#  LinguAlchemy: Fusing Typological and Geographical Elements
for Unseen Language Generalization

Muhammad Farid Adilazuarda1, Samuel Cahyawijaya3, Alham Fikri Aji1,
Genta Indra Winata4, Ayu Purwarianti2
1MBZUAI 2Institut Teknologi Bandung
3The Hong Kong University of Science and Technology 4Bloomberg
<EMAIL_ADDRESS>
###### Abstract

Pretrained language models (PLMs) have become remarkably adept at task and
language generalization. Nonetheless, they often fail dramatically when faced
with unseen languages, posing a significant problem for diversity and equal
access to PLM technology. In this work, we present LinguAlchemy, a
regularization technique that incorporates various aspects of languages
covering typological, geographical, and phylogenetic constraining the
resulting representation of PLMs to better characterize the corresponding
linguistics constraints. LinguAlchemy significantly improves the accuracy
performance of mBERT and XLM-R on unseen languages by $\sim$18% and $\sim$2%,
respectively compared to fully fine-tuned models and displaying a high degree
of unseen language generalization. We further introduce AlchemyScale and
AlchemyTune, extension of LinguAlchemy which adjusts the linguistic
regularization weights automatically, alleviating the need for hyperparameter
search. LinguAlchemy enables better cross-lingual generalization to unseen
languages which is vital for better inclusivity and accessibility of PLMs.

##  1 Introduction

Significant advancements in language processing technology have been achieved
through the development of PLMs, leading to a commendable proficiency in
language comprehension and generation Devlin et al. (2019); Liu et al. (2019);
Conneau et al. (2020); Sanh et al. (2022); Lewis et al. (2019); Raffel et al.
(2023); Li et al. (2021); Cahyawijaya et al. (2021); Wilie et al. (2020).
However, there remains a notable deficiency in the ability of these models to
generalize effectively to unseen languages, resulting in a considerable
performance reduction of PLMs across thousands of unseen languages. To
mitigate this problem, efforts to develop efficient language adaptation
approaches are underway, focusing on the incorporation of these unseen
languages to PLMs Pfeiffer et al. (2021b); Alabi et al. (2022); Ebrahimi et
al. (2022); Goyal et al. (2021).

Figure 1: LinguAlchemy enhances performance in unseen languages by allowing
the model to predict the linguistic vector and then fitting it via a
similarity loss towards the specific language’s URIEL vector.

Incorporating new unseen languages has been a longstanding problem in natural
language processing (NLP) research, especially given that most of these unseen
languages are low-resource and underrepresented making PLMs difficult to adapt
to these languages. MAD-X Pfeiffer et al. (2020b) employs a language adapter
to learn new unseen languages by incorporating language adapters that mitigate
the risk of forgetting pre-trained knowledge which is known as the curse-of-
mulitlinguality. Nonetheless, this approach requires training for generalizing
to new unseen languages which makes it costly and difficult to scale to
thousands of languages. MAD-G Ansell et al. (2021) and Udapter Üstün et al.
(2020) further generalize this approach by utilizing a linguistic-driven
contextual parameter generator (CPG) module to generate language-specific
parameters allowing the models to generalize to other languages with similar
linguistic characteristics. Recently, Rathore et al. (2023) introduced ZGUL,
which combines representations over multiple language adapters to generate the
unseen language representation. Despite the effectiveness, all these
approaches largely rely on two assumptions, i.e., 1) strict categorization of
languages and 2) knowing the language category of the query apriori. The first
assumption disregards the fact that linguistic phenomena such as code-mixing
may occur in the query. While the second assumption might cause performance
degradation due to the error propagation from the language identification
module Adilazuarda et al. (2023).

To overcome those limitations, in this work, we introduce LinguAlchemy. Unlike
adapter-based approaches which isolate the capability to understand different
languages over multiple language-specific adapters, LinguAlchemy learns a
shared representation across multiple languages, allowing the model to
leverage shared knowledge between languages. Such behavior is attainable by
eliminating the language-specific module from the model and instead utilizing
a regularization to learn language-specific knowledge. LinguAlchemy allows the
model to perform inference without knowing the language of the query prior.
Our evaluation suggests that LinguAlchemy can improve mBERT and XLM-R
generalization in unseen languages while maintaining the performance on the
high-resource languages without requiring information on the language of the
query apriori.

In summary, our contributions are as follows:

  1. 1.

We propose LinguAlchemy, a regularization method that improves unseen language
performance on language models and aligns them to arbitrary languages.

  2. 2.

We demonstrate strong performance on unseen languages for models trained with
LinguAlchemy.

  3. 3.

We introduced a dynamic scaling method to scale the classification and
auxiliary loss factors used in the fine-tuning stage.

##  2 Related Works

PLMs with their transformer-based architectures have been demonstrating
exceptional capabilities in language comprehension and generation. These
models excel in abstract linguistic generalization by capturing complex
linguistic patterns and understanding structural positions and thematic roles,
which are crucial for interpreting language semantics. Research in this area
Ganesh et al. (2021) has provided critical insights that enable these models
to process and generate human language effectively. The studies have explored
how these models grasp intricate linguistic features, including syntax and
semantics, thereby enhancing their performance across a wide range of language
tasks Rathore et al. (2023).

In parallel, the development of resources like URIEL vector and lang2vec
utility Littell et al. (2017) has been instrumental in extending the reach of
multilingual NLP, particularly for less-resourced languages. These tools
provide vector representations of languages, leveraging typological,
geographical, and phylogenetic data, thus offering a structured approach to
understanding linguistic diversity. Complementing this, recent research has
conducted a comprehensive survey on the utilization of typological information
in NLP, highlighting its potential in guiding the development of multilingual
NLP technologies Ponti et al. (2019). This survey emphasized the
underutilization of typological features in existing databases and the need
for integrating data-driven induction of typological knowledge into machine
learning algorithms.

Recent advancements in prefix tuning Li and Liang (2021) and subspace learning
Zhang et al. (2020) have contributed significantly to improving generalization
in PLMs. These methods focus on learning prefix subspaces to stabilize the
direct learning of embeddings, addressing the instability issues present in
earlier approaches. Two notable methods are the MAD-X Pfeiffer et al. (2020b)
and MAD-G Ansell et al. (2021) frameworks, which employ adapter-based
techniques for multi-task cross-lingual transfer, highlighting modularity and
parameter efficiency. However, it inherits the limitations of the pretrained
multilingual models, such as the limited capacity to adapt effectively to low-
resource and unseen languages. Furthermore, while the framework facilitates
adaptation to specific target languages, it may bias the model towards these
languages, potentially impacting its performance on other languages.

However, despite these advancements, PLMs still face significant challenges in
generalizing to unseen languages, particularly when adapting to low-resource
and unseen languages. These challenges stem from the vast structural and
semantic variation across languages Bender (2011); Jurafsky and Martin (2019),
the scarcity of resources Mohammad (2019); Lewis et al. (2020), and the
limitations inherent in the models themselves Lin et al. (2017). This
situation highlights the complexity of scaling and generalizing these models
effectively and underscores the need for more sophisticated approaches in
model training and adaptation to ensure broader and more equitable language
coverage.

##  3 Unseen Languages Adaptation with LinguAlchemy

In this section, we provide an overview of how LinguAlchemy can capture
linguistic constraint and how is the intuition behind LinguAlchemy. We also
discuss in detail how do we align model representations with the linguistic
vector.

###  3.1 Does Multilingual LMs capture Linguistic Constraint?

In this work, we define the linguistic knowledge as a vector gathered from
URIEL vector (Littell et al., 2017). We chose three distinct linguistic
knowledge from the database, namely ’syntax_knn’, ’syntax_average’111In this
work, we chose the ’knn’ and ’average syntax features. These include consensus
values (like averages) and predicted values (such as kNN regressions based on
phylogenetic or geographical neighbors), and ’geo’ features. The choice of
’syntax_knn’ and ’syntax_average’ is motivated by the typological nature of
syntax. Syntax in languages varies widely; hence, by using aggregate measures
like averages and k-nearest neighbors (kNN), we can capture a more general
representation of syntactic features across languages. These features include
consensus values, like averages, and predicted values, such as kNN regressions
based on phylogenetic or geographical neighbors.

We excluded phonological features, as our focus is not on speech but on
textual data. Phonology primarily pertains to spoken languages and would not
add significant value to our analysis of written language structures.
Additionally, we excluded language family features (’family’). These features
are typically binary, indicating whether a language belongs to a particular
subfamily. While they can be useful for language identification tasks, they
tend to be sparse and may not provide the granularity needed for our study. By
focusing on syntax and geographical features, our approach aims to encompass a
broader and more nuanced understanding of linguistic variations in a
multilingual context.

#### Syntax Feature

These feature vectors denote a typological feature that is adapted from
several sources including World Atlas of Language Structures (WALS), Syntactic
Structures of World Languages Collins (2010), and short prose descriptions on
typological features in Ethnologue Lewis (2009). Syntax vectors captures
information about the syntactic properties of languages which derived from
large-scale typological databases, which document the structural and semantic
variation across different languages. These syntax features in the URIEL
vector are utilized to represent languages in vector form, allowing for the
analysis and comparison of languages based on their syntactic properties.

#### Geographical Feature

On the other hand, geographical features represent languages in terms of their
geographical properties. The inclusion of ’geo’ features aims to capture
geographical attributes of languages. Geographic factors can significantly
influence language evolution and structure, making them crucial for
understanding linguistic variations. This feature expresses geographical
location with a fixed number of dimensions that each represents the “great
circle” distance—from the language in question to a fixed point on the Earth’s
surface. By incorporating geographical information into language vectors,
URIEL and lang2vec provide a more comprehensive view of languages, considering
not only their structural and semantic properties but also their geographical
context.

###  3.2 Proof of Concept

#### Linguistic Separability in LMs

To explore whether multilingual language models (MLMs), such as Multilingual
BERT (mBERT), capture the linguistic constraints as defined by the URIEL
vectors, we align the mBERT language embeddings to the linguistic knowledge
vectors. This projection is quantitatively measured to assess the
representation of both seen and unseen languages within the model. The
fundamental question we address is the extent to which mBERT’s embeddings
correspond to the typological and geographical features encapsulated in the
URIEL vectors.

Figure 2: Alignment between mBERT Representation with URIEL Language
Representation. The green-shaded areas indicate the sentence representations
of mBERT while the brown dots represent the URIEL representations of the
corresponding language.

Figure 2 presents a visual analysis facilitated by UMAP McInnes et al. (2018),
showing the correlation between mBERT language representation and the
linguistic vectors from the URIEL database ($R^{2}$ = 0.816). By leveraging
UMAP, the plot accentuates the principal variances within the joint feature
space of the embeddings and vectors. The spatial representation of languages
on this plot mirrors their linguistic and geographical relatedness, as
encapsulated by mBERT. This visualization underscores the model’s ability to
mirror linguistic typologies, with languages sharing common roots such as ’de-
DE’ and ’nl-NL’ naturally clustering together. The density and arrangement of
these clusters potentially reflect mBERT capacity to capture and represent
language family traits. Conversely, the presence of sparser clusters or
outliers prompts a closer examination of mBERT’s coverage and consistency in
representing diverse linguistic features. We also formally defined the
language representation alignment in the Appendix B.

###  3.3  LinguAlchemy

We introduce LinguAlchemy as an approach that intuitively aligns model
representations with linguistic knowledge, leveraging URIEL vectors. This is
operationalized through an auxiliary loss function, involving the training
process with a nuanced understanding of linguistic characteristics.

In LinguAlchemy, we enhance the fine-tuning of encoder models such as mBERT
for downstream tasks by not only using the regular classification loss but
also introducing a novel linguistic regularization term. This is achieved
through the implementation of a URIEL loss, designed to align the model’s
representations with linguistic knowledge derived from URIEL vectors.
Specifically, this process involves applying a linear projection to the
model’s pooled output, which aligns it with the URIEL vector space. The URIEL
loss is quantified as the mean squared error (MSE) between the projected model
outputs and the corresponding URIEL vectors. This dual approach, combining
classification loss and URIEL loss, allows for a more linguistically informed
model training, enhancing the model’s ability to capture and reflect complex
linguistic patterns and relationships.

$L_{uriel}(R,U)=\frac{1}{N}\sum_{i=1}^{N}\|R_{i}-U_{i}\|^{2}$

where $R$ represents the model-generated representations, $U$ denotes the
URIEL vectors, and $N$ is the number of data points. To generate the model
representation, we take the output representation from the $CLS$ token and
multiply it with a new, trainable projection layer to transform the vector
size so that they are compatible.

Note that there may be discrepancies between the scales of the standard
classification loss and the URIEL loss. To address this, we introduce an
optional hyperparameter, denoted as $\lambda$, to scale the URIEL loss
appropriately.

#### Dynamic Scaling Approaches

In addition to the fixed scaling factor, we also explore dynamic adjustment of
this scaling factor at each training step. This aims to maintain a balance
between the classification and URIEL losses, and even considers making the
scale trainable. The final loss formula when training with LinguAlchemy is
given by:

$L=\lambda_{cls}*L_{cls}+\lambda_{uriel}*L_{uriel}(R,U)$

We define two methods to implement dynamic scaling:

  1. 1.

AlchemyScale: This method dynamically adjusts the scaling factor $\lambda$
during training. It initiates with scaling factors set relative to the mean of
initial losses, ensuring proportional importance to each loss component.
Subsequently, these factors are updated periodically using an Exponential
Moving Average (EMA) method, with a conservative adjustment for specific loss
components to maintain stability. This approach ensures an optimal balance
between different loss components, adjusting the contribution of each to the
total loss based on predefined criteria.

  2. 2.

AlchemyTune: Here, $\lambda$ is conceptualized as a trainable parameter within
the model’s architecture. Initialized as part of the model’s parameters,
$\lambda$ undergoes optimization during the training process. This method
applies the scaling factors to loss components, and an additional
mini_loss—representing the deviation of the sum of scaling factors from
unity—is computed. This mini_loss is used for backpropagation, enabling the
scaling factors to adapt based on the training dynamics and dataset specifics.

Both methods aim to enhance model performance by dynamically and intelligently
scaling loss components, with the first method relying on predefined,
periodically updated scaling mechanisms, and the second integrating the
scaling factor into the model’s learning parameters for adaptive adjustments.

##  4 Experiment Setting

#### Datasets

In our experiments, we utilize the MASSIVE Xu et al. (2022), which is a
comprehensive collection of multilingual data incorporating intent
classification tasks. This dataset is particularly notable for its inclusion
of a diverse range of languages, with various language families, genera, and
scripts. Specifically, we split MASSIVE into 25 languages that are ’seen’
during finetuning and the rest 27 languages that are ’unseen’, which we
exclusively used for evaluation. This splitting is based on the language
adapters availability as outlined in the prior research of (Pfeiffer et al.,
2020a), which we utilized in the AdapterFusion experiment for our baseline
model. For a detailed breakdown of the languages used, including their
respective families, genera, and script can be found in Appendix A.

#### Models

Our research employs two state-of-the-art language models: Multilingual BERT
Base (mBERTBase) and XLM-RoBERTa Base (XLM-RBase). These models are chosen for
their robust performance across a wide range of multilingual NLP tasks, making
them ideal for our intent classification and language identification
objectives.

#### Hyperparameters

The following hyperparameters are used in our training process:

  * •

Learning rate: $5\times 10^{-5}$

  * •

Training epochs: 30

  * •

Performance metric: Accuracy

  * •

Early stopping criterion: 3 epochs without improvement

##  5 Results and Discussion

Table 1: Performance of LinguAlchemy in MASSIVE dataset for unseen languages.
These models have not seen these languages during fine-tuning. For languages
in *, mBERT model also never seen the languages during pre-training.

###  5.1 LinguAlchemy Performance

To evaluate the effectiveness of our proposed technique on unseen languages,
we trained mBERT and XLM-R on the MASSIVE dataset. Specifically, we train on
25 languages and test on 27 different, unseen languages. Our results are
summarized in Table 1. We compared our method with zero-shot generalization,
where the model is fully tuned on seen languages and then tested on unseen
languages (referred to as Full FT in the Table). Additionally, we explored
AdapterFusion Pfeiffer et al. (2021a) as another baseline. AdapterFusion has
shown better adaptation to unseen languages than naive zero-shot
generalization. Unfortunately, many language adapters that we need for
AdapterFusion is not available for XLM-R.

From Table 1, it is shown that LinguAlchemy achieves better generalization for
unseen languages. We observed a significant improvement for mBERT and a modest
average improvement for the stronger XLM-R model. For mBERT, LinguAlchemy can
significantly increase performance in truly unseen languages of am-ET, km-KH,
mn-MN, in which mBERT has never seen during the pre-training stage nor fine-
tuning. These findings show that LinguAlchemy can be useful in truly zero-shot
settings.

We note that LinguAlchemy tends to even out the performance across all unseen
languages, leading to a massive boost in weaker languages such as cy-GB or sw-
KE. However, in languages where zero-shot performance is already strong,
LinguAlchemy does not seem to provide benefits and in some cases degrade
performance. This is more evident in XLM-R, where LinguAlchemy flattens
performance to the 80-82% range, in which many of zero-shot performance is
already reached that level or more. Regardless, the potential of our method is
clear, showing it to be beneficial in cases where zero-shot performance is
poor.

###  5.2 Effect of Scaling URIEL loss

The classification and URIEL losses are not on the same scale. Therefore,
simply adding both losses together means that the model will give more weight
to the loss with the higher magnitude. When observing both the classification
and URIEL losses during the early stages of training, we note that the
classification loss is around 10 times larger than the URIEL loss. In this
part, we explore the effect of different scaling factors for the URIEL loss.

Figure 3: Average performance of unseen languages under various URIEL loss
scales.

#### Constant Scaling

We explore consistently scaling up the URIEL loss across various scaling
factors. The results can be seen in Figure 3. It is important to note that, as
we use the scale-invariant optimizer AdamW, we don’t have to worry about
gradients becoming too large due to extremely large losses. Generally, we
observe that a scaling factor of 10x slightly outperforms other scaling
factors, and the performance appears to decline with higher scale factors.

#### Dynamic and Trainable Scaling

One issue with introducing a scaling factor is the addition of another tunable
hyperparameter. Intuitively, we might aim for a balanced weight between the
classification and URIEL losses. Therefore, instead of expensively testing
different scaling factors, an adaptive scaling factor might be more cost-
effective and beneficial. Here, we explore two ideas: dynamic and trainable
factors. The results of these approaches can be seen in Table 2.

URIEL scaling | mBERT | XLM-R
---|---|---
Constant 10x | 64.68% | 80.32%
AlchemyScale | 62.97% | 80.43%
AlchemyTune | 63.24% | 79.10%
Table 2: Performance Comparison Across Different URIEL Scaling Methods.

Interestingly, these dynamic scale factors do not significantly outperform a
constant factor. In contrast, a 10x scaling achieves the best performance in
mBERT, while dynamic scaling barely outperforms the 10x scaling in XLM-R.
Therefore, in a limited budget scenario, a suggested 10x scaling factor should
suffice, and one may explore different scaling factors given more
computational resources.

Train Group | Lang. Family |  Languages | Num. Languages
---|---|---|---
1 | Indo-European |  af-ZA, bn-BD, ca-ES, cy-GB, da-DK, de-DE, el-GR, en-US, es-ES, fa-IR, fr-FR, hi-IN, hy-AM, is-IS, it-IT, lv-LV, nb-NO, nl-NL, pl-PL, pt-PT, ro-RO, ru-RU, sl-SL, sq-AL, sv-SE, ur-PK | 26
2 | Dravidian |  Train Group 1 + kn-IN, ml-IN, ta-IN, te-IN | 30
3 | Afro-Asiatic |  Train Group 2 + am-ET, ar-SA, he-IL | 33
4 | Sino-Tibetan |  Train Group 3 + my-MM, zh-CN, zh-TW | 36
Unseen Languages |  sw-KE, km-KH, vi-VN, id-ID, jv-ID, ms-MY, tl-PH, ja-JP, ka-GE, ko-KR, mn-MN, th-TH, az-AZ, tr-TR, fi-FI, hu-HU | 16
Table 3: Language family distribution used in the language family
generalization experiment (§5.3)

###  5.3 Generalization Across Language Family

We investigate LinguAlchemy across language families to further analyze the
generalization capabilities of BERT and XLM-R models. This experiment offers
insight into how adaptable LinguAlchemy is to a variety of linguistic
features. We perform our experiment by splitting the languages in MASSIVE
according to their language families and train the model on a subset of
language families while testing on the rest, unseen language families. We
explore on including different subset of language families, as seen in Table
3.

The "others unseen" category includes additional language families not
incorporated in the training set, serving as an "unseen" testbed. As
illustrated in Figure 4, LinguAlchemy demonstrates generalization towards
these unseen language families. Perhaps unsurprisingly, adding more languages
and, importantly, diversity to the training data improves generalization
performance. Interestingly, including the Afro-Asiatic language family
group—comprising "am-ET," "ar-SA," and "he-IL," all of which have distinct
scripts—resulted in a notable performance enhancement from train group 2 to
train group 3 experiment.

The performance of both models, combined with LinguAlchemy underscores the
advantage of including a broader spectrum of languages within training groups
for enhanced model generalization. However, the impact of this diversity is
not uniform across all language families: While some consistently benefit from
the expansion of training data, others do not, indicating that merely
increasing the volume of data from the same family may not necessarily improve
performance. This inconsistency indicates the potential limitations within the
models’ capacity to learn and generalize the linguistic features specific to
certain language families. Consequently, our observation shows that the degree
of generalization varies noticeably among different families, suggesting that
while some may significantly profit from these models’ capabilities, others
may require more tailored strategies to gain similar performance improvement.

Figure 4: Model performance across language families. Dotted lines indicates
language families used in training in some of the training stages (solid dots
for active use–refer to Table 3), and solid grey lines for families unseen in
all training stages, with variance shown in shading.

###  5.4 Seen Language Performance

One drawback of LinguAlchemy that we have noticed is the sacrifice in
performance for seen languages. This result is related to our findings in
unseen languages, where LinguAlchemy flattens out the performance across all
languages, thus being extremely beneficial to poorly performing ones.
Additionally, we note that the performance of seen languages is flattened to a
similar level, which, most of the time, is worse than that of standard in-
language fine-tuning. The compiled results can be seen in Table 4

Based on this finding, LinguAlchemy is suitable for enhancing the performance
of extremely low-resource languages where standard cross-lingual zero-shot
fine-tuning does not improve performance. In cases where training data is
available, normal fine-tuning is a better choice. We are exploring why
LinguAlchemy does not help with seen languages and how to boost the
performance of seen languages as future work. Nevertheless, our method is
still beneficial in under-resourced settings where multilingual models perform
poorly.

Performance | mBERT | XLM-R
---|---|---
| Full FT | Ours | Full FT | Ours
Unseen Lgs. | 45.48% | 64.68% | 78.97% | 80.43%
Seen Lgs. | 84.52% | 67.45% | 86.45% | 81.05%
Average | 64.25% | 66.01% | 82.56% | 80.62%
Table 4: Average performance of LinguAlchemy on unseen vs seen languages

##  6 Conclusion

We introduced LinguAlchemy, a novel approach that demonstrates strong
performance across 27 unseen languages in a 60-class intent classification
task. Our method hinges on the integration of linguistic knowledge through the
URIEL vectors, enhancing the language model’s ability to generalize across a
diverse set of languages. We also proposed AlchemyScale and AlchemyTune, which
employs a hyperparameter search for the URIEL scaling factor. This is achieved
by two key strategies: (1) weight-averaging classification and URIEL loss, and
(2) learning to balance the scale between classification and URIEL loss, thus
ensuring a more adaptable and robust model performance.

## Limitations

LinguAlchemy enhances performance across many unseen languages in intent
classification, yet it faces limitations. Performance on seen languages is
less than ideal, indicating room for improvement through methods like weight
freezing. Also, better generalization appears to reduce accuracy in seen
languages, pointing to a need for balanced approaches. Currently, the research
is limited to intent classification, and expanding to other NLP tasks could
reveal more about its versatility. Moreover, the choice of URIEL
features—syntax, geography, language family—is theoretically sound, as
discussed in Chapter 3, but empirical tests with different features might
refine the model further. Overcoming these limitations could greatly improve
the generalizability and effectiveness of multilingual NLP models.

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##  Appendix A Languages in Dataset

The MASSIVE Dataset, also known as the Multilingual Amazon SLU Resource
Package (SLUPR), offers a comprehensive collection of approximately one
million annotated utterances for various natural language understanding tasks
such as slot-filling, intent detection, and Virtual Assistant performance
evaluation. It is an extensive dataset that includes 51 languages, 60 intents,
55 slot types, and spans 18 different domains. The dataset is further enriched
with a substantial amount of English seed data, comprising 587k training
utterances, 104k development utterances, and 152k test utterances.

Code | Name | Script | Genus | Code | Name | Script | Genus
---|---|---|---|---|---|---|---
ar-SA | Arabic | Arab | Semitic | is-IS | Icelandic | Latn | Germanic
bn-BD | Bengali | Beng | Indic | ka-GE | Georgian | Geor | Kartvelian
el-GR | Greek | Grek | Greek | km-KH | Khmer | Khmr | Khmer
en-US | English | Latn | Germanic | lv-LV | Latvian | Latn | Baltic
es-ES | Spanish | Latn | Romance | ml-IN | Malayalam | Mlym | Southern Dravidian
fa-IR | Persian | Arab | Iranian | nb-NO | Norwegian | Latn | Germanic
fr-FR | French | Latn | Romance | ro-RO | Romanian | Latn | Romance
he-IL | Hebrew | Hebr | Semitic | sl-SI | Slovenian | Latn | Slavic
hu-HU | Hungarian | Latn | Ugric | ur-PK | Urdu | Arab | Indic
hy-AM | Armenian | Armn | Armenian | zh-CN | Mandarin | Hans | Chinese
id-ID | Indonesian | Latn | Malayo-Sumbawan | zh-TW | Mandarin | Hant | Chinese

Table 5: Statistics and description of the dataset used Xu et al. (2022). The
dataset used is a subset of the MASSIVE dataset, selecting 25 different seen
languages.

Code | Name | Script | Genus | Code | Name | Script | Genus
---|---|---|---|---|---|---|---
af-ZA | Afrikaans | Latn | Germanic | my-MM | Burmese | Mymr | Burmese-Lolo
am-ET | Amharic | Ethi | Semitic | nl-NL | Dutch | Latn | Germanic
az-AZ | Azerbaijani | Latn | Turkic | pl-PL | Polish | Latn | Slavic
cy-GB | Welsh | Latn | Celtic | pt-PT | Portuguese | Latn | Romance
da-DK | Danish | Latn | Germanic | ru-RU | Russian | Cyrl | Slavic
de-DE | German | Latn | Germanic | sq-AL | Albanian | Latn | Albanian
fi-FI | Finnish | Latn | Finnic | sv-SE | Swedish | Latn | Germanic
hi-IN | Hindi | Deva | Indic | sw-KE | Swahili | Latn | Bantoid
ja-JP | Japanese | Jpan | Japanese | ta-IN | Tamil | Taml | Southern Dravidian
kn-IN | Kannada | Knda | Southern Dravidian | te-IN | Telugu | Telu | South-Central Dravidian
ko-KR | Korean | Kore | Korean | th-TH | Thai | Thai | Kam-Tai
mn-MN | Mongolian | Cyrl | Mongolic | vi-VN | Vietnamese | Latn | Viet-Muong
ms-MY | Malay | Latn | Malayo-Sumbawan |  |  |  |

Table 6: Statistics and description of the dataset used Xu et al. (2022). The
dataset used is a subset of the MASSIVE dataset, selecting 27 different unseen
languages.

##  Appendix B Algorithm

Formally, we define the language representation alignment in Algorithm 1,
where $F_{U}$ represents the features extracted from URIEL, $S$ is the set of
sentence representations, $H_{x}$ and $N_{x}$ are the hidden states and number
of attention-masked tokens for a sentence $x$, respectively. The matrix $W$ is
used for the linear projection, and $A$ holds the final aligned
representations. Algorithm 1 outlines the process for aligning language
representations we use in Figure 2. It leverages the URIEL database for
linguistic features, processes sentences through a language model ($\Theta$),
and aligns these with mBERT representations ($M$). The algorithm iteratively
updates transformation parameters ($W$ and $b$) through a training loop to
minimize the loss between the projected mBERT representations and the target
sentence representations in set $S$, thus achieving aligned language
representations ($A$).

Algorithm 1 Language Representation and Alignment Process

0: Dataset $D$, URIEL database $U$, Language Model $\Theta$, mBERT
representations $M$

0: Aligned Language Representations $A$

$F_{U}\leftarrow$ ExtractFeatures($U$)

$S\leftarrow\\{\\}$

for each sentence $x$ in $D$ do

$H_{x}\leftarrow$ GetLastHiddenStates($x$, $\Theta$)

$N_{x}\leftarrow$ CountAttentionMasked($x$)

$R_{x}\leftarrow\frac{\textsc{Sum}(H_{x})}{N_{x}}$

$S\leftarrow S\cup\\{R_{x}\\}$

end for

$W,b\leftarrow$ InitializeParameters()

for each training epoch do

$P_{U}\leftarrow(W\times S)+b$

$loss\leftarrow$ ComputeLoss($P_{U}$, $F_{U}$)

$W,b\leftarrow$ UpdateParametersWithConstraint($W,b,loss$)

end for

$A\leftarrow\\{\\}$

for each sentence representation $s$ in $S$ do

$A_{m}\leftarrow(W\times s)+b$

$A\leftarrow A\cup\\{A_{m}\\}$

end for

##  Appendix C Language Family Experiment

Tables 7 and 8 provide a comprehensive analysis of language family performance
across different training groups. These tables compare the accuracy
percentages of the Multilingual BERT and XLM-RoBERTa models, respectively. The
results displayed in the tables elucidate the models’ capabilities in
generalizing from the training data to unseen languages. A clear trend that
can be observed is the improvement in performance as the training groups
progress from 1 to 4, which suggests that the models benefit from exposure to
a wider variety of language families during training. The ’Average’ row at the
bottom of each table indicates the mean accuracy across all language families,
providing an insight into the overall performance enhancement achieved by each
model with incremental training diversity.

Language Family | Train Group 1 | Train Group 2 | Train Group 3 | Train Group 4
---|---|---|---|---
Afro-Asiatic | 52.82% | 52.93% | 61.26% | 61.00%
Atlantic-Congo | 65.71% | 68.08% | 70.62% | 71.79%
Austroasiatic | 64.77% | 66.78% | 69.72% | 70.16%
Austronesian | 66.88% | 68.66% | 72.06% | 72.19%
Dravidian | 64.74% | 67.97% | 70.93% | 71.41%
Indo-European | 67.50% | 68.61% | 72.53% | 72.95%
Japonic | 72.11% | 71.98% | 75.80% | 75.67%
Kartvelian | 68.91% | 68.89% | 72.46% | 72.32%
Koreanic | 64.80% | 66.46% | 70.04% | 69.91%
Mongolic-Khitan | 63.11% | 66.44% | 69.71% | 69.59%
Sino-Tibetan | 62.65% | 66.29% | 68.79% | 70.33%
Tai-Kadai | 63.52% | 67.89% | 70.23% | 71.34%
Turkic | 54.69% | 56.91% | 63.54% | 64.05%
Uralic | 71.49% | 71.27% | 75.33% | 75.15%
Average | 65.54% | 67.07% | 71.04% | 71.43%
Table 7: Multilingual BERT Performance of Language Families Across Training Groups Language Family | Train Group 1 | Train Group 2 | Train Group 3 | Train Group 4
---|---|---|---|---
Afro-Asiatic | 75.74% | 76.23% | 85.56% | 85.39%
Atlantic-Congo | 70.86% | 72.38% | 83.24% | 82.73%
Austroasiatic | 74.85% | 76.04% | 83.91% | 83.59%
Austronesian | 78.94% | 79.83% | 84.77% | 84.69%
Dravidian | 81.49% | 82.20% | 85.41% | 85.43%
Indo-European | 80.31% | 81.21% | 83.26% | 83.47%
Japonic | 80.21% | 81.36% | 82.67% | 83.15%
Kartvelian | 80.40% | 81.53% | 82.79% | 83.27%
Koreanic | 79.74% | 80.91% | 82.14% | 82.61%
Mongolic-Khitan | 79.54% | 81.00% | 82.20% | 82.65%
Sino-Tibetan | 79.25% | 81.00% | 82.14% | 82.58%
Tai-Kadai | 79.08% | 80.81% | 81.90% | 82.35%
Turkic | 79.20% | 80.90% | 81.96% | 82.39%
Uralic | 79.24% | 80.91% | 81.92% | 82.47%
Average | 79.45% | 80.48% | 83.44% | 83.62%
Table 8: XLM-RoBERTa Performance of Language Families Across Training Groups


# Circle packings and total geodesic curvatures in hyperbolic background
geometry

Te Ba, Guangming Hu, Yu Sun

###### Abstract

In this paper, we study a new type of circle packings in hyperbolic background
geometry. Horocycles and hypercycles are also considered in this packing. We
give the existence and rigidity of this type of circle packing with conical
singularities in terms of the total geodesic curvature. Moreover, we introduce
the combinatorial curvature flow on surfaces to find the desired circle
packing with the prescribed total geodesic curvature.

Mathematics Subject Classification (2020): 52C26, 53A70, 53E20, 57Q15.

##  1 Introduction

###  1.1 Background

Let $(S,T)$ be a connected closed surface $S$ with triangulation $T$. Let $V$,
$E$, $F$ be the sets of vertices, edges and triangles of $T$, respectively. A
circle packing metric on $(S,T)$ is a map $r:V\to\mathbb{R}_{+}$ such that the
associated polyhedral metric on $(S,T)$ is given by

$l(uv)=r(u)+r(v),$

where $u$, $v$ are the endpoints of edge $uv$. A circle packing metric is
called Euclidean or hyperbolic if we calculate the geometry of triangles by
Euclidean or hyperbolic trigonometric identities. A circle packing metric may
have conical singularities at the center of the circles. The classical
discrete Gaussian curvature is introduced to describe the singularity at the
center of each circle, which is defined as the angle deficit at the vertex.
The notion of circle patterns was proposed in the work of Thurston [33] as a
significant tool to study the hyperbolic structure on $3$-manifolds. Over the
past decades, circle patterns have been bridging discrete conformal geometry
[17, 3, 18, 19, 20], combinatorics [25], minimal surfaces [2, 23],
combinatorial curvature flows [8, 26, 13] and others. Please refer to [32, 5]
for more background. Below, we present Thurston’s remarkable theorem regard to
the existence and uniqueness of the hyperbolic circle packing metric.

######  Theorem 1.1 (Thurston-Andreev).

Let $(S,T)$ be a closed surface with triangulation $T$. Let $V$, $F$ be the
sets of vertices and faces of $T$. Let $F_{I}$ be the set of faces having at
least one vertex in $I$ for the subset $I\subset V$. Then there exists a
hyperbolic circle packing metric on $(S,T)$ with discrete Gaussian curvatures
$x_{i}$ on $i\in V$ if and only if $(x_{1},\cdots,x_{|V|})\in\Omega$, where

$\Omega=\left\\{(x_{1},\cdots,x_{|V|})\in\mathbb{R}^{|V|}\ |\
x_{i}<2\pi,\sum\nolimits_{i\in I}x_{i}>2\pi|I|-\pi|F_{I}|\ \text{for each}\
I\subset V\right\\}.$

Moreover, the hyperbolic circle packing metric is unique if it exists.

There are many results related to Theorem 1.1 and its proof. See, for example,
the works of Andreev [1], Colin de Verdiere [9], Marden-Rodin [28], Bowers-
Stephenson, [6] Chow-Luo [8], Bobenko-Springborn [4], Guo-Luo [21], Xu [35],
Connelly-Gortler [10] and Ge-Hua-Zhou [14].

###  1.2 Set up

For simplicity of notations, we use one index to denote a vertex ($i\in V$),
two indices to denote an edge ($ij$ is the arc on $S$ joining $i$, $j$) and
three indices to denote a face ($ijk$ is the region on $S$ bounded by $ij$,
$jk$, $ik$). For each $i\in V$, denote $U(i)$ as a small open regular
neighborhood of $i$. We define

$N(I):=\cup_{i\in I}U(i)$

for $I\subset V$. Suppose $I_{1},I_{2}\subset V$ satisfying $I_{1}\cap
I_{2}=\emptyset$. Set

$S_{I_{1},I_{2}}:=S\setminus(N(I_{1})\cup I_{2}).$

The intersection

$T_{I_{1},I_{2}}:=T\cap S_{I_{1},I_{2}}$

is called the pseudo ideal triangulation of $S_{I_{1},I_{2}}$. The
intersections

$E_{I_{1},I_{2}}:=\\{ij\cap S_{I_{1},I_{2}}|ij\in E\\},\quad
F_{I_{1},I_{2}}:=\\{ijk\cap S_{I_{1},I_{2}}|ijk\in F\\}$

are called the edge and face set of $T_{I_{1},I_{2}}$, respectively. The
intersection of a face of $S_{I_{1},I_{2}}$ and $\partial S_{I_{1},I_{2}}$ is
called a $B$-arc. It is easy to see $S_{I_{1},I_{2}}=S$ and
$T_{I_{1},I_{2}}=T$ if $I_{1}=\emptyset$ and $I_{2}=\emptyset$. The pseudo
ideal triangulation on surfaces is a generalization of ideal triangulation on
surfaces. Please refer to [30, 27] for the definition of ideal triangulation.

Given $I_{1},I_{2}\in V$ satisfying $I_{1}\cap I_{2}=\emptyset$, we use
$(S_{I_{1},I_{2}},T_{I_{1},I_{2}})$ to denote the surface $S_{I_{1},I_{2}}$
with pseudo ideal triangulation $T_{I_{1},I_{2}}$. Set
$I_{3}=V\setminus(I_{1}\cup I_{2})$. A generalized hyperbolic circle packing
metric on $(S_{I_{1},I_{2}},T_{I_{1},I_{2}})$ is a map $k:V\to\mathbb{R}_{+}$
satisfying

  * ($a$)

$k(i)<1$ if $i\in I_{1}$,

  * ($b$)

$k(i)=1$ if $i\in I_{2}$,

  * ($c$)

$k(i)>1$ if $i\in I_{3}$.

The geometry of $(S_{I_{1},I_{2}},T_{I_{1},I_{2}})$ is determiend as follows:

(a) A face with $i\in I_{1}$, $j,k\in I_{3}.$

(b) A face with $i,j\in I_{1}$, $k\in I_{3}.$

(c) A face with $i,j,k\in I_{1}.$

Figure 1: Faces of $F_{I_{1},I_{2}}$ with $B$-arc

  * ($i$)

The polyhedral metric on $E_{I_{1},I_{2}}$ is defined by
$d:E_{I_{1},I_{2}}\to\mathbb{R}_{+}$, where

$d(ij)=\left\\{\begin{aligned}
&\operatorname{arctanh}k(i)+\operatorname{arctanh}k(j),&i,j\in I_{1},\\\
&\operatorname{arccoth}k(i)+\operatorname{arccoth}k(j),&i,j\in I_{3},\\\
&\operatorname{arctanh}k(i)+\operatorname{arccoth}k(j),&i\in I_{1},j\in
I_{3},\\\ &+\infty,&\ i\ \text{or}\ j\in I_{2}.\end{aligned}\right.$

  * ($ii$)

Let $\alpha$ be an inner angle of faces of $F_{I_{1},I_{2}}$. If $\alpha$ is
at the endpoint of a $B$-arc, then $\alpha$ is defined to be $\pi/2$.

Given a generalized hyperbolic circle packing metric $k$, it is obvious that
the side lengths of edges of $E_{I_{1},I_{2}}$ is determined by ($i$). We
assert that the side lengths of $B$-arcs and inner angles of each face of
$T_{I_{1},I_{2}}$ are determined by ($i$), ($ii$). Please refer to Lemma
2.1-Lemma 2.4 for proofs. See Figure 1 for three types of faces of
$T_{I_{1},I_{2}}$ with $B$-arcs. Then there exists a polyhedral metric on
$(S_{I_{1},I_{2}},T_{I_{1},I_{2}})$ determined by $k$ and ($i$), ($ii$). See
Figure 2 for an example of a generalized hyperbolic circle packing metric.

Figure 2: A generalized hyperbolic circle packing metric on some
$(S_{I_{1},I_{2}},T_{I_{1},I_{2}})$ where $|I_{1}|=2$, $|I_{2}|=1$ and
$|I_{3}|=1$.

Next, we provide a brief introduction to the geometric meaning of the
generalized hyperbolic circle packing metric on
$(S_{I_{1},I_{2}},T_{I_{1},I_{2}})$. The relationship between radii and
geodesic curvatures of circles, horocycles and hypercycles is shown in the
Table 1. Here $\theta$ is a generalized angle, which can be an angle or a
geodesic segment.

| circle | horocycle | hypercycle
---|---|---|---
radius | $0<r<+\infty$ | $r=+\infty$ | $0<r<+\infty$
geodesic curvature | $k=\coth r$ | $k=1$ | $k=\tanh r$
arc length | $l=\theta\sinh r$ | —– | $l=\theta\cosh r$
Table 1: The relationship of radii, geodesic curvatures and arc lengths

  * ($1$)

If $i,j\in I_{1}$, the geometric meaning of $d(ij)$ is the distance between
the axis of two hypercycles with curvature $k(i)$, $k(j)$.

  * ($2$)

If $i,j\in I_{3}$, the geometric meaning of $d(ij)$ is the distance between
the centers of two circles with curvature $k(i)$, $k(j)$.

  * ($3$)

If $i\in I_{1}$ and $j\in I_{3}$, the geometric meaning of $d(ij)$ is the
distance between the axis of the hypercycle with curvature $k(i)$ and the
center of the circle with curvature $k(j)$.

  * ($4$)

If $i\in I_{2}$, the geometric meaning of $d(ij)$ is the distance between the
center of the circle with curvature $k(i)=1$ (a horocycle) to the center or
axis of a circle, or a horocycle, or a hypercycle with curvature $k(j)$, which
is $+\infty$.

Figure 3: Three-circle configurations

Above analysis indicate that each $ijk\in F_{I_{1},I_{2}}$ can be embedded
into a configuration of three mutually tangent circles (with possibly
horocycles or hypercycles), as shown in the Figure 3. Then for any generalized
hyperbolic circle packing metric $k:V\to\mathbb{R}_{+}$ on
$(S_{I_{1},I_{2}},T_{I_{1},I_{2}})$, there exists a hyperbolic circle packing
(with possibly horocycles or hypercycles) on $S_{I_{1},I_{2}}$ induced by $k$.
The total geodesic curvature of $k$ at $v\in V$ is defined as the total
geodesic curvature of the circle, or horocycle, or hypercycle at $v\in V$ in
the hyperbolic circle packing induced by $k$. It can be calculated by

$L(v)=l(v)k(v),$

where $l(v)$ is the length of the circle. The total geodesic curvature was
first introduced in the work of Nie [29] as an important tool to study the
existence and rigidity of the circle patterns in spherical background
geometry.

###  1.3 Main results

Motivated by [29], the first result of the paper is to provide the following
existence and rigidity result for generalized hyperbolic circle packing
metrics where the discrete Gaussian curvature at each vertex is replaced by
the total geodesic curvature of each vertex.

######  Theorem 1.2.

Let $(S,T)$ be a connected closed surface with vertex, face set $V$, $F$. Let
$F_{I}$ be the set of faces having at least one vertex in $I$ for subset
$I\subset V$. Then there exists $I_{1},I_{2}\subset V$ such that the
generalized hyperbolic circle packing metric on
$(S_{I_{1},I_{2}},T_{I_{1},I_{2}})$ having the total geodesic curvature
$L_{1},\cdots,L_{|V|}$ on each vertex if and only if
$(L_{1},\cdots,L_{|V|})\in\Omega$, where

(1.1)
$\Omega=\left\\{(L_{1},\cdots,L_{|V|})\in\mathbb{R}^{|V|}_{+}\left|\right.\sum\nolimits_{i=1}^{|V|}L_{i}<\pi|F_{I}|\
\ \text{for each}\ I\subset V\right\\}.$

Moreover, the generalized hyperbolic circle packing metric is unique if it
exists.

######  Remark 1.3.

The Euclidean version of Theorem 1.2 is reduced to Thurston-Andreev’s Theorem
in Euclidean background geometry. Because the total geodesic curvature at
$v\in V$ can be calculated by

$L(v)=l(v)k(v)=\Theta(v)r(v)\frac{1}{r(v)}=\Theta(v),$

where $\Theta(v)$ is the the angle deficit at $v\in V$.

One may ask the following question: Can we determine the topology of
$S_{I_{1},I_{2}}$ from the total geodesic curvature at each vertex? For this
purpose, we follow the work of Chow-Luo [8] on combinatorial Ricci flows.
There are many results on combinatorial Ricci flows, for example, see [24, 15,
12].

Let $k_{i}$ be the geodesic curvature of the disk centered at $i$. We consider
the flow

(1.2) $\frac{dk_{i}}{dt}=-k_{i}(L_{i}-\hat{L}_{i}),$

for $i=1,\cdots,|V|$ with an initial radius vector
$k(0)\in\mathbb{R}^{|V|}_{+}$. Below is the second result of the paper.

######  Theorem 1.4.

The solution $k(t)$ of the flow (1.2) exists for all time. The following two
statements are equivalent:

  * ($a$)

$k(t)$ converges as $t\to\infty$.

  * ($b$)

The prescribed total geodesic curvature $\\{\hat{L}_{i}\\}_{i\in V}$ satisfies

$\hat{L}_{i}>0,\quad\sum_{i\in I}\hat{L}_{i}<\pi|F_{I}|$

for each subset $I\subset V$.

If one of the above statements holds, then the flow (1.2) converges
exponentially fast to a unique generalized hyperbolic circle packing metric
with the total geodesic curvature $\hat{L}_{i}$ at $i\in V$.

######  Remark 1.5.

The topology of $S_{I_{1},I_{2}}$ can be deduced from the limit
$\\{\hat{k}_{i}\\}_{i\in V}$ of the solution of (1.2). In addition, we can
also calculate the discrete Gaussian curvature at each conical singularity and
the length of each geodesic boundary component.

The paper is organized as follows: In Section 2, we study the structure of
three mutually tangent circles (with possibly horocycle and hypercycle). In
Section 3, applying the variational principle, we derive Theorem 1.2. In
Section 4, we introduce some properties of flows (1.2) and prove Theorem 1.4.

##  2 Three-circle configurations

###  2.1 Admissible space of generalized hyperbolic circle packing metrics

This subsection is devoted to characterizing the space of generalized
hyperbolic circle packing metrics on $(S_{I_{1},I_{2}},T_{I_{1},I_{2}})$.
Hyperbolic cosine law indicates that a hyperbolic triangle can be determined
by three side lengths of the triangle. The following three lemmas show that
some special quadrilaterals, pentagons and hexagons can be determined by three
side lengths as well. Please refer to [7, Chapter 2] for some hyperbolic
trigonometric identities.

######  Lemma 2.1.

For any $r_{1},r_{2},r_{3}>0$, there exists a unique hyperbolic quadrilateral
such that the following statements hold:

  * ($a$)

There exist two adjacent right angles.

  * ($b$)

Except for the one with right angles at both ends, the other three sides have
lengths $l_{1}$, $l_{2}$, $l_{3}$, where $l_{i}=r_{j}+r_{k}$ for
$\\{i,j,k\\}=\\{1,2,3\\}$.

###### Proof.

Without loss of generality, we assume that the side facing the side with right
angles at both ends has length $l_{2}$. Suppose $l_{1}\geq l_{3}$. It suffices
to prove that there exists $x\in(0,l_{1})$ such that there exists a hyperbolic
quadrilateral with three right angles with side length $x,y,l_{3}$ and a
hyperbolic right-angled triangle with side length $l_{1}-x,y,l_{2}$, as shown
in the Figure 4(a). This is equivalent to show that there exists
$x\in(0,l_{1})$ such that

$\sinh l_{3}=\sinh x\cosh y\ \text{and}\ \cosh l_{2}=\sinh(l_{1}-x)\cosh y,$

which is equivalent to

(2.1) $\cosh y=\frac{\sinh l_{3}}{\sinh x}=\frac{\cosh
l_{2}}{\cosh(l_{1}-x)}>1.$

Set

$f(x)=\frac{\sinh x}{\cosh(l_{1}-x)}.$

Note that $f(x)$ is strictly increasing for $x\in(0,l_{1})$. It is easy to see
$f(x)\to 0$ when $x\to 0$ and $f(x)\to\sinh l_{1}$ when $x\to l_{1}$. We need
to demonstrate there exists $x_{0}\in(0,l_{1})$ such that

(2.2) $f(x_{0})=\frac{\sinh l_{3}}{\cosh l_{2}}$

and

(2.3) $l_{3}>x_{0},\quad l_{2}>l_{1}-x_{0}.$

Notice that (2.2) can be proved by showing

$\frac{\sinh l_{3}}{\cosh l_{2}}<\sinh l_{1}.$

This is the direct result of $l_{1}\geq l_{3}$. Assume that $l_{3}\leq x_{0}$.
Then $l_{2}\leq l_{1}-x_{0}$, which follows from (2.1). Then we obtain
$l_{2}+l_{3}\leq l_{1}$, which contradict to $r_{1}>0$. Suppose $l_{1}<l_{3}$.
We can prove the lemma by finding $x\in(0,l_{3})$ that satisfies similar
properties as above. We omit the proof here.

∎

######  Lemma 2.2.

For any $r_{1},r_{2},r_{3}>0$, there exists a unique hyperbolic pentagon with
four right angles such that the following statements hold:

  * ($a$)

The two sides adjacent to the non-right angles have lengths $r_{2}+r_{3}$,
$r_{1}+r_{3}$.

  * ($b$)

The middle of the three sides with right angles at both ends has length
$r_{1}+r_{2}$.

###### Proof.

It suffices to prove that there exists $x\in(0,l_{3})$ such that there exist
two hyperbolic quadrilaterals with three right angles with side length
$x,y,l_{1}$ and $l_{3}-x,y,l_{2}$, as shown in the Figure 4(b). It is
equivalent to show that there exists $x\in(0,l_{3})$ such that

$\sinh l_{1}=\sinh x\cosh y\ \text{and}\ \sinh l_{2}=\sinh(l_{3}-x)\cosh y,$

It is equivalent to

(2.4) $\cosh y=\frac{\sinh l_{1}}{\sinh x}=\frac{\sinh
l_{2}}{\sinh(l_{3}-x)}>1.$

Set

$g(x)=\frac{\sinh x}{\sinh(l_{3}-x)}.$

It is easy to see $g(x)$ is strictly increasing for $x\in(0,l_{3})$.
Furthermore, we know that $g(x)\to 0$ as $x\to 0$ and $g(x)\to+\infty$ as
$x\to l_{3}$. Then for any $l_{1},l_{2}>0$, there exists $x_{0}\in(0,l_{3})$
satisfying

$\frac{\sinh x_{0}}{\sinh(l_{3}-x_{0})}=\frac{\sinh l_{1}}{\sinh l_{2}}.$

Observe that $x_{0}<l_{1}$, $l_{3}-x_{0}<l_{2}$. Assume that it is not true,
then (2.4) indicates that $x_{0}\geq l_{1}$, $l_{3}-x_{0}\geq l_{2}$. It
follows that $l_{3}\geq l_{1}+l_{2}$, which leads to $r_{3}\leq 0$. Then we
find $x_{0}\in(0,l_{3})$ satisfying (2.4), which completes the proof. ∎

(a)

(b)

Figure 4: Construction of hyperbolic quadrilaterals and pentagons

The following is the corollary of the classical result of the existence of
hyperbolic right-angled hexagon.

######  Lemma 2.3.

For any $r_{1},r_{2},r_{3}>0$, there exists a unique right angled hyperbolic
hexagon whose three non-pairwise adjacent edge lengths are $r_{1}+r_{2}$,
$r_{2}+r_{3}$, $r_{3}+r_{1}$.

If some $r_{i}=+\infty$ in the above lemmas, the polygon has some vertices at
infinity. The existence and uniqueness still hold for the polygon satisfying
the condition of the above lemmas.

######  Lemma 2.4.

For any $k_{1},k_{2},k_{3}>0$, there exist unique (up to isometry) mutually
externally tangent hyperbolic circles (with possibly horocycles and
hypercycles) and having $k_{1}$, $k_{2}$, $k_{3}$ as their geodesic
curvatures.

###### Proof.

Let us turn the problem into the proof of the existence of polygons that
satisfy the following conditions.

  * ($a$)

Suppose $k_{i}>1$ for $i=1,2,3$. Set $r_{i}=\operatorname{arccoth}{k_{i}}$.
There exists a unique triangle with side lengths $d_{1}$, $d_{2}$, $d_{3}$,
where $d_{i}=r_{j}+r_{k}$. Then we draw a circle of radius $r_{k}$ centered at
the vertex of the triangle if the side facing the vertex has length
$r_{i}+r_{j}$. These three circles are exactly mutually tangent to each other
on three sides of the triangle. The curvature of the three circles is exactly
$k_{1}$, $k_{2}$, $k_{3}$.

  * ($b$)

Suppose $k_{1},k_{2}>1$ and $k_{3}<1$. Set
$r_{i}=\operatorname{arccoth}{k_{i}}$ for $i=1,2$ and
$r_{3}=\operatorname{arctanh}{k_{3}}$. It suffices to prove the existence of
quadrilaterals described in Lemma 2.1. Because we can construct two circles
with radius $r_{2}$, $r_{3}$ centered at two non-right angle vertices of the
quadrilateral and construct a hypercycle of radius $r_{1}$, taking the side of
the quadrilateral where all vertices are right angles as the axis, as shown in
the Figure 5(a). These three circles are mutually tangent to each other with
curvatures $k_{1}$, $k_{2}$, $k_{3}$.

  * ($c$)

Suppose $k_{1},k_{3}<1$ and $k_{2}>1$. Set
$r_{i}=\operatorname{arctanh}{k_{i}}$ for $i=1,3$ and
$r_{2}=\operatorname{arccoth}{k_{2}}$. Similar analysis as $(a)$, $(b)$ shows
that it is equivalent to Lemma 2.2. See Figure 5(b) for an example.

  * ($d$)

Suppose $k_{i}<1$ for $i=1,2,3$. Set $r_{i}=\operatorname{arctanh}{k_{i}}$ for
$i=1,2,3$. This situation corresponds to Lemma 2.3. See Figure 5(c) for an
example.

  * ($e$)

Suppose $k_{i}=1$ for some $i\in\\{1,2,3\\}$. Then $r_{i}$ approaches to
$+\infty$. Then it suffices to prove the existence of some ideal polygons.
This can be seen as the corollary to the above results.

The uniqueness is derived from the fact that the isometry of the hyperbolic
plane maps circles (resp. horocycles, hypercycles) to circles (resp.
horocycles, hypercycles). Thus we complete the proof. ∎

(a) A quadrilateral

(b) A pentagon

(c) A hexagon

Figure 5: Hyperbolic polygons with right angles

The direct result of Lemma 2.4 is the following lemma. Lemma 2.5 implies the
space of generalized hyperbolic circle packing metric on
$(S_{I_{1},I_{2}},T_{I_{1},I_{2}})$ can be characterized by
$\mathbb{R}^{|V|}_{+}$.

######  Lemma 2.5.

Let $(S,T)$ be a closed surface with triangulation $T$. Let $V$ be the vertex
set of $T$. For any $k:V\to\mathbb{R}_{+}$, there exists unique choice of
$I_{1},I_{2}\subset V$ and exists unique polyhedral metric induced by
($i$),($ii$) on $(S_{I_{1},I_{2}},T_{I_{1},I_{2}})$.

###  2.2 Construction of convex functionals

The main target of this subsection is to define convex functionals on the
admissible space of generalized hyperbolic circle packing metric on
$(S_{I_{1},I_{2}},T_{I_{1},I_{2}})$. It can be constructed by taking the sum
of the convex functionals on the admissible space of generalized hyperbolic
circle packing metric on each face of $(S_{I_{1},I_{2}},T_{I_{1},I_{2}})$.

Let $C_{1}$, $C_{2}$, $C_{3}$ be three mutually tangent circles (with possibly
horocycles or hypercycles). Let $l_{i}$ be the length of the arc between two
points of tangency of $C_{i}$. Lemma 2.4 indicates $l_{i}(k_{1},k_{2},k_{3})$
is well-defined on $\mathbb{R}^{3}_{+}$. We begin by introducing the following
two preliminary lemmas.

######  Lemma 2.6.

Let $H$ be a horocycle and $G$ be a geodesic intersecting $H$ at two points.
Suppose that the intersection angle of $H$ and $G$ is $\alpha$, which is not
equal to $\frac{\pi}{2}$. Then the length of the horocycle segment between two
intersection points is $2\tan\alpha$.

###### Proof.

Without loss of generality, we assume the horocycle is $y=1$ on the upper
half-plane model. Then the horocycle segment can be written as

$\left\\{\begin{aligned} &x(t)=\tan\alpha(2t-1),\\\
&y(t)=1.\end{aligned}\right.$

for $0\leq t\leq 1$. Denote $l$ by the length of the horocycle segment. By the
formula of hyperbolic length of curves, we obtain

$l=\int_{0}^{1}\frac{\sqrt{x^{\prime}(t)^{2}+y^{\prime}(t)^{2}}}{y(t)}dt=2\tan\alpha.$

∎

######  Lemma 2.7.

Let $\gamma_{1}$, $\gamma_{2}$ be two circle arcs intersect each other at the
right angle. Let $k_{1}$ $l_{1}$, $k_{2}$ $l_{2}$ be their curvatures and
lengths. If $k_{2}>1$, then the differential form

$\eta=l_{1}dk_{1}+l_{2}dk_{2}$

is closed.

###### Proof.

We divide the proof into three cases.

  * ($a$)

Suppose $k_{1}>1$, as depicted in Figure 6. Let $\theta_{1}$, $\theta_{2}$ be
the angle of $\gamma_{1}$, $\gamma_{2}$. Set
$r_{i}=\operatorname{arccoth}k_{i}$. We can find a right triangle with acute
angles $\theta_{1}/2$, $\theta_{2}/2$ by connecting two centers of
$\gamma_{1}$, $\gamma_{2}$ and one of the intersection point of $\gamma_{1}$,
$\gamma_{2}$. By hyperbolic trigonometric identities, we obtain

$\cot\frac{\theta_{1}}{2}=\coth r_{2}\sinh
r_{1}=\frac{k_{2}}{\sqrt{k_{1}^{2}-1}}.$

It follows that

$\theta_{1}=2\operatorname{arccot}\frac{k_{2}}{\sqrt{k_{1}^{2}-1}}.$

Then

$\frac{\partial l_{1}}{\partial k_{2}}=\frac{\partial\theta_{1}}{\partial
k_{2}}\sinh r_{1}=\frac{2}{1-k_{1}^{2}-k_{2}^{2}}=\frac{\partial
l_{2}}{\partial k_{1}}.$

  * ($b$)

Suppose $k_{1}<1$, as depicted in Figure 6. Let $\theta_{1}$, $\theta_{2}$ be
the angle of $\gamma_{1}$, $\gamma_{2}$. Set
$r_{1}=\operatorname{arctanh}k_{1}$ and $r_{2}=\operatorname{arccoth}k_{2}$.
Then we can find a quadrilateral with three right angles satisfying the edge
angle relation. By hyperbolic trigonometric identities, we obtain

$\coth\frac{\theta_{1}}{2}=\coth r_{2}\cosh
r_{1}=\frac{k_{2}}{\sqrt{1-k_{1}^{2}}},\quad\cot\frac{\theta_{2}}{2}=\tanh
r_{1}\sinh r_{2}=\frac{k_{1}}{\sqrt{k_{2}^{2}-1}}.$

It follows that

$\theta_{1}=2\operatorname{arccoth}\frac{k_{2}}{\sqrt{1-k_{1}^{2}}},\quad\theta_{2}=2\operatorname{arccot}\frac{k_{1}}{\sqrt{k_{2}^{2}-1}}.$

Then we derive

$\frac{\partial l_{1}}{\partial k_{2}}=\frac{\partial\theta_{1}}{\partial
k_{2}}\cosh r_{1}=\frac{2}{1-k_{1}^{2}-k_{2}^{2}}.$

A similar deduction gives that

$\frac{\partial l_{2}}{\partial k_{1}}=\frac{\partial\theta_{2}}{\partial
k_{1}}\sinh r_{2}=\frac{2}{1-k_{1}^{2}-k_{2}^{2}}.$

  * ($c$)

Suppose $k_{1}=1$, as depicted in Figure 6. Set
$r_{2}=\operatorname{arccoth}k_{2}$ and $\theta_{2}$ the angle of
$\gamma_{2}$. We can find an ideal right triangle with an acute angle
$\theta_{2}/2$ by connecting two centers of $\gamma_{1}$, $\gamma_{2}$ and one
of the intersection points of $\gamma_{1}$, $\gamma_{2}$. By hyperbolic
trigonometric identities, we obtain

(2.5) $\sin\frac{\theta_{2}}{2}\cosh r_{2}=1.$

Let $C_{1}$, $C_{2}$ be the center of $\gamma_{1}$, $\gamma_{2}$ and let $P$,
$Q$ be the intersection points of $\gamma_{1}$, $\gamma_{2}$. We connect the
$P$, $Q$ by geodesic and let $D$ be the intersection point of $PQ$ and
$C_{1}C_{2}$. It is easy to see that $\triangle PDC_{2}$ is a right triangle.
By hyperbolic trigonometric identities, we obtain

(2.6) $\cosh r_{2}=\cot\angle DPC_{2}\cot\frac{\theta_{2}}{2}.$

Combing (2.5), (2.6), we obtain

$\tan\angle DPC_{2}=\frac{\cot\frac{\theta_{2}}{2}}{\cosh
r_{2}}=\frac{\sqrt{\cosh^{2}r_{2}-1}}{\cosh r_{2}}=\tanh
r_{2}=\frac{1}{k_{2}}.$

Lemma 2.6 shows that

$l_{1}=\frac{2}{k_{2}},$

which yields that

$\frac{\partial l_{1}}{\partial k_{2}}=-\frac{2}{k^{2}_{2}}.$

By $(a)$, $(b)$, we obtain

$\lim_{k_{1}\to 1}\frac{\partial l_{2}}{\partial k_{1}}=-\frac{2}{k^{2}_{2}}.$

By mean value theorem, it is easy to prove that $l_{2}(k_{1},k_{2})$ is
$C^{1}$ smooth with respect to $k_{1}$ when $k_{1}=1$ and

$\frac{\partial l_{2}}{\partial k_{1}}=-\frac{2}{k^{2}_{2}}$

when $k_{1}=1$.

Combining the analysis above, we obtain

$\frac{\partial l_{1}}{\partial k_{2}}=\frac{\partial l_{2}}{\partial k_{1}}.$

Thus we complete the proof. ∎

Figure 6: Two circle arcs intersect each other at the right angle

The proof of the following lemma is similar to the work of Colin de Verdiere
[9].

######  Lemma 2.8.

The differential $1$-form $\omega=l_{1}dk_{1}+l_{2}dk_{2}+l_{3}dk_{3}$ is
closed.

###### Proof.

For any $k_{1}$, $k_{2}$, $k_{3}>0$, Lemma 2.4 shows that there exist three
mutually tangent circles with curvatures $k_{1}$, $k_{2}$, $k_{3}$.
Furthermore, we can draw a unique circle $C$ passing through three tangency
points as shown in Figure 7. It is easy to prove that $C$ is a hyperbolic
circle. Let $k$ be the curvature of this circle. Then we obtain a continuous
map $f:\mathbb{R}^{3}_{+}\to\mathbb{R}_{+}$ from the curvature of three
mutually tangent circles to the curvature of $C$. Let $l$ be the perimeter of
$C$. Then we define

$\eta=l_{1}dk_{1}+l_{2}dk_{2}+l_{3}dk_{3}+ldk$

on $\Lambda=\\{(x,f(x))|x\in\mathbb{R}^{3}_{+}\\}.$ Note that $\Lambda$ is
diffeomorphism to $\mathbb{R}^{3}_{+}$. It suffices to show that $\eta$ is
closed on $\Lambda$. Let $C_{i}$ be the circle with curvature $k_{i}$. Set
$l^{\prime}_{i}$ as the length of arc of $C$ which lies in the $C_{i}$, as
depicted in Figure 7. Note that

$\eta=\sum_{i=1}^{3}(l_{i}dk_{i}+l^{\prime}_{i}dk).$

It suffices to prove that

$\eta_{i}=l_{i}dk_{i}+l^{\prime}_{i}dk$

is closed, which is the direct result of Lemma 2.7. ∎

Let $L_{i}=l_{i}k_{i}$ be the total geodesic curvature of the arc between two
points of tangency of $C_{i}$. It is easy to see $L_{i}$ is the smooth
regarding $k_{1}$, $k_{2}$, $k_{3}$. Set $S_{i}=\ln k_{i}$. Then

(2.7)
$\omega=l_{1}dk_{1}+l_{2}dk_{2}+l_{3}dk_{3}=L_{1}dS_{1}+L_{2}dS_{2}+L_{3}dS_{3},$

where $L_{i}=l_{i}k_{i}$ is the total geodesic curvature of the arc between
two points of tangency of $C_{i}$. Note that $l_{i}(k_{1},k_{2},k_{3})$ is
$C^{1}$-smooth. Let us define $C^{2}$-smooth function

$F(x)=\int^{x}_{x_{0}}L_{1}dS_{1}+L_{2}dS_{2}+L_{3}dS_{3}$

on $\mathbb{R}^{3}$.

######  Remark 2.9.

Let $k_{i}=\coth r_{i}>1$. Then there exists a triangle formed by connecting
the centers of three hyperbolic circles with radii $r_{i}$, $r_{j}$, $r_{k}$
and tangent to each other. Let $\theta_{i}$ be the inner angle opposite to
side with length $r_{j}+r_{k}$. Then

(2.8)
$l_{i}dk_{i}=\frac{-l_{i}dr_{i}}{\sinh^{2}r_{i}}=\frac{-\theta_{i}dr_{i}}{\sinh
r_{i}}=-\theta_{i}du_{i},$

where $u_{i}=\ln\tanh\frac{r_{i}}{2}$. Set
$\omega=\sum_{i=1}^{3}\theta_{i}du_{i}$ and define $G(x)=\int^{x}\omega$.
Colin de Verdiere [9] first introduced $G(x)=\int^{x}\omega$ to prove the
uniqueness part of Theorem 1.1 by variational principle. Then (2.7) and (2.8)
indicate $F(x)$ can be viewed as an extension of $G(x)$ with different
geometric meaning.

Figure 7: A three-circle configuration

The following lemma is directly obtained by the Gauss-Bonnet theorem.

######  Lemma 2.10.

Let $C_{1}$, $C_{2}$, $C_{3}$ be three mutually tangent circles. Let $\Omega$
be the region enclosed by three arcs between tangency points. Then

$\operatorname{Area}(\Omega)=\pi-L_{1}-L_{2}-L_{3},$

where $L_{i}$ is the total curvature of the arc between two points of tangency
of $C_{i}$.

######  Lemma 2.11.

The integral function $F(x)$ is strictly convex.

###### Proof.

It is easy to see

$\frac{\partial F(x)}{\partial x_{i}}=L_{i}.$

The Hessian matrix of $F(x)$ is

$\mathrm{M}=\left[\frac{\partial L_{i}}{\partial S_{j}}\right]_{3\times 3}.$

Recall that $C_{1}$, $C_{2}$, $C_{3}$ be three mutually tangent circles (with
possibly horocycles or hypercycles) in the hyperbolic plane. A key observation
is that the Euclidean radius of $C_{1}$ decreases as $k_{1}$ increases and
$k_{2}$, $k_{3}$ unchanged if $C_{1}$, $C_{2}$, $C_{3}$ are required to remain
tangent to each other all the time. Then $C_{1}$ will shrink to the tangency
point of $C_{2}$, $C_{3}$ as $k_{1}$ increases. A direct corollary of above
observation is that

$\frac{\partial L_{i}}{\partial
S_{j}}<0,\quad\frac{\partial\operatorname{Area}(\Omega)}{\partial S_{i}}<0.$

Lemma 2.10 shows that

$\operatorname{Area}(\Omega)=\pi-L_{1}-L_{2}-L_{3}.$

As a result, we have

$\frac{\partial L_{i}}{\partial
S_{i}}=-\frac{\partial(\operatorname{Area}(\Omega)+L_{j}+L_{k})}{\partial
S_{i}}>0.$

Then

$\left|\frac{\partial L_{i}}{\partial S_{i}}\right|-\left|\frac{\partial
L_{j}}{\partial S_{i}}\right|-\left|\frac{\partial L_{k}}{\partial
S_{i}}\right|=\frac{\partial(L_{1}+L_{2}+L_{3})}{\partial
S_{i}}=-\frac{\partial\operatorname{Area}(\Omega)}{\partial S_{i}}>0.$

Therefore, the Hessian matrix of $F(x)$ is a symmetric strictly diagonally
dominant matrix with positive diagonal entries, which yields that $\mathrm{M}$
is positive definite. ∎

###  2.3 Limit behaviors of total geodesic curvatures

To characterize the space of total geodesic curvature of generalized
hyperbolic circle packing metrics on $(S_{I_{1},I_{2}},T_{I_{1},I_{2}})$, we
need to study the limit behavior of total geodesic curvatures as generalized
hyperbolic circle packing metrics varies.

######  Lemma 2.12.

Let $C_{1}$, $C_{2}$, $C_{3}$ be three mutually tangent circles with curvature
$k_{1}$, $k_{2}$, $k_{3}$. Let $L_{i}$ be the total curvature of the arc
between two points of tangency of $C_{i}$. Let $0\leq c_{i}<+\infty$ and
$\\{r,s,t\\}=\\{1,2,3\\}$. Then the following statements hold:

(2.9a) $\displaystyle\lim_{k_{r}\to 0}L_{r}$ $\displaystyle=0,$ (2.9b)
$\displaystyle\lim_{(k_{r},k_{s},k_{t})\to(+\infty,a,b)}L_{r}$
$\displaystyle=\pi,$ (2.9c)
$\displaystyle\lim_{(k_{r},k_{s},k_{t})\to(+\infty,+\infty,c)}L_{r}+L_{s}$
$\displaystyle=\pi,$ (2.9d)
$\displaystyle\lim_{(k_{r},k_{s},k_{t})\to(+\infty,+\infty,+\infty)}L_{r}+L_{s}+L_{t}$
$\displaystyle=\pi.$

###### Proof.

We divide the proof of (2.9a) into three cases.

  * ($a$)

Suppose $k_{s}\to 0$, $k_{t}\to 0$. Denote the region enclosed by $C_{r}$,
$C_{s}$, $C_{t}$ as $\Lambda_{r,s,t}$. Then $\Lambda_{r,s,t}$ approaches to an
ideal triangle $T$. By Lemma 2.10, we have

$\operatorname{Area}(T)-\operatorname{Area}(\Lambda_{r,s,t})=L_{r}+L_{s}+L_{t}\to
0.$

Thus $L_{r},L_{s},L_{t}\to 0$.

  * ($b$)

Suppose $k_{s}\to 0$, $k_{t}\to c(c\neq 0)$. Note that $L_{r}$, $L_{s}$ are
strictly less than the corresponding total geodesic curvature in the case
($a$), see Figure 8 for an explanation. Thus $L_{r},L_{s}\to 0$.

  * ($c$)

Suppose $k_{s},k_{t}\nrightarrow 0$. Then $\Lambda_{r,s,t}$ is bounded, which
yields that $l_{r}$ is bounded. Thus $L_{r}\to 0$.

Next, we prove (2.9b)-(2.9d). Note that the area of $\Lambda_{r,s,t}$
approaches to zero as one of $k_{i}\to+\infty$, where $i=r,s,t$. By Lemma
2.10, it can be derived that

$L_{r}+L_{s}+L_{t}\to\pi$

as one of $k_{i}\to+\infty$.

  * ($1$)

Suppose $(k_{r},k_{s},k_{t})\to(+\infty,a,b)$. Then $l_{s},l_{t}\to 0$, which
implies $L_{s},L_{t}\to 0$. Thus $L_{r}\to\pi$.

  * ($2$)

Suppose $(k_{r},k_{s},k_{t})\to(+\infty,+\infty,c)$. Then $l_{t}\to 0$, which
implies $L_{t}\to 0$. Thus $L_{r}+L_{s}\to\pi$.

  * ($3$)

Suppose $(k_{r},k_{s},k_{t})\to(+\infty,+\infty,+\infty)$. Let $\tau_{r,s,t}$
be the triangle formed by connecting the centers of $C_{r}$, $C_{s}$, $C_{t}$.
Let $\theta_{i}$ be the inner angle of $\tau_{r,s,t}$ at the center of
$C_{i}$, where $i=r,s,t$. Then

$L_{i}=\theta_{i}\cosh r_{i}\to\theta_{i}.$

The area formula of the hyperbolic triangle demonstrates

$\operatorname{Area}(\tau_{r,s,t})=\pi-\theta_{r}-\theta_{s}-\theta_{t}\to 0,$

which implies that

$L_{r}+L_{s}+L_{t}\to\pi$

where $L_{r},L_{s},L_{t}\nrightarrow 0$.

∎

Figure 8: $k_{t}\to 0$, or $k_{t}\to c$, or $k_{t}\to+\infty$ when
$k_{r},k_{s}\to 0$

##  3 Proof of Theorem 1.2

The main target of this section is to prove Theorem 1.2 by the variational
principle. Recall that $T$ is the triangulation of $S$ and $V$ is the vertex
set of $T$. We define $C^{2}$-smooth function
$W:\mathbb{R}^{|V|}\to\mathbb{R}$ by

(3.1) $W(x)=\sum_{ijk\in F}F(x_{i},x_{j},x_{k}),$

where $x_{p}$ represents the value of the generalized circle packing metric at
$p\in V$. Lemma 2.11 shows that $F(x_{i},x_{j},x_{k})$ is strictly convex,
which yields that $W(x)$ is strictly convex. Note that

$\frac{\partial W(x)}{\partial x_{i}}=\frac{\partial}{\partial
x_{i}}\sum_{pqr\in F}F(x_{p},x_{q},x_{r})=\sum_{ijk\in F}L_{i}^{jk},$

where $L_{i}^{jk}$ is the total curvature of the arc of $C_{i}$ between
tangency points of $C_{i}$, $C_{j}$ and tangency points of $C_{i}$, $C_{k}$.

The following property of convex functions is important in the proof of
Theorem 1.2. Please refer to [11, 27] for a proof.

######  Lemma 3.1.

If $G:\mathbb{R}^{n}\to\mathbb{R}$ is a $C^{2}$-smooth strictly convex
function, then its gradient $\nabla G:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is a
smooth embedding.

Combining Lemma 3.1, we deduce that

$\displaystyle\nabla W:\quad\mathbb{R}^{|V|}$
$\displaystyle\to\quad\quad\mathbb{R}^{|V|}$
$\displaystyle(K_{1},\cdots,K_{|V|})$ $\displaystyle\to(L_{1},\cdots,L_{|V|})$

realizes a smooth embedding from the space of generalized circle packing
metrics to the total geodesic curvature on each vertex. Next we will give the
image space of the mapping $\nabla W$.

######  Lemma 3.2.

Let $W$ be defined as (3.1). Then $\nabla W:\mathbb{R}^{|V|}\to\Omega$ is a
homeomorphism, where

$\Omega=\left\\{(L_{1},\cdots,L_{|V|})\in\mathbb{R}^{|V|}_{+}\ |\ \sum_{i\in
I}L_{i}<\pi|F_{I}|\ \text{for any subset}\ I\subset V\right\\}.$

###### Proof.

It suffices to prove that $\nabla W(\mathbb{R}^{|V|})=\Omega$. To this end, we
need to analyze the boundary of $\nabla W(\mathbb{R}^{|V|})$ in
$\mathbb{R}^{|V|}_{+}$. Take a sequence $K^{(m)}\in\mathbb{R}^{|V|}$ so that

$\lim_{m\to\infty}K^{(m)}=a\in[-\infty,+\infty]^{|V|},$

where $a(i)=-\infty$ or $+\infty$ for some $i\in V$. We need to prove that
$\nabla W(K^{(m)})$ converges to the boundary of $\Omega$.

Let $I$ (resp. $I^{\prime}$) be a subset of $V$ such that $a(i)=+\infty$
(resp. $a(i)=-\infty$) for $i\in I$ (resp. $i\in I^{\prime}$). Let $F_{I}$ be
the set of faces having at least one vertex in $I$. Suppose $ijk\in F_{I}$.
Let us divide it into three situations.

  * ($a$)

There exists exactly one vertex, and say $i\in I$. By (2.9b), we have
$L_{i}^{jk}\to\pi$ as $K^{(m)}\to a$.

  * ($b$)

There exist exactly two vertices, and say $i,j\in I$. By (2.9c), we have
$L_{i}^{jk}+L_{j}^{ik}\to\pi$ as $K^{(m)}\to a$.

  * ($c$)

All vertices $i$, $j$, $k\in I$. By (2.9d), we have
$L_{i}^{jk}+L_{j}^{ik}+L_{k}^{ij}\to\pi$ as $K^{(m)}\to a$.

Let us compute the sum of all total curvatures of disks in $I$. Let
$F_{I_{s}}$ be the set of faces having exactly $s$ vertex in $I$. We group the
total curvatures of disks in $I$ according to the faces of $F_{I}$ in which
they lie. Then

$\sum_{i\in I}L_{i}=\sum_{ijk\in
F_{I}}L_{i}^{jk}\to\pi(|F_{I_{1}}|+|F_{I_{2}}|+|F_{I_{3}}|)=\pi|F_{I}|.$

Suppose $v\in I^{\prime}$. By (2.9a), we have

$L_{i}=\sum_{ijk\in F}L_{i}^{jk}\to 0.$

Lemma 2.10 indicates that

$L_{i}^{jk}+L_{j}^{ik}+L_{k}^{ij}<\pi$

for each $ijk\in F$, which yields that

$\sum_{i\in I}L_{i}<\pi|F_{I}|.$

In addition, it is obvious that $L_{i}>0$. According to the arbitrariness of
the choice of $I$, we derive that $\nabla W(K^{(m)})\in\Omega$ and $\nabla
W(K^{(m)})$ converges to the boundary of $\Omega$ as $m\to+\infty.$ Hence, it
can be inferred that $\nabla W(\mathbb{R}^{|V|})=\Omega$. Thus we complete the
proof. ∎

###### Proof of Theorem 1.2.

Set $L=(L_{1},\cdots,L_{|V|})\in\Omega$. Lemma 3.2 indicates that there exists

$K=(K_{1},\cdots,K_{|V|})\in\mathbb{R}^{|V|}$

such that $\nabla W(K)=L$. Set $k_{i}=e^{K_{i}}$. Let us define
$k_{L}:V\to\mathbb{R}_{+}$ where $k_{L}(i)=k_{i}$. It is obvious that $k_{L}$
is the generalized circle packing metric on
$(S_{I_{1},I_{2}},T_{I_{1},I_{2}})$ with the total geodesic curvature
$L_{1},\cdots,L_{|V|}$ on each vertex. This proves the ’if’ part. The ’only
if’ is the direct result of Lemma 3.2. Thus the theorem is proved.

∎

##  4 Combinatorial Ricci flows

By the change of variables $K_{i}=\ln{k_{i}}$, we rewrite the flow (1.2) as

(4.1) $\frac{dK_{i}}{dt}=-(L_{i}-\hat{L}_{i}).$

######  Lemma 4.1.

The flow (4.1) exists a unique solution all the time.

###### Proof.

Note that $L_{i}$ is a $C^{1}$ smooth function of $K_{i}$. By Cauchy-Lipschitz
Theorem, the flow (4.1) exists a unique solution $K(t)$ for some
$t\in(0,\varepsilon)$. Now we prove that $\varepsilon=+\infty$. Assume on the
contrary that $\varepsilon$ is finite. Then there exists a sequence
$t_{m}\to\varepsilon$ such that $K\left(t_{m}\right)\to+\infty$ or $-\infty$.
Lemma 3.2 indicates that

$\left|L_{i}\right|<c_{i}\pi$

for some $c_{i}>0$. Thus we have

$\left|\frac{dK_{i}}{dt}\right|=\left|L_{i}-\hat{L}_{i}\right|\leq\left|L_{i}\right|+\left|\hat{L}_{i}\right|\leq
c_{i}\pi+\left|\hat{L}_{i}\right|\leq N_{i}$

for some $N_{i}>0$. This yields that

$\left|K(t_{m})\right|\leq\left|K(0)\right|+N_{i}t_{m}.$

Then we obtain

$\lim_{m\to\infty}\left|K(t_{m})\right|\leq\left|K(0)\right|+N_{i}\varepsilon,$

which yields a contradiction. ∎

Let us define

$\Phi(K)=W(K)-\sum_{i=1}^{|V|}\hat{L}_{i}K_{i}.$

Note that the flow (4.1) is the negative gradient flow of $\Phi(K)$. It is
obvious that $\Phi(K)$ is smooth and strictly convex. Set
$\hat{L}=(\hat{L}_{1},\cdots,\hat{L}_{|V|})$. It can be derived that $\Phi(K)$
exists a unique critical point if and only if there exists
$\hat{K}\in\mathbb{R}^{|V|}$ such that $\nabla W(\hat{K})=\hat{L}$. It is
equivalent to $\hat{L}\in\Omega$, which follows from Lemma 3.2. Then we obtain
the following lemma.

######  Lemma 4.2.

$\Phi(K)$ exists a unique critical point if and only if $\hat{L}\in\Omega$.

The following property of convex functions plays an important role in proving
Theorem 1.4, a proof of which could be found in [16, Lemma 4.6].

######  Lemma 4.3.

Suppose $f(x)$ is a $C^{1}$ smooth convex function on $\mathbb{R}^{n}$ with
$\nabla f(x_{0})=0$ for some $x_{0}\in\mathbb{R}^{n}$. Suppose $f(x)$ is
$C^{2}$ smooth and strictly convex in a neighborhood of $x_{0}$. Then the
following statements hold:

  * ($a$)

$\nabla f(x)\neq 0$ for any $x\notin\mathbb{R}^{n}\setminus\\{x_{0}\\}$.

  * $(b)$

Then $\lim_{\|x\|\to+\infty}f(x)=+\infty$.

Before we prove Theorem 1.4, we give a brief introduction to Lyapunov’s
Stability theory. Please refer to [31, 34, 22] for more information. Let us
consider the real autonomous systems

(4.2) $\frac{dx}{dt}=f(x),$

where $f$ is continuous on the open set $D\subset\mathbb{R}^{n}$, which
contains the origin. The following result is the classic Lyapunov’s Stability
Theorem.

######  Theorem 4.4.

[22, Theorem 3.3] Let $f(x)$ be a continuous function defined on an open set
$D\subset\mathbb{R}^{n}$, which contains the origin, and $f(0)=0$. Let $V(x)$
be a continuously differentiable function defined over $D$ such that

  * ($a$)

$V(0)=0$ and $V(x)>0$ for all $x\in D\setminus\\{0\\}$,

  * ($b$)

$\dot{V}(x)\leq 0$ for all $x\in D$.

  * ($c$)

$\dot{V}(x)<0$ for all $x\in D\setminus\\{0\\}$,

Then $x(t)=0$ is an asymptotically stable equilibrium point of (4.2). Suppose
$D=\mathbb{R}^{n}$ and ($a$),($b$) hold. If $V(x)\to+\infty$ as
$\|x\|\to+\infty,$ then $x(t)=0$ is globally asymptotically stable.

###### Proof of Theorem 1.4.

Since the flow (1.2) is equivalent to the flow (4.1), we can prove this
theorem for the flow (4.1). Due to Lemma 4.1, denote $K(t)$ as the solution of
the flow (4.1).

First we prove $(a)\Rightarrow(b)$. Suppose $K(t)\to\hat{K}$ as $t\to\infty$.
By the mean value theorem, there exists $\xi_{n}\in(n,n+1)$ such that

$\Phi(K(n+1))-\Phi(K(n))=\frac{d}{dt}\Phi(K(\xi_{n}))\to 0$

as $n\to+\infty$. Note that

$\frac{d}{dt}\Phi(K(t))=\sum\nolimits_{i=1}^{|V|}\frac{\partial\Phi(K)}{\partial
K_{i}}\frac{dK_{i}}{dt}=-\sum\nolimits_{i=1}^{|V|}\left(\frac{\partial\Phi(K)}{\partial
K_{i}}\right)^{2}.$

Consequently, we have

$\left|\nabla\Phi(K(\xi_{n}))\right|\to 0$

as $n\to+\infty$. Note that $K(\xi_{n})\to\hat{K}.$ It can be concluded that
$\nabla\Phi(\hat{K})=0$, which implies that $\nabla W(\hat{K})=\hat{L}$. Then
Lemma 3.2 demonstrates $\hat{L}\in\Omega$.

Next, we prove $(b)\Rightarrow(a)$. Assume that $\hat{L}\in A$. Lemma 3.2
shows that there exists a unique $\hat{K}\in\mathbb{R}^{|V|}$ such that
$\nabla W(\hat{K})=\hat{L}$. It follows that $\nabla\Phi(\hat{K})=0$. Let us
define

$V(K)=\Phi(K)-\Phi(\hat{K}).$

Combining to lemma 4.2 and lemma 4.3, it is easy to verify that

  * ($i$)

$V(K)\geq 0$ and $V(K)>0$ for $K\neq\hat{K}$.

  * ($ii$)

$\frac{d}{dt}V(K(t))\leq 0$.

  * ($iii$)

$\frac{d}{dt}V(K(t))=0$ if and only if $K=\hat{K}$.

  * ($iv$)

$V(K)\to+\infty$ as $\|K\|\to+\infty$.

Then $K(t)=\hat{K}$ is the asymptotically stable equilibrium point of (4.1).
The convergence of $K(t)$ to $\hat{K}$ follows from the Theorem 4.4.

The next step is to prove that $K(t)$ converges exponentially fast to
$\hat{K}$. Define

$C\left(K\right)=\sum\nolimits_{i=1}^{n}(L_{i}-\hat{L}_{i})^{2}.$

Note that $W$ is strictly convex. Then there exists $\lambda>0$ such that

$\displaystyle\frac{dC\left(K(t)\right)}{dt}$
$\displaystyle=-2\left(L_{i}-\hat{L}_{i},\cdots,L_{i}-\hat{L}_{i}\right)^{\mathrm{T}}\mathrm{M}\left(L_{i}-\hat{L}_{i},\cdots,L_{i}-\hat{L}_{i}\right),$
$\displaystyle\leq-2\lambda\sum\nolimits_{i=1}^{|V|}\left(L_{i}-\hat{L}_{i}\right)^{2}$
$\displaystyle=-2\lambda C(K),$

where $\mathrm{M}$ is the Hessian matrix of $W(K)$. It means

$\sum\nolimits_{i=1}^{n}(L_{i}-\hat{L}_{i})^{2}=C\left(K(t)\right)\leq
C(K(0))e^{-2\lambda_{0}t},$

therefore, we have

$|L_{i}-\hat{L}_{i}|\leq\sqrt{C(K(0))}e^{-\lambda_{0}t}.$

Then we obtain

$|K_{i}(t)-\hat{K}_{i}|\leq\int_{t}^{\infty}\left|L_{i}-\hat{L}_{i}\right|dt\leq\int_{t}^{\infty}C(K(0))e^{-\lambda_{0}t}dt\leq\frac{C(K(0))}{\lambda_{0}}e^{-\lambda_{0}t},$

for any $t>0$. This gives the exponential convergence of the flow (4.1). ∎

##  5 Open questions

This paper leaves behind several topics that merit investigation.

  * $1.$

Explore more rigidity results of circle patterns with respect to the total
geodesic curvature. There are two generalized structures for circle packings,
allowing for the intersection or separation of circles. These two types of
generalized structures are determined by discrete Gaussian curvatures, which
are established in the works of Thurston [33], Ge-Hua-Zhou [14], Xu [35]. We
want to know if Theorem 1.2 can be extended to the above two generalized
structures.

  * $2.$

Let $(S,T)$ be a closed triangulated surface and let
$(r_{1},\cdots,r_{|v|})\in\mathbb{R}^{N}_{+}$ be a hyperbolic circle packing
metric. Let $L_{i}$, $\Theta_{i}$ be the total geodesic curvature and angle
deficit at $i\in V$. Set $A_{i}=\Theta_{i}r_{i}$. The Gauss-Bonnet theorem
indicates that

$L_{i}-A_{i}-\Theta_{i}=0.$

Theorem 1.1 indicates hyperbolic circle packing metrics are determined by
angle deficit at each vertex. Theorem 1.2 shows that hyperbolic circle packing
metrics are determined by total geodesic curvature at each vertex as well. The
viewpoints of Theorem 1.1 and Theorem 1.2 are similar to a dual thought on the
Gauss-Bonnet Formula. We are interested in studying whether $A_{i}$ can be
used to characterize the rigidity of hyperbolic circle packing metrics.

  * $3.$

Since the circle packing metric induces a special polyhedral surface, whether
we can generalize total geodesic curvatures to the polyhedral surfaces or not.
The relationship between the discrete conformal structures on polyhedral
surfaces and 3-dimensional hyperbolic geometry was first discovered by
Bobenko-Pinkall-Springborn [3], in the case of vertex scaling, which is
further studied by Zhang-Guo-Zeng-Luo-Yau-Gu [36]. We want to know if there
exists a relationship between the generalized circle packing induced by total
geodesic curvatures and $3$-dimensional hyperbolic geometry.

##  6 Acknowledgments

Te Ba is supported by NSF of China (No. 11631010). Guangming Hu is supported
by NSF of China (No. 12101275). The authors would like to thank Xin Nie, Xu Xu
and Ze Zhou for helpful discussions.

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Te Ba<EMAIL_ADDRESS>
School of Mathematics, Hunan University, Changsha 410082, P.R. China

Guangming Hu<EMAIL_ADDRESS>
College of Science, Nanjing University of Posts and Telecommunications,
Nanjing, 210003, P.R. China.

Yu Sun<EMAIL_ADDRESS>
School of Mathematics and Physics, Nanjing Institute of Technology, 211100,
P.R. China.