# CleanGraph: Human-in-the-loop Knowledge Graph Refinement and Completion

Tyler Bikaun, Michael Stewart  Wei Liu
The University of Western Australia
<EMAIL_ADDRESS>

###### Abstract

This paper presents CleanGraph, an interactive web-based tool designed to
facilitate the refinement and completion of knowledge graphs. Maintaining the
reliability of knowledge graphs, which are grounded in high-quality and error-
free facts, is crucial for real-world applications such as question-answering
and information retrieval systems. These graphs are often automatically
assembled from textual sources by extracting semantic triples via information
extraction. However, assuring the quality of these extracted triples,
especially when dealing with large or low-quality datasets, can pose a
significant challenge and adversely affect the performance of downstream
applications. CleanGraph allows users to perform Create, Read, Update, and
Delete (CRUD) operations on their graphs, as well as apply models in the form
of plugins for graph refinement and completion tasks. These functionalities
enable users to enhance the integrity and reliability of their graph data. A
demonstration of CleanGraph and its source code can be accessed at
https://github.com/nlp-tlp/CleanGraph under the MIT License.

##  1 Introduction

In the current data-centric era, where data is frequently referred to as the
‘new oil’, knowledge graphs (KGs)111Alongside knowledge bases.—structured
representations of facts, encompassing semantically defined entities and
relations—stand at the forefront of harnessing its value across various tasks,
including question-answering Huang et al. (2019), information retrieval Wise
et al. (2020), recommendation Guo et al. (2020), and reasoning Chen et al.
(2020a).

Text-based knowledge graphs, a product of information extraction methods such
as entity recognition, relation extraction, and entity linking Ji et al.
(2022), have seen immense growth through automation, leading to expansive,
encyclopedic graphs predominantly from high-quality web data Bollacker et al.
(2008); Mitchell et al. (2018) using techniques like distance supervision
Mintz et al. (2009). Despite these advances, creating domain-specific
knowledge graphs from diverse sources, including user-generated content or
specialised fields like medicine, law enforcement, and engineering, has
received less focus Abu-Salih (2021). These domains present unique challenges
due to resource scarcity, potential data quality issues, and the need for
expert verification of factual accuracy. These difficulties have hindered the
production of reliable, cost-effective knowledge graphs in these domains,
leaving potential downstream applications underexplored compared to general-
domain counterparts.

Figure 1: Schematic overview of the CleanGraph tool illustrating (A) graph
data input, along with the use of optional model plugins for knowledge graph
refinement (KGR) and completion (KGC), (B) the inclusion of human-in-the-loop
(HITL) operations in the process, and (C) graph data output.

While the need for high-quality and complete knowledge graphs predates the
advent of large language models (LLMs) Paulheim (2017), the widespread
availability and capabilities of LLMs for zero or few-shot learning Brown et
al. (2020) have further underscored this necessity. LLMs have democratised
knowledge graph construction from various text corpora Ye et al. (2022), thus
lowering the barriers to creating domain-specific graphs; however, they are
not guaranteed to be factually correct. Consequently, maintaining the quality
and completeness of these graphs remains paramount, especially for downstream
applications like question-answering and recommendation, where factuality is
crucial. As a result, there has been a concerted effort among researchers to
improve knowledge graph quality and coverage via knowledge graph refinement
(KGR) and completion (KGC) Paulheim (2017); Rossi et al. (2021); Ji et al.
(2022).

However, there remains a deficiency in task-specific software capable of
interactive, human-in-the-loop operations for these tasks. Presently, the
available software predominantly supports visualisation and querying but lacks
open accessibility or the capacity to incorporate KGR and KGC models easily.
As a solution, such software could employ domain experts—or oracles—for
quality assurance and validation, enhancing data accuracy before integration
into downstream applications.

To address this gap, we introduce CleanGraph, a powerful yet intuitive web-
based tool developed for introspection of, and interaction with, knowledge
graphs222Represented as untyped and typed property graphs. and results of KGR
and KGC models (Figures 1 and 2). CleanGraph simplifies knowledge graph
verification and refinement, making it user-friendly for technical and non-
technical users. With built-in features that support visualisation and
interaction, along with customisable KGR and KGC model plugins, CleanGraph
assures an effortless quality control experience (Figure 1). This paper
details the design, functionality, and architecture of CleanGraph,
demonstrating its potential to enhance the integrity and reliability of
knowledge graphs.

##  2 Preliminaries - Knowledge graph refinement and completion

Knowledge graph refinement (KGR) and completion (KGC) are distinct, essential
techniques used to enhance the quality and comprehensiveness of knowledge
graphs Paulheim (2017); Chen et al. (2020b). Despite extensive research in
this domain, there remains a scarcity of accessible software allowing the
direct involvement of an oracle in these processes. This section outlines both
techniques, their subtasks, and their latest advancements.

###  2.1 Knowledge graph refinement (KGR)

KGR enhances the accuracy and relevance of the existing facts within a
knowledge graph Paulheim (2017). This includes modifying or deleting incorrect
or outdated entities (nodes), relationships (edges), or properties. The goal
of KGR is to maintain the graph’s accuracy, reliability, and overall quality.
KGR tasks encompass:

  * •

Entity correction: Checking and rectifying incorrect entity representations
(e.g., resolving ambiguities, synonyms, etc.) and types.

  * •

Relationship correction: Examining and fixing erroneous relationships between
entities, as well as types and directions.

  * •

Property correction: Validating and correcting the property values associated
with entities or relationships.

  * •

Noise reduction: Removing irrelevant or redundant entities and relationships.

KGR challenges stem from the dynamic nature of knowledge leading to outdated
information, misinterpretations, data input errors, and lack of context.
Extensive research effort has been devoted to KGR, resulting in the
development of statistical, machine learning, and deep learning techniques
Paulheim (2017). Current advancements in KGR focus on utilising knowledge
representations like knowledge graph embeddings Ji et al. (2022).

###  2.2 Knowledge graph completion (KGC)

In contrast, KGC aims to fill in the missing gaps within the knowledge graph
Chen et al. (2020b). This involves predicting and adding missing entities,
relationships, or properties, enhancing the graph’s comprehensiveness. KGC
tasks include:

  * •

Entity prediction: Inferring and adding new missing entities from the graph.

  * •

Relationship prediction: Discovering and adding missing relationships between
existing entities.

  * •

Property prediction: Predicting and adding missing entity or relationship
properties.

KGC challenges arise from incomplete source data, the complexity of inferring
unknown relations or properties, and the inherent uncertainty of predictions.
Ensuring the accuracy of new information to prevent polluting the graph with
incorrect data is also a major concern. Similar to KGR, KGC has received
significant research attention, with recent advancements made in applying
graph embeddings Rossi et al. (2021); Choudhary et al. (2021); Ji et al.
(2022).

Figure 2: User interface of CleanGraph: Starting clockwise from the top right,
(1) the action tray and subgraph pagination, (2) a secondary sidebar showing
details, properties, errors, and suggestions for the chosen node or edge, (3)
an interactive graph visualisation, and finally, (4) a primary sidebar
displaying a progress overview and subgraphs.

##  3 Key features

The key features of CleanGraph, designed to simplify the introspection of
knowledge graphs, are elaborated upon as follows.

###  3.1 Graph creation and data structure

Building a graph in CleanGraph is a user-friendly process. Users simply upload
or input a JSON array of semantic triples, minimally in the {head, relation,
tail} format333For an untyped graph.. Models for error detection (KGR) and
completion (KGC) can optionally be selected at this stage. To create more
intricate property graphs, additional optional fields such as types and
properties can be supplied (Figure 1A). Once the initial graph is constructed,
any selected error detection and completion models are applied. This process
populates the errors and properties fields of the relevant nodes and edges in
the graph which are subsequently rendered in the user interface for human
interaction (Figure 2).

###  3.2 Graph visualisation and interaction

##### Visualisation

While viewing graph data in a tabular structure like (head, relation, tail) is
sufficient for simple, untyped graphs, this approach is inefficient for
larger, complex graphs. To address this, CleanGraph adopts a visually
intuitive approach by representing graph data in its native format: as
frequency-weighted nodes and edges (Figure 2).

Graphs are displayed as directed acyclic graphs using a simulated force-
directed layout, facilitated by the d3 visualisation library444Implemented via
react-force-graph.. This layout, compared to tabular formats, allows users to
visually parse the graph, identify patterns, and interact with nodes and edges
easily through mouse actions and shortcuts. Users can quickly grasp the
frequencies and types of nodes and edges, aided by colour coding.

To manage information overload, CleanGraph partitions the entire graph into
focused subgraphs; each centred on a single node. Users can then ‘paginate’
through these subgraphs, effectively managing large graph data (Figure 3). In
addition, errors and suggestions identified by the EDM and CM plugins are
displayed adjacent to the relevant nodes and edges, offering instant feedback
to users (Figure 2).

Figure 3: Illustration of CleanGraph’s subgraph pagination process: A subgraph
centred on the node (A) with 12 connected edges is split into 3 ‘pages’ of 5
triples (size) for manageable viewing.

##### Interaction

CleanGraph enables intuitive interaction with the graph. Users can manipulate
the entire graph view through panning and zooming. Direct engagement with
nodes and edges is possible via mouse clicks and shortcuts. Clicking on a node
or edge reveals its details, properties, and any errors and suggestions
(Figure 2). Users can toggle the reviewed state of these elements or remove
them from the graph entirely.

###  3.3 Comprehensive graph CRUD operations

CleanGraph is designed to provide comprehensive Create, Read, Update, and
Delete (CRUD) operations across graphs, specifically accommodating the
property graph format555This allows any arbitrary graph structure to be
supported, even RDF-graphs, where the minimum amount of information is a head,
relation and tail.. It effectively handles nodes and edges, permitting them to
support arbitrary key-value pair properties of any data type except objects
and arrays.

The core CRUD operations that are supported to make the manipulation of graph
data666item(s) refer to nodes and edges. seamless include (1) the Addition of
new nodes or edges to the graph, (2) Item review for error detection and graph
refinement (Figure 2), (3) Subgraph merging (section 3.3.2), (4)
Acknowledgement and action on detected errors and suggested modifications
(Figure 2), (5) Reversal of edge direction to correct relational flow, and (6)
Item deletion (section 3.3.1). These operations facilitate the integration of
detected errors by KGR models and predictions made by KGC models into the
graph. The subgraph merging and item deletion functionalities are in further
detail below.

####  3.3.1 Item deletion

CleanGraph facilitates KGR by enabling users to perform item deletions. To
maintain the directed acyclic graph structure, it automatically propagates
1-hop edge removal and orphan node elimination when nodes or edges are deleted
(Figure 4).

Figure 4: CleanGraph’s 1-hop Item Deletion Illustrated: The removal of node
(A) consequently eliminates all its corresponding edges and any nodes (C, D)
that would become orphaned due to this operation.

####  3.3.2 Subgraph merging

CleanGraph’s subgraph merging feature resolves entity errors. When a node’s
name or type changes, the system checks for potential conflicts with existing
nodes. On detecting a conflict, it prompts the user to approve a proposed node
merge. If approved, the respective subgraphs of the conflicting nodes merge,
incorporating not only nodes and edges, but also associated properties,
errors, suggestions, and frequencies for a comprehensive and accurate
unification of the node data (Figure 5).

Figure 5: CleanGraph’s Node Merge Illustrated: The merging of node (E) into
(G) increments the node frequency and redistributes corresponding edges,
resulting in a new node (I).

###  3.4 Plugin architecture

CleanGraph’s plugin architecture enables flexible support for KGR and KGC
tasks. By accommodating a broad spectrum of Error Detection Models (EDMs for
KGR) and Completion Models (CMs for KGC), users can integrate any model that
complies with the plugin interface, enabling seamless integration with the
CleanGraph user interface (Figure 2). Note: plugins are optional and only
expedite the graph quality assurance process.

####  3.4.1 Authoring plugins

CleanGraph’s plugin authoring process is designed to promote collaboration and
encourage the exchange of plugins via its open-source repository. These
plugins may span heuristic, statistical, and deep-learning models, which we
hope to evolve over time, leading to a shared resource community. To create a
plugin, authors simply add a Python script that complies with the predefined
plugin interface and adheres to the specified Pydantic input/output data
models, into the server’s /plugins directory. The input model is fed the graph
as a set of semantic triples, and the output model returns a list of errors or
suggestions, each containing relevant information and potential actions
(defined and validated as Pydantic data models). These results are then
rendered interactively on the user interface. Plugins are easily accessible
from the user interface due to CleanGraph’s plugin manager, ensuring
reproducible usage across different graph construction processes.

Figure 6: Display of CleanGraph’s Error and Suggestion Features: (A) shows errors associated with a particular item (node), offering an optional corrective action (yellow triangle), while (B) presents informational suggestions (purple triangle). Both errors and suggestions can be acknowledged by the user. Table 1: Comparative analysis of CleanGraph with existing tools for knowledge graph management and manipulation. Abbreviations: HITL, KGR, and KGC refer to Human-in-the-Loop, Knowledge Graph Refinement, and Knowledge Graph Completion, respectively. Tool | Visualisation | Querying |  HITL CRUD |  HITL KGR |  HITL KGC |  Open- Source |  RDF- based |  Property Graph-based
---|---|---|---|---|---|---|---|---
AllegroGraph | ✓ | ✓ | - | - | - | - | ✓ | -
Blazegraph | - | ✓ | - | - | - | - | ✓ | -
Neo4J | ✓ | ✓ | - | - | - | ✓ | - | ✓
JanusGraph | ✓ | ✓ | - | - | - | ✓ | - | ✓
TigerGraph | ✓ | ✓ | - | - | - | - | - | ✓
CleanGraph | ✓ | - | ✓ | ✓ | ✓ | ✓ | - | ✓

####  3.4.2 Error detection models

Error detection model outputs, ‘errors’, are displayed in the UI as easily
identifiable ‘notifications’ (Figures 2 and 6A) attached to graph nodes and
edges. Each notification can optionally include a corrective action, such as
recommendations to merge nodes, update node types or labels, reverse edge
directions, modify edge types, and update or delete node/edge properties.
These actions assist the user in undertaking pertinent actions during the
quality assurance process. CleanGraph defines a finite set of actions — Update
and Delete — embodied in Pydantic data models, which can be freely used by EDM
plugins. Nevertheless, actions are not mandatory; reviewers can simply
‘acknowledge’ them within the UI.

####  3.4.3 Completion models

Analogous to EDMs, completion model outputs, ‘suggestions’, are also displayed
in the UI similarly as ‘notifications’ on nodes and edges (Figures 2 and 6B).
Suggestions notifications can have Create and Update actions, corresponding to
the addition of new nodes/edges (Create) or updated node/edge properties
(Update).

##  4 System architecture

CleanGraph is built using the FARM777FARM - FastAPI, React, MongoDB. web
application stack and comprises a server and a client. The server, implemented
using FastAPI in Python, collaborates seamlessly with the client developed
using a blend of HTML, Javascript (React), and CSS. The choice of a Python-
based server is so that there is native support for KGR and KGC models
typically implemented in Python libraries such as PyTorch and TensorFlow.

A NoSQL database, MongoDB, is used for data management, and stores graph data
in four collections (Graphs, Triples, Nodes, and Edges). The choice of NoSQL
over graph databases such as Neo4J owes to its support for graph-structured
data and its ability to store arbitrarily structured data on nodes and edges.
This flexibility circumvents the need for intricate representations often
required by graph databases when dealing with complex data, such as nested
arrays of complex objects for errors, suggestions, and properties.

##  5 Comparison with existing tools

Table 1 compares CleanGraph with existing tools for knowledge graph
management. While most tools, such as AllegroGraph and Neo4J, are designed to
handle large-scale knowledge graphs and lack Human-in-the-Loop (HITL) support,
CleanGraph focuses on managing smaller text-based knowledge graphs with robust
HITL functionalities.

Existing tools excel in visualising, storing, and querying extensive graph
structures, but their support for HITL refinement and completion is often
limited. Integration with external environments like Python is possible (e.g.
Py2Neo for Neo4J), but these solutions generally fall short of a seamless
iterative workflow involving human interaction. CleanGraph, however, offers
visualisation, property graph-based support, and comprehensive HITL CRUD, KGR,
and KGC functionalities. Its unique design emphasises flexibility and
adaptability, enabling continuous human-involved refinement and completion.

It’s important to recognize that this comparison may not be entirely fair, as
the tools serve different purposes. Large graph databases aim to efficiently
handle substantial data, while CleanGraph prioritises interactive, iterative
development. This distinction, along with variations in open-source
availability and feature sets, reflects the diverse goals and target audiences
of these tools.

##  6 Conclusion and future work

This paper presented CleanGraph, a versatile and interactive tool designed for
the refinement and completion of knowledge graphs. CleanGraph’s unique
contributions include providing comprehensive capabilities for graph CRUD
operations and review that does not require any programming experience,
alongside arbitrary knowledge graph refinement and completion models through
its plugin infrastructure.

While CleanGraph is a robust, ready-to-use tool, it continues to evolve.
Future enhancements include: i) expanding the range of available Error
Detection and Completion plugins to incorporate graph embedding models, ii)
supporting graph queries, iii) optimising performance for large-scale
knowledge graphs, and iv) accommodating RDF-based semantic knowledge graphs.

## Acknowledgements

This research is supported by the Australian Research Council through the
Centre for Transforming Maintenance through Data Science (grant number
IC180100030), funded by the Australian Government. Additionally, Bikaun
acknowledges funding from the Mineral Research Institute of Western Australia.
Bikaun and Liu acknowledge the support from ARC Discovery Grant DP150102405.

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PASA 2024

# Mass of prominences experiencing failed eruptions

B. Filippov E-mail<EMAIL_ADDRESS>Pushkov Institute of Terrestrial
Magnetism, Ionosphere and Radio Wave Propagation of the Russian Academy of
Sciences (IZMIRAN),
Troitsk, Moscow 108840, Russia

###### Abstract

A number of solar filaments/prominences demonstrate failed eruptions, when a
filament at first suddenly starts to ascend and then decelerates and stops at
some greater height in the corona. The mechanism of the termination of
eruptions is not clear yet. One of the confining forces able to stop the
eruption is the gravity force. Using a simple model of a partial current-
carrying torus loop anchored to the photosphere and photospheric magnetic
field measurements as the boundary condition for the potential magnetic field
extrapolation into the corona, we estimated masses of 15 eruptive filaments.
The values of the filament mass show rather wide distribution in the range of
$4\times 10^{15}$ – $270\times 10^{16}$ g. Masses of the most of filaments,
laying in the middle of the range, are in accordance with estimations made
earlier on the basis of spectroscopic and white-light observations.

###### doi:

10.1017/pas.2024.xxx

###### keywords:

Sun: activity – Sun: filaments, prominences – Sun: magnetic fields

##  1 INTRODUCTION

Early models of prominence support were aimed at the search of an agent able
to withstand the gravity force high in the corona. It was magnetic field that
was considered as most suitable to resolve this problem (Menzel, 1951;
Kippenhahn & Schlüter, 1957; Kuperus & Raadu, 1974). Later there were found
magnetic configurations that can provide equilibrium of coronal structures
without taking into account their weight, and that this equilibrium can be
suddenly lost leading to eruption (van Tend & Kuperus, 1978; Molodenskii &
Filippov, 1987; Priest & Forbes, 1990; Forbes & Isenberg, 1991; Lin et al.,
1998; Schmieder et al., 2013). On the other hand, resent researches showed
that mass may be able to influence the local and global properties of coronal
magnetic configuration (Low et al., 2003; Petrie et al., 2007; Seaton et al.,
2011; Gunár et al., 2013; Bi et al., 2014; Reva et al., 2017; Jenkins et al.,
2018; Tsap et al., 2019). Therefore, estimation of prominence mass and its
role in equilibrium and initiation of an eruption is significant for theory,
modeling, and prediction of eruptive events on the Sun.

There are several methods of measuring prominence mass. Mass estimation using
typical prominence neutral hydrogen density and geometrical dimensions gives
the total prominence mass $M$ in the range $5\times 10^{12}-10^{15}$ g
(Labrosse et al., 2010). Measurements of continuum absorption observed in EUV
yield total mass values ranging from $10^{14}$ to $2\times 10^{15}$ g (Heinzel
et al., 2003; Gilbert et al., 2006, 2011). Carlyle (2016) estimated the mass
of the intermediate filament before an eruption on 2015 March 15 as $2.4\times
10^{15}$ g. Prominence mass can be calculated on the base of cloud models by
comparing observations with the results of non-LTE magneto-hydrostatic models.
This method gives values in the range $2\times 10^{15}-6\times 10^{15}$ g
(Koutchmy et al., 2008; Grechnev et al., 2014). White light coronagraph
observations are used to calculate the contribution of erupting prominences to
the mass of coronal mass ejections (CMEs). Athay & Illing (1986) took into
account both white light and H$\alpha$ emissions and obtained a mass of
$10^{16}$ g for the eruptive prominence on 1980 August 18. Some authors
reported even greater values of eruptive prominence masses. Rusin & Rybansky
(1982) estimated the mass of the quiescent prominence before the eruption on
1980 August 18 as $2.5\times 10^{16}$ g. Gopalswamy & Hanaoka (1998)
determined the mass of the quiescent prominence, which erupted on 1994 April
4, as $6\times 10^{16}$ g. It should be noted that all evaluations of
prominence masses strongly depend on several poorly known parameters such as
ionization degree, filling factor, optical thickness, chemical composition,
etc. These uncertainties together with difficulties in the interpretation of
the prominence spectrum permit only order of magnitude estimates to be made.

Different approach to estimate the prominence mass used Low et al. (2003).
They treated solar quiescent prominences as a thin plasma sheet suspended in
an axisymmetric, hydromagnetic atmosphere. They suggested that the magnetic
energy needed for driving CMEs is stored prior to an eruption in a magnetic
flux rope held in equilibrium by the weight of a quiescent prominence. For
coronal fields of 5–10 G, hydromagnetic solutions suggest that a prominence
mass of $10^{16}-26\times 10^{16}$ g is needed to hold detached magnetic
fields of intensity comparable to the coronal fields.

In this paper, we estimate mass of prominences experiencing failed eruptions
with the help of a model of a partial current-carrying torus loop anchored to
the photosphere. We suppose that in most considered cases the gravity force
plays a major role in the confinement of the eruptions. However there are
examples showing other reasons for the termination of eruptions.

Figure 1:  H$\alpha$ filtergrams showing the failed filament eruption on 2014
March 20. (Courtesy of the Udaipur Solar Observatory.)

##  2 Observational data

We selected a number of confined filament eruptions from the AIA Filament
Eruption Catalog (http://aia.cfa.harvard.edu/filament/) (McCauley et al.,
2015) observed by the Atmospheric Imaging Assembly [AIA: (Lemen et al., 2012)]
onboard the Solar Dynamics Observatory [SDO: (Pesnell et al., 2012)]. Since
eruptive events are more frequent at solar maximum, most events belong to the
maximum phase.

It is evident that the necessary condition is the absence of the associated
CME. The need to measure the final height of the filament above the
photosphere prescribed the selection of the events that were not too close to
the centre of the solar disk, however in some cases the calculation of the
neutral surface can be helpful for the estimation of height (Filippov, 2016).
Events located close to the east limb were not suitable, because the initial
filament position relative photospheric magnetic fields would be difficult to
determine. We selected 15 events observed during the period from May 2013 to
July 2014 and studied in a paper by Filippov (2020b) (hereafter referred as
Paper I). For comparison, we also consider several successful eruptions.

H$\alpha$ filtergrams from the Big Bear Solar Observatory, the Kanzelhohe
Solar Observatory, the Udaipur Solar Observatory, and the National Solar
Observatory (NSO)/Global Oscillation Network Group (GONG) were used in
addition to the AIA data. Observation in EUV with the Sun Earth Connection
Coronal and Heliospheric Investigation (SECCHI) Extreme Ultraviolet Imager
[EUVI: (Wuelser et al., 2004; Howard et al., 2008)] ) onboard the Solar
Terrestrial Relations Observatory (STEREO) allowed the measurements of the
final eruptive prominence height and in some cases its initial height before
the eruption if a filament was close to the limb for one of the STEREO
spacecraft. For the potential magnetic field calculations, magnetograms taken
by the Heliospheric and Magnetic Imager [HMI: (Schou et al., 2012)] onboard
the SDO were used as the boundary conditions.

Figure 2: Fragment of the SDO/HMI magnetogram around the filament on 2014
March 20 at 06:20 UT with superposed PILs at the height of 6 Mm (thick red
lines) and at 16 and 30 Mm (blue lines) (a); the same PILs superposed on the
H$\alpha$ filtergram (b). Blue dots show the points above which the potential
magnetic field was calculated. Blue circles indicate the endpoints of the
filament and corresponding flux rope. (Courtesy of the NASA/SDO HMI science
team and the Udaipur Solar Observatory.)

##  3 Prominence mass estimation method

In Paper I, a simple model for the flux-rope equilibrium was analysed in which
the flux rope was considered as a section of a torus with its ends embedded in
the photosphere, which keeps its partly-circular shape during all evolution
and remains in the vertical plane that pass through the ends. Similar models
were proposed by Chen (1989), Cargill et al. (1994),Isenberg & Forbes
(2007),Olmedo & Zhang (2010), and some other authors. The equilibrium of the
flux rope is determined by the balance of forces acting on the flux rope

$F=F_{R}+F_{I}+F_{B}+F_{g}=0.$ (1)

The first term is the radial self-force per unit length acting on the circular
flux rope (Shafranov, 1966)

$F_{R}=\frac{I^{2}}{c^{2}R}\left(\ln\frac{8R}{r}-\frac{3}{2}+\frac{l_{i}}{2}\right).$
(2)

where $I$ is the toroidal electric current, $R$ is the major radius of the
torus, $r$ is the minor radius of the torus, and $l_{i}$ is the internal
inductance per unit length. The equilibrium along the minor radius of the
torus $r$ is taken into account in Equation (2), which suggests the force-free
internal magnetic field structure. We chose thereafter $l_{i}=1$, which
corresponds to the linear force-free internal magnetic structure (Lundquist,
1951; Lin et al., 1998; Isenberg & Forbes, 2007). The major radius of the
torus $R$ depends of the height of the flux-rope apex $h$ and the half-
footpoint-separation $a$:

$R=\frac{a^{2}+h^{2}}{2h}.$ (3)

The self-force (2) vanishes at low heights, since the axis curvature becomes
large, and a nearly straight electric current does not influence itself. On
the other hand, a low-height flux rope is under the strong influence of the
diamagnetic photosphere creating the upward force usually modelled by acting
of the mirror current(Kuperus & Raadu, 1974)

$F_{I}=\frac{I^{2}}{c^{2}h}.$ (4)

Both $F_{R}$ and $F_{I}$ are directed upwards, and the flux rope is held by
the external constraining poloidal magnetic field $B_{e}$

$F_{B}=\frac{IB_{e}}{c},$ (5)

and the gravity force

$F_{g}=mg\frac{R_{\odot}^{2}}{(h+R_{\odot})^{2}},$ (6)

where $m$ is the mass of the filament per unit length and $g$ is the free-fall
acceleration at the level of the photosphere (at the distance of $R_{\odot}$
from the centre of the Sun).

The internal linear force-free field structure and the conservation of the
toroidal flux within the flux rope lead to the dependence of the flux-rope
radius $r$ on the current $I$ (Lin et al., 1998):

$r=r_{0}\frac{I_{0}}{I},$ (7)

while changes of the electric current $I$ during flux-rope evolution in the
corona are determined by the conservation of the poloidal magnetic flux
between the photosphere and the flux rope:

$\Phi_{p}=\Phi_{I}+\Phi_{s}=\frac{I}{c}L_{e}+\int_{S}B_{e}ds=\mbox{const},$
(8)

where $S$ is the area between the photosphere and the flux rope and $L$ is the
self-inductance of the torus fraction above the photosphere. The self
-inductance of a thin circular flux rope is defined as (Landau & Lifshits,
1984)

$L_{e}=\int\limits_{\varphi_{0}}^{\varphi}\int\limits_{\varphi_{0}}^{\varphi}\frac{R\cos\varphi}{2\sin\varphi}d\varphi^{\prime}d\varphi=\\\
=2R\varphi\left(\ln\tan\frac{\varphi}{4}+2\cos\frac{\varphi}{2}+\ln{\frac{8R}{r}}-2\right),$
(9)

where

$\varphi=\left\\{\begin{array}[]{ll}2\arcsin\frac{a}{R}\>,&h\leq a\;,\\\ &\\\
2\pi-2\arcsin\frac{a}{R}\>,&h>a\;,\end{array}\right.$ (10)

and

$\varphi_{0}\approx\frac{r}{2R}.$ (11)

The value of the flux $\Phi_{s}$ can be calculated numerically:

$\Phi_{s}=\int\limits_{-l}^{l}\int\limits_{z_{1}}^{z_{2}}B_{e}dxdz,$ (12)

where $x$ is the horizontal coordinate along the line connecting the
footpoints with the origin halfway between them, $z$ is the vertical
coordinate with the origin at the photosphere,

$l=\left\\{\begin{array}[]{ll}a\>,&h\leq a\;,\\\ &\\\
R\>,&h>a\;,\end{array}\right.$ (13)

$z_{1}=\left\\{\begin{array}[]{ll}0\>,&|x|\leq a\;,\\\ &\\\
\frac{h^{2}-a^{2}-\left((a^{2}+h^{2})^{2}-4h^{2}x^{2}\right)^{1/2}}{2h}\>,&|x|>a\;,\end{array}\right.$
(14)

$z_{2}=\frac{h^{2}-a^{2}+\left((a^{2}+h^{2})^{2}-4h^{2}x^{2}\right)^{1/2}}{2h}.$
(15)

It was found in Paper I that for typical dependence of the external magnetic
field $B_{e}$ on height $h$, the constraining force $F_{B}$ cannot balance the
upward forces $F_{R}+F_{I}$ at a greater height after a catastrophic loss of
equilibrium. The gravity force $F_{g}$ is significant for the termination of
ascending motion in the model. However, the external field was modelled in
Paper I by the field of two 2D horizontal dipoles located at different depths
below the photosphere. Now we calculate the mass of a filament needed to stop
eruptions in real events using real distribution of the external field in the
corona in potential approximation based on the photospheric magnetic field
measurements. We assume that a flux rope is initially in stable equilibrium
and then experiences a catastrophic loss of equilibrium at the height $h_{c}$.
After a failed eruption it stops at the height $h_{m}$. Therefore three
equations:

$\displaystyle F(h_{c})=0,$ (16)
$\displaystyle\left.\frac{\mathrm{d}F}{\mathrm{d}h}\right|_{h_{c}}=0,$
$\displaystyle F(h_{m})=0$

should be solved numerically to find three unknown values $h_{c},I_{c}$, and
$m$. In so doing, changes of the electric current with height according to
Equation (8) should be taken into account.

##  4 Example of the failed filament eruption on 2014 March 20

Failed filament eruption happened on 2014 March 20 not far from the centre of
the solar disk (Figure 1). Coordinates of the middle of the filament were N23,
W21, therefore the line-of-sight component of the photospheric magnetic field
rather well represents the radial magnetic field needed for potential field
calculation. The eruption terminated in about 50 minutes after beginning.
Suggesting the filament rose in nearly radial direction, the final height was
about 200 Mm.

As usual, we cut a rectangular area around the filament from a full-disc
magnetogram and transform it into the array with pixels of equal area. We use
the modified arrays as the boundary conditions for solving the Neumann
external boundary-value problem [see (Filippov, 2013) and references therein].
Figure 2(a) presents a fragment of the SDO/HMI magnetogram around the filament
on 2014 March 20 at 06:20 UT with superposed polarity inversion lines (PILs)
at the height of 6 Mm (thick red lines) and at 16 and 30 Mm (blue lines)
taking into account the projection shift. In Figure 2(b), the same PILs are
superposed on the H$\alpha$ filtergram taken at the Udaipur Solar Observatory
showing the filament just before the eruption. The height at which the PIL is
the nearest one to the filament spine can be considered as the estimate of the
height of the filament top (Filippov, 2016). Calculation of PILs at greater
heights shows that in the top view the PILs become arranged one above the
other, or the neutral surface is nearly radial. This fact permits to suppose
that the eruptive filament also moves in the radial direction and for the
estimation of its final height from the horizontal displacement, only the
spherical projection effect should be taken into account.

The ends of the filament marked by the blue circles determine the flux-rope
footpoint separation $2a$ = 120 Mm. We draw a line through this two points and
calculate the horizontal potential magnetic field $B_{e}$ perpendicular to the
vertical plane that passes through this line at different heights. Actually,
we calculate the vertical distribution of $B_{e}$ above 19 points equally
spread along the line within the interval of $4a$, thus covering the distance
of $a$ away from either footpoint. Then values of $B_{e}$ in any point of the
vertical plane passing through the line are found by interpolation or
extrapolation. We assume that the erupting flux rope remains in the vertical
plane all the time.

Figure 3 shows the vertical profiles of the parameters of the potential
magnetic field averaged over the interval of $a$ around the central point. The
magnetic decay index $n$ defined as usual (Bateman, 1978; Filippov & Den,
2000, 2001; Kliem & Török, 2006)

$n=-\frac{\partial\ln B_{e}}{\partial\ln h}$ (17)

shows monotonic behaviour [Figure 3(b)], as it is typical for dipole-like
field. Indeed, there is no significant patches of different polarities apart
from two major areas of opposite polarity [Figure 2(a)]. While the field
rotates with height, it does not change direction to opposite even at great
heights [Figure 3(c)]. We use here for the decay-index calculation only the
component of the horizontal potential magnetic field perpendicular to the
plane of the flux rope $B_{e}$ in contrast to the whole horizontal field as in
some other studies (e.g. in Paper I).

This values of $B_{e}$ are used in solving of the set of equations (16). The
radius of the flux-rope cross-section is chosen as $r_{0}$ = 6 Mm. The
solution gives $h_{c}$ = 18 Mm, $I_{c}$ = $4\times 10^{11}$ A, $m$ = $10^{7}$
g cm-1. The values of the decay index at the heights $h_{c}$ and $h_{m}$ are
$n_{c}$ = 0.43 and $n_{m}$ = 2.5. Changes of the electric current according to
Equation (8) are presented in Figure 3(d). The current value decreases below
the point of catastrophic loss of equilibrium $h_{c}$, increases a little
above it, and declines significantly at greater heights. Figure 4 demonstrates
the height dependence of the total force acting on the flux rope and the
contributions to it according Equations (2), (4), (5), and (6). The curve of
the total force $F$ touches the abscissa axis at $h_{c}$ demonstrating an
unstable equilibrium and crosses the axis at $h_{m}$ from up to down
evidencing a stable equilibrium. It should be noted that in the absence of the
gravity force, the curve would approach to abscissa not crossing it, which
implies a successful eruption. Of course, all obtained values depend on the
assumptions taken in the model and can be considered only as estimates. On
account of many uncertainties in data, all obtained values have errors no less
than 50%.

To illustrate the process of failed eruption, we solve the equation of motion

$M\frac{\mathrm{d}^{2}z}{\mathrm{d}t^{2}}=F+F_{d},$ (18)

where $F_{d}$ is the drag force, which is evidently needed to prevent long
oscillations of the flux rope about the upper equilibrium point or even a
successful eruption, if the gained kinetic energy is sufficient to overcome
the confining forces. The drag force in hydrodynamics is assumed to be
proportional to the first degree of velocity for small values of the Reynolds
number, $F_{d}=-k_{1}v$, and proportional to the second degree of velocity for
great values of the Reynolds number, $F_{d}=-k_{2}v^{2}$ (Landau & Lifshitz,
1987). We chose the values of the coefficients by the trial-and-error method
in order to have only small oscillations of the flux rope after reaching the
upper equilibrium position. Figure 5 shows the height-time and velocity-time
plots for $k_{1}=1.8\times 10^{4}$ g cm-1 s-1, $k_{2}=1.1\times 10^{-2}$ g
cm-2, and $F_{d}=0$. The acceleration time needed to reach the maximum
velocity is approximately the same in all cases and comprises about 15 min.
However the ascending time needed to reach the maximum height is different.
For linear dependence of the drag force on velocity, it is about 30 min, for
squared dependence it is about 1 hour, and in the absence of the drag force
the flux rope does not stop at all, although its speed is decreasing. The
duration of the fast motion in the SDO/AIA observations is also about 1 hour.

Figure 3: Dependence on height of the horizontal potential magnetic field $B_{e}$ (a), the decay index $n$ (b), the rotation angle $\alpha$ of $B_{e}$ above the middle of the filament (c), and the value of electric current $I$ in the flux rope (d).  Table 1: Failed filament eruptions in the regions with the monotonic decay-index height-dependence Data | Coordinates | Beginning | End | $2a$ | $h_{0}$ | $h_{m}$ | $B_{e}$ | $\alpha_{m}$ | $h_{c}$ | $n_{c}$ | $n_{m}$ | $I_{c}$ | m | M
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| (deg) | of the fast | of the fast | (Mm) | (Mm) | (Mm) | (G) | (deg) | (Mm) |  |  | ($10^{11}$ A) | ($10^{6}$ | ($10^{16}$ g)
|  | motion UT | motion UT |  |  |  |  |  |  |  |  |  | g cm-1) |
2013/05/21 | S15 W62 | 10:00 | 11:00 | 150 | 15 | 150 | 120 | 8 | 17 | 0.5 | 2.1 | 12 | 180 | 270
2013/06/11 | S05 W35 | 20:20 | 22:00 | 180 | 50 | 120 | 4 | 20 | 34 | 0.4 | 1.3 | 1.4 | 3.2 | 6
2013/07/03 | N18 W40 | 05:55 | 06:10 | 55 | 10 | 70 | 4 | 40 | 12 | 0.2 | 1.6 | 0.3 | 0.18 | 1
2013/09/20 | S34 W12 | 04:05 | 04:50 | 170 | 12 | 170 | 110 | 45 | 24 | 0.6 | 1.8 | 2 | 5.4 | 9
2014/02/17 | S03 W02 | 02:45 | 03:00 | 100 | 18 | 180 | 90 | 80 | 12 | 0.5 | 6.1 | 6 | 17 | 17
2014/03/20 | N23 W21 | 06:27 | 07:20 | 120 | 30 | 200 | 40 | 80 | 18 | 0.4 | 2.5 | 4 | 10 | 12
2014/07/05 | S12 W07 | 22:30 | 22:45 | 75 | 25 | 75 | 45 | 50 | 9 | 0.4 | 2.3 | 1.8 | 7 | 5
Table 2: Failed filament eruptions in the regions with the non-monotonic decay-index height-dependence Data | Coordinates | Beginning | End | $2a$ | $h_{0}$ | $h_{m}$ | $B_{e}$ | $\alpha_{m}$ | $h_{c}$ | $n_{c}$ | $n_{m}$ | $I_{c}$ | m | M
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| (deg) | of the fast | of the fast | (Mm) | (Mm) | (Mm) | (G) | (deg) | (Mm) |  |  | ($10^{11}$ A) | ($10^{6}$ | ($10^{16}$ g)
|  | motion UT | motion UT |  |  |  |  |  |  |  |  |  | g cm-1) |
2013/05/14 | S12 E70 | 06:03 | 07:32 | 160 | 20 | 150 | 5 | 160 | 16 | 0.6 | \- 0.8 | 0.5 | 0.3 | 0.5
2013/06/12 | N33 W10 | 15:13 | 15:55 | 150 | 20 | 180 | 8 | 140 | 15 | 0.6 | 1 | 0.6 | 0.4 | 0.6
2013/08/04 | N20 E61 | 06:45 | 07:00 | 120 | 10 | 130 | 3 | 170 | 40 | 0.04 | 0.8 | 0.6 | 0.3 | 0.4
2013/12/08 | S03 W90 | 04:10 | 05:20 | 130 | 30 | 120 | 3 | 70 | 22 | 0.6 | 19 | 0.7 | 0.7 | 0.9
2014/03/27 | S24 W60 | 12:40 | 13:30 | 110 | 15 | 140 | 11 | 210 | 12 | 0.8 | 0.3 | 1 | 1.4 | 1.5
2014/03/28 | S10 W66 | 01:40 | 02:10 | 100 | 15 | 120 | 110 | 150 | 8 | 0.7 | \- 1.2 | 5.8 | 44 | 44
2014/05/03 | N13 E63 | 07:35 | 08:10 | 80 | 15 | 130 | 30 | 90 | 12 | 0.47 | 4.7 | 2 | 2.7 | 2
2014/07/02 | N05 W52 | 17:20 | 18:20 | 190 | 25 | 160 | 20 | 50 | 17 | 0.9 | 1.8 | 2 | 6.4 | 12
Table 3: Successive filament eruptions in the regions with the monotonic decay-index height-dependence Data | Coordinates | Beginning | $2a$ | $h_{0}$ | $B_{e}$ | $h_{c}$ | $n_{c}$ | $I_{c}$ | m | M
---|---|---|---|---|---|---|---|---|---|---
| (deg) | of the fast | (Mm) | (Mm) | (G) |  |  | ($10^{11}$ A) | ($10^{6}$ | ($10^{16}$ g)
|  | motion UT |  |  |  |  |  |  | g cm-1) |
2013/06/22 | N25 E26 | 15:00 | 380 | 100 | 10 | 42 | 0.6 | 2.6 | < 2.7 | < 10
2013/09/23 | S31 W60 | 08:00 | 300 | 30 | 11 | 30 | 0.6 | 1.4 | < 0.6 | < 1.8
2016/01/26 | S22 W29 | 16:55 | 170 | 60 | 45 | 17 | 0.44 | 3.5 | < 2.1 | < 3.6
Table 4: Successive filament eruptions in the regions with the non-monotonic decay-index height-dependence Data | Coordinates | Beginning | $2a$ | $h_{0}$ | $B_{e}$ | $h_{c}$ | $n_{c}$ | $I_{c}$ | m | M
---|---|---|---|---|---|---|---|---|---|---
| (deg) | of the fast | (Mm) | (Mm) | (G) |  |  | ($10^{11}$ A) | ($10^{6}$ | ($10^{16}$ g)
|  | motion UT |  |  |  |  |  |  | g cm-1) |
2012/06/23 | N10 W90 | 06:30 | 300 | 60 | 50 | 21 | 0.6 | 4.3 | < 3.2 | < 10
2013/09/29 | N20 W24 | 21:25 | 450 | 80 | 30 | 33 | 0.67 | 3 | < 2 | < 10
2014/03/29 | N52 E53 | 01:53 | 70 | 20 | 65 | 7 | 0.4 | 0.5 | < 0.3 | < 0.2
Figure 4: Vertical distributions of the forces acting on the flux rope.
Figure 5: Height and velocity of the erupting flux rope experiencing the
action different kinds of the drag force.  Figure 6: Dependence on height of
the horizontal potential magnetic field $B_{e}$ (a), the decay index $n$ (b),
the rotation angle $\alpha$ of $B_{e}$ (c) above the middle of the previous
filament position on 2013 August 7.  Figure 7: Dependence on height of the
forces acting on the flux rope on 2013 August 7 (a, c) and the value of
electric current $I$ in the flux rope (b, d) for opposite directions of the
electric current in the unstable equilibrium point.

##  5 Results

The described procedure was applied to the analysis of 15 failed eruptions.
Since some eruptions happened close to the limb, magnetograms on a day, when a
source region was close to the central meridian, were used for calculations.
According to characteristic properties of the coronal magnetic field, the
results are presented in two tables. In Table 1, events that happened in the
regions with the monotonic decay-index height-dependence of the ambient
magnetic field are listed. Table 2 presents events in the regions with the
non-monotonic decay-index height-dependence. Values of the horizontal
potential magnetic field, perpendicular to the plane of the flux rope, at the
height of about 10 Mm halfway between the flux-rope endpoints are demonstrated
in the eighth column. The ninth column shows the values of the rotation angle
of the potential magnetic field $\alpha$ at the height $h_{m}$ comparative to
the initial direction near the photosphere. The total mass of a filament $M$
is displayed in the last column.

The mass needed to terminate the eruption in the monotonic decay-index
magnetic field could be expected to be less then in the non-monotonic decay-
index field, because the field acts downwards on the flux rope at any height,
while in the latter case it can act upwards due to the change of direction to
opposite. However, the values of mass in Table 1 are systematically greater
than in Table 2. One reason can be the strength of the external field $B_{e}$,
which is typically greater in Table 1. The correlation between the field
strength, electric current, and filament mass is evident. In most cases, the
value of the filament mass (of the order of 106 g cm-1) seems reasonable,
corresponding to the ion number density of the order of $10^{11}-10^{12}$ cm-3
in filaments. Nevertheless, in several events (on 2013.05.21, 2014.03.28,
2014.02.17) the mass is too large.

For comparison, we considered six successful eruptions in regions with two
different types of the decay-index behaviour (Tables 3 and 4). We can find
only the upper limit of the filament mass assuming the upper equilibrium at
the height of about the solar radius $R_{\odot}$. The values of the upper
estimates of the mass are similar to the values obtained for the failed
eruptions. The magnitudes of the magnetic field and critical electric current
are also similar to those of the failed eruptions. Masses of filaments needed
to terminate these eruptions should be much greater than the upper estimates
shown in Tables 3 and 4.

One event falls out of line. This is the failed eruption on 2013 August 04.
The magnetic field, although it can be analysed only on August 07 when the
region comes closer to the central meridian, changes direction showing the
presence of a null point at a very low height of about 10 Mm (Figure 6), in
contrast to the other analysed regions. Formally, we can derive the reasonable
mass of the filament shown in Table 2, but the point of the loss of
equilibrium is too high (about 40 Mm, above the null point) unlike the clearly
observed low initial height of the filament of about 10 Mm [Figure 7(a)].
However, there is another possibility for unstable equilibrium below the null
point with the opposite initial direction of the electric current [Figure
7(c)]. In this case, the current changes direction to opposite during the
eruption due to inductance [Figure 7(d)]. Strong current changes lead to
termination of the eruption at a rather low height of about 50 Mm even in the
absence of gravity. Unfortunately, this height is significantly lower than the
observed maximum height of the filament. Therefore, our model cannot fit the
observed parameters of the eruption on 2013 August 04. Possibly, additional
agents acting on the filament should be taken into account (Myers et al.,
2015; Gronkiewicz et al., 2016; Filippov, 2020a).

##  6 Discussion and conclusions

We estimated the mass of eruptive prominences using the model of a partial
current-carrying torus loop anchored to the photosphere. The gravity force
seems the most suitable agent to stop the eruption at a height comparable with
the initial length of the eruptive filament. The obtained estimates of mass
range from $4\times 10^{15}$ to $270\times 10^{16}$ g. For most events, the
masses are in accordance with the measurements based on spectroscopic and
white-light methods. In several events, the mass is much larger that it is
usually expected in prominences. However, in the failed eruptions we deal with
rather short and low laying prominences, which are not too convenient targets
for a detailed spectroscopic study. High quiescent prominences are more
suitable for this purpose and since plasma within them is more rarefied than
in lower lying active-region and intermediate filaments, the results in optics
could show lower magnitudes than obtained from the balance of forces.

The estimations of the mass of three filaments (namely, observed on 2013 May
21, 2014 March 28, 2014 February 17) much over $10\times 10^{16}$ g seem not
too reliable. It looks that some other forces not taken into account in our
model play significant roles in the termination of eruptions of these
filaments. Gronkiewicz et al. (2016) pointed out several various mechanisms
that can decelerate and stop eruptions in the corona. The first one is the
action of gravitation on a filament, which possesses the amount of energy not
enough to escape the gravitational potential of the Sun. This is the mechanism
considered in this paper.

The second cause is the pressure of overlying magnetic field (Török & Kliem,
2005; Wang & Zhang, 2007; Amari et al., 2018). The amount of stored free
magnetic energy in the region before the eruption can be also not sufficient
to overcome coronal magnetic confinement (Liu et al., 2018). We consider only
the potential field in the corona. While it also can be rather peculiar as in
2013 August 04 event, in all three ‘overloaded’ events the potential field
structure looks not too different from others. However, the field strength and
consequently the initial value of the flux-rope electric current are
significantly greater. We can also underestimate the value of the magnetic
field at greater heights because we use only limited area of the photosphere
for the boundary condition in solving the Neumann external boundary-value
problem, while distant magnetic concentrations may add to the field strength
at great heights. There is also possibility that the coronal magnetic field,
or a ‘magnetic cage’ for an eruptive flux rope, may be strengthen by field-
aligned electric currents, which add the excess of a force-free field over the
potential field (Amari et al., 2018).

It is widely accepted that the strapping magnetic field plays the crucial role
in the prospects of a filament eruption to be successful or failed. Sometimes
the strapping magnetic field is manifested by a EUV arcade (Chen et al.,
2013). Many authors analyse whether the decay index in the interesting height
range is below the threshold for the torus instability ($n_{c}=1.5$) or above
it. Failed filament eruptions on 2005 May 27 (Guo et al., 2011), 2011 March 9
(Li et al., 2018), and 2014 April 7 (Xue et al., 2016) were observed at
heights where the decay index was below the threshold. However, sometimes
eruptive filaments stop at heights where the decay index is greater that the
threshold for the torus instability. It happens in most of our events in the
regions with the monotonic decay index behaviour. In the regions with the non-
monotonic decay-index height-dependence, the distribution of the index values
at the final height is rather wide, including values below the threshold, but
the direction of the horizontal field at this height is turned through more
than 100∘ relative its direction at the initial height. This rotation of the
coronal-field direction was not taken into account in many studies. On the
other hand, Zhou et al. (2019) relate the failure of filament eruptions with
strong writhing motions during the eruptions with the rotation of the filament
axis in the range of 50∘ — 130∘.

Magnetic tension within the erupting flux rope is considered as a restricting
force for its motion (Vrsnak, 1990). Ji et al. (2003) argued that observed in
the failed filament eruption on 2002 May 27 deceleration, exceeding the
gravitational deceleration by a factor of ten, suggests that the filament
material is pulling back by magnetic tension. We take into account the
magnetic tension in our model, but for force-free internal structure of the
flux rope the tension cannot overcame the hoop force stretching the curved
flux-rope axis, but only reduces it. However, if the internal field is not
force-free, the tension may play more significant role. For example, enlarged
coronal plasma pressure can keep a greater longitudinal field than in the
force-free case resulting in the increased tension.

Laboratory experiments (Myers et al., 2015) showed that the presence of a
strong toroidal magnetic field along the flux-rope axis in the ambient space
can prevent a flux rope eruption. This guide field interacts with the flux-
rope electric current and produces a dynamic toroidal field tension force that
is able to terminate the eruption. Precise measurements of forces acting on
low-aspect-ratio, line-tied magnetic flux ropes (Myers et al., 2016) revealed
that the hoop force is systematically smaller than predicted by analytical
expressions for large-aspect ratio flux ropes. On the other hand, the toroidal
field tension force in the laboratory experiments is larger than it is
expected from the analytical theory. Both this factors reduces the role of
gravity in preventing successful eruptions.

Kink instability is widely believed to be able to destabilize a flux rope but
it does not lead itself to a full successful eruption if the magnetic field
above the flux rope does not decrease with height sufficiently steep(Török &
Kliem, 2005; Fan & Gibson, 2007; Fan, 2010). In our events, the overlaying
potential field changes with height rather fast and in many cases it changes
its direction to nearly opposite to the initial direction. Moreover, there is
no significant rotation of the filament arch plane observed during the ascent
in all studied failed eruptions. Therefore, there is no evidence of writhing
and kink-instability in this sample of events, although we cannot rule out
absolutely the kink-instability as a trigger of eruptions.

Amari & Luciani (1999) presented the results of the three-dimensional
numerical simulations where a twisted flux rope is destabilized and after a
phase of dynamic evolution disrupts into two almost untwisted flux tubes
confined within a closed overlaying arcade. During the relaxation of the
magnetic configuration, the magnetic field of the flux tube reconnects with
the overlaying arcade, and some part of the free magnetic energy is released.
The model shows some possible scenario of a failed flux-rope eruption and
confined flare. Reconnection of coronal field lines above filaments before
eruptions can either strengthening or weakening the magnetic confinement of
filaments facilitating either failed or successful eruptions(Wang et al.,
2018)).

The existence of a high-altitude stable equilibrium point is only a necessary
condition for a failed eruption. There should be a drag force that prevents
the flux rope from large-amplitude and long-time oscillation about the high
equilibrium position. In the absence of the drag force, the flux rope could
reach too high height where the strapping external field becomes too small to
bring the flux rope back to the equilibrium point. In other words, if the
kinetic energy of the eruptive filament does not dissipate during the
ascending motion, it could be enough to support a successful eruption.

In the illustrative example shown in Figure 5, we used two functional
dependences of the drag force on velocity. The linear dependence is usually
accepted in hydrodynamics for small values of the Reynolds number

$\mathrm{R}=\frac{\rho vl}{\eta},$ (19)

where $\rho$ is the density, $l$ is a characteristic size, and $\eta$ is the
coefficient of dynamic viscosity. For great values of the Reynolds number, the
drag force is proportional second degree of velocity. We chose the values of
the coefficients of proportionality rather arbitrarily as $k_{1}=1.8\times
10^{4}$ g cm-1 s-1, $k_{2}=1.1\times 10^{-2}$ g cm-2 in order only to damp
oscillations of the flux rope about the upper equilibrium position. Evidently,
at the beginning of eruption the velocity is low, and the Reynolds number is
small. After the main phase of acceleration, the velocity is of the order of
$10^{7}$ cm s-1. Assuming the typical coronal density $\rho=2\times 10^{-15}$
g cm-3, $l=2r_{0}=10^{9}$ cm, and $\eta\approx 1$ g s-1 cm-1, (Zaitsev &
Stepanov, 2018) we obtain R = 20. Zaitsev & Stepanov (2018) considered motion
of a smaller flux rope with a lower speed. The Reynolds number for their case
is about unity. They found $k_{1}\approx 10$ g cm-1 s-1, which is three orders
less than used in our calculations. The corresponding decay time of the flux-
rope oscillation was in their estimations about 10 hours. Of course, it is too
long for failed eruptions. True enough, they use the expression for the drag
force action on a long cylinder valid strictly only for R $\ll$ 1\. In
eruption of filaments with lesser mass, the coefficient $k_{1}$ is also needed
lesser, but still greater than typical for low-Reynolds-number motion.

Expression for the drag force for large Reynolds-numbers has the form (Landau
and Lifshitz 1987)

$F_{d}=-C\rho Sv^{2},$ (20)

where $C$ is the dimensionless coefficient dependent on the Reynolds number
and the body shape and $S$ is the cross-section area of the body. The value of
$C$ decreases from $\approx$ 100 to $\approx$ 0.5, when R increases from 0.1
to 1000. For the eruption described in Section 4, $k_{2}=C\rho S\approx
2\times 10^{-5}$ g cm-2. This value is also three orders less than used in our
calculations.

Wide discrepancy of the drag coefficients needed to damp oscillations of the
flux rope about the upper equilibrium position with the theoretical
estimations can be for reasons of inapplicability of hydrodynamic formulas
derived for solid bodies in viscous liquid for coronal conditions or presence
of additional dissipative processes. Anyway, our kinematics calculations have
only illustrative character.

Although the filament mass seems to be the major factor able to terminate
eruptions of rather short filaments, the example of the eruption on 2013
August 04 shows that there are other possibilities. The specific distribution
of the coronal magnetic field can induce strong changes of the flux-rope
electric current due to inductance, which can turn over the action of the
external field from restraining to pushing upwards and backwards [Figure 7(c,
d)]. There can be also other mechanisms of confining eruptions.

###### Acknowledgements.

The author thanks the Big Bear Solar Observatory, the Kanzelhohe Solar
Observatory, the Udaipur Solar Observatory, the SOHO, the STEREO, and the SDO
scientific teams for the high-quality data they supply. SOHO is a project of
international cooperation between ESA and NASA. STEREO is the third mission in
NASAs Solar Terrestrial Probes program. SDO is a mission of NASAs Living With
a Star Program. This work utilizes GONG data from NSO, which is operated by
AURA under a cooperative agreement with NSF and with additional financial
support from NOAA, NASA, and USAF.

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# Surfaces of genus $g\geq 1$ in 3D contact sub-Riemannian manifolds

Ugo Boscain  CNRS, Laboratoire Jacques-Louis Lions, Sorbonne Université,
Université de Paris, Inria, Boîte courrier 187, 75252 Paris Cedex 05 Paris,
France<EMAIL_ADDRESS>Eugenio Bellini  SISSA, Via
Bonomea 265, 34136 Trieste, Italy<EMAIL_ADDRESS>Dipartimento di
Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, Milano,
Italy.

###### Abstract

We consider surfaces embedded in a 3D contact sub-Riemannian manifold and the
problem of the finiteness of the induced distance (i.e., the infimum of the
length of horizontal curves that belong to the surface). Recently it has been
proved that for a surface having the topology of a sphere embedded in a tight
co-orientable structure, the distance is always finite. In this paper we study
closed surfaces of genus larger than 1, proving that such surfaces can be
embedded in such a way that the induced distance is finite or infinite. We
then study the structural stability of the finiteness/not-finiteness of the
distance.

##  1 Introduction

Consider a three dimensional contact sub-Riemannian manifold
$(M,\mathcal{D},\bf{g})$, where $M$ is a smooth manifold of dimension three,
$\mathcal{D}$ is a contact distribution (i.e., such that
$\operatorname{\mathrm{rank}}{\mathcal{D}}=2$ and
$\mathcal{D}+\left[\mathcal{D},\mathcal{D}\right]=TM$) and $\bf{g}$ is a sub-
Riemannian metric [ABB20, Mon02]. Let $S$ be a surface embedded in $M$. The
intersection $\mathcal{D}\cap TS$ defines a field of directions with
singularities on $S$, where the singularities are the points $q\in S$ such
that $\mathcal{D}_{q}=T_{q}S$. Such points are called _characteristic points_.
In the following the set of characteristic points is denoted with $\Sigma(S)$.
The set of integral curves of $\mathcal{D}\cap TS$ on $S\setminus\Sigma(S)$ is
called the _characteristic foliation_ of $S$, and it is denoted with
$\mathcal{D}S$. For an example of characteristic foliation of a surface in the
Heisenberg group see Figure (1).

Figure 1: Characteristic foliation of the Euclidean unitary sphere centered at
the origin, embedded in the Heisenberg group
$(\mathbb{R}^{3},\mathcal{D}=\text{span}\\{\partial_{x}+\frac{y}{2}\partial_{z},\partial_{y}-\frac{x}{2}\partial_{z}\\})$.

Each leaf is a one-dimensional Riemannian manifold, the metric being the
restriction of $\bf{g}$. A lipschitz curve $\gamma:[0,T]\to S$ is called
_admissible_ if $\dot{\gamma}(t)\in\mathcal{D}_{\gamma(t)}$ for almost every
$t\in[0,T]$. We can use the metric $\bf{g}$ to measure the length of
admissible curves on $S$

$L(\gamma):=\int_{0}^{T}\sqrt{{\bf{g}}(\dot{\gamma}(t),\dot{\gamma}(t))}\,dt,$

and we define the _induced distance_ between two points $q_{0},q_{1}\in S$ as

$d_{S}(q_{0},q_{1})=\inf\\{\,L(\gamma)\,:\,\gamma\,\,\text{admissible},\,\gamma(0)=q_{0},\,\gamma(T)=q_{1}\,\\},$
(1.1)

with the understanding that the induced distance is infinite whenever there is
no admissible curve connecting $q_{0}$ to $q_{1}$. In this paper we are
interested in studying the finiteness of $d_{S}$. We say that the induced
distance is finite if it is finite for every pair of points, in that case the
couple $(S,d_{S})$ defines a metric space and in particular a length space.
Notice moreover that this distance does not coincide with the restriction of
the sub-Riemannian distance on $S$, moreover $d_{S}$ is never continuous with
respect to the topology of the surface [D ̵B22]. If the sub-Riemannian
manifold admits a global orthonormal frame $(F_{1},F_{2})$, or equivalently if
$\mathcal{D}$ is a trivial vector bundle over $M$, then the problem of finding
the curve realizing the distance between two points $q_{0}$ and $q_{1}$ can be
written as the optimal control problem

$\displaystyle\dot{\gamma}(t)=u_{1}(t)F_{1}(\gamma(t))+u_{2}(t)F_{2}(\gamma(t)),$
$\displaystyle\gamma(0)=q_{0},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \gamma(T)=q_{1},$
$\displaystyle\int_{0}^{T}\sqrt{u_{1}^{2}(t)+u_{2}(t)^{2}}\,dt\to\min,$ with
the state constraint $\gamma(t)\in S$.

In the study of contact distributions on 3-manifolds from a topological
perspective, embedded surfaces play a central role [Ben83, Gir91, Gir00].
Recently related topics have been studied when the contact distribution is
endowed with a Riemannian metric (surfaces embedded in 3D contact sub-
Riemannian manifolds). See for instance [DGN12, DGN07] for Carnot groups, [D
̵B22] for generic structures, [BTV17, BTV20, Vel20] for Gauss-Bonnet theorems
and [Bar+21] for stochastic evolution equations. In particular in [D ̵B22] the
authors have proved that the induced distance is always finite for spheres
embedded in a co-oriented tight 3D-contact sub-Riemannian manifolds. Recall
that a 3D-contact sub-Riemannian manifold $(M,\mathcal{D},\bf{g})$ is co-
orientable if the distribution $\mathcal{D}$ can be globally expressed as the
kernel of a 1-form $\omega$. This form is called the contact form, and in the
context of sub-Riemannian geometry is normalized in such a way that

$d\omega_{|\mathcal{D}}=\text{Vol}_{\textbf{g}},$ (1.2)

where $\text{Vol}_{\textbf{g}}$ is the area form induced on $\mathcal{D}$ by
the metric $\bf{g}$. The contact condition
$\mathcal{D}+[\mathcal{D},\mathcal{D}]=TM$ can be expressed in term of the
contact form $\omega$ as $\omega\wedge d\omega\neq 0$, thus a co-orientable
contact manifold $M$ is necessarily orientable. Recall moreover that a contact
distribution is called overtwisted (see 4.5 of [Gei08]) if admits an
overtwisted disk, i.e., an embedding of a disk with horizontal boundary whose
characteristic foliation has a unique singular point: an elliptic point in the
interior of the disk, see Figure (2).

Figure 2: The image depicts the overtwisted disk
$\\{\,(r,\theta,z)\,:\,z=\frac{1}{2}(\pi^{2}-r^{2}),\,r\leq\pi,\,\theta\in[0,2\pi)\,\\}$
in the contact structure
$(\mathbb{R}^{3},\mathcal{D}=\text{span}\\{\partial_{r},\frac{\cos
r}{r}\partial_{\theta}-\sin r\partial_{z}\\})$, expressed in cylindrical
coordinates $(r,\theta,z)$.

A contact distribution is called tight if it is not overtwisted. Since every
overtwisted contact structure contains a sphere $S$ having closed orbits in
its characteristic foliation, which necessarily implies that $d_{S}$ is not
finite, the result obtained in [D ̵B22] can be stated as a metric
characterization of tightness.

######  Theorem 1.

([D ̵B22]) Let $(M,\mathcal{D},\bf{g})$ be a co-oriented $3D$-contact sub-
Riemannian manifold, then $(M,\mathcal{D})$ is tight if and only if every
embedded surface homemorphic to a sphere has finite induced distance.

In this paper we prove that the characterization given by Theorem (1) works
only with spheres. In particular we obtain the following result.

######  Corollary 1.

Let $(M,\mathcal{D},\bf{g})$ be a co-orientable 3D-contact sub-Riemannian
structure. Let $S$ be a closed orientable surface of genus $g\geq 1$. Then
there exist an embedding of $S$ in $M$ for which the induced distance is
finite, and one for which the induced distance is not finite.

The latter result is a corollary of the following theorem, stating that
generically the finiteness (not finitess) of the distance is a structurally
stable property.

######  Theorem 2.

Let $(M,\mathcal{D},\bf{g})$ be a co-orientable 3D-contact sub-Riemannian
manifold, let $S$ be a closed orientable surface of genus $g\geq 1$, and let
$\textbf{Emb}(S,M)$ be the space of embeddings of $S$ in $M$ endowed with the
$C^{\infty}$-topology. Then there exist two disjoint non empty open subsets,
$\mathcal{U}_{f},\mathcal{U}_{\infty}\subset\textbf{Emb}(S,M)$, having dense
union, such that the induced distance is finite on $\mathcal{U}_{f}$ and not
finite in $\mathcal{U}_{\infty}$.

The proof of this theorem is largely based on the theory of Morse-Smale vector
fields, which are structurally stable. In Section (2) we recall the basic
facts concerning these vector fields. We prove that the only obstruction to
the finiteness of the induced distance on Morse-Smale characteristic
foliations is the existence of closed orbits. We then recall the elimination
lemma, a result from contact topology which we use to break up the closed
orbits of a Morse-Smale foliation by means of an arbitrarily small $C^{0}$
perturbation. These facts together ensure the existence of the set
$\mathcal{U}_{f}$ of Theorem (2). In Section (3) we prove the existence of the
set $\mathcal{U}_{\infty}$ and we conclude the proof of Theorem (2).

##  2 Morse-Smale vector fields

The characteristic foliation of an oriented surface $S$ embedded in a co-
oriented contact manifold $(M,\omega)$ is driven by a global vector field:
there exists a smooth vector field $X$ on $S$ vanishing on $\Sigma(S)$,
satisfying

$T_{q}S\cap\mathcal{D}_{q}=\text{span}_{\mathbb{R}}X_{q},\,\,\forall\,q\in
S\setminus\Sigma(S).$ (2.1)

A vector field with the properties described above is called a characteristic
vector field of $S$. Characteristic vector fields are not unique, and can be
obtained one from the other via multiplication by a positive function. In
particular the choice of a characteristic vector field corresponds to the
choice of an area form $\Omega$ on the orientable surface $S$, compatible with
the orientation of the latter. Indeed, once the area form $\Omega$ is chosen,
the corresponding characteristic vector field is the unique solution to the
equation

$i_{X}\Omega=\omega_{|S}.$ (2.2)

The contact condition $\omega\wedge d\omega>0$ implies that the divergence
$\operatorname{\mathrm{div}}_{\Omega}X(q)$ is non zero for every
$q\in\Sigma(S)$:

$d\circ i_{X}\Omega_{q}=\operatorname{\mathrm{div}}_{\Omega}X(q)\Omega_{q}\neq
0.$ (2.3)

The sign of a critical point $q\in\Sigma(S)$ is defined as

$\text{sign}(q):=\text{sign}(\operatorname{\mathrm{div}}_{\Omega}X(q)).$ (2.4)

For a generic surface $S$, the characteristic vector field is Morse-Smale: a
type of vector field with particularly simple dynamical features, recalled in
the definition below.

######  Definition 1.

Let $S$ be a closed orientable surface, a vector field $X$ on $S$ is called
Morse-Smale if

  * •

$X$ has finitely many critical points and closed orbits, all of which are non
degenerate,

  * •

the $\alpha$-limit of every trajectory is either a critical point or a closed
orbit, and the same holds for $\omega$-limits,

  * •

there are no saddle connections.

In its celebrated stability theorems, M.M. Peixoto showed that such vector
fields are generic and structurally stable on closed orientable surfaces
([Pei62],[M ̵C59]). These results are summarized in the following theorem.

######  Theorem 3.

Let $X$ be a Morse-Smale vector field on a closed orientable surface $S$. The
dynamic of any vector field $X^{\prime}$ sufficiently $C^{1}$-close to $X$ is
topologically conjugated to the dynamic of $X$. The homeomorphism realizing
the topological equivalence can be chosen $C^{0}$-close to the identity.
Morse-Smale vector fields of class $C^{r}$, with $1\leq r\leq\infty$, on a
closed orientable surface form an open and dense subset of the set o vector
fields on $S$ endowed with the $C^{r}$-topology.

Actually, Morse-Smale vector fields are generic among the characteristic
vector fields of closed orientable surfaces in contact 3-manifolds, as it
follows from the following proposition.

######  Proposition 1.

[Proposition 4.6.11 of [Gei08]] Let $S$ be a closed orientable surface
embedded in a co-oriented contact manifold. There exists a surface
$S^{\prime}$, isotopic and $C^{\infty}$ close to $S$, having a Morse-Smale
characteristic foliation.

One clearly necessary condition for a surface to have finite induced distance
is the absence of closed orbits. For a surface with a Morse-Smale
characteristic foliation this condition is also sufficient.

######  Lemma 1.

Let $S$ be a closed orientable surface embedded in a co-oriented 3D-contact
sub-Riemannian manifold $M$, having characteristic foliation driven by a
Morse-Smale vector field $X$. Then the induced distance $d_{S}$ is finite if
and only if $X$ does not have closed orbits.

###### Proof.

As we have already said, if $X$ has closed orbits, then the induced distance
is not finite. Assume now that $X$ does not have closed orbits. The Morse-
Smale property ensures that any leaf of the characteristic foliation is
admissible, and hence has finite length: the finiteness of these lengths is
proved in [D ̵B22] Proposition 1.3, and it is a consequence of the fact that
the integral curves of $X$ contained in the stable manifold of a non-
degenerate critical point $q$ converge sub-exponentially to $q$. Denoting with
$\mathcal{A}_{x,y}$ the set of admissible curves joining $x$ to $y$, we only
need to show that $\mathcal{A}_{x,y}\neq\emptyset$ for any $x,y\in S$. Let
$S^{\prime}$ be the complement of the hyperbolic points (saddles) in $S$,
which is open and connected. By definition of Morse-Smale vector field, every
saddle is connected to some elliptic point, thus it is sufficient to show that
$\mathcal{A}_{x,y}\neq\emptyset$ for any $x,y\in S^{\prime}$. For every
$x,y\in S$ we define the equivalence relation

$x\sim y\iff\mathcal{A}_{x,y}\neq\emptyset,$ (2.5)

and we denote with $[x]$ the equivalence class of $x$. By definition of Morse-
Smale vector field, for every $x\in S^{\prime}$ the limits

$\omega(x)=\lim_{t\to\infty}e^{tX}(x),\,\,\,\alpha(x)=\lim_{t\to\infty}e^{-tX}(x),$
(2.6)

exist and are critical points of $X$. Moreover at least one of the two limits
must be an elliptic point, because there are no trajectories joining
hyperbolic points. Therefore for every $x\in S^{\prime}$ the set $[x]\cap
S^{\prime}$ contains an open neighbourhood of $x$, which is the stable
(unstable) manifold of some elliptic point. It follows that the equivalence
classes are open, thus, $S^{\prime}$ being connected, there is only one
equivalence class. ∎

Given any orientable surface $S$ embedded in a 3D-contact manifold, with an
arbitrarily small $C^{\infty}$ perturbation, we can achieve an isotopic
surface with characteristic foliation of Morse-Smale type. The new surface may
well present closed orbits, even if the original surface $S$ did not have any.
The elimination lemma, a result due to Giroux which we are about to state,
describes a procedure which allows us to destroy closed orbits preserving the
Morse-Smale property, by means of an arbitrarily $C^{0}$-small perturbation of
the surface, see Figure (3) and (4).

######  Lemma 2.

[Lemma 4.6.26 of [Gei08]] Let $(M,\mathcal{D})$ be a three dimensional co-
orientable contact manifold and $S$ and embedded closed oriented surface with
characteristic foliation of Morse-Smale type. Assume that there exist two
singular point of the same sign, one elliptic point $q_{e}$ and one hyperbolic
point $q_{h}$, connected by a a separatrix $\gamma$ of the latter. Let $U$ be
an arbitrarily small neighbourhood of $\gamma$. Then there exists an isotopy
$\varphi_{t}:S\to M$, $t\in[0,1]$, with the following properties:

  * •

$\varphi_{0}:S\to M$ is the inclusion of $S$ in $M$,

  * •

$\varphi_{t}$ can be chosen arbitrarily $C^{0}$-close to $\varphi_{0}$,

  * •

The isotopy is fixed on $\gamma$ and outside $U$,

  * •

$\varphi_{1}(S)$ has a characteristic foliation of Morse-Smale type,

  * •

the characteristic foliation of $\varphi_{1}(S)$ has no singularities in
$\varphi_{1}(U)$.

Figure 3: An elliptic point $q_{e}$ and a saddle point $q_{h}$ in elimination
position, as described by Lemma (2).

The elimination lemma allows us to reduce the number of singularities in the
characteristic foliation, without destroying the Morse-Smale property. Of
course, this procedure can be reversed: we can use the elimination procedure
to increase the number of singular points. In particular by means of an
arbitrarily small $C^{0}$ perturbation of the surface $S$, it is possible to
break any closed orbit introducing a pair of singular points along it, both
positive if the orbit is repelling and negative if the orbit is attracting. We
summarize this immediate consequence of the lemma in the following corollary.

Figure 4: Elimination of the critical point $q_{e}$, $q_{h}$ depicted in
Figure (3)

.

######  Corollary 2.

Let $S$ be a closed orientable surface embedded in an co-oriented contact
three manifold $(M,\mathcal{D})$, having characteristic foliation of Morse-
Smale type. There exists a surface $S^{\prime}$ arbitrarily $C^{0}$-close to
$S$, having a characteristic foliation of Morse-Smale type without closed
orbits.

Putting together Proposition (1), Lemma (1) and Corollary (2), we deduce the
following fact.

######  Proposition 2.

Let $(M,\mathcal{D},\bf{g})$ be a contact sub-Riemannian manifold, let $g$ be
a positive integer and let $\bf{Emb}$$(S,M)$ be the space of embeddings of the
genus $g$ orientable surface in $M$, endowed with the $C^{0}$ topology. The
set of surfaces having finite induced distance is dense in $\bf{Emb}$$(S,M)$.

##  3 Proof of Theorem (2)

It is left to show that the subset of $\textbf{Emb}(S,M)$ of embedded surfaces
with infinite induced distance is non empty, for any contact $3$-manifold $M$.
The result is a consequence of the following lemma, paired with Darboux
theorem.

######  Lemma 3.

For every positive integer $g$ there exist a closed surface of genus $g$
embedded in the three dimensional Heisenberg group, having infinite induced
distance.

###### Proof.

The contact structure of the 3-dimensional Heisenberg group can be written as
$(\mathbb{R}^{3},\mathcal{D}=\text{span}\\{\partial_{x}+\frac{y}{2}\partial_{z},\partial_{y}-\frac{x}{2}\partial_{z}\\})$.
Let us denote the characteristic foliation of a surface $S$ embedded in
$(\mathbb{R}^{3},\mathcal{D})$ by $\mathcal{D}S$. Consider first the case
$g=1$. There exists an embedded torus in $(\mathbb{R}^{3},\mathcal{D})$ such
that at least one leaf of $\mathcal{D}T$ is an embedded closed curve. Indeed
given $0<r<R$, consider the torus $T_{R,r}$ given by the embedding

$\mathbb{T}^{2}\ni(\theta_{1},\theta_{2})\mapsto((R+r\cos\theta_{1})\cos\theta_{2},(R+r\cos\theta_{1})\sin\theta_{2},r\sin\theta_{1}).$
(3.1)

The characteristic foliation on $T_{R,r}$ is driven by the vector field

$X=\frac{(R+r\cos\theta_{1})^{2}}{2}\partial_{\theta_{1}}-\frac{r\cos\theta_{1}}{2}\partial_{\theta_{2}}.$
(3.2)

Thanks to Proposition (1) we know that with a $C^{\infty}$ small perturbation
of $T_{r,R}$ we can obtain a torus $T$ having a Morse-Smale characteristic
foliation, which can be assumed to be devoid of singular points, since the
field $X$ in (3.2) is nowhere vanishing. A Morse-Smale characteristic
foliation without singular points must contain closed orbits. This follows
from the second property of Morse-Smale vector fields listed in Definition
(1). The lemma is proved for $g=1$. Let now $\Sigma_{g-1}$ be any embedded
surface of genus $g-1$, not intersecting the torus $T$ mentioned above. Let
$\ell\in\mathcal{D}T$ be a closed leaf and let $B\subset T$ be a ball not
intersecting $\ell$, $\ell\cap B=\emptyset$. We define
$\Sigma_{g}:=T\\#\Sigma_{g-1}$, where the connected sum is made deleting $B$
from $T$, deleting a small ball from $\Sigma_{g-1}$, and gluing together
(smoothly) the resulting boundary circles, see Figure (5). Since
$B\cap\ell=\emptyset$, $\mathcal{D}\Sigma_{g}$ still has the closed curve
$\ell$ among its leaves, thus $\Sigma_{g}$ has infinite induced distance. ∎

Figure 5: Connected sum between the torus $T$ containing the closed leaf
$\ell$, on the left, and the surface $\Sigma_{g-1}$, on the right.

We conclude the proof of Theorem (2).

###### Proof.

We define $\mathcal{U}_{\infty},\mathcal{U}_{f}\subset\textbf{Emb}(S,M)$ to be
set of embeddings of $S$ having a characteristic foliation of Morse-Smale type
with and without closed orbits respectively. They are both open because of
structural stability of Morse-Smale vector fields. Moreover, it follows from
Proposition (1) that the union $\mathcal{U}_{f}\cup\mathcal{U}_{\infty}$ is
dense in $\textbf{Emb}(S,M)$. Finally Lemma (3) ensures that
$\mathcal{U}_{\infty}\neq\emptyset$, while Proposition (2) proves that
$\mathcal{U}_{f}\neq\emptyset$. ∎

Aknowledgements
This work has been partly supported by the ANR-DFG projects “CoRoMo”
ANR-22-CE92-0077-01 and received financial support from the CNRS through the
MITI interdisciplinary programs 80 Prime.
The authors would like thank Andrei A. Agrachev for very helpful discussions.

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