$
$Dedup[\sum T^{2}_{\text{metrics}}]+T^{12}_{-}$ | $40.8$ | $32.0$ | $27.2$
$Dedup[T^{4}_{\text{BLEU}}+T^{4}_{\text{score}}]+T^{1}_{\text{BLEU}}+T^{1}_{\text{score}}$ | $43.5$ | $34.7$ | $29.2$
$Dedup[T^{4}_{\text{BLEU}}+T^{4}_{\text{score}}]+Dedup[T^{1}_{\text{BLEU}}+T^{1}_{\text{score}}]$ | $42.6$ | $34.9$ |  $\boldsymbol{29.6}$ $\star$
$Dedup[T^{4}_{\text{BLEU}}+T^{4}_{\text{score}}]$ | $43.5$ | $35.0$ | $\boldsymbol{29.3}$
Table 6: BLEU scores for students trained on datasets made of combination of
sampling methods. $\sum_{\text{metric}}$ sums over all used metrics (BLEU,
ChrF, TER, SP, score).

#### Bigger student model.

To demonstrate the data sampling method behaviour on slightly larger models,
the common distillation baseline ($T^{1}_{\text{score}}+\text{Original}$) and
the best performing proposed sampling method
($S^{4,3,2,1}_{\text{BLEU}}+4\times\text{Original}$) were used to train a
student of the same size as the used teacher (embedding vector dimension 512
and 8 attention heads). The results are shown in Table 7. They are
systematically higher than for the smaller models, and the difference between
the baseline and the best sampling is preserved.

Dataset | CS$\rightarrow$EN | EN$\rightarrow$CS | EN$\rightarrow$DE
---|---|---|---
$T^{1}_{\text{score}}+\text{Original}$ | $44.7$ | $36.2$ | $28.3$
$Dedup[T^{4}_{\text{BLEU}}+T^{4}_{\text{score}}]+$ Original | $44.3$ | $36.2$ | $28.5$
$S^{4,3,2,1}_{\text{BLEU}}+2\times\text{Original}$ | $\boldsymbol{46.9}$ | $\boldsymbol{38.5}$ | $\boldsymbol{28.8}$
$S^{4,3,2,1}_{\text{BLEU}}+4\times\text{Original}$ |  $\boldsymbol{47.4}$ $\star$ |  $\boldsymbol{38.9}$ $\star$ | $\boldsymbol{28.9}$
Table 7: BLEU scores for students trained on datasets made of combination of
top hypothesis and original data. Trained with parameters equal to the
teacher’s: embedding vector dimension 512 and 8 attention heads.

##  4 Summary

Although widely used, taking only the highest-scoring sentence (with respect
to the decoder score or any reference-based metrics, such as BLEU) does not
lead to the best results. In the context of the proposed experiments, these
are achieved by a combination of top hypotheses and the original data, such as
$S_{\text{BLEU}}^{4,3,2,1}+4\times\text{Original}$ (upsampling according to
BLEU and joining with the original data duplicated four times). Here, an
improvement of an average $+2$ BLEU points against $T^{1}_{\text{score}}$ \+
Original was achieved.

The choice of the sampling metric does not significantly influence the
results, especially in cases where more than the top one hypothesis is
sampled. Because of this, in most scenarios the decoder score can be used
instead, reducing the need for translation references.

#### Future work.

We worked with only two upsampling schemes: $S^{4,3,2,1}$ and $S^{2,2,1,1}$.
However, the two vectors are arbitrary and more of the vast vector space
should be explored, especially with more than the top four hypotheses
considered or more skewed towards the best hypothesis. More sophisticated
methods based on the value of the metric instead of just the ordering could
also be tried out.

The effects of large models (both teacher and student) and data access should
be explored to verify the transferability of the results of the current setup.
Specifically, the teacher model should not be a distilled model itself. The
robustness of the training should also be established.

Even though this paper focused solely on MT, the importance sampling methods
could also be applied and verified on other NLP tasks, possibly even on more
general machine learning problems.

## Acknowledgements

Sincere thanks to Philipp Zimmermann, Martin Popel and both PBML reviewers for
their helpful suggestions and comments.

This study was supported by the Czech Science Foundation (grant n. 19-26934X,
NEUREM3).

The work described also used services provided by the LINDAT/CLARIAH-CZ
Research Infrastructure (lindat.cz), supported by the Ministry of Education,
Youth and Sports of the Czech Republic (Project No. LM2018101).

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# Compact approach to the positivity of Brown-York mass and rigidity of
manifolds with mean-convex boundaries in flat and spherical contexts

Sebastián Montiel  Departamento de Geometría y Topología
Universidad de Granada
18071 Granada
Spain<EMAIL_ADDRESS>
(Date: September, 2022)

###### Abstract.

In this article we develope a spinorial proof of the Shi-Tam theorem for the
positivity of the Brown-York mass without necessity of building non smooth
infinite asymptotically flat hypersurfaces in the Euclidean space and use the
positivity of the ADM mass proved by Schoen-Yau and Witten. This same compact
approach provides an optimal lower bound [HMZ] for the first non null
eigenvalue of the Dirac operator of a mean convex boundary for a compact spin
manifold with non negative scalar curvature, an a rigidity result for mean-
convex bodies in flat spaces. The same machinery provides analogous, but new,
results of this type, as far as we know, in spherical contexts, including a
version of Min-Oo’s conjecture.

###### Key words and phrases:

Dirac Operator, Spectrum, APS boundary condition, Brown-York mass, Mean-
convexity, Min-Oo conjecture

###### 1991 Mathematics Subject Classification:

Differential Geometry, Global Analysis, 53C27, 53C40, 53C80, 58G25

Declaration: The author declares to be free of conflicts of interests
competing interests of any type retive to this article.

Research partially supported by a Junta de Andalucia FQM-325

.

##  1\. Introduction

In a beautiful work by Shi and Tam [ST, Theorem 4.1], the positivity of the
ADM mass theorem, proved by Schoen and Yau [SY] and [Wi], independently, was
used in a new way to get nice results on the boundary behaviour of a compact
Riemannian manifold with non-negative scalar curvature. They proved that if
$\Omega$ is a three-dimensional compact Riemannian manifold with non-negative
scalar curvature and strictly convex boundary $\Sigma$, then

$\int_{\Sigma}H_{0}-\int_{\Sigma}H,$

that is, the so called Brown-York mass enclosed in $\Omega$, is greater than
or equal to zero, where $H$ is the inner mean curvature of $\Sigma$ in
$\Omega$ and $H_{0}$ is the Euclidean inner mean curvature of $\Sigma$ in
${\mathbb{R}}^{3}$ corresponding to the (unique up to rigid motions) Weyl
embedding of $\Sigma$ into ${\mathbb{R}}^{3}$ obtained by Pogorelov [Po] and
Nirenberg [Ni] independently. Moreover, the equality holds for some boundary
component $\Sigma$. The proof uses two fundamental facts. First, following an
idea by Bartnik [Bar], the construction of a suitable infinite asymptotically
flat extension of $\Omega$ with non negative scalar curvature by deforming the
exterior of $\Sigma$ in ${\mathbb{R}}^{3}$ in order to have the same mean
curvature as $\Sigma$ in $\Omega$ with the intention to glue it to manifold
$\Omega$ along its boundary $\Sigma$. This construction is equivalent to solve
a non linear parabolic equation in a $C^{1}$ context, because this is the
degree of differentiability after the glueing. The second point is that the
difference of integrals

$\int_{\Sigma_{r}}H_{0}-\int_{\Sigma_{r}}H_{r},$

where $\Sigma_{r}$ is an expansion to infinity of the original boundary
$\Sigma$ into ${\mathbb{R}}^{3}$, where $H_{0}$ is the mean curvature with
respect to the Euclidean metric and $H_{r}$ is the mean curvature with respect
to the Bartnik metric goes in a non decreasing way to the ADM mass of the
built asymptotically flat manifold and, so, one can use the non negativity of
its mass. This approach was successfully used to prove the positivity of other
quasi local masses proposed by Liu and Yau [LiY1, LiY2] and Wang and Yau
[WaY]. Along these comments one can see that positivity of quasi local masses
and rigidity of compact manifolds with non empty boundary are different,
although similar, aspects of a quasi same question.

It can be seen, since one starts to study elementary differential geometry,
for example, when one begins to prove the Cohn-Vossen rigidity of ovaloids in
${\mathbb{R}}^{3}$, that there is a close connexion between rigidity of these
manifolds and the fact that the integral its mean curvatures coincide [MR]
(pages 218-219). In this paper, we will obtain a compact approach of the
positivity of the Brown-York mass for the compact spin manifolds of arbitrary
dimension and its corresponding rigidity theorem for mean-convex bodies in the
Euclidean spaces. These two type of results are relied with an estimate for
the lower eigenvalues of the Dirac operator of these bodies and this, in turn,
to theorem type Min-Oo. In fact, in the flat case the corresponding Min-Oo
conjecture was obtained by Miao [Mi1] (see Remark 1), although it was an easy
consequence of the estimate obtained in [HMZ] for this eigenvalue of the Dirac
operator.

Also, in this article, for the sake of completeness, we explain the relation
between the spin structures of the Euclidean space or the sphere and their
hypersurfaces and we will see that their Dirac operator relate basically
through the mean curvature of these hypersurfaces and the scalar curvature of
the ambient spaces. This fact will allow us, for a $(n+1)$-dimensional compact
spin Riemannian manifold $\Omega$ with non negative scalar curvature and mean-
convex boundary $\Sigma$, to unify a compact proof to obtain a lower estimate
for the spectrum of the Dirac of $\Sigma$, the positivity of the Brown-York
mass for mean-convex non necessarily convex domains, the aforementioned
resolution of the flat version of Min-Oo’s conjecture and the rigidity of
these mean-convex bodies in the Euclidean spaces.

In the final section of the article, we will show that this same scheme, with
the important difference of establish the exact value $R=n(n+1)$ of the scalar
curvature of the bulk manifold, works to obtain exactly the same sequence of
results. In this case, all of them, as far as we know, are new.

##  2\. Riemannian spin manifolds and hypersurfaces

Consider an $(n+1)$-dimensional spin Riemannian manifold $\Omega$ with non-
empty boundary $\partial\Omega=\Sigma$ and denote by $\langle\;,\;\rangle$ its
scalar product and by $\nabla$ its corresponding Levi-Civita connection on the
tangent bundle $T\Omega$. We fix a spin structure (and so a corresponding
orientation) on the manifold $\Omega$ and denote by $\hbox{\bb S}\Omega$ the
associated spinor bundle, which is a complex vector bundle of rank
$2^{\left[\frac{n+1}{2}\right]}$. Then let

$\gamma:{\mathbb{C}}\ell(\Omega)\longrightarrow{\rm
End}_{{\mathbb{C}}}(\hbox{\bb S}\Omega)$

be the Clifford multiplication, which provides a fibre preserving irreducible
representation of the Clifford algebras constructed over the tangent spaces of
$\Omega$. When the dimension $n+1$ is even, we have the standard chirality
decomposition

(1) $\hbox{\bb S}\Omega=\hbox{\bb S}\Omega^{+}\oplus\hbox{\bb S}\Omega^{-},$

where the two direct summands are respectively the $\pm 1$-eigenspaces of the
endomorphism $\gamma(\omega_{n+1})$, with
$\omega_{n+1}=i^{\left[\frac{n+2}{2}\right]}e_{1}\cdots e_{n+1}$, the complex
volume form. It is well-known (see [LM]) that there are, on the complex spinor
bundle $\hbox{\bb S}\Omega$, a natural Hermitian metric $(\;,\;)$ and a
spinorial Levi-Civita connection, denoted also by $\nabla$, which is
compatible with both $(\;,\;)$ and $\gamma$ in the following sense:

(2) $\displaystyle X(\psi,\phi)=(\nabla_{X}\psi,\phi)+(\psi,\nabla_{X}\phi)$
(3)
$\displaystyle\nabla_{X}\left(\gamma(Y)\psi\right)=\gamma(\nabla_{X}Y)\psi+\gamma(Y)\nabla_{X}\psi$

for any tangent vector fields $X,Y\in\Gamma(T\Omega)$ and any spinor fields
$\psi,\phi\in\Gamma(\hbox{\bb S}\Omega)$ on $M$. Moreover, with respect to
this Hermitian product on $\hbox{\bb S}\Omega$, Clifford multiplication by
vector fields is skew-Hermitian or equivalently

(4) $(\gamma(X)\psi,\gamma(X)\phi)=|X|^{2}(\psi,\phi).$

Since the complex volume form $\omega_{n+1}$ is parallel with respect to the
spinorial Levi-Civita connection, when $n+1=\dim\Omega$ is even, the chirality
decomposition (1) is preserved by $\nabla$. Moreover, from (4), one sees that
it is an orthogonal decomposition.

In this setting, the (fundamental) Dirac operator $D$ on the manifold $M$ is
the first order elliptic differential operator acting on spinor fields given
locally by

$D=\sum_{i=1}^{n+1}\gamma(e_{i})\nabla_{e_{i}},$

where $\\{e_{1},\dots,e_{n+1}\\}$ is a local orthonormal frame in $T\Omega$.
When $n+1=\dim\Omega$ is even, $D$ interchanges the chirality subbundles
$\hbox{\bb S}\Omega^{\pm}$.

The boundary hypersurface $\Sigma$ is also an oriented Riemannian manifold
with the induced orientation and metric. If $\nabla^{\Sigma}$ stands for the
Levi-Civita connection of this induced metric we have the Gauss and Weingarten
equations

$\nabla_{X}Y=\nabla^{\Sigma}_{X}Y+\langle AX,Y\rangle
N,\qquad\nabla_{X}N=-AX,$

for any vector fields $X,Y$ tangent to $\Sigma$, where $A$ is the shape
operator or Weingarten endomorphism of the hypersurface $\Sigma$ corresponding
to the unit normal field $N$ compatible with the given orientation. As the
normal bundle of the boundary hypersurface is trivial, the Riemannian manifold
$\Sigma$ is also a spin manifold and so we will have the corresponding spinor
bundle $\hbox{\bb S}\Sigma$, the Clifford multiplication $\gamma^{\Sigma}$,
the spinorial Levi-Civita connection $\nabla^{\Sigma}$ and the intrinsic Dirac
operator $D^{\Sigma}$. It is not difficult to show (see [Bä2, BFGK, Bur, Tr,
Mo]) that the restricted Hermitian bundle

${\bf S}:=\hbox{\bb S}\Omega_{|\Sigma}$

can be identified with the intrinsic Hermitian spinor bundle $\hbox{\bb
S}\Sigma$, provided that $n+1=\dim\Omega$ is odd. Instead, if $n+1=\dim\Omega$
is even, the restricted bundle ${\bf S}$ could be identified with the sum
$\hbox{\bb S}\Sigma\oplus\hbox{\bb S}\Sigma$. With such identifications, for
any spinor field $\psi\in\Gamma({\bf S})$ on the boundary hypersurface
$\Sigma$ and any vector field $X\in\Gamma(T\Sigma)$, define on the restricted
bundle ${\bf S}$, the Clifford multiplication $\gamma^{{\bf S}}$ and the
connection $\nabla^{{\bf S}}$ by

(5) $\displaystyle\gamma^{{\bf S}}(X)\psi=\gamma(X)\gamma(N)\psi$ (6)
$\displaystyle\nabla^{{\bf S}}_{X}\psi=\nabla_{X}\psi-\frac{1}{2}\gamma^{{\bf
S}}(AX)\psi=\nabla_{X}\psi-\frac{1}{2}\gamma(AX)\gamma(N)\psi\,.$

Then it easy to see that $\gamma^{{\bf S}}$ and $\nabla^{{\bf S}}$ correspond
respectively to $\gamma^{\Sigma}$ and $\nabla^{\Sigma}$, for $n+1$ odd, and to
$\gamma^{\Sigma}\oplus-\gamma^{\Sigma}$ and
$\nabla^{\Sigma}\oplus\nabla^{\Sigma}$, for $n+1$ even. Then, $\gamma^{{\bf
S}}$ and $\nabla^{{\bf S}}$ satisfy the same compatibilty relations (2), (3)
and (4) and together with the following additional identity

$\nabla^{{\bf S}}_{X}\left(\gamma(N)\psi\right)=\gamma(N)\nabla^{{\bf
S}}_{X}\psi.$

As a consequence, the hypersurface Dirac operator ${\bf D}$ acts on smooth
sections $\psi\in\Gamma({\bf S})$ as

${\bf D}\psi:=\sum_{j=1}^{n}\gamma^{{\bf S}}(u_{j})\nabla^{{\bf
S}}_{u_{j}}\psi=\frac{n}{2}H\psi-\gamma(N)\sum_{j=1}^{n}\gamma(u_{j})\nabla_{u_{j}}\psi,$

where $\\{u_{1},\dots,u_{n}\\}$ is a local orthonormal frame tangent to the
boundary $\Sigma$ and $H=(1/(n)){\rm trace}\,A$ is its mean curvature
function, coincides with the intrinsic Dirac operator $D^{\Sigma}$ on the
boundary, for $n+1$ odd, and with the pair $D^{\Sigma}\oplus-D^{\Sigma}$, for
$n+1$ even. In the particular case where the field $\psi\in\Gamma({\bf S})$ is
the restriction of a spinor field $\psi\in\Gamma(\Sigma)$ on $\Omega$, this
means that

(7) ${\bf D}\psi=\frac{n}{2}H\psi-\gamma(N)D\psi-\nabla_{N}\psi.$

Note that we always have the anticommutativity property

(8) ${\bf D}\gamma(N)=-\gamma(N){\bf D}$

and so, when $\Sigma$ is compact, the spectrum of ${\bf D}$ is symmetric with
respect to zero and coincides with the spectrum of $D^{\Sigma}$, for $n+1$
odd, and with ${\rm Spec}(D^{\Sigma})\cup-{\rm Spec}(D^{\Sigma})$, for $n+1$
even [see HMR]. In fact, we and other authors also have remarked in several
papers that the spectrum Spec($\bf D$) is a ${\mathbb{Z}}$-symmetric sequence

$-\infty\swarrow\cdots\leq-\lambda_{k}\leq\cdots\leq-\lambda_{1}<\lambda_{0}=0<\lambda_{1}\leq\cdots\leq\cdots\leq\lambda_{k}\leq\cdots\nearrow+\infty$

(each eigenvalue repeated according to its corresponding multiplicity). This
is because $D$ is an elliptic operator of order one which is self-adjoint due
to the compacity of $\Sigma$ and the fact that $\gamma(N)$ maps the eigenvalue
$\lambda_{k}$ into $-\lambda_{k}$, being $\gamma$ the Clifford multiplication
and $N$ the inner unit normal of $\Sigma$ in $\Omega$. Let’s choose an
$L^{2}(\mathbb{S}\Sigma)$-orthonormal basis $\\{\psi_{k}\\}_{k\in{Z}}$ of the
Hilbert space $L^{2}(\mathbb{S}\Sigma)$ constituted by eigenspinors of ${\bf
D}$, that is, ${\bf D}\psi_{k}=\lambda_{k}\psi_{k}$ (and so $\psi_{k}\in
C^{\infty}(\Sigma))$, for all $k\in\mathbb{Z}$. We have to remark that the
presence of the eigenvalue $\lambda_{0}=0$ (repeated with its corresponding
multiplicity) is not compulsory. It is a standard fact (Perceval equality)
that, if $\phi$ is an $L^{2}$ spinor field on $\Sigma$, the series

$\sum^{+\infty}_{-\infty}\phi_{k}=\lim_{k\geq
0,k\rightarrow\infty}\sum^{k}_{-k}\phi_{k}=\phi,$

converges in the strong $L^{2}(\mathbb{S}\Sigma)$-topology, where each
$\phi_{k}$ is the projection of $\phi$ onto the eigenspace corresponding to
the eigenvalue $\lambda_{k}$. As a consequence of Hölder inequalities and
well-known facts, we have $L^{1}(\mathbb{S}\Sigma)$-convergence as well and
moreover, as an easy consequence of the completeness of
$L^{2}(\mathbb{S}\Sigma)$ or $L^{1}(\mathbb{S}\Sigma)$, pointwise convergence
almost everywhere (and so everywhere if $\phi$ is continous) for a suitable
subseries of $\sum^{+\infty}_{-\infty}\phi_{k}$.

##  3\. A spinorial Reilly inequality

A basic tool to relate the eigenvalues of the Dirac operator and the geometry
of the manifold $\Omega$ and those of its boundary $\Sigma$ will be, as in the
closed case (see [Fr1]), the integral version of the Schrödinger-Lichnerowicz
formula

(9) $D^{2}=\nabla^{*}\nabla+\frac{1}{4}R,$

where $R$ is the scalar curvature of $\Omega$. In fact, given a spinor field
$\psi$ on $\Omega$, taking into account the formula above, if we compute the
divergence of the one-form $\alpha$ defined by

$\alpha(X)=\left<\gamma(X)D\psi+\nabla_{X}\psi,\psi\right>,\qquad\forall X\in
T\Omega$

and integrate, one gets

$-\int_{\Sigma}\left<\gamma(N)D\psi+\nabla_{N}\psi,\psi\right>=\int_{\Omega}\left(|\nabla\psi|^{2}-|D\psi|^{2}+\frac{1}{4}R|\psi|^{2}\right),$

which by (7), could be written as

$\int_{\Sigma}\left(({\bf
D}\psi,\psi)-\frac{n}{2}H|\psi|^{2}\right)=\int_{\Omega}\left(|\nabla\psi|^{2}-|D\psi|^{2}+\frac{1}{4}R|\psi|^{2}\right).$

Finally, we will use the pointwise spinorial Schwarz inequality

$|D\psi|^{2}\leq(n+1)|\nabla\psi|^{2},\qquad\forall\psi\in\Gamma(\hbox{\bb
S}\Omega),$

where the equality is achieved only by the so-called twistor spinors, that is,
those satisfying the following over-determined first order equation

$\nabla_{X}\psi=-\frac{1}{n+1}\gamma(X)D\psi,\qquad\forall X\in T\Omega.$

Then we get the following integral inequality, called Reilly inequality [see
HMZ, for example], because of its similarity with the corresponding one
obtained in [Re] for the Laplace operator,

(10) $\int_{\Sigma}\left(({\bf
D}\psi,\psi)-\frac{n}{2}H|\psi|^{2}\right)\geq\int_{\Omega}\left(\frac{1}{4}R|\psi|^{2}-\frac{n}{n+1}|D\psi|^{2}\right),$

with equality only for twistor spinors on $\Omega$.

##  4\. The APS boundary condition

It is a well-known fact that the Dirac operator $D$ on a compact spin
Riemannian manifold $\Omega$ with boundary, $D:\Gamma(\hbox{\bb
S}\Omega)\rightarrow\Gamma(\hbox{\bb S}\Omega)$ has an infinite dimensional
kernel and a closed image with finite codimension. People has looked for
conditions $B$ to be imposed on the restrictions to the boundary $\Sigma$ of
the spinor fields on $\Omega$ so that this kernel becomes finite dimensional
and then the boundary problem

(BP) $\left\\{\begin{array}[]{lll}D\psi&=\Phi&\hbox{on $\Omega$}\\\
B\psi_{|\Sigma}&=\chi&\hbox{along $\Sigma$},\end{array}\right.$

for $\Phi\in\Gamma(\hbox{\bb S}\Omega)$ and $\chi\in\Gamma({\bf S})$, is of
Fredholm type. In this case, we will have smooth solutions for any data $\Phi$
and $\chi$ belonging to a certain subspace with finite codimension and these
solutions will be unique up to a finite dimensional kernel.

To our knowledge, the study of boundary conditions suitable for an elliptic
operator $D$ (of any order, although for simplicity, we only consider first
order operators) acting on smooth sections of a Hermitian vector bundle
$F\rightarrow\Omega$ has been first done in the fifties of past century by
Lopatinsky and Shapiro ([Hö, Lo]), but the main tool was discovered by
Calderón in the sixties: the so-called Calderón projector

${\mathcal{P}}_{+}(D):H^{\frac{1}{2}}(F_{|\Sigma})\longrightarrow\\{\psi_{|\Sigma}\,|\,\psi\in
H^{1}(F),D\psi=0\\}.$

This is a pseudo-differential operator of order zero (see [BW, Se]) with
principal symbol ${\mathfrak{p}}_{+}(D):T\Sigma\rightarrow{\rm
End}_{\mathbb{C}}(F)$ depending only on the principal symbol $\sigma_{D}$ of
the operator $D$ and can be calculated as follows

(11) ${\mathfrak{p}}_{+}(D)(X)=-\frac{1}{2\pi
i}\int_{\Gamma}\left[(\sigma_{D}(N))^{-1}\sigma_{D}(X)-\zeta
I\right]^{-1}\,d\zeta,$

for any $p\in\Sigma$ and $X\in T_{p}\Sigma$, where $N$ is the inner unit
normal along the boundary $\Sigma$ and $\Gamma$ is a positively oriented cycle
in the complex plane enclosing the poles of the integrand with negative
imaginary part. Although the Calderón projector is not unique for a given
elliptic operator $D$, its principal symbol is uniquely determined by
$\sigma_{D}$. One of the important features of the Calderón projector is that
its principal symbol detects the ellipticity of a boundary condition, or in
other words, if the corresponding boundary problem (BP) is a well-posed
problem (according to Seeley in [Se]). In fact (cfr. [Se] or [BW, Chap. 18]),
a pseudo-differential operator

$B:L^{2}(F_{|\Sigma})\longrightarrow L^{2}(V),$

where $V\rightarrow\Sigma$ is a complex vector bundle over the boundary, is
called a (global) elliptic boundary condition when its principal symbol
$b:T\Sigma\rightarrow{\rm Hom}_{\mathbb{C}}(F_{|\Sigma},V)$ satisfies that,
for any non-trivial $X\in T_{p}\Sigma$, $p\in\Sigma$, the restriction

$b(X)_{|{\rm image}\,{\mathfrak{p}}_{+}(D)(X)}:{\rm
image}\,{\mathfrak{p}}_{+}(D)(X)\subset F_{p}\longrightarrow V_{p}$

is an isomorphism onto ${\rm image}\,b(X)\subset V_{p}$. Moreover, if ${\rm
rank}\,V=\dim\\-{\rm image}\\-{\mathfrak{p}}_{+}(D)(X)$, we say that $B$ is a
local elliptic boundary condition. When $B$ is a local operator this
definition yields the so-called Lopatinsky-Shapiro conditions for ellipticity
(see for example [Hö]).

When these definitions and the subsequent theorems are applied to the case
where the vector bundle $F$ is the spinor bundle $\hbox{\bb S}\Omega$ and the
elliptic operator $D$ is the Dirac operator $D$ on the spin Riemannian
manifold $\Omega$, we obtain the following well-known facts in the setting of
the general theory of boundary problems for elliptic operators (see for
example [BrL, BW, GLP, Hö, Se]):

It is easy to see that the principal symbol $\sigma_{D}$ of the Dirac operator
$D$ on $\Omega$ is given by

$\sigma_{D}(X)=i\gamma(X),\qquad\forall X\in T\Omega.$

Then by (11), the principal symbol of the Calderón projector of the Dirac
operator is given by

${\mathfrak{p}}_{+}(D)(X)=-\frac{1}{2|X|}(i\gamma(N)\gamma(X)-|X|I)=\frac{1}{2|X|}(i\gamma^{\bf
S}(X)+|X|I),$

for each non-trivial $X\in T\Sigma$ and where $\gamma^{\bf S}$ is identified
in (5) as the intrinsic Clifford product on the boundary. As the endomorphism
$i\gamma(N)\gamma(u)=-i\gamma^{\bf S}(X)$ is self-adjoint and its square is
$|X|^{2}$ times the identity map, then it has exactly two eigenvalues, say
$|X|$ and $-|X|$, whose eigenspaces are of the same dimension
$\frac{1}{2}\dim\hbox{\bb S}\Omega_{p}=2^{[\frac{n+1}{2}]-1}$, since they are
interchanged by $\gamma(N)$. Hence the symbol ${\mathfrak{p}}_{+}(D)(X)$ is,
up to a constant, the orthogonal projection onto the eigenspace corresponding
to the eigenvalue $-|X|$ and so

$\displaystyle{\rm image}\,{\mathfrak{p}}_{+}(D)(X)=\\{\eta\in\hbox{\bb
S}\Omega_{p}\,|\,i\gamma(N)\gamma(X)\eta=-|X|\eta\\},$ $\displaystyle\dim{\rm
image}\,{\mathfrak{p}}_{+}(D)(X)=\frac{1}{2}\dim\hbox{\bb
S}\Omega_{p}=2^{[\frac{n+1}{2}]-1}.$

From these equalities and from the definition of ellipticity for the boundary
condition represented by the pseudo-differential operator $B$, we have that
the first equation in the statement above is equivalent to the injectivity of
the map $b(X)_{|{\rm image}\;{\mathfrak{p}}_{+}(D)(X)}$. The second one
implies that $\dim{\rm image}\;b(X)=\dim{\rm image}\;{\mathfrak{p}}_{+}(D)(X)$
and so, together with the injectivity, this means that $b(X)_{|{\rm
image}\;{\mathfrak{p}}_{+}(D)(X)}$ is surjective. So we have proved that the
two claimed conditions are equivalent to the ellipticity of a given boundary
condition $B$ for the Dirac operator $D$ on $\Omega$. Now, from this
ellipticity, one may deduce that the problem (BP) and the spectral
corresponding one are of Fredholm type and the remaining assertions on
eigenvalues and eigenspaces follow in a standard way (see [BW, Hö]).

At the risk of being too long, we will explain what is the Atiyah, Patodi and
Singer condition in this context. This well-known boundary condition was
introduced in [APS] in order to establish index theorems for compact manifolds
with boundary. Later, this condition has been used to study the positive mass
and the Penrose inequalities (see [He2, Wi]). Such a condition does not allow
to model confined particle fields since, from the physical point of view, its
global nature is interpreted as a causality violation. Although it is a well-
known fact that the APS condition is an elliptic boundary condition, we are
going to sketch the proof in the setting established above, for three reasons:
first, for completeness, second for pointing out that the APS condition for an
achiral Dirac operator covers both cases of odd and even dimension, although
the latter case is not referred to the spectral resolution of the intrinsic
Dirac operator $D^{\Sigma}$ but to the system $D^{\Sigma}\oplus-D^{\Sigma}$
and third, because we will use it with some little modifications which may not
familiar to some readers.

Precisely, this condition can be described as follows. Choose the
aforementioned Hermitian bundle $V$ over the boundary hypersurface $\Sigma$ as
the restricted spinor bundle ${\bf S}$ defined in Section 2, and define, for
each ${a\in\mathbb{Z}}$, as $B^{a}_{{\rm APS}\;}:=B_{\geq a}:L^{2}({\bf
S})\rightarrow L^{2}({\bf S})$ as the orthogonal projection onto the subspace
spanned by the eigenvalues of the self-adjoint intrinsic operator ${\bf D}$
greater or equal to $a$. Atiyah, Patodi and Singer showed in [APS] (see also
[BW, Prop. 14.2]) that $B_{{\rm APS}\;}^{a}$ is a zero order pseudo-
differential operator whose principal symbol $b_{{\rm APS}\;}$ (independent of
$a$) satisfies the following fact: for each $p\in\Sigma$ and $X\in
T_{p}\Sigma-\\{0\\}$, the map $b_{{\rm APS}\;}(X)$ is the orthogonal
projection onto the eigenspace of $\sigma_{{\bf D}}(X)=i\gamma^{\bf S}(X)$
corresponding to the positive eigenvalue $|X|$. That is

(12) $b_{{\rm APS}\;}(X)=\frac{1}{2}\left(i\gamma^{\bf
S}(X)+|X|I\right)=\frac{1}{2}\left(-i\gamma(N)\gamma(X)+|X|I\right),$

and so the principal symbol $b_{{\rm APS}\;}$ of the APS operator coincides,
up to a constant, with the principal symbol ${\mathfrak{p}}_{+}(D)$ of the
Calderón projector of $D$, for all $a\in{\mathbb{Z}}$. From this, it is
immediate to see that the two ellipticity required conditions are satisfied.

Analytically, this APS boundary condition can be formulated as follows. If
$\phi$ is a spinor on $\Sigma$ and $a\in\mathbb{Z}$ is an integer number, we
will denote by $b_{APS}^{a}(\phi)=\phi_{\geq a}$ the orthonormal projection
referred to the diagonalization of ${\bf D}$ given above just after (8)

$b_{APS}^{a}=\phi_{\geq a}=\sum^{+\infty}_{k\geq a}\phi_{k}=\lim_{l\geq
a}\sum^{l}_{m=a}\phi_{m},$

where the above subseries convergences remain to be valid, as we had already
commented for this truncated series, and moreover

$\int_{\Sigma}|\phi_{\geq a}|^{2}\leq\int_{\Sigma}|\phi|^{2}\qquad\forall
a\in{\mathbb{Z}}$

(Bessel inequality). It was long ago known the fact [see APS, Se, BB, HMR,
BäBa1,BäBa2] that this spectral projection $b_{APS}^{a}$ trough ${\bf D}$
provided an elliptic self-adjoint boundary condition on $\Omega$ for $D$ and
for all $a\in{\mathbb{R}}$, if we take the spectral projection on eigenspaces
corresponding to its non negative eigenvalues. (Be careful, in some papers, as
[APS and BäBa1,BäBa2], the adapted Dirac operator $\bf D$ on the boundary is
taken to be $-\bf D$ and, so, it is necessary to reverse positive and
negative). But, we strongly advise to study all those questions about first
order differential operators, existence of solutions and their regularity in
the two recent works by Ballmann and Bär [BäBa1,BäBa2]. They are much more
well clearly written and in a much more modern language. We have procured
attain at the maximum number of readers.

Anyway, we conclude that, for each integer number $a\leq 0$, we can pose the
elliptic boundary problem

(13) $\left\\{\begin{array}[]{ll}D\psi_{a}=\xi\\\ \psi_{\geq a}=\phi_{\geq
a},\end{array}\right.$

where $\psi$ and $\xi$ are spinor fields on the bulk manifold $\Omega$, $\phi$
is a spinor field on the boundary $\Sigma$ and the meaning of $\psi_{\geq a}$
should be already evident. This problem will be a Fredholm type problem.
Analogously, the corresponding eigenvalue problem

(14) $\left\\{\begin{array}[]{ll}D\psi_{a}=\lambda\psi_{a}\\\ \psi_{\geq
a}=0,\end{array}\right.$

[see HMR, BaBä1,BaBä2], is also well posed and has nice existence and
regularity properties.

##  5\. A compact approach spinor proof of the Shi-Tam theorem valid for mean-
convex domains

We assume that the scalar curvature $R$ of the compact spin Riemannian
manifold $\Omega$ is non-negative and consider a solution to the following
homogeneous problem on $\Omega$

(15) $\left\\{\begin{array}[]{ll}D\psi_{a}=0\\\ \psi_{\geq
a}=0,\end{array}\right.$

for any $a\in{\mathbb{Z}}$. Then, if we put it in the inequality (10) and use
that $R\geq 0$. It only remains the terms on the boundary $\Sigma$ of
$\Omega$. We get

(16) $0\leq\int_{\Sigma}\left(\left<{\bf
D}\psi_{a},\psi_{a}\right>-\frac{n}{2}H|\psi_{a}|^{2}\right),$

and the equality is attained if and only if $\psi_{a}$ is a parallel (twistor
plus harmonic) spinor on $\Omega$. Now, choose $a\leq 0$ in the problem above
and substitute the corresponding solution in the integral inequality above.
Since $\left<{\bf D}\psi_{<a},\psi_{<a}\right>$ and $\left<{\bf D}\psi_{\geq
a},\psi_{\geq a}\right>$ are clearly $L^{2}$-orthogonal, we have

$\int_{\Sigma}\left<{\bf D}\psi_{a},\psi_{a}\right>=\int_{\Sigma}\left<{\bf
D}\psi_{<a},\psi_{<a}\right>=\sum^{-\infty}_{k<a\leq
0}\lambda_{k}\int_{\Sigma}|\psi_{k}|^{2}\leq 0,$

for each integer $a\leq 0$. If, moreover, we add the hypothesis that the inner
mean curvature $H$ is non negative along the boundary $\Sigma$, (16) implies

(17) $0\leq\int_{\Sigma}\left(\left<{\bf
D}\psi_{a},\psi_{a}\right>-\frac{n}{2}H|\psi_{a}|^{2}\right)\leq 0.$

Hence, the first term in the integral above implies $\psi_{<a}=0$ for all
$a\leq 0$. Since, from the boundary condition in (15), $\psi_{\geq a}=0$, for
all $a\in{\mathbb{Z}}$, $a\leq 0$, we conclude

$\psi_{a}=\psi_{<a}+\psi_{\geq a}=0.$

for all $a\in{\mathbb{Z}}$, $a\leq 0$ along $\Sigma$. So, the spinor
$\psi_{a}$ is identically zero along $\Sigma$. As $\psi_{a}$ was harmonic on
$\Omega$, its length must be identically zero because the Hausdorff measure of
$\Sigma$ is greater than two in the bulk manifold as was shown by Bär in
[Bä3]. Then, the unique solution to the homogeneous equation (15) is
$\psi_{a}=0$. Using that this equation is of type Fredholm, we need to study
its cokernel, that is, that the adjoint problem to (15) has also trivial
kernel. Thus, consider the homogeneous problem

(18) $\left\\{\begin{array}[]{ll}D\psi_{a}=0\\\
\psi_{>a}=0,\end{array}\right.$

where it is easy to imagine what $\psi_{>a}$ denotes. Working as in (15), we
also obtain the same inequality (17) for all $a\leq 0$. But, in this case,
when the equality is attained, we are not sure that $\psi_{\leq a}=0$ for all
$a\leq 0$, because $\psi_{0}$ could be vanish, that is, $\psi$ could contain a
non trivial harmonic component $\psi_{0}$. So, the homogeneous problem (18)
could have non trivial harmonic solutions, unless $H>0$, that is, the boundary
$\Sigma$ be strictly mean convex. In any of these two cases, the unique
solutions to (15) and its adjoint (18) would be the null ones. Then, the non
homogeneous problems would always unique solutions.

######  Proposition 1.

Let $\Omega$ be a compact spin manifold with non negative scalar curvature and
non-empty boundary $\Sigma$ such that its inner mean curvature is non-
negative. Then, if we suppose that $\Sigma$ either does not admit harmonic
spinors or is strictly mean convex ($H>0$) in $\Omega$, both homogeneous
problems (15) and (18) have as unique solution the zero spinor. As a
consequence of this, the inhomogeneous problem

(19) $\left\\{\begin{array}[]{ll}D\psi_{a}=\xi\\\ \psi_{\geq a}=\phi_{\geq
a},\end{array}\right.$

has a unique solution $\psi_{a}\in H^{1}(\Omega)\cap H^{\frac{1}{2}}(\Sigma)$
for prescribed spinor fields $\xi\in L^{2}(\Omega)$ and $\phi\in
L^{2}(\Sigma)$ for each integer number $a\leq 0$.

We will apply this result à la Reilly, thought for solving equations on the
bulk manifold, to obtain results on its boundary.

######  Theorem 2.

Let $\Omega$ be a compact spin Riemannian manifold of dimension $n+1$ with
non-negative scalar curvature $R\geq 0$ and having a non-empty boundary
$\Sigma$ whose inner mean curvature $H\geq 0$ is also non-negative (mean-
convex). Suppose also that $\Sigma$ either does not carry harmonic spinors or
$H>0$. Then, for every spinor $\phi\in H^{1}(\Sigma)$, we have

$\int_{\Sigma}|{\bf D}\phi||\phi|\geq\frac{n}{2}\int_{\Sigma}H|\phi|^{2}.$

The equality occurs if and only if $\phi$ is the restriction to $\Sigma$ of a
parallel spinor on $\Omega$ and so

${\bf D}\phi=\frac{n}{2}H\phi.$

Proof. Using the Proposition above, for all integer $a\leq 0$, solve the type
(19) equation

(20) $\left\\{\begin{array}[]{ll}D\psi_{a}=0\\\ \psi_{\geq a}=\phi_{\geq
a}.\end{array}\right.$

By working in same manner as in the introduction of this section, we can
arrive to the inequality (16) because, until this moment, we did not use the
boundary conditions. On the other hand, by using the orthogonal projections
defined above, we know that

$\int_{\Sigma}\left<{\bf D}\psi_{a},\psi_{a}\right>\leq\int_{\Sigma}\left<{\bf
D}\psi_{\geq a},\psi_{\geq a}\right>,$

because the summand

$\int_{\Sigma}\left<{\bf D}\psi_{<a},{\bf\psi}_{<a}\right>$

only contains non positive eigenvalues of ${\bf D}$. So, since $\psi_{\geq
a}=\phi_{\geq a}$, using the pointwise Cauchy-Schwarz inequality, the unique
solution $\psi_{a}$ of (20) satisfies

(21) $\int_{\Sigma}\left<{\bf
D}\psi_{a},\psi_{a}\right>\leq\int_{\Sigma}\left<{\bf D}\phi_{\geq
a},\phi_{\geq a}\right>\leq\int_{\Sigma}|{\bf D}\phi_{\geq a}||\phi_{\geq
a}|,$

for all $a\in{\mathbb{Z}},a\leq 0$. Analogously, we have the $L^{2}$
decomposition

$0\leq\int_{\Sigma}|\psi_{a}|^{2}=\int_{\Sigma}|\psi_{\geq
a}|^{2}+\int_{\Sigma}|\psi_{<a}|^{2}.$

So, in a similar way as above,

(22) $\int_{\Sigma}|\psi_{a}|^{2}\leq\int_{\Sigma}|\psi_{\geq
a}|^{2}=\int_{\Sigma}|\phi_{\geq a}|^{2},$

for each non positive integer $a$. From the fact that $\psi_{\geq
a}=\phi_{\geq a}$ lies in $H^{\frac{1}{2}}(\Sigma)$, we know that this
sequence with index $a$ converges strongly in the $L^{2}(\Sigma)$ topology
and, so, in the $L^{1}$ topology as well. As the non negative mean curvature
function $H$ is smooth on the compact manifold $\Sigma$, $\sqrt{H}\in
L^{2}(\Sigma)$. Then,

$\lim_{a\rightarrow-\infty}\int_{\Sigma}H|\phi_{\geq
a}|^{2}\,d\Sigma=\int_{\Sigma}H|\phi|^{2}$

strongly in $L^{2}(\Sigma)$. From (16), (20) and this last limit, we have

$0\leq\lim_{a\rightarrow-\infty}\int_{\Sigma}\left(|{\bf D}\phi_{\geq
a}||\phi_{\geq a}|-\frac{n}{2}H|\phi_{\geq
a}|^{2}\right)=\int_{\Sigma}\left(|{\bf
D}\phi||\phi|-\frac{n}{2}H|\phi|^{2}\right).$

Hence, the inequality is true.

Let us see that the solution $\psi_{a}$ to the boundary problem (10) is smooth
on ${\bar{\Omega}}$ for all integer $a\leq 0$. In fact, the classical theory
for the Dirac operator [see BäBa1,BäBa2] assures the smoothness of solutions
in the interior $\Omega$. With respect to the regularity at $\Sigma$, we can
apply Theorems 6.11 and 7.17 and Corollary 7.18 in the first on the papers
cited above (using a bootstrapping process) and obtain smoothness on $\Sigma$
as well. Then
$\psi_{a}\in\cap_{k=1}^{\infty}H^{k}({\bar{\Omega}})=C^{\infty}({\bar{\Omega}})$.
For all non positive integer number $a$, the limit function $\psi$ will also
be smooth on the compact manifold ${\bar{\Omega}}$. This implies that the
limit function

$\psi=\lim_{a\rightarrow-\infty}\psi_{a}$

is also strongly harmonic on $\Omega$. Taking limit in (15) and taking into
account that we also have the equality in the right side of 10, we see that
$\psi$ is also a parallel spinor on the whole of ${\bar{\Omega}}$. And it is
clear that

$\phi=\lim_{a\rightarrow-\infty}\phi_{\geq a}=\psi_{|\Sigma}$

as well. But the relation between the operators $D$ and its restriction ${\bf
D}$ is

(23) ${\bf D}\phi=\frac{n}{2}H\phi-\gamma(N)D\psi-\nabla_{N}\psi.$

Then, [see Bä1,BFGK,Bur] the spinor $\phi$ has to satisfy on the whole
$\Sigma$ the equality

${\bf D}\phi=\frac{n}{2}H\phi.$

The converse is obvious.

######  Remark 1.

Suppose that, in Theorem 2 above, we choose the spinor $\phi$ on the boundary
$\Sigma$ as an eigenvalue corresponding to the first non negative eigenvalue
$\lambda_{1}({\bf D})$ of the intrinsic Dirac operator ${\bf D}$ of the
boundary. A direct application of Theorem 2 gives

$\int_{\Sigma}\left(\lambda_{1}({\bf D})-\frac{n}{2}H\right)|\phi|^{2}\geq 0.$

As a consequence, we obtain the following lower bound which we firstly found
in [HM]:

$\lambda_{1}({\bf D})\geq\frac{n}{2}\max_{\Sigma}H$

and that improves, in the immersed case, the well-known lower bound by
Fiedrich [F] for boundary manifolds in ${\mathbb{R}}^{n+1}$.

######  Remark 2.

It is also worthy to note that, as a consequence of this estimate, if $\Sigma$
is a compact boundary in an Euclidean space and we know that $\lambda_{1}({\bf
D})\leq n/2$ and $H\geq 1$, then have the equality and a fortiori and the mean
curvature $H$ of $\Sigma$ must be constant. Hence, from the Alexandrov
Theorem, $\Omega$ is a round disc. This is a Euclidean analogue to the Min-Oo
conjecture and was remarked to be true by Miao in the paper [Mi2], by
supposing that $\Sigma$ was isometric to a unit $n$-sphere.

######  Theorem 3 (Brown-York mass for mean-convex surfaces).

Let $\Omega$ be a compact spin Riemannian manifold of dimension $n+1$ with
non-negative scalar curvature $R\geq 0$ and having a non-empty boundary
$\Sigma$ whose inner mean curvature $H\geq 0$ is strictly (strictly mean-
convex). Suppose that there is an isometric and isospin immersion from
$\Sigma$ into another spin manifold $\Omega_{0}$ carrying on a non-trivial
parallel spinor field and let $H_{0}$ its mean curvature with respect to any
of its orientations. Then, we have

$\int_{\Sigma}H\leq\int_{\Sigma}|H_{0}|.$

The equality implies that $H=|H_{0}|=H_{0}$. Then, if $n=2$, $\Omega_{0}$ is a
domain in ${\mathbb{R}}^{3}$ and the two embeddings differ by a direct rigid
motion

######  Remark 3.

Note that, in the original Shi-Tam original result, the authors assume that
the boundary $\Sigma$ is strictly convex. Then a well-known by Pogorelov [Po]
and Nirenberg [Ni] guarantees the existence of an isometric embedding into the
Euclidean space. Here, we need suppose the existence of this second isometric
immersion. Moreover we have $H_{0}>0$ a fortiori.

Proof. Denote by $\psi$ the parallel spinor on $\Omega_{0}$ and let
$\phi=\psi_{|\Sigma}$ its restriction onto $\Sigma$ through the existent
immersion. Let’s recall that the parallelism of $\psi$ (see (22)) gives

${\bf D}\phi=\frac{n}{2}H_{0}\phi\qquad\hbox{and}\qquad|\phi|=1.$

Now, we apply Theorem 2 and have the desired inequality

$\int_{\Sigma}H\leq\int_{\Sigma}|H_{0}|\,d\Sigma.$

If the equality is attained, then

$\frac{n}{2}H_{0}=D\phi=\frac{n}{2}H$

and so $H=H_{0}>0$. Then, the immersion of $\Sigma$ into the second ambient
space $\Omega_{0}$ is strictly mean-convex as well (with respect to the choice
of inner normal to $\Omega$).

When $n=2$, from this equality and the fact that $K=K_{\phi}$ (because the two
embeddings are isometric and preserve the Gauss curvatures), we deduce that
the two second fundamental forms coincide. The Fundamental Theorem of the
Local Theory of Surfaces allows us to conclude that the two boundaries differ
by a direct rigid motion of the Euclidean space.

######  Remark 4.

It is obvious that the result above is a generalization of the positivity
theorem for the Brown-York mass previously proved for strictly convex surfaces
by Shi and Tam. In their proof, the solution of difficult boundary equations
and the positivity of the ADM-mass obtained by Shoen-Yau and Witten [SY, Wi]
in the context of asymptotically flat manifolds are essential components.
Here, these difficulties are avoided and, as we have already remarked
somewhere above, this compact version of the theorem implies the
asymptotically flat version for the ADM mass (see [HMRa]).

######  Corollary 4.

Let $\Omega$ be a compact spinor manifold of dimension $3$ with non-negative
scalar curvature $R\geq 0$ and having a mean-convex boundary $\Sigma$
isometric to a sphere of any radius. Then, we have

$\int_{\Sigma}H\leq\sqrt{\pi A(\Sigma)}$

where $A(\Sigma)$ is the area of $\Sigma$. It the equality holds, then the two
boundaries are spheres of the same radius.

Proof. It is clear that the boundary ${\mathbb{S}}^{2}$ of $\Omega$ admits and
isometric and isospin (the sphere supports a unique spin stricture) embedding
into the Euclidean space ${\mathbb{R}}^{3}$ with $|H_{0}|=1/r$ and area
$A(\Sigma)=\pi r^{2}$, where $r>0$ is the radius of the sphere. The fact that
the two embeddings of ${\mathbb{S}}^{2}$ are isometric allows us to finish.

######  Corollary 5 (Cohn-Vossen rigidity theorem for mean-convex domains).

Two isometric and isospin strictly mean-convex compact surfaces in the
Euclidean space ${\mathbb{R}}^{3}$ must be congruent.

Proof. Let $\Omega$ and $\Omega_{0}$ be the two domains determined in
${\mathbb{R}}^{3}$ by two corresponding surfaces identified by means of an
isometry. Then, we can apply Theorem 3 interchanging the roles of $\Omega$ and
$\Omega_{0}$ and applying the case of the equality.

######  Remark 5.

The integral inequality in Corollary 4, for strictly convex surfaces of
${\mathbb{R}}^{3}$, is attributed to Minkowski (1901), although its very
probable that it were previously known to Alexandrov and Fenchel. Recently
have proved that the Minkowski inequality is not valid for any compact
surface, although they proved it is for the axisymmetric ones. Its is also
worthy to remark the following conjecture by Gromov: If $\Sigma$ is the
boundary of a compact Riemannian manifold $\Omega$, then, if $R\geq\sigma$,
for a certain constant $\sigma$, where $R$ is the scalar function of $\Omega$,
then there exists a constant $\Lambda(\Sigma,\sigma)$ such that

$\int_{\Sigma}H\,d\Sigma\leq\Lambda(\Sigma,\sigma).$

######  Remark 6.

Much more recently, in the context [SWWZ] of fill in problems posed firstly by
Bartnik, proved that, if $\Omega$ is the hemisphere $B^{n+1}$ and $\gamma$ is
a metric on the boundary ${\mathbb{S}}^{n}$ isotopic to the standard one with
mean curvature $H>0$, then there is a constant $h_{0}=h_{0}(\gamma)$ such that

$\int_{\Sigma}H\leq h_{0}.$

It is clear that this result and our Corollary 4 belong to a same family.

##  6\. Ambients with positive scalar curvature

Until now we have suppose that the scalar curvature of our compact spin
Riemannian manifold $\Omega$ satisfied $R\geq 0$ (Euclidean context). Let’s
enhance this positivity assumption to $R\geq n(n+1)$ (spherical context). This
lower bound is just the constant value of the scalar curvature of the
$(n+1)$-dimensional unit sphere. Then, by putting this assumption and the
Schwarz inequality

(24) $|D\psi|^{2}\leq(n+1)|\nabla\psi|$

(already used in Section 3) into the right side of the Weitzenbök-Lichnerowicz
formula, we obtain

(25) $\int_{\Sigma}\left(\left<{\bf
D}\psi,\psi\right>-\frac{n}{2}H|\psi|^{2}\right)\geq\int_{\Omega}\left(-\frac{1}{n+1}|{D}\phi|^{2}+\frac{n+1}{4}|\psi|^{2}\right),$

for all compact spin manifold $\Omega$, with equality only for the twistor
spinor fields on $\Omega$. From now on, by using this integral inequality, we
will work in a similar, but a few more elaborated, way as in Theorem 2, and
will get the following result.

######  Theorem 6.

Let $\Omega$ be a $(n+1)$-dimensional compact spin Riemannian manifold whose
scalar curvature is constant $R=n(n+1)$ and having a non-empty boundary
$\Sigma$ whose inner mean curvature $H\geq 0$ is non negative (mean-convex).
Suppose also that $\Sigma$ does not admit harmonic spinors. Then, for every
spinor $\phi\in H^{1}(\Sigma)$, we have

$\int_{\Sigma}|{\bf
D}\phi||\phi|\geq\frac{n}{2}\int_{\Sigma}|\phi|^{2}\sqrt{H^{2}+1}.$

The equality holds if and only if $\phi$ is a spinor field coming from a real
Killing spinor $\psi$ defined on $\Omega$, and so

${\bf D}\phi=\frac{n}{2}H\phi-\frac{n}{2}\gamma(N)\phi.$

######  Remark 7.

Note that our hypotheses impose the equality $R=n(n+1)$ and not the maybe
expected inequality $R\geq n(n+1)$, conjectured by Min-Oo and based on the
flat case. Obviously the condition $R\geq n(n+1)$ is necessary, because one
can otherwise perturb the hemisphere at an interior point so that $R\geq
n(n+1)-\varepsilon$, for some small $\varepsilon>0$, without changing the
assumptions on the boundary [HW1]. Moreover the other possibility $R>n(n+1)$
had to be invalidated by the counterexamples built by Brendle, Marques and
Neves [BMN], since all of them require at least one point with this strict
inequality in the bulk manifold. Hence, we are brought to suppose the
equality.

Proof. Given $a\in{\mathbb{Z}}$, $a\leq 0$, consider now the self-adjoint
eigenvalue problem for the Dirac operator $D$ on the bulk manifold $\Omega$

(26) $\left\\{\begin{array}[]{ll}D\psi_{a}=\mu_{a}\,\psi_{a}\\\ \psi_{\geq
a}=0,\end{array}\right.$

corresponding to the Dirac operator on $\Omega$ subjected to the usual APS
elliptic boundary condition as in the section above. Since the boundary does
not carry harmonic spinors, then the corresponding $D$ is symmetric in
$\Omega$ and its eigenvalues are real numbers. If $\psi_{a}$ is a solution to
(25), from Schwarz inequality (23), we get from the spinorial Reilly
inequality

$0\geq\int_{\Sigma}\left(\left<{\bf
D}\psi_{a},\psi_{a}\right>-\frac{n}{2}H|\psi_{a}|^{2}\right)\geq\int_{\Omega}\left(-\frac{\mu^{2}_{a}}{n+1}+\frac{n+1}{4}\right)|\psi_{a}|^{2},$

Since we have the $L^{2}$-orthogonal decomposition $\psi_{|\Sigma}=\psi_{\geq
a}+\psi_{<a}=0$ on $\Sigma$, $\psi_{\geq a}=0$, from (20) and $H\geq 0$ in the
hypotheses, the integral on the right side is non positive. Then

$\mu_{a}^{2}\geq\frac{(n+1)^{2}}{4},$

with equality if and only $\psi_{a}$ is a twistor spinor and, in this case, a
real Killing spinor on $\Omega$ with

${\nabla_{X}}\psi_{a}=-{\mu_{a}}\gamma(X)\psi_{a},\qquad\forall X\in T\Omega.$

The equality would imply as well

$\int_{\Sigma}\left(\left<{\bf
D}\psi_{<a},\psi_{<a}\right>-\frac{n}{2}H|\psi|^{2}\right)=0,$

and then ${\psi_{a}}_{|\Sigma}=0$. But a real Killing spinor has constant
length and so $\psi_{a}$ would identically null on the whole of $\Omega$. As a
consequence of this, the first eigenvalue $\mu_{1}(a)$ of (25) satisfies

$\mu_{1}(a)>\frac{n+1}{2},$

for all $a$ integer non positive. Henceforth, there exists a unique solution
with $\psi_{a}\in H^{1}(\Omega)\cap H^{\frac{1}{2}}(\Sigma)$ for the problem

(27) $\left\\{\begin{array}[]{ll}D\psi_{a}=\frac{n+1}{2}\,\psi_{a}\\\
\psi_{\geq a}=\phi_{\geq a},\end{array}\right.$

for each $a\in{\mathbb{Z}}$, $a\leq 0$ and for all $\phi\in L^{2}(\Sigma)$.
Take a such solution to (26) and put it in (24). Then

(28) $0\leq\int_{\Sigma}\left(\left<{\bf
D}\psi_{a},\psi_{a}\right>-\frac{n}{2}H|\psi_{a}|^{2}\right),\qquad\forall
a\in{\mathbb{Z}},a\leq 0,$

which is exactly the same expression that we obtained in the flat ambient
case, but here the equality would attain only in the case where $\psi_{a}$ is
a real Killing spinor (and not a parallel one)

$\nabla_{X}\psi_{a}=-\frac{1}{2}\gamma(X)\psi_{a},\qquad\forall X\in T\Omega.$

From this point, by working in the very exact way as in the proof of Theorem
2, we arrive at

(29) $0\leq\lim_{a\rightarrow-\infty}\int_{\Sigma}\left(|{\bf D}\phi_{\geq
a}||\phi_{\geq a}|-\frac{n}{2}H|\phi_{\geq a}|^{2}\right).$

But, with this choice of $\psi_{a}$ as a solution to (26), the left side in
the integral Reilly inequality after (25) vanishes. Hence, $\psi_{a}$ is a
twistor spinor and also

$D\psi_{a}=\frac{n+1}{2}\psi_{a}$

on the whole $\Omega$. That is, $\psi_{a}$ is a real Killing spinor. More
precisely, we arrive just to the expression which had already rejected some
times to avoid several reductionis ad absurdum, more precisely,

(30) $\nabla_{X}\psi_{a}=-\frac{1}{2}\gamma(X)\psi_{a},\qquad\forall X\in
T\Omega.$

By using (7) and the equality above, we have

(31) ${\bf D}\psi_{a}=\frac{n}{2}H\psi_{a}-\frac{n}{2}\gamma(N)\psi_{a},$

along the boundary $\Sigma$. Taking into account the orthogonal $L^{2}$
decomposition $\psi_{a}=\psi_{\geq a}+\psi_{<a}$, the fact that ${\bf D}$
preserves this decomposition and that $\gamma(N)$ reverses it pointwise, if we
multiply scalarly by the spinor $\gamma(N)\psi_{a}$ and integrate on $\Sigma$,
we get

$\int_{\Sigma}\left<{\bf
D}\psi_{a},\gamma(N)\psi_{a}\right>=-\frac{n}{2}\int_{\Sigma}|\psi_{a}|^{2}.$

By working in an analogous, but not exactly, equal way as in (20), we obtain

(32) $\lim_{a\rightarrow-\infty}\int_{\Sigma}|{\bf D}\phi_{\geq a}||\phi_{\geq
a}|\geq\lim_{a\rightarrow-\infty}\frac{n}{2}\int_{\Sigma}|\phi_{\geq a}|^{2},$

for all $a\in{\mathbb{Z}}$, $a\leq 0$ and the convergence is in
$H^{\frac{1}{2}}(\Sigma)$ and, so, strongly in $H^{2}(\Sigma)$.

In order to obtain the inequality in the Theorem, note that (28) and (31)
provide us integral sequences which converge strongly in $L^{2}(\Sigma)$.
Then, we can extract from each one of them a subsequence converging a.e. (and,
so, since $\phi$ is continuous converging on the whole) to the corresponding
function limit. These two subsequences will be (probably different), but have
been extracted from a same converging sequence, so, we will label them with
the same indices as the original ones, and have

$\lim_{a\rightarrow-\infty}|D\phi_{\geq a}||\phi_{\geq
a}|\geq\frac{n}{2}H|\phi|^{2}\quad\mbox{and}\quad\lim_{a\rightarrow-\infty}|D\phi_{\geq
a}||\phi_{\geq a}|\geq\frac{n}{2}|\phi|^{2}.$

As a consequence, we obtain

$\lim_{a\rightarrow-\infty}|D\phi_{\geq a}||\phi_{\geq
a}|\geq\frac{n}{2}\sqrt{1+H^{2}}|\phi_{\geq a}|^{2},$

which was the inequality we are looking for.

For the case of the equality, as in the proof of Theorem 2, we can use the
standard and well written results of regularity results in [BaBä1,BaBä2] to
show that the solution $\psi_{a}$ to the boundary problem (26) is smooth on
${\bar{\Omega}}$ for all integer $a\leq 0$. As before, with respect to the
boundary $\Sigma$, we can apply Theorem 7.17 and Corollary 7.18 in [BaBä]
repeatedly in a bootstraping process and obtain smoothness on $\Sigma$ as
well. Then
$\psi_{a}\in\cap_{k=1}^{\infty}H^{k}({\bar{\Omega}})=C^{\infty}({\bar{\Omega}})$.
For all non positive integer number $a$, the limit function $\psi$ is also
smooth on the compact manifold ${\bar{\Omega}}$. This implies that the limit
function

$\psi=\lim_{a\rightarrow-\infty}\psi_{a}$

satisfies (15) on $\Omega$ as well. That is, $D\psi=(n+1)/\psi$. Taking limit
in (29) and substituting in the integral Reilly inequality we have, from the
left side, that $\psi$ is a parallel spinor on the whole of ${\bar{\Omega}}$.
Also it is clear that

$\phi=\lim_{a\rightarrow-\infty}\phi_{\geq a}=\psi_{|\Sigma}.$

But the relation between the operators $D$ and its restriction ${\mathcal{D}}$
is

(33) ${\bf D}\phi=\frac{n}{2}H\phi-\gamma(N)D\psi-\nabla_{N}\psi$

and the spinor $\phi$ comes from a real Killing spinor on $\Omega$ with
constant $-1/2$. Hence, it has to satisfy on the whole $\Sigma$ the equality

${\bf D\phi}=\frac{n}{2}H\phi-\frac{n}{2}\phi.$

The converse is obvious.

As far as we know, we obtain from Theorem 6 a lower estimate for the first
eigenvalue of the Dirac operator in a hypersurface in a context on scalar
curvature positive.

######  Corollary 7.

Consider a compact spin Riemannian manifold $\Omega$ of dimension $n+1$ and
scalar curvature $R=n(n+1)$ and mean-convex boundary. Suppose that $\Sigma$
does not support harmonic spinors and let $\phi$ be the eigenspinor
corresponding to the first eigenvalue $\lambda_{1}({\bf D})$ positive of the
intrinsic Dirac operator ${\bf D}$ of the boundary. A direct application of
Theorem 6 gives

$\int_{\Sigma}\left(\lambda_{1}({\bf
D})-\frac{n}{2}\sqrt{1+H^{2}}\right)|\phi|^{2}\geq 0.$

As a consequence, we obtain the following lower bound:

$\lambda_{1}({\bf D})\geq\frac{n}{2}\max_{\Sigma}\sqrt{1+H^{2}}$

that improves the well-known lower bound by Fiedrich [F] for non immersed
manifolds boundary manifolds (the Gauß for the curvature in the unit
$(n+1)$-dimensional sphere gives

$\sqrt{\frac{nR}{4(n-1}}\leq\frac{n}{2}\sqrt{1+H^{2}}$

and the equality is attained only by the umbilical hypersurfaces).

######  Remark 8.

It is also worthy to note that, as a consequence of this estimate, if $\Sigma$
is a compact boundary in an Euclidean space and we know that $\lambda_{1}({\bf
D})\leq n/2$ and $H\geq 0$, then we have the equality and a fortiori the mean
curvature $H$ of $\Sigma$ must be identically null. Moreover, $\Omega$
supports the existence of a non trivial real Killing spinor (see [Bä]). Hence,
$\Omega$ is close to being a spherical domain bounded by a minimal
hypersurface. This would be the most similar to the solution to the spherical
analogue to the Min-Oo conjecture.

As in the flat case, we can obtain a kind of positivity for a possible quasi-
local mass in this new context.

######  Theorem 8 (Brown-York in mean-convex and spherical case).

Let $\Omega$ be a spin Riemannian manifold of dimension $n+1$ with scalar
curvature $R=n(n+1)$ and having a non-empty boundary $\Sigma$ whose inner mean
curvature $H\geq 0$ is also non-negative (mean-convex). Suppose that there is
an isometric and isospin immersion from $\Sigma$ into another spin manifold
$\Omega_{0}$ carrying on a non-trivial real Killing spinor field and let
$H_{0}$ its mean curvature with respect to any of its orientations. Then, we
have

$\int_{\Sigma}\sqrt{1+H^{2}}\leq\int_{\Sigma}\sqrt{1+H_{0}^{2}},$

provided that the boundaries do not admit harmonic spinors. The equality
implies that $H=H_{0}$. Then, if $n=2$, $\Omega_{0}$ is a domain in
${\mathbb{S}}^{3}$ and the two embeddings differ by a direct rigid motion.

Proof. Denote by $\psi$ the spinor on $\Omega_{0}$ and let
$\phi=\psi_{|\Sigma}$ its restriction onto $\Sigma$ through the existent
immersion. Let’s recall that the parallelism of $\psi$ gives

${\bf D}\phi=\frac{n}{2}\sqrt{1+H_{0}}\,\phi\qquad\hbox{and}\qquad|\phi|=1.$

Now, we apply Theorem 2 and have the desired inequality

$\int_{\Sigma}\sqrt{1+H^{2}}\leq\int_{\Sigma}\sqrt{1+H_{0}}.$

If the equality is attained, then

$\frac{n}{2}\sqrt{1+H_{0}^{2}}={\bf D}\phi=\frac{n}{2}\sqrt{1+H^{2}}$

and so $H=H_{0}\geq 0$. Then, the immersion of $\Sigma$ into the second
ambient space $\Omega_{0}$ is mean-convex as well (with respect to the choice
of inner normal to $\Omega$).

When $n=2$, from this equality and the fact that $K=K_{\phi}$ (because the two
embeddings are isometric and preserve the Gauss curvatures), we deduce that
the two second fundamental forms coincide. The Fundamental Theorem of the
Local Theory of Surfaces allows us to conclude that the two boundaries differ
by a direct rigid motion of the Euclidean space.

######  Corollary 9 (Cohn-Vossen rigidity theorem in the sphere).

Two isometric and isospin mean-convex compact surfaces embedded in a sphere
${\mathbb{S}}^{3}$ must be congruent, provided they do not admit harmonic
spinors.

Proof. Let $\Omega$ and $\Omega_{0}$ be the two domains determined in
${\mathbb{S}}^{3}$ by two corresponding surfaces identified by means of an
isometry. Then, we can apply Theorem 8 interchanging the roles of $\Omega$ and
$\Omega_{0}$ and applying the case of the equality.

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