# Im2Vec: Synthesizing Vector Graphics without Vector Supervision

Pradyumna Reddy1 Michaël Gharbi2 Michal Lukáč2 Niloy J. Mitra1,2 1University
College London 2Adobe Research

###### Abstract

Vector graphics are widely used to represent fonts, logos, digital artworks,
and graphic designs. But, while a vast body of work has focused on generative
algorithms for raster images, only a handful of options exists for vector
graphics. One can always rasterize the input graphic and resort to image-based
generative approaches, but this negates the advantages of the vector
representation. The current alternative is to use specialized models that
require explicit supervision on the vector graphics representation at training
time. This is not ideal because large-scale high-quality vector-graphics
datasets are difficult to obtain. Furthermore, the vector representation for a
given design is not unique, so models that supervise on the vector
representation are unnecessarily constrained. Instead, we propose a new neural
network that can generate complex vector graphics with varying topologies, and
only requires indirect supervision from readily-available _raster_ training
images (i.e., with no vector counterparts). To enable this, we use a
differentiable rasterization pipeline that renders the generated vector shapes
and composites them together onto a raster canvas. We demonstrate our method
on a range of datasets, and provide comparison with state-of-the-art SVG-VAE
and DeepSVG, both of which require explicit vector graphics supervision.
Finally, we also demonstrate our approach on the MNIST dataset, for which no
groundtruth vector representation is available. Source code, datasets and more
results are available at http://geometry.cs.ucl.ac.uk/projects/2021/Im2Vec/.

##  1 Introduction

In vector graphics, images are represented as collections of parametrised
shape primitives rather than a regular raster of pixel values. This makes for
a compact, infinitely scalable representation with appearance that may be
varied at need simply by modifying stroke or colour parameters. As a result,
it is favoured by graphic artists and designers.

Figure 1: We present Im2Vec that can be trained with only image supervision to
produce a latent space for vector graphics output. The learned space supports
reprojection, sampling (i.e., generation), and interpolation.

Unfortunately, creating vector graphics still remains a difficult task largely
limited to manual expert workflows, because the same irregular structure makes
it ill-suited for today’s convolution-based generative neural architectures.
There is demand for a generative approach suitable for this domain, but it is
not yet well served by research because of the difficult design requirements.
Suitable approaches should:  (i) produce output in vector format;  (ii)
establish correspondence across elements of the same family;  (iii) support
reconstruction, sampling, and interpolation;  (iv) give user control over
accuracy versus compactness of the representation; and finally,  (v) be
trainable directly using images without the need for vector supervision.

SVG-VAE [24] and DeepSVG [5], the two leading generative algorithms for vector
graphics, cast synthesis as a sequence prediction problem, where the graphic
is a sequence of drawing instructions, mimicking how common formats actually
represent vector art. Training these methods therefore requires supervision
from ground truth vector graphics sequences, which are difficult to collect in
large volumes. Furthermore, the mapping from sequences of parametrised drawing
instruction to actual images is highly non-linear with respect to the
parameters and also non-injective, allowing a variety of different sequences
to produce the same visual result. This makes it difficult to consider
appearance as a criterion, and also causes the produced results to inherit any
structural bias baked into the training sequences.

An approach aiming to do away with such vector supervision would need to
overcome a number of challenges. First, the relationship between the
representation and its appearance must be made explicit and differentiable.
Second, it must operate on an internal representation that directly maps to a
vector graphics representation and is flexible enough to support a large range
of topologies and shape complexities. Finally, it should extract
correspondences between related shapes, directly from unlabelled images.

In this paper, we propose such a method, called Im2Vec, based on a
representation that mimics the compositing behaviour of complex vector
graphics. It uses a variable-complexity closed Bézier path as the fundamental
primitive, with the capability to composite a variable number of these to
create shapes of arbitrary complexity and topology (shown in Figure 2).

\begin{overpic}[width=433.62pt,tics=5]{images/decomposition.pdf}
\put(9.0,1.0){$I$ } \put(29.0,1.0){$L_{1}$, $d_{1}$ } \put(47.0,1.0){$L_{2}$,
$d_{2}$ } \put(65.0,1.0){$L_{3}$, $d_{3}$ } \put(78.5,2.0){\ldots}
\put(85.0,1.0){$L_{T}$, $d_{T}$ } \end{overpic} Figure 2:  Im2Vec encodes a
shape as a layered set of filled curves (or shapes). Each shape is obtained by
deformation of a topological disk, differentiably rasterized into images
$L_{i}$, then differentiably composited back-to-front according to scalar
depth variables $d_{i}$.

The key insight that allows the handling of arbitrary complexity is that we
can treat any primitive closed shape as a deformation of a unit circle, which
is modelled as 1D convolution on samples from this circle conditioned on a
common latent vector. By recombining these primitive paths through a
differentiable rasterizer [22] and differentiable compositing [28], we can
natively represent vector art while learning to generate it purely based on
appearance, obviating the need for vector supervision.

We evaluate Im2Vec on a variety of examples with varying complexity and
topology including fonts, emojis, and icons. We demonstrate that Im2Vec, even
without any vector supervision, consistently performs better reconstruction
compared to SVG-VAE and DeepSVG when trained on the same dataset. We also
compare our approach to a purely raster-based autoencoder, which we dub
ImageVAE. While ImageVAE and Im2Vec produce comparable reconstruction quality,
Im2Vec outputs vector graphics and hence enjoys the associated editability and
compactness benefits. Finally, we quantify the compactness versus
approximation power of our method, and demonstrate Im2Vec can be used to
vectorize the MNIST dataset for which no groundtruth vector representation is
available.

##  2 Related Work

Deep learning techniques for parametric vector shapes have recently garnered
significant interest from the machine learning community [19, 11, 13, 40, 27].

##### Learning-based image vectorization.

Our autoencoder encodes raster images. It can therefore address the single-
image vectorization problem [3, 9, 31, 20, 1, 17], for which learning-based
solutions have been proposed. Egiazarian et al. [7] vectorize technical line
drawings. They predict the parameters of vector primitives using a
transformer-based network, and refine them by optimization. DeepSpline [11]
produces parametric curves of variable lengths from images using a pre-trained
VGG network [33] for feature extraction followed by a hierarchical recurrent
network. Guo et al. [14] use neural networks sub-divide line drawings and
reconstruct the local topology at line junctions. The network predictions are
used in a least squares curve fitting step to estimate Bézier curve
parameters. Liu et al. [23] focus on vectorization of rasterized floorplans.
They use a network to extract and label wall junctions, and use this
information to solve an integer program that outputs the vectorized floor
plans as a set of architectural primitives. These works produce high-quality
vectorizations but, unlike ours, focus on the single image case. In contrast,
our objective is to train a latent representation which can serve both for
vectorization of existing raster images, and for generating new graphics by
sampling with no post-processing.

##### Parametric shape estimation.

Deep learning methods for parametric shape estimation typically encode shapes
as an assembly of primitives, often with fixed topology and cardinality [13].
Smirnov et al. [36] fit rasterized fonts using quadratic Bézier curves, and 3D
signed distance fields using cuboids. Their outputs have predetermined, fixed
topologies that are specified as class-dependent templates. Zou et al. [41]
train a recurrent network that predict shapes as a collection of cuboids from
depth maps; they supervise directly on the shape parameters. Tulsiani et al
[39] also use hierarchies of cuboids, but from occupancy volumes. Similar
techniques have explored other primitives like superquadrics [27] and Coon
patches [35] as primitives. Sinha et al. [34] represents watertight 3D shapes
as continuous deformation of a sphere. This is analogous to our representation
of closed 2D curves.

##### Shape-generating programs.

Ganin et al. [10], Huang et al. [18], and Nakano [25] train Reinforcement
Learning (RL) drawing agents. They circumvent the need for direct supervision
on the drawing program by simulating a rendering engine to produce images from
which they compute a reward signal. Ellis et al. [8] use program synthesis to
generate graphics expressed using a subset of the LaTeX language from hand
drawings. They do not work with complex parametric shapes like Bézier curves,
which are the basic building block of most vector designs. Another notable
work is the CSGNet [32] that present impressive performance in estimating
constructive solid geometry programs. It uses the REINFORCE [37] algorithm to
learn in an unsupervised manner, but runs into issues like drawing over
previous predictions in the later stages of the generation process. Further,
it can only output $32\times 32$ raster images, which lacks the flexibility of
vector graphics and is insufficient for applications that require high
fidelity. Strokenet [40] trains an agent that draws strokes after observing a
canvas image and a generator that maps stroke parameters to a new image.

##### Generative vector graphics model.

Our goal is to obtain a generative model for vector graphics. Previous works
in this area have focused predominantly on the case where direct vector
supervision is available. In contrast, our model can be trained from raster
data alone. SketchRNN [15] introduces a model for both conditional and
unconditional sketch generation. Sketches are encoded as a sequence of pen
position and on/off states. An LSTM is then trained to predict the parameters
of a density function over the sketch parameter space, which can then be
sampled to produce a new sketches. Similarly, Sketchformer [29] proposed a
transformer based architecture for encoding vector form sketches. They show
how the encoding can be used for sketch classification, image retrieval, and
interpolation.

SVG-VAE [24] is the first method that attempts to estimate vector graphics
parameters for generative tasks. They follow a two stage training process.
First, they train an image Variational Auto Encoder (VAE). Second, they freeze
the VAE’s weights and train a decoder that predicts vector parameters from the
latent variable learned on images. They show a style-transfer application from
one vector graphic to another. Unlike ours, their method is not end-to-end,
and it requires vector supervision. More recently, DeepSVG [5] showed that
models operating on vector graphics benefit from a hierarchical architecture;
they demonstrate interpolation and generation tasks. Prior works [2, 12] can
generate new font glyphs from partial observations, but they only work in a
low-resolution raster domain. Li et al. [22] have recently proposed a
differentiable rasterizer that enables gradient based optimization and
learning on vector graphics, using raster-based objectives. This is a key
building block for our method. However, we go beyond the generative models
they demonstrate. In particular, our network can generate graphics made up of
closed curves with complex and varying topologies; it does not produce
artifacts like overlapping paths.

##  3 Method

\begin{overpic}[width=433.62pt]{images/arch.pdf} \put(9.5,36.0){$I$ }
\put(89.5,36.0){$O$ } \put(29.3,27.8){$z$ } \par\put(43.4,32.7){$z_{1}$ }
\put(43.4,28.0){$z_{2}$ } \put(43.4,23.3){$z_{T}$ }
\put(71.0,33.7){\scriptsize$L_{1}$ } \put(71.0,31.7){\scriptsize$d_{1}$ }
\put(71.0,28.9){\scriptsize$L_{2}$ } \put(71.0,27.0){\scriptsize$d_{2}$ }
\put(71.0,24.4){\scriptsize$L_{T}$ } \put(71.0,22.4){\scriptsize$d_{T}$ }
\put(13.7,11.5){$z_{t}$ } \put(34.0,6.5){$z_{t}$ } \put(36.3,6.5){$c_{i}$}
\put(38.6,6.5){$p_{i}$} \put(57.7,6.5){$z_{t}$ } \put(60.1,6.5){$c_{i}$}
\put(62.4,6.5){$p_{i}+\delta p_{i}$} \put(86.1,12.0){$x_{i}$} \end{overpic}
Figure 3:  Architecture overview. We train an end-to-end variational
autoencoder that encodes a raster image to a latent code $z$, which is then
decoded to a set of ordered closed vector paths (top). We then rasterize the
paths using DiffVG [22] and composite them together using DiffComp to obtain a
rasterized output, which we compare to the ground truth raster target for
supervision at training time. Our model can handle graphics with multiple
component paths. It uses an RNN to produce a latent code $z_{t}$ for each
path, from the global latent code $z$ representing the graphic as a whole. Our
path decoder (bottom) decodes the path codes into closed Bézier paths. Our
representation ensures the paths are closed by sampling the path control
points uniformly on the unit circle. These control positions are then deformed
using a 1D convolutional network with circular boundary conditions to enable
adaptive control over the point density. Finally, another 1D circular CNN
processes the adjusted points on the circle to output the final path control
points in the absolute coordinate system of the drawing canvas. The auxiliary
network that predicts the optimal number of control points per path is trained
independently from our main model; it is not shown here.

Our goal is to build a generative model for vector graphics that does not
require vector supervision, i.e., that only requires raster images at training
time. Our model follows an encoder–decoder architecture (Fig. 3). The encoder
has a standard design [16]; it maps a raster image $I$ to a latent variable
$z\in\mathbb{R}^{d}$, which is then decoded into a vector graphic structure.
Our decoder has been carefully designed so that it can generate complex
graphics, made of a variable number $T$ of paths, with varying lengths and no
predetermined topology (§ 3.1). We also train an auxiliary model to predict
the optimal number of control points for each path (§ 3.2). Finally, each
vector shape is rasterized using a differentiable rasterizer [22] and
composited into a final rendering [28], which we compare to a raster ground
truth for training (§ 3.3).

###  3.1 Vector Graphics Decoder

We choose to represent a vector graphic as a depth-ordered set of $T$ closed
Bézier paths, or equivalently, a set of $T$ simply connected solid 2D shapes.
The first operator in our decoder is a recurrent neural network (RNN) that
consumes the global latent code $z$ representing the graphic as a whole (§
3.1.3). At each time step $t$, the RNN outputs a per-path latent code $z_{t}$.
This mechanism lets us generate graphics with arbitrary numbers of paths, and
arbitrary topology (using fill rules to combine the shapes). The path-specific
codes are then individually processed by a _path decoder_ module (§ 3.1.1)
which outputs the parameters of a closed path of arbitrary length using cubic
Bézier segments.

####  3.1.1 Single path decoder with circular convolutions

To ensure the individual paths are closed, we obtain them by continuous
deformation of the unit circle. Specifically, for each shape, we sample $3k$
points along the circle, corresponding to the control points of $k$ cubic
Bézier segments. We compute the 2D cartesian coordinates $p_{i}$ of each of
these points, and annotate them with a 1-hot binary variable $c_{i}$ to
distinguish between the segment endpoints — every third point, which the
Bézier path interpolates — and the other control points.

We replicate the path’s latent code $z_{t}$ and concatenate it with the sample
position and point type label, so that each sample on the circle is
represented as a vector $\begin{bmatrix}p_{i}&c_{i}&z_{t}\end{bmatrix}$,
$i\in\\{1,\ldots,3k\\}$, which we call a _fused latent vector_. These are then
arranged into a cyclic buffer, which is then processed by a neural network
performing 1D convolutions with cyclic boundary conditions (along the sample
dimension) to obtain the final spatial locations of the path’s control points:
$x_{1},\ldots,x_{3k}$. The cyclic convolution along the sample axis
corresponds to convolution along the perimeter of the unit circle. It is a
crucial component of our method because it enables information sharing between
neighbouring samples, while respecting the closed topology of the shape. We
use 3-tap filters for all convolutions and ReLU activations.

Sampling the unit circle rather than using a fixed-length input array allows
us to adjust the complexity (i.e., the number of segments $k$) of the Bézier
path by simply changing the sampling density. In Section 3.2, we show this
sampling density can be determined automatically, based on complexity of the
shape to match, using an auxiliary network. Figure 4 shows the impact of the
number of segments on the reconstruction quality.

(a) visual fidelity vs. number of segments

(b) error vs. number of segments

Figure 4:  Uniform vs. adaptive sampling. Our decoder provides a natural
control over the complexity of the vector graphics it produces. By adjusting
the sampling density on the unit circle, we can increase the number of Bézier
segments and obtain a finer or vector representation of a target raster image
4(a). Our adaptive sampling mechanism (§ 3.1.2) improves reconstruction
accuracy, compared to a uniform distribution of the control points with the
same number of segments 4(b). This adaptive scheme achieves good
reconstructions with as few as 7–8 segments, while uniform sampling requires
12–14.

####  3.1.2 Adaptive control point density

The most natural choice for our control point parameterization would be to
choose equally spaced sample points along the unit circle (in angle). We found
this uniform control points allocation was often sub-optimal. Ideally, more
control points should be allocated to sections of the path with higher
complexity (e.g., sharp creases or serifs for fonts). To address this, we
propose an adaptive sampling mechanism, which we call the _sample deformation_
subnetwork. This module is a 1D convolutional network with cyclic boundary
condition acting on the fused latent vectors
$\begin{bmatrix}p_{i}&c_{i}&z_{t}\end{bmatrix}$, where the $p_{i}$ are
uniformly spaced along the circle. It outputs a displacement $\delta p_{i}$
for each sample point. We parameterize this output in polar coordinates so
that $p_{i}+\delta p_{i}$ remains on the circle.

With our adaptive sampling mechanism turned on, the path decoder now operates
on the fused latent vector with sample deformation,
$\begin{bmatrix}p_{i}+\delta p_{i}&c_{i}&z_{t}\end{bmatrix}$, instead of the
regularly-spaced positions. In Figure 4(b), we show the sample deformation
module improves the reconstruction accuracy, especially when few segments are
used. The benefit over the uniform sampling distribution diminishes as more
curve segments are added.

####  3.1.3 Decoding multi-part shapes using an RNN

So far, we have discussed a decoder architecture for a single shape, but our
model can represent vector graphics made of multiple parts. This is achieved
using a bidirectional LSTM [30] that acts on the graphic’s latent code $z$. To
synthesize a graphic with multiple component shapes, we run the recurrent
network for $T$ steps, in order to obtain shape latent codes for each shape:
$z_{1},\ldots,z_{T}$. We set $T$ to a fixed value, computed before training,
equal to the maximum number of components a graphic in our training dataset
can have. When a graphic requires fewer than $T$ shapes, the extra paths
produced by the RNN are degenerate and collapse to a single point; we discard
them before rendering.

In addition to the shape latent codes $z_{i}$, the recurrent network outputs
an unbounded scalar depth value $d_{i}$ for each path which is used by our
differentiable compositing module when rasterizing the shapes onto the canvas.

###  3.2 Predicting the number of path control points

Each path (shape) in our vector output can be made of a variable number of
segments. Figure 4(a) shows how the reconstruction loss decreases as we
increase the number of curve segments from 6-25, for multiple designs. It also
shows that, depending on the design’s complexity, not all paths need many
segments to be represented accurately. We train an auxiliary network
conditioned on a path latent variable $z_{t}$ to model the complexity–fidelity
trade-off and automatically determine the optimal number of segments for a
path. This auxiliary network has 3 fully connected layers. It outputs 3
parameters $a$, $b$, and $c$ of a parametric curve $x\mapsto ae^{-bx}+c$ that
approximates the loss graph of a given shape, with respect to the number of
segments. Given this parametric approximation, we allow the user to set the
quality trade-off as a threshold on the derivative of the parametric curve.
Specifically, we solve for $x$ in the derivative expression and round up to
obtain the number of segments to sample. This threshold defines what
improvement in the reconstruction error is worth the added complexity of an
additional Bézier segment. Please refer to our supplementary for more
information on the auxiliary network.

Figure 5:  Latent space correspondences. Im2Vec encodes shapes as deformation
of a topological disk. This naturally gives a point-to-point correspondence
between shapes across graphics design once we encode them in our latent space.
Graphics can be made of a single path (top), or multiple paths (bottom). In
both cases, our model establish meaningful geometric correspondences between
the designs, indicated by the blue–orange color coding.

###  3.3 Multi-resolution raster loss

Given a raster input image $I$, our model encodes the design into a global
latent code $z$, which the RNN decomposes into path latent codes
$z_{1},\ldots,z_{T}$. Our path decoder maps each path latent code to a closed
Bézier path. We rasterize each path individually, as a solid shape using the
differentiable rasterizer of Li et al. [22], and composite them together into
a final raster image $O$ using the differentiable compositing algorithm of
Reddy et al [28]. Since every step of the pipeline is differentiable, we can
compute a loss between input image $I$ and raseterized generated vector
graphic $O$, and backpropagate the error to train our model using gradient
descent.

When we differentiate $O$ with respect to the Bézier parameters, the gradients
have a small area of influence, corresponding to the support of the
rasterization prefiltering kernel. This adversely affects convergence
especially when the mismatch between $I$ and $O$ is high (e.g., at the early
stages of the training). We alleviate this issue by rasterizing our graphics
at multiple resolutions. That is, we render an image pyramid instead of a
single image, and aggregate the loss at each pyramid level. We obtain the
ground truth supervision for each level by decomposing the target image into a
Gaussian pyramid, where each level is downsampled by a factor 2 along each
dimension from the previous level. The gradients at the coarsest level are
more stable and provide a crucial signal when the images differ significantly,
while the fine-scale gradients are key to obtaining high spatial accuracy. The
loss we minimize is given by:

$\mathbb{E}_{I\sim\mathcal{D}}\sum_{l=1}^{L}\|\textit{pyr}_{l}(I)-O_{l}\|^{2},$
(1)

where $L$ is the number of pyramid levels, $\textit{pyr}_{l}(I)$ the $l$-th
pyramid level, $O_{l}$ our output rasterized at the corresponding spatial
resolution, and $\mathcal{D}$ the training dataset.

###  3.4 Shape correspondences by segmentation

When specializing a generative models to a single class, e.g., the same glyph
or digit across multiple fonts, it is often desirable that the model’s latent
space capture correspondences between parts of the instance, like the opening
in the capital letter ‘A’, or the eyes and mouth of an emoji face. To enable
this, we segment our raster training dataset using an automatic off-the-shelf
tool [20]. We cluster these segments across the dataset based on spatial
position, and assign to each cluster a unique RGB colour. This consistent
labeling helps learn a more interpretable latent space for purposes of
interpolation, but is not itself critical; we show in supplementary material
that our reconstruction is robust to inconsistent labeling thanks to the
differentiable compositing step.

###  3.5 Training details

We train our model end-to-end for 100 – 1000 epochs, using a batch size
between 2 – 256 and the Ranger optimizer [38] with learning rate between
$10^{-3}$ and $10^{-4}$, depending on the dataset. To evaluate path decoder’s
generalization to variable number of segments, we randomly chose the number of
segments $k\in\\{7,\ldots,25\\}$ at every iteration.

##  4 Evaluation

We demonstrate Im2Vec’s quantitative performance in 3 tasks: reconstruction,
generation, and interpolation. We compare it with raster based ImageVAE and
vector based SVG-VAE, DeepSVG on all the tasks.

Figure 6:  Reconstructions on Fonts. Our model, Im2Vec, captures complex
topologies and produces vector outputs. ImageVAE has good fidelity but
produces raster outputs with limited resolution (see Table 1). SVG-VAE and
DeepSVG produce vector outputs but often fail to accurately reproduce complex
fonts. All the methods were trained on the same set of fonts. Please use
digital zoom to better appreciate the quality of the vector graphics.

##### Reconstruction

We measure the reconstruction performance of the baselines and Im2Vec using
$L_{2}$ loss in image space. This quantifies how accurately the latent space
of the different methods captures the training dataset. Since SVG-VAE and
DeepSVG work in vector domain, we rasterize their vector estimates using
CairoSVG [4].

Table 1 shows reconstruction quality of the Im2Vec and other baselines on
Fonts [24], MNIST [21], Emojis [26], and Icons [6]. While vector based methods
have the advantage of being able reproduce the exact intended vector
parametrization, they are adversely effected by the non-linear relationship
between vector parameters and image appearance. Therefore what seems like a
small error in the vector parameters estimated by SVG-VAE and DeepSVG may
result in dramatic changes in appearance. Unlike vector domain methods, Im2Vec
is not affected by the objective mismatch between the vector parameter and
pixel spaces, thereby achieving significant improvement in the reconstruction
task.

Refer to our supplementary for a chamfer distance based reconstruction
comparison between SVG-VAE, DeepSVG and our method.

Table 1:  Reconstruction quality. Comparison of pixel-space reconstruction losses for various methods and datasets. Note that neither SVG-VAE nor DeepSVG operate on datasets without vector supervision.  | Fonts | MNIST | Emojis | Icons
---|---|---|---|---
ImageVAE | 0.0116 | 0.0033 | 0.0016 | 0.0002
SVG-VAE | 0.1322 | ✕ | - | -
DeepSVG | 0.0938 | ✕ | - | -
Im2Vec (Ours) | 0.0284 | 0.0036 | 0.0014 | 0.0003

We show qualitative comparisons of input shape reconstruction between methods
in Figures 6 and 7(a). We also show reconstruction output of Im2Vec on Emojis
and Icons in Fig. 8.

(a) reconstruction

(b) interpolation

Figure 7:  MNIST results. The MNIST dataset only provides raster data. Since
no vector graphics ground truth is available, neither SVG-VAE nor DeepSVG can
be trained on this dataset. We trained both ImageVAE and Im2Vec on the full
dataset, with no digit class specialization or conditioning. Our model
produces vector outputs, while ImageVAE is limited to low-resolution raster
images (top). Both models produce convincing interpolation (bottom).  Figure
8:  Reconstructions. Results on the Emojis and the Icons datasets. In each
case, we show the input image ($128\times 128$) and the corresponding vector
graphics output, which can be rasterized at arbitrary resolutions.

##### Generation and Interpolation

We present a random sample of font glyphs generated using Im2Vec in Figure 10.
A qualitative comparison of latent space interpolation between baselines and
Im2Vec is presented in Figures 9(b) and 7(b). We also present latent space
interpolation between 4 input images of Emojis and Icons in Fig. 9(a).

Table 2:  Generation and Interpolation quality. Results on the Fonts and the MNIST are more accurate than both previous techniques that require vector supervision, and an image-based baseline autoencoder.  | Generation | Interpolation
---|---|---
| Fonts | MNIST | Fonts | MNIST
ImageVAE | 0.171 | 0.058 | 0.184 | 0.072
SVG VAE | 0.206 | ✕ | 0.206 | ✕
DeepSVG | 0.210 | ✕ | 0.202 | ✕
Im2Vec (Ours) | 0.187 | 0.069 | 0.188 | 0.0872

To quantitatively evaluate our generation results with others, we quantify how
realistic the intermediate shapes in the latent shape as the average closest
distance between the intermediate shapes to any sample in the training
dataset:

$\sum_{O\in O_{G}}\operatorname{min}_{I\in\mathrm{dataset}}(\|I,O\|^{2}),$ (2)

where $O_{G}$ is the set of all generated shapes. We variationally sample 1000
shapes from all the methods and present the quality of the generated shapes in
Table 2.

We perform similar evaluation to quantify the quality of our interpolations.
For comparison we sample 4 evenly spaced interpolations between 250 random
pairs of images from the training dataset to create interpolation samples. The
results of the quality of interpolation between different methods is presented
in Table 2.

(a) Emojis and Icons interpolations using Im2Vec

(b) Comparison to baselines on Fonts

Figure 9:  Interpolations. Our learned latent space enables plausible
interpolation between samples. In 9(a), we show interpolations between
source–target pairs on the Emojis and Icons datasets. In 9(b) we show
interpolations on the Fonts dataset. Unlike previous work, Im2Vec enables
plausible interpolation even across significant changes in shape. For
instance, the stem of the digit ‘9’ naturally curls along the interpolation
path.  Figure 10:  Random samples. We show a random selection of digits
generated by Im2Vec. The latent space was trained on the full Fonts dataset.
Our model is capable of generating samples with significant topological
variations across the different font types. In the supplemental material, we
include 1000 random samples from the latent space. Please use digital zoom to
better evaluate the quality.

##  5 Limitations

Figure 11:  Limitations. Im2Vec is only supervised by an image-space loss, so
it can sometimes miss small topological features (Left), or produce
semantically meaningless or degenerate geometries (Right). While the former
can be resolved by providing higher resolution supervision, the later could be
mitigated by using local geometric priors.

The raster-based nature of the training imposes the principal limitations of
our method (see Figure 11). It is possible for some very fine features to
underflow the training resolution, in which case they may be lost. This could
be addressed by increasing the resolution at the expense of computational
efficiency, or perhaps by developing a more involved image-space loss.
Secondly, in particularly difficult cases it is possible for the generated
shape to go to a local optimum that contains degenerate features or
semantically non-meaningful parts which nonetheless still result in a
plausible rasterised image. This is a consequence of lack of vector
supervision, but could possibly be addressed by imposing geometric constraints
on the generated paths.

##  6 Conclusion

We presented Im2Vec as a generative network that can be trained to produce
vector graphics output of varying complexity and topology using only image
supervision, without requiring vector sequence guidance. Our generative setup
supports projection (i.e., converting images to vector sequences), sampling
(i.e., generating new shape variations directly in vector form), as well as
interpolation (i.e., morphing from one vector sequence to another, even with
topological variations). Our evaluations show that Im2Vec achieves better
reconstruction fidelity compared to methods requiring vector supervision.

We hope that this method can become the fundamental building block for neural
processing of vector graphics and similar parametric shapes.

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Renada Konstantinova-Antova, Stefan Georgiev, Agnes Lebre et al.
11institutetext: Institute of Astronomy and NAO, Bulgarian Academy of
Sciences, BG-1784, 72 Tsarigradsko shosse, Sofia, Bulgaria 22institutetext:
LUPM, University of Montpellier, CNRS, Montpellier, France 33institutetext:
Department of Astronomy, University of Geneva, Chemin des Maillettes 51,
CH-1290 Versoix, Switzerland 44institutetext: IRAP, Université de Toulouse,
CNRS, Observatoire Midi Pyrénés, 57 Avenue d’Azereix, 65008 Tarbes, France
44email<EMAIL_ADDRESS>Review, Submitted on xx.xx.xxxx; Accepted on
xx.xx.xxxx

#  The origins of the magnetic field in tip-RGB, AGB and post-AGB stars

Renada Konstantinova-Antova 11 Stefan Georgiev 1122 Agnès Lèbre 22 Corinne
Charbonnel 3344 Ana Palacios 22 Michel Aurière 44

###### Abstract

During the last decade and a half, the new generation spectropolarimeter
Narval at Pic du Midi, France allowed the study of weak magnetic fields in
cool giant stars that are fairly evolved after main sequence. We present a
short summary on the recent knowledge on the magnetic fields and activity in
giants situated in the upper right part of the Hertzsprung-Russel (H–R)
diagram and discuss on the possible mechanisms for magnetic field generation
in the asymptotic giants branch (AGB) and post-AGB stars.

###### keywords:

magnetic activity AGB post-AGB stars

## Introduction

When a star evolves after the main sequence, it expands. A convective envelope
begins to develop in intermediate mass stars, and in low-mass stars the
existing one deepens toward the stellar core. The stars ascending the red
giant branch (RBG) possess vigorous convection in a deep convective envelope.
Their internal structure is a complex one, consisting of a hydrogen burning
shell and a contracting core. After the star reaches the tip of the RGB, the
He-burning follows. Later on, after the He–burning phase, the star continues
to climb up and right on the Herzsprung-Russel (H–R) diagram, on the
asymptotic giant branch (AGB) and its structure becomes even more complex,
with hydrogen and helium burning shells and contracting core.

During the course of evolution, at certain phases additional angular momentum
as a result of core-envelope interaction or eventual planet engulfment could
speed up the stellar rotation and operation of an $\alpha$ \- $\omega$ dynamo
could become possible there (Schröder & Konstantinova-Antova, 2022). Also,
when the star crosses the instability strip, pulsations begin to occur in its
envelope and propagate trough the atmosphere. They could compress the existing
weak magnetic field (MF) at certain phases of the pulsation.

We present here the first findings on how all these structural changes in tip-
RGB, AGB and post-AGB stars are related with their magnetic fields and
activity. This knowledge is based on more than 15 years of study of the
magnetic fields and activity in such stars in the framework of a Bulgarian-
French collaboration using the spectropolarimeter Narval at TBL, Pic du Midi,
France (Aurière, 2003) and the Least Square Deconvolution (LSD) method (Donati
et al., 1997), that enabled detection of weak magnetic fields of the order of
1 G and below in cool stars, as a result of averaging more than 10 000
absorption lines in their spectrum.

##  1 First studies of magnetic field in tip-RGB and AGB stars. Discovery of
the second magnetic strip on the H-R diagram where $\alpha-\omega$ dynamo
operates.

Until a decade ago, magnetic fields in evolved giants situated at the tip of
the RGB, AGB and beyond on the H–R diagram were poorly studied. Some
suspicions that M giants could be magnetically active appeared in Hünsch et
al. (1998) on the basis of enhanced X-ray emission in four M giants. Herpin et
al. (2009) found evidence for a MF in the circumstellar envelopes of four AGB
stars on the basis of IRAM-30m observations. From a theoretical point of view,
Soker (2000), Soker & Zoabi (2002), Brandenburg & Blackman (2002) and Nordhaus
et al. (2008) predicted that magnetic dynamo of different types could operate
in AGB stars.

We started our study with a sample of 9 apparently single M giants selected
for their faster rotation and/or X-ray emission in 2008. The first Zeeman-
detected M giant is EK Boo = HD 130144, a fast rotator ($V\sin i$ = 8.5 ) with
an unusual X-ray emission. A longitudinal magnetic field ($B_{l}$, the
magnetic field component projected along the line of sight) reaching up to 8 G
was measured for it (Konstantinova-Antova et al., 2010). Later on, 6 more M
giants from this sample were also Zeeman-detected and weak magnetic fields of
the order of 1 G were measured for them with one exception, the M5 giant RZ
Ari, which possesses a magnetic field even stronger than EK Boo
(Konstantinova-Antova et al., 2013). All these objects are tip-RGB or AGB
stars (Konstantinova-Antova et al., 2010). An interesting fact for them is
that their magnetic field was not easily detected, because they display both
periods of detection and periods of non-detection, contrary to the magnetic G
and K giants, that begin to climb the red giant branch (RGB) or those in the
He-burning phase (Aurière et al., 2015). A typical example of a star
exhibiting alternating periods of detection and non-detection is EK Boo. Its
long-term MF and activity indicators behavior is presented in Figure 1
(Georgiev et al., 2020). In this figure, different symbols represent the
different levels of magnetic field detection based on a reduced $\chi^{2}$
test calculated from the LSD profiles as described in (Donati et al., 1997).
The results of this test are converted into a False Alarm Probability (FAP),
based on which three different levels of detection are introduced: definite
detection (DD, FAP $<~{}10^{-4}$), marginal detection (MD, $10^{-2}>$ FAP
$>~{}10^{-4}$) and no detection (ND, FAP $>~{}10^{-2}$. These periods of MF
detection and non-detection in the studied M giants could be a result of the
convective cells lifetime (Freytag et al., 2017). These cells are believed to
be the transporter of the magnetic field.

Figure 1: Longitudinal magnetic field $B_{l}$ and line activity indicators
variability in EK Boo. Periods of MF detection and non-detection are visible
(filled and open symbols, respectively). Radial velocity long-term variation
indicates for a companion at a distant orbit. Such one is not visible in the
stellar spectrum and is likely a dwarf. Figure from Georgiev et al. (2020).

However, this first sample is a biased one and cannot answer the question of
what percentage of the tip-RGB and AGB stars are magnetically active. For this
purpose we decided to study all such stars up to 4th magnitude in V-band in
the Solar vicinity which are not binaries and are available for observations
at Pic du Midi observatory, France. We limited the brightness up to 4th
magnitude because we wanted to be able to perform a deep study, down to 0.2 G
accuracy. In this way, we studied 17 tip-RGB and AGB stars and about 60
percent of them were Zeeman-detected (Konstantinova-Antova et al., 2014). Weak
magnetic fields of a few Gauss and even at the sub-gauss level were detected
for them. It appeared that all these stars form a strip on the H–R diagram
which we called “second magnetic strip” (Figure 2). The first magnetic strip
is observed in G and K giants, situated at the base of the RGB and in the He-
burning phase (Aurière et al., 2015). Later on, Charbonnel et al. (2017) show
with evolutionary models that a dynamo of the $\alpha-\omega$ type could be
the explanation for the M giants clumping in a strip (and earlier, in the
first magnetic strip for the G and K giants situated at the base of the RGB
and the He-burning phase). They found that these areas on the H–R diagram are
favorable for the $\alpha-\omega$ dynamo, because of the conditions in the
convective envelopes, in combination with the stellar rotation that enable a
Rossby number, $R_{o}$ ($R_{o}$ = $P_{rot}$/$\tau_{conv}$, where $P_{rot}$ is
the rotation period of the star and $\tau_{conv}$ is the convective turnover
time) less than one there and, hence, an efficient action for this type of
dynamo.

Figure 2: Situation of the Zeeman detected giants from Solar vicinity sample
on the H-R diagram. Longitudinal magnetic field $B_{l}$ strength is designated
by symbols with different size. Open circles designate stars with no
detection. Dashed lines mark the first dredge-up phase. Colors stand for the
maximal convective turnover time at each point of the evolutionary tracks.
Figure from Konstantinova-Antova et al. (2014). The locus of the second
”magnetic strip” is designated by an ellipse.

##  2 Cool giants with magnetic activity outside the magnetic strips

Besides the case of $\alpha-\omega$ dynamo operation in giants situated in the
first and second magnetic strips and depending mostly on rotation and the
convective envelope properties (like the convective turnover time,
$\tau_{conv}$), a few fairly evolved giants outside the magnetic strips were
found. These are mostly AGB and post-AGB stars with a magnetic field of a few
Gauss. Their magnetic field cannot be explained by the $\alpha-\omega$ dynamo,
because of the big Rossby number, $R_{o}$. A different type of dynamo, or even
different mechanism for the magnetic field production, could operate in these
giants.

###  2.1 Pulsating magnetic giants

Some AGB and post-AGB stars are known to exhibit radial pulsations, which
generate strong shockwaves - mechanical waves with velocities larger than the
sound velocity in the given medium. These shockwaves propagate through the
stellar atmosphere, causing a sharp change in temperature, pressure and
density at the shock front. As the shock propagates, it creates complex
ballistic motions in its host environment, causing different layers to move
with different velocities and in different directions with respect to the
stellar restframe, which in turn affects the spectral lines that form there.
Layers affected by the shock are initially caused to move away from the
photosphere and towards the observer, resulting in blueshifted (with respect
to the restframe of the star) spectral lines; after sufficient amount of time
has passed since the shock’s appearance, these same layers then begin to fall
ballistically towards the photosphere and away from the observer, resulting in
redshifted spectral lines.

The first detection of a surface magnetic field as a result of shockwave
propagation was in a Mira star (in the AGB evolutionary stage) was reported by
Lèbre et al. (2014). They measured a longitudinal magnetic field $B_{l}$ of
2-3 G for the star $\chi$ Cyg from spectropolarimetric observations with
Narval. Lèbre et al. (2014) found that in $\chi$ Cyg, the circularly polarized
signatures found in the LSD profiles obtained from high resolution spectra of
the star are clearly associated to the blue lobe of the atomic spectral lines,
i.e. are being accelerated by the shockwave. From this observation, the
authors concluded that the shockwaves propagating in the atmospheres of Mira
stars may be locally amplifying the weak surface magnetic field already
present in the affected atmospheric layers. An example is shown in Figure 3
(from Lèbre et al. 2014), where the LSD profile of an observation of $\chi$
Cyg near maximum light is displayed, and it can be seen that the Stokes V
profile is clearly associated to the blueshifted lobe, which is related to the
matter affected by the shock.

Figure 3: LSD profile of $\chi$ Cyg, computed from Narval observations taken
in March 2012. From bottom to top, the three panels in the figure show the
intensity profile (Stokes $I$), the diagnostic null profile and the circularly
polarized profile (Stokes $V$). It can be seen that the signal in the Stokes
$V$ profile is clearly associated to the blue wing of the intensity profile.
Figure from Lèbre et al. (2014).

Later, Sabin et al. (2015) reported the first detections of a weak magnetic
field in two pulsating post-AGB stars, R Sct and U Mon. Lèbre et al. (2015)
found indications of the same connection between the observed surface magnetic
field and the atmospheric dynamics in R Sct, which is a pulsating variable of
the RV Tau type. The authors observed R Sct in circular polarization with
Narval and discovered that when circularly polarized profiles were present in
the LSD spectrum of the star, they were always associated to the blue lobe of
the intensity profile, as in $\chi$ Cyg.

To investigate the hypothesis of a connection between the atmospheric dynamics
and surface magnetism in pulsating post-AGB stars, Georgiev et al. (2023)
present the longest monitoring of the pulsating variable star R Sct done using
high resolution spectropolarimetry. The authors analyze observations of R Sct
obtained with Narval on 67 different nights in the period between July 2014
and August 2019, of which 31 observations contain Stokes V measurements that
allow studying the surface magnetic field. The authors confirm the stellar and
Zeeman origin of the observed circularly polarized profiles, and show that
their timescale of variation is similar to that of the atmospheric dynamics.
Using a new approach for the calculation of LSD profiles for this star,
Georgiev et al. (2023) constrain the LSD analysis to the lowest parts of the
stellar atmosphere and are able to study the longitudinal magnetic field at
the level of the photosphere, where the shock is just emerging. The authors
found that for all observations where the LSD Stokes I profile displays a
double peak (indicating that the atmosphere is strongly affected by a
shockwave), the Stokes V signature, if present, is associated to the blue
intensity lobe in radial velocity space.

In Figure 4 (from Georgiev et al. 2023), the temporal evolution of $B_{l}$ of
R Sct between 2014 and 2019 is presented together with the visual lightcurve
(from AAVSO). In this figure, it can be seen that the surface magnetic field
varies with time. However, there is no clear correlation between the field
strength and the photometric phase of the star, suggesting that the
variability observed in the longitudinal magnetic field is not entirely caused
by the propagation of shockwaves and the resulting local amplification of the
surface magnetic field, but also by a variable and non-homogeneous magnetic
structure at the surface level of R Sct.

Figure 4: Temporal evolution of the longitudinal component of the magnetic
field $B_{l}(t)$ (upper panel) and lightcurve (bottom panel) of R Sct. In the
upper panel, the different symbols represent the different types of
detections, described in Section 1. The red vertical lines in the bottom panel
indicate the time of Narval observations that contain Stokes V measurements,
and the red star indicates the date considered as the start of reference
period $P_{0}$ (see Section 2.1 of Georgiev et al. 2023). The dashed vertical
red lines indicate the approximate beginning and end of the irregular part of
the lightcurve. Figure and caption adapted from Georgiev et al. (2023)

###  2.2 The role of planet engulfment for magnetic field generation

When a star expands during the giant stages, its radius could reach the orbit
of an eventual planet and cause the planet to be engulfed. As a result, an
increase of the rotation rate of the star and change in its chemical
composition (especially Lithium) occurs. The first idea of the result of
planet engulfment in cool giants is presented in Siess & Livio (1999). Later,
more studies were presented by different authors, among them the detailed
models by Privitera (2016) and Privitera et al. (2016a, b) who modeled the
rotation rate from the base of the RGB up to the AGB for stars of different
mass in the case of planet engulfment. According to them, after this event the
giants should rotate faster for some interval of time. The degree to which the
rotation speeds up depends on the mass and orbital momentum of the engulfed
companion, a planet or brown dwarf, and also on the mass of the star. Since
for each star the planetary orbits are situated at different distances, the
engulfment could occur at any different evolutionary stage on the giant
branches. Hence, the type of dynamo and its efficiency depend on the
properties of the convective envelope and the rotation there. In the case
where the Rossby number is less than 1, operation of the $\alpha-\omega$
dynamo is possible. Otherwise, other types of dynamo could operate there, like
the $\alpha^{2}-\omega$ one, or the turbulent dynamo (which does not depend on
the rotation). The strength of the magnetic field depends on the dynamo type.
For example, it was shown by Soker & Zoabi (2002) that the turbulent dynamo
could produce a surface magnetic field of the order of one Gauss in cool
giants. In any case, our experience up to now shows that the most effective
dynamo in evolved cool giants is the $\alpha-\omega$ dynamo (Schröder &
Konstantinova-Antova, 2022). The loci on the H–R diagram where this type
dynamo operates are shown in Konstantinova-Antova et al. (2014) and in Aurière
et al. (2015) and are supported by the theoretical models in Charbonnel et al.
(2017). They explain the so-called “first and second magnetic strips” on the
giant branches with the convective envelope properties, namely the large
convective turnover time. Together with the rotation in intermediate mass
stars in these strips on the H–R diagram, these values of $\tau_{conv}$ yield
a Rossby number less than 1 and hence, conditions for an efficient
$\alpha-\omega$ dynamo action. If a giant inside a magnetic strip engulfed a
planet, it will gain additional rotation and will appear as an extremely
magnetically active star. However, a gain of angular momentum could also come
from the inside during the dredge-up phases at the bottom of the RGB and after
the He-burning, i.e. early-AGB (Schröder & Konstantinova-Antova, 2022). Hence,
M-giants with faster rotation and strong magnetic field that are outside the
second magnetic strip could indicate for a planet engulfment episode.

A typical example for magnetic activity triggered by planet engulfment is the
M5 giant RZ Ari (Konstantinova-Antova et al., 2024). It seems to be a single
early AGB star outside the second magnetic strip (Figure 5). What is unusual
in this star is its faster rotation ($V\sin i$ = $6\kms$, $P_{rot}$ = 530
days) and higher Lithium content (A(Li) = 1.2) that could not be explained by
the theory of stellar evolution. A variable surface magnetic field on the
order of a few Gauss up to 14 Gauss was Zeeman detected. It could not be
explained in the term of $\alpha-\omega$ dynamo, because the Rossby number is
of the order of 20. However, a different type of dynamo, like
$\alpha^{2}-\omega$ which is efficient for $R_{o}$ larger than 1 could operate
there. These results are also in a good agreement with the models for planet
engulfment by Privitera et al. (2016a). Nevertheless, RZ Ari is a semiregular
pulsating star with a secondary period of about 480 days, and no relation
between the pulsations and the magnetic field was found. The situation is
similar in two other magnetic giants, EK Boo and $\beta$ Peg (Georgiev et al.,
2020). RZ Ari was also studied in search for the presence of large convective
cells, because of the long periods in the magnetic field and photometric
datasets of about 1200 days that could not be explained by rotational
modulation (Konstantinova-Antova et al., 2024). No evidence for linear
polarization was found in this star in 2015 when clear Zeeman detections and
magnetic field of a few Gauss were observed. In the future, interferometry
could shed more light on the surface structure and the existence of large
convective cells. In any case, if large convective cells exist in RZ Ari, an
eventual local dynamo is not the dominant reason for the magnetic field and
observed activity in this giant (Konstantinova-Antova et al., 2024).

Figure 5: Situation of RZ Ari on the H–R diagram. Colors stand for the
convective turnover time at each point of the evolutionary tracks. Figure from
Konstantinova-Antova et al. (2024). The locus of the second ”magnetic strip”
is designated by an ellipse.

Another giant candidate for planet engulfment is IRAS 12556–7731 (Alcalá et
al., 2011), where planet engulfment is also suspected to cause its relatively
fast rotation and high lithium content. However, its magnetic field and
activity are not yet studied.

## 3\. The global picture. Conclusions.

After more than a decade and a half of studies of the magnetic field in giants
after the He-burning phase, two main mechanisms for the magnetic activity are
evident: 1) dynamo of different types and 2) compression of a weak magnetic
field as a result of shockwave propagation during pulsation. The dynamo should
be initiated due to different reasons: evolutionary (the second magnetic
strip) as it is in the models for the convective envelope and rotation, and
$R_{o}$ by Charbonnel et al. (2017), or by the event of a planet engulfment
that speeds up the surface rotation and could also initiate a dynamo. Since
the engulfment could happen anywhere on the H–R diagram, the easiest way to
find such stars is to look for magnetic giants outside the magnetic strips.
Such a giant is already found – RZ Ari. It also has a high Lithium abundance
in support of the planet engulfment hypothesis. While in the case of the
second magnetic strip an $\alpha-\omega$ dynamo is possible, outside the
strips (both the first and the second one) different kinds of dynamo could
operate, depending on the properties of the convective envelope and the
orbital momentum gained from the engulfed planet.

On the other hand, strong shocks during pulsations could compress the existing
weak magnetic field. As a result, a magnetic field of a few Gauss is observed
in some pulsating Mira-type stars (Lèbre et al., 2014) and in post-AGB stars
(Sabin et al., 2015; Georgiev et al., 2023). No magnetic field compression is
observed in the studied early AGB stars (see Sect. 2.2). Perhaps the
shockwaves are not sufficiently strong there. The absence of linear
polarization in the spectra of $\beta$ Peg and RZ Ari, two semi-regular
pulsating variables, is also in support of this assumption (Georgiev et al.,
2020; Konstantinova-Antova et al., 2024).

It remains unclear, however, if a local dynamo like the one operating in
Betelgeuse (Mathias et al., 2018) could operate in these intermediate mass AGB
and post-AGB stars. Large convective cells are observed in the 2 M${}_{\sun}$
AGB star $\pi^{1}$ Gru by means of interferometry (Paladini et al., 2018).
However, in our spectropolarimetric studies we do not detect linear
polarization in the few AGB stars we observed. In the case of RZ Ari, the star
from our selection studied in the most detail, simultaneous linear and
circular polarization observations in 2015 did not confirm a local dynamo
operation. However, long periods of the order of 1200 days exist in the
magnetic field and in the photometry datasets (Konstantinova-Antova et al.,
2024). These periods could be explained by the lifetime of giant convective
cells, according to (Freytag et al., 2017). If large convective cells exist in
these giants, then we can assume that they are not as large as in the
supergiant Betelgeuse, and so if a local dynamo does operate there, it is of
smaller efficiency and does not dominate the magnetic field we detect. In the
future, interferometric observations of these giants could reveal their
surface structure. Further interferometric and spectropolarimetric studies are
highly desired.

Further studies could expand or modify this first knowledge on the mechanisms
for MF and activity in ABG and post-AGB stars and their magnetic properties.
However, such studies will take a decade or even more. Nevertheless, they
deserve the efforts, because the magnetic activity in these stars is poorly
studied and the results discussed above are just the beginning of an exiting
new era in magnetism in fairly evolved stars.

## Acknowledgements

We thank the TBL team for service observations during all these years. Thanks
also to the French PNPS program for the observations after 2014, to the
OPTICON for the observations in semesters 2008B and 2011B and to the Bulgarian
NSF grant DSAB 02/3 for the observations in 2010. Part of the study was
carried out under Bulgarian-French collaboration (program RILA) and Bulgarian
NSF project DN 18/2. SG and AL thank the University of Montpellier for its
support through the I-SITE MUSE project MAGEVOL.

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# Triangulation complexity and systolic volume of hyperbolic manifolds

L. Chen  School of Mathematics and Statistics, Lanzhou University
Lanzhou 730000, P.R. China<EMAIL_ADDRESS>
###### Abstract.

Let $M$ be a closed $n$-manifold with nonzero simplicial volume. A central
result in systolic geometry from Gromov is that systolic volume of $M$ is
related to its simplicial volume. In this short note, we show that systolic
volume of hyperbolic manifolds is related to triangulation complexity. The
proof is based on Jørgensen and Thurston’s theorem of hyperbolic volume.

###### Key words and phrases:

Complexity, systolic volume, hyperbolic manifolds

###### 2020 Mathematics Subject Classification:

Primary 57K31, Secondary 53C23.

##  1\. Introduction

In systolic geometry, the systole of a Riemannian manifold is defined to be
the shortest length of a noncontractible loop. Gromov’s systolic inequality
implies that systole of many manifolds (including hyperbolic manifolds) is
bounded from above by Riemannian volume. The optimal constant in systolic
inequality is usually called systolic volume in literature. Let $M$ be a
closed $n$-dimensional manifold with nonzero simplicial volume. Gromov [Gro83,
Section 6.4.D] proved that systolic volume is related to the topological
complexity of $M$. More explicitly, Gromov proved that systolic volume relates
to simplicial volume and other topological invariants. In this paper, we
verify that systolic volume of hyperbolic manifolds is related to
triangulation complexity. Triangulation complexity of a closed manifold is
defined to be the minimum number of simplices in a triangulation. Hence the
triangulation complexity naturally represents how complicated a manifold is.
Our result is a supplement of Gromov’s work.

Let $M$ be a closed $3$-manifold. If $M$ is irreducible and not homeomorphic
to $S^{3},\mathbb{R}\mathbb{P}^{3}$ or $L(3,1)$, the triangulation complexity
coincides with the complexity defined by Matveev [Mat90]. In the following, we
use $c(M)$ to denote the triangulation complexity of a closed $3$-manifold
$M$. We refer to [JRT13, JRST20, LP21] for recent developments of
triangulation complexity of 3-manifolds. The systolic volume of $M$, denoted
by $\operatorname{\operatorname{\mathsf{SR}}\hskip 0.55557pt}(M)$, is defined
to be

$\inf_{\operatorname{\operatorname{\mathcal{G}}}}\frac{\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.55557pt}_{\operatorname{\operatorname{\mathcal{G}}}}(M)}{\operatorname{\operatorname{\mathsf{Sys}}\hskip
0.55557pt}\pi_{1}(M,\operatorname{\operatorname{\mathcal{G}}})^{n}},$

where the infimum is taken over all Riemannian metrics
$\operatorname{\operatorname{\mathcal{G}}}$ on $M$. The systolic volume
$\operatorname{\operatorname{\mathsf{SR}}\hskip 0.55557pt}(M)$ is positive if
$M$ is a closed essential $3$-manifold, see [Gro83, Theorem 0.1.A.].

Let $s_{0}$ be any sufficiently large positive number.

######  Theorem 1.1.

Suppose that $M$ is a closed hyperbolic $3$-manifold with
$\operatorname{\operatorname{\mathsf{SR}}\hskip 0.55557pt}(M)\leqslant s_{0}$.
There exists a positive constant $C(s_{0})$ only depending on $s_{0}$, such
that

$\operatorname{\operatorname{\mathsf{SR}}\hskip 0.55557pt}(M)\geqslant
C(s_{0})\hskip 1.11115pt\frac{c(M)}{\log^{3}{c(M)}}.$

Simplicial volume $\|M\|$ of an oriented closed connected $n$-dimensional
manifold $M$ is a homotopy invariant defined by Gromov [Gro82]. The definition
is

$\|M\|=\left\\{\left.\sum_{i=1}^{k}|r_{i}|\right|\sum_{i=1}^{k}r_{i}\sigma_{i}\,\text{represents
the fundamental class}\,[M]\right\\},$

where the infimum is taken over all real singular chains representing the
fundamental class of $M$. Moreover, theorem of Gromov and Thurston implies
that simplicial volume of hyperbolic $n$-manifolds $M$ ($n\geqslant 2$)
satisfies

(1.1) $\nu_{n}\hskip
1.11115pt\|M\|=\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.55557pt}_{\operatorname{\operatorname{\mathsf{hyp}}\hskip 0.3889pt}}(M),$

where $\nu_{n}$ is a positive constant only depending on $n$. Gromov indicated
that systolic volume is related to simplicial volume.

######  Theorem 1.2 (Gromov, see [Gro83, Section 6.4.D.] and [Gro96, Section
3.C.3.]).

Let $M$ be a closed $n$-dimensional manifold with non-zero simplicial volume.
Then the systolic volume $\operatorname{\operatorname{\mathsf{SR}}\hskip
0.55557pt}(M)$ of $M$ satisfies

(1.2) $\|M\|\leqslant C_{n}\operatorname{\operatorname{\mathsf{SR}}\hskip
0.55557pt}(M)\hskip 1.11115pt\log^{n}{\left(C_{n}^{\prime}\hskip
1.11115pt\operatorname{\operatorname{\mathsf{SR}}\hskip
0.55557pt}(M)\right)},$

where $C_{n}$ and $C_{n}^{\prime}$ are two positive constants only depending
on $n$.

######  Remark 1.3.

In [Gro83, Section 6.4.D.], there is a typo of missing the exponent $n$ in the
logarithm part of (1.2). In literature the estimate (1.2) in Theorem 1.2 is
also written to be

$\operatorname{\operatorname{\mathsf{SR}}\hskip 0.55557pt}(M)\geqslant
C_{n}\frac{\|M\|}{\log^{n}{\|M\|}},$

where $C_{n}$ is a positive constant only depending on $n$, see [BK19, Section
4.3.].

Theorem 1.2 builds a bridge between systolic geometry and hyperbolic geometry.
We refer to [Gut10] for more explanations about this interplay. Let $M$ be a
hyperbolic manifold with volume $V$. Gromov and Thurston’s theorem (see (1.1))
yields that any triangulation of $M$ has at least $C\cdot V$ number of
simplices for some constant $C$. Therefore, Gromov’s theorem (Theorem 1.2)
implies relation between topological complexity and systolic volume. More
explicitly, if a hyperbolic manifold is topologically complicated, the
systolic volume also should be sufficiently large. Our results in this paper
can be seen as an attempt to interpret this relation with more details.

The main technique used in the proof of Theorem 1.1 is the connection between
triangulation and volume for hyperbolic manifolds. Jørgensen and Thurston
proved that any complete hyperbolic $3$-manifold admits a triangulation with
the number of tetrahedra bounded from above by its volume. A detailed proof of
this theorem is provided by Kobayashi and Rieck in [KR11].

The triangulation complexity of manifolds in higher dimensions is studied in
[FFM12]. Let $M$ be a closed manifold of dimension $n$. The triangulation
complexity of $M$, denoted $\sigma(M)$, is defined to be the minimum number of
$n$-simplices in any triangulation of $M$. When $n=3$, $\sigma(M)$ coincides
with $c(M)$.

######  Theorem 1.4.

Let $M$ be a closed hyperbolic manifold of dimension $n$ with $n\geqslant 4$.
The triangulation complexity $\sigma(M)$ and systolic volume of $M$ is related
by

$\operatorname{\operatorname{\mathsf{SR}}\hskip 0.55557pt}(M)\geqslant
D_{n}\frac{\sigma(M)}{\log^{n}{\sigma(M)}},$

where $D_{n}$ is a positive constant only depending on $n$.

######  Remark 1.5.

It is proved in [FFM12] that under assumptions of Theorem 1.4,
$\|M\|<\sigma(M)$ holds. Hence above theorem is indeed a generalization of
Gromov’s theorem (Theorem 1.2).

The embolic volume defined in terms of injectivity radius is another geometric
quantitiy representing the topological complexity of manifolds. We refer to
[Ber03, Section 11.2.3.] for a general description.

######  Definition 1.6.

The embolic volume of $M$, denoted
$\operatorname{\operatorname{\mathsf{Emb}}\hskip 0.55557pt}(M)$, is defined to
be

$\inf_{\operatorname{\operatorname{\mathcal{G}}}}\frac{\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.55557pt}_{\operatorname{\operatorname{\mathcal{G}}}}(M)}{\operatorname{\operatorname{\mathsf{Inj}}\hskip
0.55557pt}(M,\operatorname{\operatorname{\mathcal{G}}})^{n}},$

where the infimum is taken over all Riemannian metrics
$\operatorname{\operatorname{\mathcal{G}}}$ on $M$.

Embolic volume is positive for all compact $n$-manifolds $M$ ($n\geqslant 2$),
see [Ber03, Section 7.2.4.]. Note that on a Riemannian manifold
$(M,\operatorname{\operatorname{\mathcal{G}}})$,
$\operatorname{\operatorname{\mathsf{Sys}}\hskip
0.55557pt}\pi_{1}(M,G)\geqslant
2\operatorname{\operatorname{\mathsf{Inj}}\hskip
0.55557pt}(M,\operatorname{\operatorname{\mathcal{G}}})$. Hence we always have
$\operatorname{\operatorname{\mathsf{Emb}}\hskip
0.55557pt}(M)\geqslant\operatorname{\operatorname{\mathsf{SR}}\hskip
0.55557pt}(M)$. Then for closed manifolds with nonzero simplicial volume,
Theorem 1.2 implies that the embolic volume is related to simplicial volume.
This result is proved by Katz and Sabourau [KS05] by a different approach.
Moreover, a direct implication of Theorem 1.4 is that any closed hyperbolic
$n$-manifold with $n\geqslant 4$ has

(1.3) $\operatorname{\operatorname{\mathsf{Emb}}\hskip 0.55557pt}(M)\geqslant
D_{n}\hskip 1.11115pt\frac{\sigma(M)}{\log^{n}{(\sigma(M))}},$

where $D_{n}$ is the positive constant in Theorem 1.4 and only depending on
$n$. In this note, we also show that embolic volume of any closed $n$-manifold
is related to its triangulation complexity. Compared with Katz and Sabourau’s
above result, our result is more general, since it includes all closed
manifolds with zero simplicial volume.

######  Theorem 1.7.

Let $M$ be a closed $n$-dimensional manifold. Then there exists a positive
constant $E_{n}$ only depending on $n$, such that

(1.4) $\operatorname{\operatorname{\mathsf{Emb}}\hskip 0.55557pt}(M)\geqslant
E_{n}\hskip 1.11115pt\sqrt{\sigma(M)}.$

### Organization

This short note is organized as follows: In Section 2, we discuss Jørgensen
and Thurston’s theorem of hyperbolic 3-manifolds. Proof of Theorem 1.1 is
given in this section. The case of hyperbolic manifolds of high dimensions is
contained in Section 3. Then Theorem 1.4 is proved. Section 4 concerns embolic
volume and triangulation complexity. Proof of Theorem 1.7 is included in this
section.

### Acknowledgement

This work is supported by the Young Scientists Fund of NSFC (award No.
11901261). Part of this work was done when the author was visiting Tianyuan
Mathematical Center in Southwest China. The author wishes to thank Professor
Bohui Chen for his invitation and support.

##  2\. Triangulation and volume of hyperbolic 3-manifolds

We prove Theorem 1.1 in this section. The proof is based on Jørgensen and
Thurston’s theorem of hyperbolic 3-manifolds. Jørgensen and Thurston’s work
([Thu79], also see [KR11]) implies that triangualtion of a hyperbolic
$3$-manifold is related to its volume.

######  Theorem 2.1 (Jørgensen, Thurston).

Let $M$ be a closed hyperbolic $3$-manifold, and $a_{0}$ be a positive
constant. Assume that $\operatorname{\operatorname{\mathsf{Inj}}\hskip
0.55557pt}(M,\operatorname{\operatorname{\mathsf{hyp}}\hskip
0.55557pt})\geqslant a_{0}$. Then there exists a triangulation of $M$, with
the number $t$ of tetrahedra satisfying

$t\leqslant K\hskip 1.11115pt\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.55557pt}_{\operatorname{\operatorname{\mathsf{hyp}}\hskip 0.3889pt}}(M),$

where $K$ is a positive constant only depending on $a_{0}$.

###### Proof.

For completeness, we show an outline of the proof. More details can be found
in [Thu79, Chapter 5], [KR11] and also [MP19, Section 4.2].

Let $R=\frac{1}{2}\operatorname{\operatorname{\mathsf{Inj}}\hskip
0.55557pt}(M,\operatorname{\operatorname{\mathsf{hyp}}\hskip 0.55557pt})$.
Assume that $X$ is a maximal subset of points in $M$, so that any two of them
having distance at least $R$. The set $X$ is maximal under inclusion. Voronoi
cell associated to $x_{0}\in X$ is defined to be the the subset

$V(x_{0})=\\{y\in M|dist(y,x_{0})\leqslant dist(y,x),\,\text{for any}\,x\in
X\,\text{and}\,x\neq x_{0}\\}.$

Since the ball $B(x_{0},\frac{R}{2})\subset V(x_{0})$, the total number of
Voronoi cells is at most
$\frac{\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.3889pt}(M)}{\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.3889pt}(B(x_{0},\frac{R}{2}))}$, and thus bounded from above by
$C(a_{0})\operatorname{\operatorname{\mathsf{Vol}}\hskip 0.55557pt}(M)$, with
$C(a_{0})$ a constant only depending on $a_{0}$. After triangulating each
Voronoi cell, we get a triangulation of $M$ with the number $t$ of tetrahedra
bounded from above by $K\cdot\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.55557pt}(M)$, where the constant $K$ is a multiplicative number of
$C(a_{0})$. ∎

Sabourau proved that there are only finitely many hyperbolic $3$-manifolds
with bounded systolic volume.

######  Theorem 2.2 (see [Sab07, Theorem B]).

For a sufficiently large positive number $s_{0}$, there are only finitely many
hyperbolic $n$-manifolds ($n\geqslant 3$) $M$ with
$\operatorname{\operatorname{\mathsf{SR}}\hskip 0.55557pt}(M)\leqslant s_{0}$.

######  Remark 2.3.

When $n\geqslant 4$, Theorem 2.2 is a direct implication of Wang’s finiteness
theorem and Theorem 1.2. Sabourau showed the case of $n=3$ with a different
approach. When $n=2$, the theorem is not true, since on each closed orientable
surface of genus $g\geqslant 2$, there exist infinitely many pairwise non-
isometric hyperbolic metrics.

Proof of Theorem 1.1:
Let $M$ be a closed hyperbolic $3$-manifold with
$\operatorname{\operatorname{\mathsf{SR}}\hskip 0.55557pt}(M)\leqslant s_{0}$,
where $s_{0}$ is a sufficiently large positive constant. According to
Sabourau’s theorem (see Theorem 2.2), there are only finitely many closed
hyperbolic $3$-manifolds $M$ with
$\operatorname{\operatorname{\mathsf{SR}}\hskip 0.55557pt}(M)\leqslant s_{0}$.
Then we know that the injectivity radii of all these hyperbolic $3$-manifolds
$M$ have a common lower bound. Denote this lower bound by $\delta_{0}$. The
constant $\delta_{0}$ is only depending on $s_{0}$. Then Theorem 2.1 implies
there exists a positive constant $K(\delta_{0})$ such that $M$ admits a
triangulation with the number $t$ of tetrahedra satisfying

$t\leqslant K(\delta_{0})\hskip
1.11115pt\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.55557pt}_{\operatorname{\operatorname{\mathsf{hyp}}\hskip 0.3889pt}}(M).$

The triangulation complexity $c(M)$ of a closed hyperbolic $3$-manifold $M$
satisfies $c(M)\leqslant t$. Hence we have

$\displaystyle c(M)$ $\displaystyle\leqslant t$ $\displaystyle\leqslant
K(\delta_{0})\hskip 1.11115pt\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.55557pt}_{\operatorname{\operatorname{\mathsf{hyp}}\hskip 0.3889pt}}(M)$
$\displaystyle\leqslant K(\delta_{0})\hskip 1.11115pt\nu_{3}\|M\|$
$\displaystyle\leqslant K(\delta_{0})\hskip
1.11115pt\nu_{3}C_{3}\operatorname{\operatorname{\mathsf{SR}}\hskip
0.55557pt}(M)\hskip 1.11115pt\log^{3}{\left(C_{3}^{\prime}\hskip
1.11115pt\operatorname{\operatorname{\mathsf{SR}}\hskip
0.55557pt}(M)\right)},$

where $\nu_{3}$, $C_{3}$ and $C_{3}^{\prime}$ are all fixed positive
constants. Therefore,

$\operatorname{\operatorname{\mathsf{SR}}\hskip 0.55557pt}(M)\geqslant
C(s_{0})\hskip 1.11115pt\frac{c(M)}{\log^{3}{c(M)}},$

where $C(s_{0})$ is a positive constant only depending on $s_{0}$. $\square$

##  3\. Hyperbolic manifolds in higher dimensions

When $n\geqslant 4$, hyperbolic manifolds of dimension $n$ have more rigidity.
There are only finitely many hyperbolic $n$-manifolds ($n\geqslant 4$) with
volume bounded from above (Wang’s finiteness theorem), but this is not true in
$n=3$. Moreover, for hyperbolic $n$-manifolds with $n\geqslant 4$, if
hyperbolic volumes are bounded from above, the injectivity radii have a common
lower bound. We generalize Jørgensen and Thurston’s theorem to hyperbolic
manifolds of dimension at least four. Then we prove Theorem 1.4 in this
section.

######  Proposition 3.1.

Let $M$ be a closed hyperbolic manifold of dimension $n$, with $n\geqslant 4$.
There exists a positive constant $K_{n}$ depending only on $n$, such that the
manifold $M$ admits a triangulation with the number $t$ of $n$-simplices is
bounded from above by its volume as follows,

(3.1) $t\leqslant K_{n}\hskip
1.11115pt\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.55557pt}_{\operatorname{\operatorname{\mathsf{hyp}}\hskip 0.3889pt}}(M).$

###### Proof.

Let $(M,\operatorname{\operatorname{\mathsf{hyp}}\hskip 0.55557pt})$ be a
closed hyperbolic manifold of dimension $n\geqslant 4$. Set
$R=\operatorname{\operatorname{\mathsf{Inj}}\hskip
0.55557pt}(M,\operatorname{\operatorname{\mathsf{hyp}}\hskip 0.55557pt})$.
Suppose that $X\subset M$ is a maximal set of points with any two of them
having distance at least $R$. We consider Voronoi cells corresponding to
points in $X$. Recall that a Voronoi cell associated to $x\in X$ is

$V(x)=\\{y\in M|dist(x,y)\leqslant dist(x^{\prime},y),\text{for
any}\,x^{\prime}\in X\,\text{and}\,x^{\prime}\neq x\\}.$

For any two distinct points $p,q\in X$, $B(p,\frac{R}{2})\cap
B(q,\frac{R}{2})=\emptyset$ since $dist(p,q)\geqslant R$. The total number of
Voronoi cells is thus bounded from by above by

$\displaystyle\frac{\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.55557pt}_{\operatorname{\operatorname{\mathsf{hyp}}\hskip
0.3889pt}}(M)}{\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.55557pt}_{\operatorname{\operatorname{\mathsf{hyp}}\hskip
0.3889pt}}(B(x,\frac{R}{2}))}$
$\displaystyle=\frac{1}{\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.55557pt}_{\operatorname{\operatorname{\mathsf{hyp}}\hskip
0.3889pt}}(B(x,\frac{R}{2}))}\hskip
1.11115pt\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.55557pt}_{\operatorname{\operatorname{\mathsf{hyp}}\hskip 0.3889pt}}(M)$
$\displaystyle\leqslant\frac{1}{c_{1}(n)e^{(n-1)R/2}}\hskip
1.11115pt\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.55557pt}_{\operatorname{\operatorname{\mathsf{hyp}}\hskip 0.3889pt}}(M)$
$\displaystyle\leqslant\frac{1}{c_{1}(n)}\hskip
1.11115pt\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.55557pt}_{\operatorname{\operatorname{\mathsf{hyp}}\hskip 0.3889pt}}(M),$

where $c_{1}(n)$ is a positive constant depending only on $n$. Note that it is
proved in [BT11] there exist hyperbolic $n$-manifolds $M$ with arbitrarily
small injectivity radius $R$. Hence the above constant $\frac{1}{c_{1}(n)}$
cannot be further improved.

A triangulation of the hyperbolic manifold $M$ is obtained by triangulating
each Voronoi cell $V(x)$ into $n$-simplices. Since two Voronoi cells $V(x)$
and $V(y)$ have a common face if and only if $B(y,R)\subset
B(x,\frac{5}{2}R)$, the number of faces of any Voronoi cell $V(x)$ has upper
bound of $\frac{\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.3889pt}_{\operatorname{\operatorname{\mathsf{hyp}}\hskip
0.27779pt}}(B(x,\frac{5}{2}R))}{\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.3889pt}_{\operatorname{\operatorname{\mathsf{hyp}}\hskip
0.27779pt}}(B(x,\frac{R}{2}))}$. Hence, the total number of faces for all
Voronoi cells is finite and only depending on $n$. In the triangulation,
maximal number of $n$-simplices in each Voronoi cell is uniformly bounded from
above by the number of its faces. Therefore, there exists a positive constant
$K_{n}$ only depending on $n$, such that the total number of $n$-simplices in
the triangulation is bounded from above by $K_{n}\hskip
1.11115pt\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.55557pt}_{\operatorname{\operatorname{\mathsf{hyp}}\hskip 0.3889pt}}(M)$. ∎

Now we apply Proposition 3.1 to Proof Theorem 1.4.
Proof of Theorem 1.4: Theorem 1.4 is proved by using Proposition 3.1 and
Gromov’s theorem (Theorem 1.2). Let $M$ be a closed hyperbolic manifold with
dimension at least $4$, and $t$ be the number of $n$-simplices of
triangulation obtained from Proposition 3.1. Then the triangulation complexity
$\sigma(M)$ is bounded from above as follows,

$\displaystyle\sigma(M)$ $\displaystyle\leqslant t$ $\displaystyle\leqslant
K_{n}\hskip 1.11115pt\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.55557pt}_{\operatorname{\operatorname{\mathsf{hyp}}\hskip 0.3889pt}}(M)$
$\displaystyle=K_{n}\hskip 1.11115pt\nu_{n}\|M\|$ $\displaystyle\leqslant
K_{n}\nu_{n}\hskip
1.11115ptC_{n}\operatorname{\operatorname{\mathsf{SR}}\hskip
0.55557pt}(M)\log^{n}{\left(C_{n}^{\prime}\operatorname{\operatorname{\mathsf{SR}}\hskip
0.55557pt}(M)\right)}.$

Hence,

$\displaystyle\operatorname{\operatorname{\mathsf{SR}}\hskip
0.55557pt}(M)\geqslant D_{n}\hskip
1.11115pt\frac{\sigma(M)}{\log^{n}{\sigma(M)}},$

where $D_{n}$ is a positive constant only depending on $n$. $\hfill\square$

##  4\. Embolic volume of compact manifolds

We generalize the above results for systolic volume to embolic volume. The
proof of Theorem 1.7 is given in this section.

Embolic volume is defined in Definition 1.6. Berger’s embolic inequality
([Ber80] or [Ber03, Section 7.2.4.]) states that for any Riemannian metric
$\operatorname{\operatorname{\mathcal{G}}}$ defined on a compact manifold of
dimension $n$,

$\operatorname{\operatorname{\mathsf{Inj}}\hskip
0.55557pt}(M,\operatorname{\operatorname{\mathcal{G}}})^{n}\leqslant C\hskip
1.11115pt\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.55557pt}_{\operatorname{\operatorname{\mathcal{G}}}}(M)$

holds, where $C$ is a positive constant. Hence the embolic volume of compact
$n$-manifolds is always positive. The relation between embolic volume and
other topological invariants is described in [Ber03, Section 11.2.3.]. On any
closed manifold $M$, we always have
$\operatorname{\operatorname{\mathsf{Emb}}\hskip
0.55557pt}(M)\geqslant\operatorname{\operatorname{\mathsf{SR}}\hskip
0.55557pt}(M)$. Hence Gromov’s theorem (Theorem 1.2) includes a relation
between embolic volume and simplicial volume for closed manifolds with nonzero
simplicial volume. Moreover, Katz and Sabourau [KS05] used a different method
to show this result. Then the estimates in Theorem 1.1 and 1.4 also hold if
the systolic volume is replaced by embolic volume.

Let $E$ be a sufficiently large positive constant. In [Yam88, GPW90, GPW91]
the finiteness theorems are proved for compact $n$-manifolds $M$ with
$\operatorname{\operatorname{\mathsf{Emb}}\hskip 0.55557pt}(M)\leqslant E$.
Therefore, from this point of view, the embolic volume of compact manifolds
functions like volume of hyperbolic manifolds. Our theorem in this section
provides more evidence to this viewpoint.

The following local embolic inequality of Croke will be used in the proof of
Theorem 1.7.

######  Lemma 4.1 (see Croke [Cro80]).

For a Riemannian metric $\operatorname{\operatorname{\mathcal{G}}}$ defined on
compact $n$-dimensional manifold $M$, any metric ball $B(p,r)$ with center $p$
and radius
$r\leqslant\frac{1}{2}\operatorname{\operatorname{\mathsf{Inj}}\hskip
0.55557pt}(M,\operatorname{\operatorname{\mathcal{G}}})$ satisfies

(4.1) $\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.55557pt}(B(x,r))\geqslant\alpha_{n}r^{n},$

where $\alpha_{n}$ is a positive constant only depending on $n$.

Proof of Theorem 1.7: For a Riemannian metric
$\operatorname{\operatorname{\mathcal{G}}}$ defined on $M$, let
$\operatorname{\operatorname{\mathsf{Inj}}\hskip
0.55557pt}(M,\operatorname{\operatorname{\mathcal{G}}})$ be injectivity
radius. The distance function induced by
$\operatorname{\operatorname{\mathcal{G}}}$ is denoted
$dist_{\operatorname{\operatorname{\mathcal{G}}}}(,)$. Assume that
$R=\frac{1}{5}\operatorname{\operatorname{\mathsf{Inj}}\hskip
0.55557pt}(M,\operatorname{\operatorname{\mathcal{G}}})$. We say a subset
$A\subset M$ is $R$-separated if
$dist_{\operatorname{\operatorname{\mathcal{G}}}}(x,y)\geqslant R$ for all
distinct $x,y\in A$. Now let $S$ be a maximal $R$-separated subset of $M$.
Here the maximal is under inclusion relation. For $x_{0}\in S$, denote by
$V(x_{0})$ the following Voronoi cell,

$V(x_{0})=\\{y\in S|dist(y,x_{0})\leqslant dist(y,x),\forall x\in
S\,\text{and}\,x\neq x_{0}\\}.$

We have $B\left(x_{0},\frac{R}{2}\right)\subset V(x_{0})$. Hence the number of
Voronoi cells $V(x_{0})$ in $M$ is bounded from above by

$\frac{\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.55557pt}_{\operatorname{\operatorname{\mathcal{G}}}}(M)}{\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.55557pt}_{\operatorname{\operatorname{\mathcal{G}}}}(B(x_{0},\frac{R}{2}))}.$

We obtain a triangulation of $M$ by triangulating each Voronoi cell $V(x)$
into $n$-simplices. In order to let the triangulations on each face of $V(x)$
match, we choose the triangulation of $M$ which induces a triangulation on
each face symmetric with respect to combinatorial isomorphisms of the face.
The combinatorial types of Voronoi cell $V(x)$ are determined by $R$ and $n$.
Hence there are only finitely many such combinatorial types. We let $T$ be the
maximal number of simplices in all Voronoi cells $V(x)$. Then the finiteness
of combinatorial types of $V(x)$ implies that $T$ is a constant only depending
on $R$ and $n$. In fact, two Voronoi cells $V(x)$ and $V(y)$ are adjacent, if
and only if $B(x,\frac{R}{2})\subset B(y,\frac{5R}{2})$. Therefore, the number
$T$ has upper bound $\frac{\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.3889pt}(B(y,\frac{5R}{2}))}{\inf_{x}\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.3889pt}(B(x,\frac{R}{2}))}$, which is bounded from above by
$C_{n}\operatorname{\operatorname{\mathsf{Emb}}\hskip 0.55557pt}(M)$ according
to Lemma 4.1, where $C_{n}$ is a positive constant only depending on $n$, and
the infimum is taken over all $x$ such that $B(x,\frac{R}{2})\subset
B(x,\frac{5}{2}R)$ holds. The number of $n$-simplices in the triangulation is
equal to

$\frac{\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.55557pt}_{\operatorname{\operatorname{\mathcal{G}}}}(M)}{\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.55557pt}_{\operatorname{\operatorname{\mathcal{G}}}}(B(x_{0},\frac{R}{2}))}\hskip
1.11115ptT.$

By using Croke’s local embolic inequality (4.1) again, we have

$\displaystyle\frac{\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.55557pt}_{\operatorname{\operatorname{\mathcal{G}}}}(M)}{\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.55557pt}_{\operatorname{\operatorname{\mathcal{G}}}}(B(x_{0},\frac{R}{2}))}\hskip
1.11115ptT$
$\displaystyle\leqslant\frac{2^{n}}{\alpha_{n}^{n}}\frac{\operatorname{\operatorname{\mathsf{Vol}}\hskip
0.55557pt}_{\operatorname{\operatorname{\mathcal{G}}}}(M)}{R^{n}}\hskip
1.11115ptT$
$\displaystyle\leqslant\beta_{n}\operatorname{\operatorname{\mathsf{Emb}}\hskip
0.55557pt}(M,\operatorname{\operatorname{\mathcal{G}}})^{2},$

where $\beta_{n}$ is a positive constant only depending on $n$. Therefore we
find a triangulation on $M$ with the number of $n$-simplices bounded from
above by $\beta_{n}\operatorname{\operatorname{\mathsf{Emb}}\hskip
0.55557pt}(M)^{2}$. $\hfill\square$

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