# Beam Profiling With Noise Reduction From Computer Vision and Principal
Component Analysis for the MAGIS-100 Experiment

Joseph Jachinowski  Center for Fundamental Physics
Department of Physics and Astronomy
Northwestern University
Evanston, IL 60208, USA
ORCID: 0000-0002-1053-5688  Natasha Sachdeva  Center for Fundamental Physics
Department of Physics and Astronomy
Northwestern University
Evanston, IL 60208, USA
ORCID: 0000-0001-9305-4208  Tim Kovachy  Center for Fundamental Physics
Department of Physics and Astronomy
Northwestern University
Evanston, IL 60208, USA
<EMAIL_ADDRESS>
###### Abstract

MAGIS-100 is a long-baseline atom interferometer that operates as a quantum
sensor. It will search for dark matter, probe fundamental quantum science, and
serve as a prototype gravitational wave detector in the 0.3 to 3 Hz frequency
range. The experiment uses light-pulse atom interferometry where pulses of
light create the atom optics equivalents of beamsplitters and mirrors. Laser
beam aberrations are a key source of systematic error for MAGIS-100, and
accurately characterizing the laser beam spatial profile is therefore
essential. In this paper, we describe a new and efficient beam profiling
technique. We use a low-cost CMOS camera affixed to a translating and rotating
optomechanical mount to image the beam, then employ computer vision and
principal component analysis to minimize background noise and produce accurate
beam profiles for a laser incident on a variety of aberration-inducing optical
elements.

##  I Introduction

The Matter-wave Atomic Gradiometer Interferometric Sensor (MAGIS-100),
currently under construction at Fermilab, is a 100-m tall atomic fountain that
will be used to explore fundamental physics on a macroscopic scale.
Functioning as a high-precision quantum sensor, it will search for ultralight
dark matter and exotic forces, as well as test quantum mechanics in a new
regime. MAGIS-100 will also serve as a testbed for new technologies required
to eventually develop a km-scale atomic interferometer capable of detecting
gravitational waves in the currently unexplored midband region, corresponding
to 0.3 to 3 Hertz, bridging the gap between the peak sensitivities of LIGO and
LISA [1].

In atom interferometry, atomic wavefunctions are made to coherently propagate
over two paths and are then combined using a beamsplitter. The relative phase
between the two paths is encoded in the probability of detecting the atoms at
either output port. The phase shift contains information on the physical
environment and relevant interactions to be studied. In light-pulse atom
interferometry, the beamsplitters and mirrors along the interferometry paths
are created by pulses of light. Aberrations in the atom optics laser beams
often limit the sensitivity of the sensor [2].

Laser beam aberrations impact atom phase through two mechanisms: (1)
imprinting of the laser wavefront and (2) light shifts coupled to intensity
perturbations. Diffraction and expansion of the atom cloud can cause the two
interferometer arms to experience these phase shifts differently, ultimately
resulting in dephasing of the atoms. These effects have been studied
analytically [1, 3, 4, 5] and would benefit from the addition of accurate
simulations of beam propagation through the experimental apparatus as well as
in situ beam profiling. Moreover, intensity and phase perturbations are
dynamically related and, thus, must be studied holistically [6]. MAGIS-100
plans to use free-space propagation over four meters to clean the beam;
therefore, only laser aberrations with a transverse spatial wavelength greater
than approximately 8.8 mm will influence the interferometry sequence [1].

Beam profiling is used to study laser beam aberrations and propagation. Many
different techniques may be used to profile laser beams, including both non-
electrical and electrical methods. Examples of non-electrical methods include
burn paper, fluorescing plates, acrylic mode burns, and the human eye.
Examples of electrical methods include knife-edge, slit, or pinhole imaging
which measure the marginal power of the laser as a function of position,
mechanically integrating over the entire beam profile. For such methods,
diffraction around the knife, slit, or pinhole sets a lower limit on precision
[7].

Camera-based imaging systems are desirable due to their convenience, ease-of-
use, and versatility. The primary image sensor options for a camera-based
imaging system are a charge-coupled device (CCD) and a complementary metal
oxide sensor (CMOS). While the significantly lower cost of cameras equipped
with CMOS image sensors suggests a practical alternative to cameras equipped
with CCD image sensors, CMOS technology also introduces additional sources of
noise. CMOS image sensors are prone to both temporal and fixed pattern noise
(FPN). FPN is non-Gaussian and requires advanced techniques to denoise [8]. On
the other hand, because CMOS sensors can operate with a higher frame rate they
are preferred over CCD sensors for video capture [9, 10].

In addition to sensor noise, physical defects in optical elements such as
surface defects on lenses, mirrors, and neutral density (ND) filters, as well
as dust particles can result in diffraction fringes. In some cases, in order
to properly characterize aberrations in the optical apparatus, such
diffraction fringes are the primary focus of beam profiling efforts. However,
extraneous aberrations, often a result of imperfect optical elements necessary
for beam profiling, contribute to non-sensor noise. While non-sensor noise can
be mitigated by using high-quality optical elements with precise
specifications as well as operating in a dust-controlled environment, this is
not always practical.

Sensor and non-sensor noise often appear stationary relative to the image
boundary for a camera using a CMOS image sensor. If the camera is
translated/rotated, then the desired image, in our case the beam profile, is
also translated/rotated with respect to the image boundary. Stationary noise
is invariant under such translation/rotations. Thus, if a series of images are
captured during camera translations/rotations, then the appearance of the
stationary noise relative to the beam profile will change. Using a video on
the order of tens of seconds, a standard CMOS camera will be able to capture
hundreds of images per video. Assuming that the beam profile is constant in
time, post-processing of the image set can isolate the beam profile from the
stationary noise.

One such technique of post-processing is principal component analysis (PCA).
PCA is a statistical method to reduce the dimensionality of a dataset while
retaining the most information possible [11]. Applied to images,
dimensionality reduction has been effectively implemented in denoising
algorithms [12, 13, 14, 8]. Moreover, PCA has been effectively used in a
variety of applications across many fields in physics [15]. For example, PCA
has been used extensively in digital holographic microscopy, an important tool
in biomedical imaging [16, 17], phase-shift interferometry [18, 19, 20],
laser-induced breakdown spectroscopy [21], imaging of ultra-cold atoms [22,
23, 24], as well as measurements of atom number densities [25]. PCA has also
been used in atom interferometry to remove noise in images of atomic wave-
packets [26, 27].

In this paper, we present a method for beam profiling which uses a camera-
based imaging system equipped with a low-cost CMOS sensor to resolve accurate,
low-noise beam profiles by translating and rotating the camera and post-
processing with computer vision and principal component analysis (PCA). The
technique removes noise from laser profile measurements to resolve features of
aberrated beams, which enables the experimenter to identify a particular
aberration of interest. We begin by briefly summarizing the method. Then, we
provide a description of the experimental implementation and analysis and
conclude with a discussion of our results.

We summarize the method as follows: (1) align camera detector with beam so
that it is contained within the image boundaries, (2) begin video capture, (3)
translate or rotate the camera while ensuring that the beam remains within
image boundaries, (4) qualitatively identify key regions of beam profile from
a sample frame as a precursor to (5) implementing a computer vision algorithm
to identify key regions for the rest of the image dataset, (6) omit
misidentified images and use identified regions to align the centers of the
images, and (7) apply PCA and reconstruct beam profile using one to several
principal components. Fig. 1 displays the implementation of this method for
two different beam profiles.

Figure 1: Region identification and PCA for two different beam profiles. The
left beam profile corresponds to a laser incident on a 3.65 mm circular
aperture propagating 177 cm. The right corresponds to a laser clipped by a
hard-edge propagating 229 cm. For the beam profile on the right, a logarithmic
color scale is used to show low-intensity details. The second row displays the
identified ”key regions” of the raw beam profiles circled in green. The third
row displays the PCA beam profiles obtained using the first two PCs.

##  II Methods

###  II-A Experimental Setup

We used an experimental setup comprised of a laser, telescope, aberration-
inducing element, and a CMOS imaging system as described at the top of Fig. 2.
We used an AOSense AOS-SP-461 diode laser to provide the 461 nm beam. We used
two aberration-inducing elements for the two test cases. For the first, we
used a 3.65 mm diameter Newport ID-0.5 iris diaphragm as a circular aperture
and a telescope with a magnification factor of five for a beam width of 3.75
mm incident on the aperture. The iris was composed of 10 leaves, but we
approximated it as circular and treated the beam profile as azimuthally
symmetric. For this test case, we rotated the camera during the measurements.
For the second test case, we used a Thorlabs D365 iris diaphragm to function
as a hard edge by setting the iris diameter (16 mm) to be much larger than
beam width (1.5 mm). We used a telescope with a magnification factor of two
for these measurements. The iris was mounted on a translation stage for
translation perpendicular to the beam path. The camera was translated along
two axes perpendicular to the beam path for this test case. Unlike for a
camera rotation during the measurement, performing the measurements with a
translated camera does not assume any symmetry of the beam profile. All beam
widths are given as the $1/e^{2}$ diameter and the iris diameters have a
measurement error of $\pm$0.1 mm.

The imaging system shown on the bottom of Fig. 2 consisted of three main
components: (1) an optomechanical mount providing two degrees of translational
motion in the plane perpendicular to the direction of beam propagation and one
degree of rotational motion about the propagation axis, (2) a Thorlabs
CS165MU1 CMOS camera with 3.726 mm by 4.968 mm image sensor and 3.45 $\mu$m
pixel size, and (3) a 60 mm lens. The quantum efficiency of the Thorlabs
CS165MU1 CMOS camera we used was 70% at 461 nm. By appropriately setting the
laser power, we maintained an exposure time of less than 0.5 ms. The video
frame rate was 20–30 fps and was limited by the speed of the USB connection to
the host computer. The raw datasets contained between 300–600 images. The 60
mm lens was mounted approximately 20 mm before the camera for slight
demagnification of the beam before imaging. The effective propagation distance
was changed with the addition of this lens, but this was accounted for in the
simulations discussed in Section II-C. The longest propagation distance after
the aberration-inducing element that we used was 236 cm.

Figure 2: Experimental setup (upper) and optomechanical camera-based imaging
system (lower) with two degrees of translational motion and one rotational
degree of motion. In order to generate a variety of different aberrated beam
profiles, the imaging system (F) in the experimental setup was moved along the
beam path. A telescope with 15 mm and 75 mm lenses was used in the aperture
experiment to achieve a magnification of five, while a telescope with 25.4 mm
and 50 mm lenses was used in the hard-edge experiment to achieve a
magnification of two.

###  II-B Data Collection

For the measurement, the beam profile was centered on the sensor and video
capture was initiated. For the circular aperture test the camera was rotated
360∘ and for the hard-edge test the camera was translated on the order of 1 mm
in each direction perpendicular to the beam path, while ensuring the beam
profile remained within the image boundaries. For the circular aperture, we
tested five propagation distances ranging from 177 cm to 236 cm from the
aberration-inducing element by translating the camera along the beam path. The
intensity of the center of the beam profile and surrounding pattern changed
according to the propagation distance. We chose not to change the aperture
diameter as it resulted in qualitatively redundant beam profile patterns when
compared to changing the propagation distance. For the hard edge, we tested
six degrees of clipping of the beam, specified by the position of the
translation stage on which the hard edge was mounted. We translated the hard
edge in increments of $\sim$0.4 mm across the beam profile for a total
translation distance of 2 mm.

Figure 3: A sample image, normalized PCA image, simulated profile, and
difference between the PCA result and simulation for two different kinds of
aberrations. The first row displays the beam profile of a laser incident on a
3.65 mm circular aperture propagating 177 cm. The RMSE value of the difference
image is 0.017 with a maximum pixel value difference 0.15. The second row
displays the beam profile of a laser clipped on a hard-edge propagating 229
cm. The RMSE value of the difference image is 0.014 with a maximum pixel value
difference of 0.13. Mean and range of RMSE values for all experiments are
provided in the text. We use a logarithmic color scale here to show low-
intensity details.

###  II-C Analysis

To prepare the captured images for analysis, we applied a centering algorithm
consisting of region identification realized by a maximally stable extremal
regions (MSER) algorithm and image translation by array padding and cropping.
First, we qualitatively identified key regions in the beam profile from a
sample frame of the captured video. For our tests on circular apertures, we
identified the center region, either a bright or dark spot, as the key beam
profile region. For our tests on hard-edges, we identified the region of
highest intensity as the key region because each beam profile retained a clear
maximum. The key regions for the rest of each dataset were then detected and
extracted using a MSER algorithm. MSER algorithms are most effective at
detecting “blobs,” a common characteristic of beam profiles. In our case, the
MSER algorithm was executed by the MATLAB detectMSERFeatures function [28, 29,
30]. Since the MSER algorithm generally detected many overlapping regions, we
used the mean of the identified region locations. The purpose of region
identification was to provide an accurate measurement of the location of the
beam profile relative to the image boundaries. Misidentified frames were
omitted from the final dataset by imposing a limit on the standard deviation
of identified region locations. If the standard deviation of detected region
locations was above a five pixel threshold, then the image was omitted from
the final dataset. Typically the final datasets contained 100 frames less then
the original datasets, depending on noise in the beam profile.

For the raw images, the centering algorithm used a combination of padding and
cropping of the image array in order to translate the image such that the
identified center of the beam profile coincided with the center of the image.
In order to ensure that we retained representative beam profiles after
applying the centering algorithm, during data collection we ensured there was
enough empty space around the beam profile such that the array manipulation
would only affect low pixel value, near-black pixels. This requirement
motivated the use of the demagnifying lens placed prior to the camera as shown
in Fig. 2. The highest translation performed by the centering algorithm was
approximately 375 pixels horizontally and 225 pixels vertically corresponding
to 1.3 mm and 0.78 mm respectively.

Figure 4: Application of singe-image denoising algorithms described in Ref.
[12] (first row) and Ref. [8] (second row) as applied to the beam profile
presented in the first row of Fig. 3 (beam incident on 3.65 mm circular
aperture with 177 cm of free-space propagation). In the Adaptive PCA denoising
algorithm, a significant amount of noise is identified and removed; however,
it misidentifies as noise a significant portion of the profile to be studied.
In the PCA Wavelet Transform algorithm, the algorithm struggles to remove
noise which is not purely columnar and random. This fact is reflected in the
small magnitude of the noise removed. Both algorithms fail to address the
diagonal fringes visible on the top right corner of the profile which are
successfully removed by our method as presented in Fig. 3.

After the final dataset was compiled, we performed PCA on the data. Following
an approach similar to that described in Ref. [31], we constructed a data
matrix $\mathbf{X}$ from our dataset. Let $F$ be the number of images each of
dimension $M\times N$. The rows of $\mathbf{X}$ corresponded to pixel values
while the columns corresponded to images in the dataset, forming an $MN\times
F$ array. Each image was normalized by its maximum pixel value. Next, to
perform PCA we calculated the $F\times F$ covariance matrix
$\boldsymbol{\sigma}$. The eigenvector of $\boldsymbol{\sigma}$ corresponding
to the $k$-th largest eigenvalue is called the $k$-th principal component (PC)
of $\mathbf{X}$. The $F$ eigenvectors of $\boldsymbol{\sigma}$ are mutually
orthogonal; therefore, the PCs of $\mathbf{X}$ form an orthogonal basis for
$F$-dimensional image space. Assuming that large variances communicate
important structure, only a few PCs are needed to communicate a large amount
of information about the dataset [11]. The PCA images were obtained through a
linear transformation. The new pixel values were defined as a linear
combination of the pixel values from the images of the original dataset, with
weights prescribed by the PCs. In our case, PCA was executed by the MATLAB pca
function which uses an algorithm based on singular value decomposition.

We quantified the accuracy of the PCA result by comparing it to a simulated
beam profile. The beam propagation simulation evaluates the Rayleigh-
Sommerfeld diffraction integral in order to determine the beam profile after
propagation. Moreover, it can incorporate various optical elements such as
lenses and circular apertures. For short distances, the integral is evaluated
using the angular spectrum method as described in Ref. [32], and for long
distances it is evaluated using convolution method as described in Ref. [33].
These methods have been demonstrated to be accurate and reliable [1]. The
simulation generated a higher resolution beam profile than the camera, with
4,095 pixels per 4.968 mm as opposed to 1,440 for the camera. Therefore, the
simulated beam profile was downsampled in order to compare it to the PCA
result. The precision with which certain experimental parameters could be
measured affected our ability to match the measured beam profile pattern with
the simulation and fine-tuning of some simulation input parameters was
required. For example, the aperture diameter was determined to about $\pm$0.1
mm, the hard-edge position $\pm$0.1 mm, and lens positions $\pm$1 mm. Because
the beam pattern was highly sensitive to these input parameters, this proved
to be a limitation of the comparison.

We used the root-mean-squared error (RMSE) of the PCA-simulation difference
profile as a metric to estimate the accuracy of the PCA reconstruction. The
experimental results with the lowest RMSE values of the difference profiles
are presented in Fig. 3. We analyzed the twelve total beam profiles by
normalizing the PCA and simulated arrays by z-score, scaling the arrays to lie
within the interval $[0,1]$, and then calculating the absolute value of the
difference of the arrays. The RMSE values calculated from the normalized
difference profile are presented in Table I.

TABLE I: RMSE values of the data-simulation difference profiles for a single representative raw image from each data set and for the PCA results of each data set. | RMSE | # of Datasets
---|---|---
| Mean | Max | Min |
Raw Circular aperture | 0.050 | 0.088 | 0.031 | 5
PCA Circular aperture | 0.025 | 0.029 | 0.017
Raw Hard-edge | 0.028 | 0.034 | 0.024 | 7
PCA Hard-edge | 0.020 | 0.035 | 0.014

The beam patterns created by the hard-edge proved to be the most sensitive to
the simulation input parameters and resulted in higher RMSE values for the
comparison with the PCA profile and a smaller difference between the RMSE
values for the raw image and the PCA result. Nevertheless, the method did
effectively remove the fringes that are apparent in the raw image.

##  III Discussion

We expect that the error observed is dominated by the discrepancy between
experimental and simulation parameters due to measurement error. Fine-tuning
of the circular aperture diameter and hard-edge position helped to a certain
degree, but parameters such as the beam envelope, precise shape of the
aberration-inducing element, or lens thickness were not optimized. For
example, we modeled the beam as an ideal Gaussian beam, assumed that the
irises were perfect circles, and used the thin-lens approximation for the
telescope and demagnifying lenses.

As a benchmark, we compared our method to the single-image denoising
algorithms described in Ref. [12] and [8]. Fig 4 presents results as applied
to one of our raw images. Ref. [12] describes an adaptive PCA denoising
algorithm in which “training” sections of the image are sampled and analyzed
via PCA before being reassembled into smaller “denoising” sections. The
algorithm best addresses zero mean Gaussian noise. Ref. [8] describes a PCA
wavelet decomposition denoising algorithm in which a three-level
multiresolution wavelet decomposition is performed on the image, the dc
frequency component is extracted from the vertical component of the wavelet
decomposition, and then PCA is applied to extract only the denoised vertical
component. The algorithm addresses columnar FPN commonly observed in images
captured by CMOS image sensors. Both algorithms use PCA to target noise while
retaining the important structure of the image. However, while both algorithms
recognize noise present in CMOS sensors, they do not sufficiently target and
remove fringes or other stationary noise present in our data.

We determine the number of PCs by comparison to the simulated beam profile.
Taking RMSE values of the PCA-simulation difference profiles for 1 through 10
PCs, we found 1–3 PCs to be sufficient for the two kinds of aberrations we
studied. The number of necessary PCs varies with the details of the beam
profile pattern, as well as the relative intensity of noise. For various
classes of aberrations, we can use results from this type of analysis to guide
selection of PCs when profiling for the MAGIS-100 experiment. In general,
however, the highest number of PCs should be used while sensor noise and
unwanted fringes are still mitigated.

For our measurements, PCA operated similar to averaging but could also respond
to non-uniformities in the dataset. All of the components of the covariance
matrix were approximately equal to each other with small fluctuations caused
by the movement of stationary noise with respect to the image boundary.
Moreover, the sensor noise was relatively isotropic, and so for all regions in
a image the centering algorithm did not alter the amount or character of
sensor noise but rather shifted its detailed local structure. Given such
conditions, the first PC will be a vector with components approximately equal
to each other. This corresponds to PCA pixel values which are approximately an
equal superposition of image pixel values.

##  IV Conclusions

In MAGIS-100, beam profiling will be used primarily to (1) measure and
characterize aberrations in the experimental apparatus, and to (2) test beams
for a sufficiently Gaussian profile after free-space propagation. Our results
suggest that the experimental beam profiling technique described is a suitable
option for such applications, providing an efficient, cost-effective, and
precise experimental method. We show that a CMOS image sensor combined with
basic optical equipment and computer is sufficient for effective beam
profiling. Furthermore, the implementation of computer vision and PCA is made
simple by readily available computational packages and functions that can run
easily on a laptop or tablet. Therefore, the experimental beam profiling
technique described is a viable option for beam profiling efforts related to
MAGIS-100 and other experiments.

##  V Acknowledgments

We acknowledge support from Gordon and Betty Moore Foundation Grant GBMF7945,
Office of Naval Research Grant Number N00014-19-1-2181, National Institute of
Standards and Technology Grant Number 60NANB19D168, and the David and Lucile
Packard Foundation through a Packard Fellowship for Science and Engineering.

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t\,dt\Big{(}\sec t\csc t-\csc t\Big{)}\,dt-dt\,\cot t\,dt\,\Big{(}\csc t-\cot
t\Big{)}\,dt$ $\displaystyle\,-\Big{(}1-\sin t\Big{)}\Big{(}\sec t\csc
t\,dt\circ\sec t-\csc t\,dt\Big{)}\,\Big{(}\csc t-\cot t\Big{)}\,dt\bigg{)}$
$\displaystyle=$
$\displaystyle\,\displaystyle\frac{2}{\pi}\displaystyle\int_{0}^{\pi/2}\bigg{(}dt\,\csc
t\,dt\Big{(}\sec t\csc t-\csc t\Big{)}\,dt-dt\,\cot t\,dt\,\Big{(}\csc t-\cot
t\Big{)}\,dt$ $\displaystyle\,-\Big{(}\sec t\csc t-\sec
t\Big{)}\,dt\,\Big{(}\sec t\csc t-\csc t\Big{)}\,dt+\Big{(}\csc
t-1\Big{)}\,dt\,\Big{(}\csc t-\cot t\Big{)}\,dt\bigg{)}$ $\displaystyle=$
$\displaystyle\,\displaystyle\frac{2}{\pi}\displaystyle\int_{0}^{1}\bigg{(}i({\texttt{a}}+2{\texttt{x}}_{-1}){\texttt{d}}_{-1,1}{\texttt{d}}_{-i,i}+i(2{\texttt{x}}_{-1}-{\texttt{x}}_{-i}-{\texttt{x}}_{i}){\texttt{y}}{\texttt{d}}_{-i,i}-({\texttt{a}}+2{\texttt{x}}_{-1})({\texttt{x}}_{-1}+{\texttt{x}}_{1})$
$\displaystyle\,\hskip
56.9055pt+(2{\texttt{x}}_{-1}-{\texttt{x}}_{-i}-{\texttt{x}}_{i})({\texttt{d}}_{-1,1}-i{\texttt{d}}_{-i,i})\bigg{)}$
$\displaystyle=$
$\displaystyle\,\displaystyle\frac{2}{\pi}\left(16\operatorname{{{Im}}}\operatorname{Li}_{3}\Big{(}\displaystyle\frac{1+i}{2}\Big{)}-\displaystyle\frac{19\pi^{3}}{48}+\displaystyle\frac{\pi^{2}}{8}+\displaystyle\frac{\pi}{2}\log
2(\log 2-2)-2\log^{2}2+G(4\log 2+2)\right)$ $\displaystyle\approx$
$\displaystyle\,0.0441495729.$

Similarly,

$\displaystyle Y_{2}:=$
$\displaystyle\,\displaystyle\sum_{n>k>m>0}\displaystyle\frac{a_{n}^{2}}{(2n-1)^{2}(2k)(2m)}$
$\displaystyle=$
$\displaystyle\,\displaystyle\frac{2}{\pi}\displaystyle\int_{0}^{\pi/2}\sin
t\,dt\,\cot^{2}t\,dt\,d(\sec t)\Big{(}\csc t\,dt\circ\sec t-\cot
t\,dt\Big{)}\,\Big{(}\csc t-\cot t\Big{)}\,dt$ $\displaystyle=$
$\displaystyle\,\displaystyle\frac{2}{\pi}\displaystyle\int_{0}^{\pi/2}\bigg{(}-\sin
t\,dt\,dt\,d(\sec t)\Big{(}\csc t\,dt\circ\sec t-\cot
t\,dt\Big{)}\,\Big{(}\csc t-\cot t\Big{)}\,dt$ $\displaystyle\,-\cos
t\,dt\,d(\sec t)\Big{(}\csc t\,dt\circ\sec t-\cot t\,dt\Big{)}\,\Big{(}\csc
t-\cot t\Big{)}\,dt$ $\displaystyle\,+\sin t\,dt\,\sec t\,dt\Big{(}\csc
t\,dt\circ\sec t-\cot t\,dt\Big{)}\,\Big{(}\csc t-\cot t\Big{)}\,dt$
$\displaystyle=$
$\displaystyle\,-Y_{1}+\displaystyle\frac{2}{\pi}\displaystyle\int_{0}^{\pi/2}\bigg{(}-dt\Big{(}\csc
t\,dt\circ\sec t-\cot t\,dt\Big{)}\,\Big{(}\csc t-\cot t\Big{)}\,dt$
$\displaystyle\,+\cos t\,dt\Big{(}\sec t\csc t\,dt\circ\sec t-\csc
t\,dt\Big{)}\,\Big{(}\csc t-\cot t\Big{)}\,dt$ $\displaystyle\,+dt\Big{(}\csc
t\,dt\circ\sec t-\cot t\,dt\Big{)}\,\Big{(}\csc t-\cot t\Big{)}\,dt$
$\displaystyle=$
$\displaystyle\,-Y_{1}+\displaystyle\frac{2}{\pi}\displaystyle\int_{0}^{\pi/2}\bigg{(}\Big{(}1-\sin
t\Big{)}\Big{(}\sec t\csc t\,dt\circ\sec t-\csc t\,dt\Big{)}\,\Big{(}\csc
t-\cot t\Big{)}\,dt$ $\displaystyle=$
$\displaystyle\,-Y_{1}+\displaystyle\frac{2}{\pi}\displaystyle\int_{0}^{\pi/2}\bigg{(}\Big{(}\sec
t(\csc t-1)\,dt\circ\sec t+(1-\csc t)\,dt\Big{)}\,\Big{(}\csc t-\cot
t\Big{)}\,dt$ $\displaystyle=$
$\displaystyle\,-Y_{1}+\displaystyle\frac{2}{\pi}\displaystyle\int_{0}^{1}\bigg{(}({\texttt{a}}+2{\texttt{x}}_{-1})({\texttt{x}}_{-1}+{\texttt{x}}_{1})+(2{\texttt{x}}_{-1}-{\texttt{x}}_{-i}-{\texttt{x}}_{i})(i{\texttt{d}}_{-i,i}-{\texttt{d}}_{-1,1})\bigg{)}$
$\displaystyle=$
$\displaystyle\,\displaystyle\frac{2}{\pi}\left(-16\operatorname{{{Im}}}\operatorname{Li}_{3}\Big{(}\displaystyle\frac{1+i}{2}\Big{)}+\displaystyle\frac{19\pi^{3}}{48}-\displaystyle\frac{\pi^{2}}{4}+\displaystyle\frac{\pi}{2}(4\log
2-\log^{2}2)+4\log^{2}2-4G(\log 2+2)\right)$ $\displaystyle\approx$
$\displaystyle\,0.002234785.$

For all $s\geq 3$, by (5.39) and (6.51)

$\displaystyle Y_{s}:=$
$\displaystyle\,\displaystyle\sum_{n>k>m>0}\displaystyle\frac{a_{n}^{2}}{(2n-1)^{s}(2k)(2m)}$
$\displaystyle=$
$\displaystyle\,\displaystyle\frac{2}{\pi}\displaystyle\int_{0}^{\pi/2}\sin
t\,dt\,(\cot t\,dt)^{s-2}\cot^{2}t\,dt\,d(\sec t)\Big{(}\csc t\,dt\circ\sec
t-\cot t\,dt\Big{)}\,\Big{(}\csc t-\cot t\Big{)}\,dt$ $\displaystyle=$
$\displaystyle\,-Y_{s-1}+\displaystyle\frac{2}{\pi}\displaystyle\int_{0}^{\pi/2}\sin
t\,dt\,(\cot t\,dt)^{s-2}\Big{(}\sec t\,dt-dt\,d(\sec t)\Big{)}\Big{(}\csc
t\,dt\circ\sec t-\cot t\,dt\Big{)}\Big{(}\csc t-\cot t\Big{)}\,dt$
$\displaystyle=$
$\displaystyle\,-Y_{s-1}+\displaystyle\frac{2}{\pi}\displaystyle\int_{0}^{\pi/2}\sin
t\,dt\,(\cot t\,dt)^{s-2}\,dt\,\Big{(}\sec t\csc t\,dt\circ\sec t-\csc
t\,dt\Big{)}\Big{(}\csc t-\cot t\Big{)}\,dt$ $\displaystyle=$
$\displaystyle\,-Y_{s-1}+\displaystyle\frac{2}{\pi}M_{s-2}=(-1)^{s}Y_{2}+\displaystyle\frac{2}{\pi}\displaystyle\sum_{k=1}^{s-2}(-1)^{s-k}M_{k},$

where

$\displaystyle M_{0}=$ $\displaystyle\,\displaystyle\int_{0}^{\pi/2}\sin
t\,dt\,\,dt\,\Big{(}\sec t\csc t\,dt\circ\sec t-\csc t\,dt\Big{)}\Big{(}\csc
t-\cot t\Big{)}\,dt$ $\displaystyle=$
$\displaystyle\,\displaystyle\int_{0}^{\pi/2}\cos t\,dt\,\Big{(}\sec t\csc
t\,dt\circ\sec t-\csc t\,dt\Big{)}\Big{(}\csc t-\cot t\Big{)}\,dt$
$\displaystyle=$ $\displaystyle\,\displaystyle\int_{0}^{\pi/2}\Big{(}1-\sin
t\Big{)}\Big{(}\sec t\csc t\,dt\circ\sec t-\csc t\,dt\Big{)}\Big{(}\csc t-\cot
t\Big{)}\,dt$ $\displaystyle=$
$\displaystyle\,\displaystyle\int_{0}^{\pi/2}\Big{(}(\sec t\csc t-\sec
t)\,dt\circ\sec t+(1-\csc t)\,dt\Big{)}\Big{(}\csc t-\cot t\Big{)}\,dt$
$\displaystyle=$
$\displaystyle\,\displaystyle\int_{0}^{1}({\texttt{a}}+2{\texttt{x}}_{-1})({\texttt{x}}_{-1}+{\texttt{x}}_{1})+(2{\texttt{x}}_{-1}-{\texttt{x}}_{-i}-{\texttt{x}}_{i})(i{\texttt{d}}_{-i,i}-{\texttt{d}}_{-1,1})$

and for all $s\geq 1$

$\displaystyle M_{s}=$ $\displaystyle\,\displaystyle\int_{0}^{\pi/2}\sin
t\,dt\,(\cot t\,dt)^{s}\,dt\,\Big{(}\sec t\csc t\,dt\circ\sec t-\csc
t\,dt\Big{)}\Big{(}\csc t-\cot t\Big{)}\,dt$ $\displaystyle=$
$\displaystyle\,\displaystyle\int_{0}^{\pi/2}\cos t\cot t\,dt(\cot
t\,dt)^{s-1}\,dt\,\Big{(}\sec t\csc t\,dt\circ\sec t-\csc
t\,dt\Big{)}\Big{(}\csc t-\cot t\Big{)}\,dt$ $\displaystyle=$
$\displaystyle\,-M_{s-1}+\displaystyle\int_{0}^{\pi/2}\csc t\,dt\,(\cot
t\,dt)^{s-1}dt\,dt\,\Big{(}\sec t\csc t\,dt\circ\sec t-\csc
t\,dt\Big{)}\Big{(}\csc t-\cot t\Big{)}\,dt$ $\displaystyle=$
$\displaystyle\,-M_{s-1}+(-1)^{s}i\displaystyle\int_{0}^{1}\Big{(}({\texttt{a}}+2{\texttt{x}}_{-1})({\texttt{a}}+{\texttt{x}}_{-1}+{\texttt{x}}_{1})-(2{\texttt{x}}_{-1}-{\texttt{x}}_{-i}-{\texttt{x}}_{i}){\texttt{d}}_{-1,1}\Big{)}{\texttt{d}}_{-i,i}{\texttt{y}}^{s-1}{\texttt{d}}_{-1,1}$
$\displaystyle=$
$\displaystyle\,(-1)^{s}M_{0}-(-1)^{s}\displaystyle\sum_{j=0}^{s-1}\displaystyle\int_{0}^{1}\Big{(}({\texttt{a}}+2{\texttt{x}}_{-1})({\texttt{a}}+{\texttt{x}}_{-1}+{\texttt{x}}_{1})-(2{\texttt{x}}_{-1}-{\texttt{x}}_{-i}-{\texttt{x}}_{i}){\texttt{d}}_{-1,1}\Big{)}{\texttt{d}}_{-i,i}{\texttt{y}}^{j}{\texttt{d}}_{-1,1}.$

Hence

$\displaystyle Y_{s}=$
$\displaystyle\,(-1)^{s}Y_{2}+\displaystyle\frac{2}{\pi}\displaystyle\sum_{k=1}^{s-2}(-1)^{s-k}M_{k}=(-1)^{s}Y_{2}+(-1)^{s}\displaystyle\frac{2}{\pi}\bigg{(}(s-2)M_{0}$
$\displaystyle\,\hskip
28.45274pt-\displaystyle\sum_{j=0}^{s-3}(s-2-j)i\displaystyle\int_{0}^{1}\Big{(}({\texttt{a}}+2{\texttt{x}}_{-1})({\texttt{a}}+{\texttt{x}}_{-1}+{\texttt{x}}_{1})-(2{\texttt{x}}_{-1}-{\texttt{x}}_{-i}-{\texttt{x}}_{i}){\texttt{d}}_{-1,1}\Big{)}{\texttt{d}}_{-i,i}{\texttt{y}}^{j}{\texttt{d}}_{-1,1}\bigg{)}.$

In particular,

$\displaystyle Y_{3}=$
$\displaystyle\,-Y_{2}-\displaystyle\frac{2}{\pi}\left(M_{0}-\displaystyle\int_{0}^{1}\Big{(}({\texttt{a}}+2{\texttt{x}}_{-1})({\texttt{a}}+{\texttt{x}}_{-1}+{\texttt{x}}_{1})-(2{\texttt{x}}_{-1}-{\texttt{x}}_{-i}-{\texttt{x}}_{i}){\texttt{d}}_{-1,1}\Big{)}{\texttt{d}}_{-i,i}{\texttt{d}}_{-1,1}\right)$
$\displaystyle=$
$\displaystyle\,\displaystyle\frac{2}{\pi}\bigg{(}\displaystyle\frac{3\pi^{2}}{8}-10{\beta}(4)+16\operatorname{{{Im}}}\operatorname{Li}_{3}\Big{(}\displaystyle\frac{1+i}{2}\Big{)}+8\operatorname{{{Im}}}\operatorname{Li}_{4}\Big{(}\displaystyle\frac{1+i}{2}\Big{)}+8\operatorname{{{Im}}}\operatorname{Li}_{3}\Big{(}\displaystyle\frac{1+i}{2}\Big{)}\log
2$ $\displaystyle\,-\displaystyle\frac{3\pi^{3}}{96}(38+15\log
2)-6\log^{2}2+G\Big{(}6-\displaystyle\frac{\pi^{2}}{4}+4\log^{2}2+4\log
2\Big{)}$
$\displaystyle\,+\pi\Big{(}\displaystyle\frac{1}{2}\log^{2}2-\displaystyle\frac{5\log^{3}2}{24}-3\log
2+\displaystyle\frac{7}{4}\zeta(3)\Big{)}\bigg{)}\approx 0.000241778756.$

######  Example B.13.

We consider series given by a ${\gamma}^{\circ d}$-${\beta}^{\star}$-block
chain, with all ${\gamma}$-blocks having weight one. When $d=1$, by (5.41) and
(6.50)

$\displaystyle X_{1}:=$ $\displaystyle\,\displaystyle\sum_{n>m\geq
0}\displaystyle\frac{a_{n}^{2}}{(2n-1)(2m+1)}=\displaystyle\frac{2}{\pi}\displaystyle\int_{0}^{\pi/2}\cos
t\,dt\,d(\tan t)\,dt$ $\displaystyle=$
$\displaystyle\,\displaystyle\frac{2}{\pi}\displaystyle\int_{0}^{\pi/2}\bigg{(}\sin
t\,dt\,dt-\cos t\,dt\,\tan
t\,dt\bigg{)}=\displaystyle\frac{2}{\pi}\displaystyle\int_{0}^{\pi/2}\Big{(}\cos
t\,dt-(1-\sin t)\,\tan t\,dt\Big{)}$ $\displaystyle=$
$\displaystyle\,\displaystyle\frac{2}{\pi}\displaystyle\int_{0}^{\pi/2}(\sec
t-\tan
t)\,dt=\displaystyle\frac{-2}{\pi}\Big{(}\displaystyle\int_{0}^{1}({\texttt{x}}_{-i}+{\texttt{x}}_{i})\Big{)}=\displaystyle\frac{2\log
2}{\pi}.$

Suppose $d\geq 2$. Then by (2.5), (5.41) and (6.50)

$\displaystyle X_{d}:=$
$\displaystyle\,\displaystyle\sum_{n_{1}>\cdots>n_{d}>m\geq
0}\displaystyle\frac{a_{n_{1}}^{2}}{(2n_{1}-1)\cdots(2n_{d}-1)(2m+1)}=\displaystyle\frac{2}{\pi}\displaystyle\int_{0}^{\pi/2}\cos
t\,dt\,(\tan t\,dt)^{d-1}\,d(\tan t)\,dt$ $\displaystyle=$
$\displaystyle\,\displaystyle\frac{2}{\pi}\displaystyle\int_{0}^{\pi/2}\bigg{(}\cos
t\,dt\,(\tan t\,dt)^{d-2}(\sec^{2}t-1)\,dt\,dt-\cos t\,dt\,(\tan
t\,dt)^{d}\bigg{)}$ $\displaystyle=$
$\displaystyle\,X_{d-1}-\displaystyle\frac{2}{\pi}(E_{d-2}+A_{d}),$

where $A_{d}$ is defined by (B.4), $E_{0}=\pi^{2}/8-1$ and for all $d\geq 1$

$\displaystyle E_{d}=$ $\displaystyle\,\displaystyle\int_{0}^{\pi/2}\cos
t\,dt\,(\tan t\,dt)^{d}\,dt\,dt=\displaystyle\int_{0}^{\pi/2}(1-\sin t)\tan
t\,dt\,(\tan t\,dt)^{d-1}\,dt\,dt$ $\displaystyle=$
$\displaystyle\,E_{d-1}+\displaystyle\int_{0}^{\pi/2}(\tan t-\sec
t)\,dt\,(\tan
t\,dt)^{d-1}\,dt\,dt=E_{d-1}+(-1)^{d}\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}^{2}{\texttt{z}}^{d-1}({\texttt{x}}_{-i}+{\texttt{x}}_{i})$
$\displaystyle=$
$\displaystyle\,E_{0}-\displaystyle\sum_{j=0}^{d-1}(-1)^{j}\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}^{2}{\texttt{z}}^{j}({\texttt{x}}_{-i}+{\texttt{x}}_{i}),$

Thus for all $d\geq 2$, by (B.4)

$\displaystyle X_{d}=$
$\displaystyle\,X_{1}-\displaystyle\frac{2}{\pi}\left(\displaystyle\sum_{k=0}^{d-2}E_{k}+\displaystyle\sum_{k=2}^{d}A_{k}\right)=\displaystyle\frac{2}{\pi}\bigg{(}d\log(2)-(d-1)\displaystyle\frac{\pi^{2}}{8}$
$\displaystyle\,+\displaystyle\sum_{j=0}^{d-3}(-1)^{j}(d-2-j)\displaystyle\int_{0}^{1}{\texttt{d}}_{-i,i}^{2}{\texttt{z}}^{j}({\texttt{x}}_{-i}+{\texttt{x}}_{i})-\displaystyle\sum_{j=1}^{d-1}(-1)^{j}(d-j)\displaystyle\int_{0}^{1}{\texttt{z}}^{j}({\texttt{x}}_{-i}+{\texttt{x}}_{i})\bigg{)}.$

We see the weight drops by 1, as predicted by Thm. 6.1(b). In particular,

$\displaystyle X_{2}=\displaystyle\frac{2}{\pi}\left(2\log
2-\displaystyle\frac{\pi^{2}}{12}-\displaystyle\frac{1}{2}\log^{2}2\right)\approx
0.20601068.$


# Every closed surface of genus at least $18$ is Loewner

Qiongling Li and Weixu Su  Qiongling Li: Chern Institute of Mathematics and
LPMC, Nankai University, Tianjin 300071, China<EMAIL_ADDRESS>Weixu Su: School of Mathematics, Sun Yat-sen University, Guangzhou 510275,
China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we obtain an improved upper bound involving the systole and
area for the volume entropy of a Riemannian surface. As a result, we show that
every orientable and closed Riemannian surface of genus $g\geq 18$ satisfies
Loewner’s systolic ratio inequality. We also show that every closed orientable
and nonpositively curved Riemannnian surface of genus $g\geq 11$ satisfies
Loewner’s systolic ratio inequality.

Keywords: Systole; Loewner’s inequality; entropy.

MSC2020: 53C23, 37C35.

Q. Li is partially supported by the National Key R$\&$D Program of China No.
2022YFA1006600, the Fundamental Research Funds for the Central Universities
and Nankai Zhide foundation. W. Su is partially supported by NSFC No.
12371076.

##  1\. Introduction

Unless otherwise stated, all surfaces we consider in this paper are assumed to
be orientable and closed, endowed with Riemannian metrics.

Let $M$ be a surface of genus $g\geq 1$. Denote the _systole_ of $M$ by
$\operatorname{sys}(M)$. By definition, $\operatorname{sys}(M)$ is the length
of shortest non-contractible closed geodesics on $M$. Denote the area of $M$
by $\operatorname{Area}(M)$. We say that the surface $M$ is _Loewner_ if the
systolic ratio satisfies the inequality

(1)
$\frac{\operatorname{sys}^{2}(M)}{\operatorname{Area}(M)}\leq\frac{2}{\sqrt{3}}.$

Around 1949, Loewner (ref. [11]) first proved that every surface of genus $1$
satisfies the inequality (1). Gromov [4] showed that, for any surface of genus
$g$,

$\frac{\operatorname{sys}^{2}(M)}{\operatorname{Area}(M)}<\frac{64}{4\sqrt{g}+27}.$

Thus all surfaces of genus $g\geq 50$ is Loewner. For details on history of
Loewner’s inequality and related developments in the systolic geometry, we
refer to the book of Katz [9]. See also Guth [5].

######  Problem 1.1.

Whether or not all surfaces of genus at least $1$ are Loewner?

For surfaces of genus $2$ and genus $g\geq 20$, the answer is affirmative,
shown by Katz and Sabourau [7, 8].

This aim of this short note is to prove

######  Theorem 1.2.

Every closed Riemannian surface of genus $g\geq 18$ is Loewner.

Our proof of Theorem 1.2 uses the argument in [7]. We improve the upper bound
for the volume entropy, which gives a better bound on the genus when the
surface is not Loewner.

For $3\leq g\leq 17$, the problem remains unsolved. In Proposition 2.2, we
show that all surfaces of of genus at least $17$ satisfying

(2) $\operatorname{inj}(M)=\operatorname{sys}(M)/2$

are Loewner. The hypothesis (2) is true for nonpositively curved surfaces, see
Proposition 4.2. We can do even better for nonpositively curved surfaces using
the method of Gromov and Bishop-Günther volume comparison theorem.

######  Theorem 1.3.

Every closed nonpositively curved Riemannnian surface of genus $g\geq 11$ is
Loewner.

Acknowledgements. We are grateful to Christpher Croke, Mikhail Katz and
Stéphane Sabourau for their very useful comments.

##  2\. An upper bound for the entropy

###  2.1. Entropy

Let $M$ be a surface of genus $g$. Let $\widetilde{M}$ be the universal cover
of $M$. The _volume entropy_ of $M$ is defined by

$h(M)=\lim_{R\to+\infty}\frac{\log\operatorname{Area}(B(\tilde{x}_{0},R))}{R}.$

The limit exists and does not depend on the choice of
$\tilde{x}_{0}\in\widetilde{M}$ (see [10]).

By a classical result of Katok [6], we have

$h(M)^{2}\geq\frac{2\pi(2g-2)}{\operatorname{Area}(M)}.$

The following equivalent definition of $h(M)$ is very useful.

######  Lemma 2.1.

Let $N(T)$ be the number of homotopy classes of non-contractible closed curves
on $M$, based at some fixed point $x_{0}\in M$ and which can be represented by
loops of length at most $T$. Then

$h(M)=\lim_{T\to+\infty}\frac{\log N(T)}{T}.$

See [7, Lemma 3.6] for the proof.

###  2.2. Upper bound

The next inequality is the key and technical part of this paper. It relates
the volume entropy to the systolic ratio. It improves [7, Proposition 3.1].

######  Proposition 2.2.

Every surface $M$ of genus $g\geq 1$ satisfies

$h(M)\leq\frac{1}{\beta\operatorname{sys}(M)}\log\frac{\operatorname{Area}(M)-\min\limits_{q\in
M}\operatorname{Area}(B(q,(\beta-3\alpha)\operatorname{sys}(M)))}{\min\limits_{q\in
M}\operatorname{Area}(B(q,\alpha\operatorname{sys}(M)))},$

whenever $0<5\alpha<\beta,\beta+4\alpha<\frac{1}{2}$.

###### Proof.

Let $s$ be the systole of $M$. Let $\mathcal{S}$ be a maximal set of centers
of disjoint disks of radius $\alpha s$. So $\\{B(q,2\alpha s)\ |\
q\in\mathcal{S}\\}$ forms a cover of $M$.

For each closed geodesic path $c$ based at $x_{0}$ of length $\leq T$, we can
always construct a homotopic path connected by at most
$m:=\lceil\frac{T}{\beta s}\rceil$ points in the set $\mathcal{S}$ in the
following way.

Let $t\in[2,3)$ to be determined. We will choose
$\underline{c}(k)=c(\gamma_{k}s)\in c$ and $q_{k}\in\mathcal{S}$ in each step.
We start with $\underline{c}(0)=c(0)=x_{0}$ (thus $\gamma_{0}=0$). Choose
$q_{0}\in\mathcal{S}$ such that $c(0)\in B(q_{0},2\alpha s)$.

We define by induction. Suppose we have defined $\gamma_{k}$ and $q_{k}$. By
the construction, $\underline{c}(k)=c(\gamma_{k}s)\in B(q_{k},2\alpha s)$. Now
we are going to define $\gamma_{k+1}$ and $q_{k+1}$.

We separate into two cases. Note that in the following discussion, the
constants $\alpha,\beta$ satisfy

$\beta>5\alpha,\beta+4\alpha<\frac{1}{2}.$

Case 1: Suppose that any $q\in\mathcal{S}$ such that
$d(c((\gamma_{k}+\beta+t\alpha)s),q)<2\alpha s$ satisfies

$d(q_{k},q)\geq(\beta-t\alpha)s.$

Choose $\hat{q}_{1}\in\mathcal{S}$ such that

$d(c((\gamma_{k}+\beta)s),\hat{q}_{1})<2\alpha s.$

Let $r_{1}\in[(\gamma_{k}+\beta)s,(\gamma_{k}+\beta+t\alpha)s]$ be the minimal
number such that $d(\underline{c}(r),\hat{q}_{1})<2\alpha s$ does not hold. So
$d(c(r_{1}),\hat{q}_{1})=2\alpha s$. Then choose $\hat{q}_{2}\in\mathcal{S}$
such that $d(c(r_{1}),\hat{q}_{2})<2\alpha s.$ We continue the process to
choose $r_{k}$’s and $\hat{q}_{k}$’s. Note that

$d(\hat{q}_{l},\hat{q}_{l+1})\leq
d(c(r_{l}),\hat{q}_{l})+d(c(r_{l}),\hat{q}_{l+1})<4\alpha s.$

$\underline{c}(k)$$c((\gamma_{k}+\beta)s)$$c((\gamma_{k}+\beta+t\alpha)s)$$q_{k}$$\hat{q}_{1}$$\hat{q}_{2}$
Figure 1. Picture in Case 1. The disks $B(\hat{q}_{k})$’s cover the segment
connecting $c((\gamma_{k}+\beta)s)$ and $c((\gamma_{k}+\beta+t\alpha)s)$.
There is a smallest $l_{0}$ such that
$d(q_{k},\hat{q}_{l_{0}})\geq(\beta-t\alpha)s.$

Let $l_{0}$ be the smallest integer such that
$d(q_{k},\hat{q}_{l_{0}})\geq(\beta-t\alpha)s.$ In fact, the distance between
$c((\gamma_{k}+\beta+t\alpha)s)$ and $q_{k}$ is not less than
$\beta+t\alpha-2\alpha$, and if $c((\gamma_{k}+\beta+t\alpha)s)$ is contained
in some disk $B(\hat{q}_{j},2\alpha)$ where $\hat{q}_{j}\in\mathcal{S}$, then
$d(q_{k},\hat{q}_{j})>(\beta+t\alpha-4\alpha)s\geq(\beta-t\alpha)s$. Thus such
an $l_{0}$ exists.

Set $q_{k+1}=\hat{q}_{l_{0}}$ and $\underline{c}(k+1)=c(r_{l_{0}}).$

If $l_{0}=1$,then $\underline{c}(k+1)=c((\gamma_{k}+\beta)s).$ The length of
the circle joining $\underline{c}(k),q_{k},\underline{c}(k+1),q_{k+1}$ is less
than

$2\alpha s+(\beta+4\alpha)s+2\alpha s+\beta s=(2\beta+8\alpha)s<s.$

So the circle is null-homotopic.

If $l_{0}>1$,then $d(q_{k},\hat{q}_{l_{0}-1})<(\beta-t\alpha)s.$ So

$d(q_{k},\hat{q}_{l_{0}})\leq
d(q_{k},\hat{q}_{l_{0}-1})+d(\hat{q}_{l_{0}-1},\hat{q}_{l_{0}})\leq(\beta-t\alpha)s+4\alpha
s=(\beta-t\alpha+4\alpha)s.$

The length of the circle joined by
$\underline{c}(k),q_{k},\underline{c}(k+1),q_{k+1}$ is less than

$2\alpha s+(\beta-t\alpha+4\alpha)s+2\alpha
s+(\beta+t\alpha)s=(2\beta+8\alpha)s<s.$

Thus the circle is also null-homotopic.

Case 2: Suppose there exists $q\in\mathcal{S}$ such that
$d(c((\gamma_{k}+\beta+t)s),q)<2\alpha s$ and $d(q_{k},q)<(\beta-t\alpha)s.$
Then set $\underline{c}(k+1)=c((\gamma_{k}+\beta+t\alpha)s)$ and $q_{k+1}=q.$
The length of the circle joined by
$\underline{c}(k),q_{k},\underline{c}(k+1),q_{k+1}$ is less than

$2\alpha s+(\beta-t\alpha)s+2\alpha s+(\beta+t\alpha)s=(2\beta+4\alpha)s<s.$

So the circle is null-homotopic.

Let $\mathcal{R}$ denote the set of all closed geodesics based at $x_{0}$ with
length less than $T$. Now the above process allows us to construct for each
closed geodesic $c\in\mathcal{R}$ a homotopic piecewise geodesic loop by
connecting points in $\mathcal{S}$. From the construction, each such
representation of $c$ satisfies:

  * •

Either $d(q_{k},q_{k+1})\geq(\beta-t\alpha)s$ and $\beta s\leq
d(\underline{c}(k),\underline{c}(k+1))\leq(\beta+t\alpha)s,$ or,

  * •

$d(q_{k},q_{k+1})<(\beta-t\alpha)s$ and
$d(\underline{c}(k),\underline{c}(k+1))\geq(\beta+t\alpha)s.$

In the latter case, we have

$\displaystyle d(q_{k},q_{k+1})$ $\displaystyle\geq$ $\displaystyle
d(\underline{c}(k),\underline{c}(k+1))-4\alpha s$ $\displaystyle\geq$
$\displaystyle(\beta+t\alpha-4\alpha)s.$

If we take $t=2$, then in both cases we have

(3) $d(q_{k},q_{k+1})\geq(\beta-2\alpha)s.$

Since $d(q_{k},q_{k+1})\geq(\beta-2\alpha)s,$ we have $B(q_{k+1},\alpha s)\in
M\setminus B(q_{k},(\beta-3\alpha)s).$ Let $S^{(k)}$ be the set of $q_{l}$
satisfying $B(q_{l},\alpha s)\in M\setminus B(q_{k},(\beta-3\alpha)s).$ So
after $q_{k}$ is determined, the choices of $q_{k+1}$ is at most

$\displaystyle|S^{(k)}|$ $\displaystyle\leq$
$\displaystyle\frac{\operatorname{Area}(M)-\operatorname{Area}(B(q_{k},(\beta-3\alpha)s))}{\min\limits_{q\in\mathcal{S}}\operatorname{Area}(B(q,\alpha
s))}$ $\displaystyle\leq$
$\displaystyle\frac{\operatorname{Area}(M)-\min\limits_{p\in
M}\operatorname{Area}(B(q,(\beta-3\alpha)s))}{\min\limits_{q\in
M}\operatorname{Area}(B(q,\alpha s))}.$

Denote the last term by $I.$

Let $m=\lceil\frac{T}{\beta s}\rceil.$ Then

$\displaystyle|\mathcal{R}|$ $\displaystyle\leq$ $\displaystyle I^{m}.$

Therefore, by Lemma 2.1,

$h(M)\leq\lim_{m\rightarrow\infty}\frac{1}{m\beta
s}\log|\mathcal{R}|\leq\frac{1}{\beta s}\log I.$

∎

We say that a metric disk $B(x,R)$ on $M$ is _small_ if it is diffeomorphic to
an two dimensional Euclidean disk and all geodesics in $B(x,R)$ connecting two
boundary points of $B(x,R)$ minimize the distance between the endpoints.

Denote by $\operatorname{inj}(M)$ the injectivity radius of $M$. Note that all
metric disks of radius less than $\operatorname{inj}(M)/2$ are small.

We recall the following result of Croke [3, Corollary 1.2].

######  Lemma 2.3.

If $B(x,R)$ is small metric disk of $M$, e.g. $R\leq\operatorname{inj}(M)/2$,
then

$\operatorname{Area}(B(x,R))\geq\frac{8-\pi}{2}R^{2}.$

The following corollary gives the upper bound of volume entropy involving the
area and the injectivity radius. It improves [6, Proposition 3.6].

######  Corollary 2.1.

Every surface $M$ of genus $g\geq 1$ satisfies

$h(M)\leq\frac{1}{(1-4\eta)\operatorname{inj}(M)}\log\frac{\frac{2\operatorname{Area}(M)}{\operatorname{inj}^{2}(M)}-(8-\pi)(\min\limits\\{1-7\eta,\frac{1}{2}\\})^{2}}{(8-\pi)\eta^{2}},$

whenever $0<\eta<\frac{1}{9}$.

###### Proof.

Note that $\operatorname{sys}(M)\geq 2\operatorname{inj}(M)$.

From Proposition 2.1, every surface $M$ of genus $g\geq 1$ satisfies

$h(M)\leq\frac{1}{(1/2-4\alpha)\operatorname{sys}(M)}\log\frac{\operatorname{Area}(M)-\min\limits_{q\in
M}\operatorname{Area}(B(q,(1/2-7\alpha)\operatorname{sys}(M)))}{\min\limits_{q\in
M}\operatorname{Area}(B(q,\alpha\operatorname{sys}(M)))},$

whenever $0<\alpha<1/18$.

Let $\eta$ satisfying $\alpha\operatorname{sys}(M)=\eta\operatorname{inj}(M)$.
It follows from $\operatorname{sys}(M)\geq 2\operatorname{inj}(M)$ that
$\eta\geq 2\alpha$. Assume that $0<\eta<1/9,$ then $0<\alpha<1/18.$ It also
follows from $\operatorname{sys}(M)\geq 2\operatorname{inj}(M)$ that

$(1/2-7\alpha)\operatorname{sys}(M)\geq(1-7\eta)\operatorname{inj}(M).$

By Lemma 2.3, we have

$\operatorname{Area}(B(q,\alpha\operatorname{sys}(M))\geq\frac{8-\pi}{2}\eta^{2}\operatorname{inj}(M)^{2},$
$\operatorname{Area}(B(q,(1/2-7\alpha)\operatorname{sys}(M)))\geq\frac{8-\pi}{2}(\min\\{1-7\eta,\frac{1}{2}\\})^{2}\operatorname{inj}(M)^{2}.$

Note that
$(1/2-4\alpha)\operatorname{sys}(M)\geq(1-4\eta)\operatorname{inj}(M).$ The
statement follows. ∎

We then have the following result.

######  Proposition 2.2.

Suppose $M$ is a surface of genus at least $17$ that satisfies
$\operatorname{inj}(M)=\frac{1}{2}\operatorname{sys}(M)$. Then $M$ is Loewner.

###### Proof.

By Corollary 2.1 and the assumption
$\operatorname{inj}(M)=\frac{1}{2}\operatorname{sys}(M)$, we obtain

$\displaystyle h(M)$ $\displaystyle\leq$
$\displaystyle\frac{1}{(1-4\eta)\operatorname{inj}(M)}\log\frac{\frac{2\operatorname{Area}(M)}{\operatorname{inj}^{2}(M)}-(8-\pi)(\min\limits\\{1-7\eta,\frac{1}{2}\\})^{2}}{(8-\pi)\eta^{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{(1/2-2\eta)\operatorname{sys}(M)}\log\frac{\frac{8\operatorname{Area}(M)}{\operatorname{sys}^{2}(M)}-(8-\pi)(\min\limits\\{1-7\eta,\frac{1}{2}\\})^{2}}{(8-\pi)\eta^{2}}$

whenever $0<\eta<\frac{1}{9}.$ Denote the last term by
$\frac{1}{(1/2-2\eta)\operatorname{sys}(M)}\log\tilde{I}$.

Using Katok’s inequality of entropy, we have

$\sqrt{\frac{2\pi(2g-2)}{\operatorname{Area}(M)}}\leq
h(M)\leq\frac{1}{\beta\operatorname{sys}(M)}\log\tilde{I}.$

Assume that $M$ is not Loewner, then
$\frac{\operatorname{Area}(M)}{\operatorname{sys}(M)^{2}}<\frac{\sqrt{3}}{2}.$
Therefore,

$\displaystyle g-1$ $\displaystyle\leq$
$\displaystyle\frac{\sqrt{3}}{8\pi}\cdot\frac{(\log\frac{4\sqrt{3}-(8-\pi)(\min\limits\\{1-7\eta,\frac{1}{2}\\})^{2}}{(8-\pi)\eta^{2}})^{2}}{(1/2-2\eta)^{2}}$
$\displaystyle\approx$ $\displaystyle 15.9493,$

where $\eta=0.065734.$ Hence $g\leq 16.$ ∎

##  3\. Proof of Theorem 1.2

Let $(M,g)$ be a surface of genus $g\geq 1$. Gromov [4, Proposition 5.1.B]
showed that, there is a non-negative function $\rho(x)$ defined on $M$ such
that every metric disk $B(x,R)$ with radius $\rho(x)/2\leq
R\leq\operatorname{sys}(M)/2$ satisfies

(4) $\operatorname{Area}(B(x,R))\geq\frac{1}{2}(2R-\rho(x))^{2}.$

The function $\rho(x)$ is called the _height function_ of $M$. It is less than
the diameter of $M$ for all surfaces of genus at least $1$. See [4, §5.1] for
details.

Given any surface $M$ of genus $g\geq 1$, one can make conformal deformation
of $M$ such that the height function becomes arbitrary small. Meanwhile, the
systole is preserved and the area is non-increasing.

######  Lemma 3.1.

For an arbitrary $\epsilon>0,$ there exists a Riemannian metric $\overline{M}$
conformal to $M$ such that

  * •

$\operatorname{sys}(\overline{M})=\operatorname{sys}(M).$

  * •

$\operatorname{Area}(\overline{M})\leq\operatorname{Area}(M).$

  * •

The height function of $\overline{M}$ is less than $\epsilon.$

This is proved by Gromov [4, §5.6.C”]. Note that

$\frac{\operatorname{sys}(\overline{M})}{\operatorname{Area}(\overline{M})}\geq\frac{\operatorname{sys}(M)}{\operatorname{Area}(M)}.$

Now we can proceed the proof of Theorem 1.2.

###### Proof.

Assume $M$ is not Loewner, then

$\frac{\operatorname{sys}(M)^{2}}{\operatorname{Area}(M)}>\frac{2}{\sqrt{3}}.$

Let $\delta>0$ be arbitrarily small. By Lemma 3.1, up to a conformal
deformation, we may assume that
$\frac{\operatorname{sys}(M)^{2}}{\operatorname{Area}(M)}>\frac{2}{\sqrt{3}}$
and the height function of $M$ is less than $\delta\operatorname{sys}(M)$.
Thus for all $q\in M$ and $\delta\operatorname{sys}(M)/2\leq
R\leq\operatorname{sys}(M)/2$,

$\operatorname{Area}(B(q,R))\geq\frac{1}{2}(2R-\delta\operatorname{sys}(M))^{2}.$

Assume that $\delta/2\leq\alpha,\beta-3\alpha\leq 1/2$, then

$\operatorname{Area}(B(q,\alpha\cdot\operatorname{sys}(M)))\geq\frac{1}{2}(2\alpha-\delta)^{2}\operatorname{sys}(M)^{2},$
$\operatorname{Area}(B(q,(\beta-3\alpha)\cdot\operatorname{sys}(M)))\geq\frac{1}{2}(2\beta-6\alpha-\delta)^{2}\operatorname{sys}(M)^{2}.$

By Proposition 2.2, we have

$\displaystyle h(M)$ $\displaystyle\leq$
$\displaystyle\frac{1}{\beta\operatorname{sys}(M)}\log\frac{\frac{\operatorname{Area}(M)}{\operatorname{sys}(M)^{2}}-\frac{1}{2}(2\beta-6\alpha-\delta)^{2}}{\frac{1}{2}(2\alpha-\delta)^{2}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{\beta\operatorname{sys}(M)}\log\frac{\sqrt{3}-(2\beta-6\alpha-\delta)^{2}}{(2\alpha-\delta)^{2}},$

whenever $2\delta<\alpha<\frac{1}{5}\beta,\beta+4\alpha<\frac{1}{2}$. Denote
the last term by $\frac{1}{\beta\operatorname{sys}(M)}\log\tilde{I}$.

Using Katok’s inequality of entropy, we have

$\sqrt{\frac{2\pi(2g-2)}{\operatorname{Area}(M)}}\leq
h(M)\leq\frac{1}{\beta\operatorname{sys}(M)}\log\tilde{I}.$

Let
$\delta=0.000001,\alpha=0.026377,\beta=0.394491<0.394492=\frac{1}{2}-4\alpha.$
Using
$\frac{\operatorname{Area}(M)}{\operatorname{sys}(M)^{2}}<\frac{\sqrt{3}}{2}$
again, we obtain

$\displaystyle g-1$ $\displaystyle<$
$\displaystyle\frac{\sqrt{3}}{8\pi}\frac{(\log\tilde{I})^{2}}{\beta^{2}}$
$\displaystyle=$
$\displaystyle\frac{\sqrt{3}}{8\pi}\frac{(\log\frac{\sqrt{3}-(2\beta-6\alpha-\delta)^{2}}{(2\alpha-\delta)^{2}})^{2}}{\beta^{2}}$
$\displaystyle\approx$ $\displaystyle 16.8728.$

Therefore, if $M$ is not Loewner, then $g\leq 17.$

∎

##  4\. Proof of Theorem 1.3

Assume that $M$ is nonpositively curved. By the Bishop-Günther volume
comparison theorem (ref [1, Theorem III.4.2]), we have for all $q\in M$ and
$0<R\leq\operatorname{inj}(M),$

$\operatorname{Area}(B(q,R))\geq\pi R^{2}.$

Note that $\pi R^{2}$ is the area of the disk of radius $R$ in the Euclidean
plane.

######  Lemma 4.1.

Let $M$ be a compact Riemannian manifolds. Then the injectivity radius of $M$
equals the minimium of the conjugate radius and half the length of a shortest
nontrivial closed geodesic on $M$.

###### Proof.

This is a corollary of Cheeger-Ebin [2, Lemma 5.6]. We give a proof for the
sake of completeness.

For $p\in M$, denote the injectivity radius and conjugate radius at $p$ by
$\operatorname{inj}_{p}(M)$ and $\operatorname{conj}_{p}(M)$, respectively.
And denote the shortest length of closed nontrivial geodesic on $M$ by
$\ell(M)$. It is obvious that
$\operatorname{inj}_{p}(M)\leq\operatorname{conj}_{p}(M)$ and thus
$\operatorname{inj}(M)\leq\operatorname{conj}(M).$

Suppose $\operatorname{inj}(M)<\operatorname{conj}(M)$. Assume that at some
$p\in M$, $\operatorname{inj}_{p}(M)=\operatorname{inj}(M)$. By Cheeger-Ebin
[2, Lemma 5.6], there is a geodesic loop starting and ending at $p$, with
length $2\operatorname{inj}_{p}(M)=2\operatorname{inj}(M)$. Let $q\in\gamma$
that separates $\gamma$ into two segments with the same length
$\operatorname{inj}(M)$. Note that $p$ and $q$ are not conjugate since
$\operatorname{inj}_{p}(M)<\operatorname{conj}(M)$. The existence of two
geodesic segments between $p$ and $q$ implies that
$\operatorname{inj}_{q}(M)=\operatorname{inj}(M)$. If we look at $\gamma$ as a
loop starting and ending at $q$, then $\gamma$ is also smooth at $p$ by using
Cheeger-Ebin [2, Lemma 5.6] again. Thus $\gamma$ is a smooth nontrivial closed
geodesic. This shows that $\operatorname{inj}(M)\geq\frac{\ell(M)}{2}$.

Therefore,

$\operatorname{inj}(M)=\min\\{\operatorname{conj}(M),\ell(M)/2\\}.$

∎

######  Proposition 4.2.

Under the assumption that $M$ is nonpositively curved, we have

(5) $\operatorname{inj}(M)=\frac{\operatorname{sys}(M)}{2}.$

###### Proof.

Since $M$ is nonpositively curved, there is no conjugate points on $M$.
Moreover, any (nontrivial) closed geodesic $\gamma$ on $M$ is not
contractible. Otherwise, $\gamma$ lifts to a closed geodesic on the universal
cover $\widetilde{M}$. This contradicts the fact that on a Cartan-Hadamard
manifold, any two points are connected by a unique geodesic.

Now (5) is an immediate corollary of Lemma 4.1.

∎

Note that for a nonpositively curved surface $M$, we already know that the
area of $M$ is at least
$\pi\operatorname{inj}(M)^{2}=\frac{\pi}{4}\operatorname{sys}(M)^{2}.$ Thus
$\frac{\operatorname{Area}(M)}{\operatorname{sys}(M)^{2}}\geq\frac{\pi}{4},$
while $\frac{\pi}{4}$ is strictly less than $\frac{\sqrt{3}}{2}$. One cannot
conclude the Loewner property directly from the nonpositively curved
condition. This is different from the case of hyperbolic metric, where we have
a better lower bound of the area of $M$ to ensure it is Loewner.

Now we can proceed the proof of Theorem 1.3.

###### Proof.

We will use the argument of Gromov [4, Section 5.3].

Fix an arbitrary point $x\in M$. Choose some $p_{0}\in M\setminus
B(x,\operatorname{sys}(M)/2)$ and choose $\gamma_{1},\cdots,\gamma_{2g}$ in
$M\setminus B(x,\operatorname{sys}(M)/2)$ as a canonical basis of of
$\pi_{1}(M,p_{0})$. Denote

$\Gamma=\bigcup_{i=1}^{2g}\gamma_{i}.$

We consider $\Gamma$ as a connected graph. The Betti number of a connected
graph is given by $1+E-V$ where $E$ is the number of edges and $V$ is the
number of vertices. Thus the Betti number of $\Gamma$, denoted by
$b_{1}(\Gamma)$, is equal to $2g$.

Let $\mathcal{S}$ be a maximal set of centers on $\Gamma$ of disjoint disks of
radius $\operatorname{sys}(M)/8$ such that

$\\{B(q,\operatorname{sys}(M)/4)\ |\ q\in\mathcal{S}\\}$

forms a cover of $\Gamma$. We assume that $p_{0}\in\mathcal{S}$.

We first construct a covering of each $\gamma_{i}$ as following. Denote
$\gamma=\gamma_{i}$ for simplicity.

Start with $p_{0}=\gamma(0)$. Set $q_{0}=p_{0}$. Let $t_{1}$ be the smallest
positive number such that
$d(\gamma(t_{1}),p_{0}=q_{0})<\operatorname{sys}(M)/4$ does not hold. Define
$p_{1}=\gamma(t_{1})$ and thus $d(p_{1},q_{0})=\operatorname{sys}(M)/4$.
Denote by $\overline{p_{0}p_{1}}$ the geodesic connecting $p_{0}$ to $p_{1}$
and so on. Since $\gamma([0,t_{1}])\cup\overline{p_{0}p_{1}}$ is contained in
$B(p_{0},\operatorname{sys}(M)/4)$, the path $\gamma([0,t_{1}])$ is homotopic
to $\overline{p_{0}p_{1}}$.

Choose $q_{1}\in\mathcal{S}$ such that
$d(p_{1},q_{1})<\operatorname{sys}(M)/4.$ Let $t_{2}$ be the smallest positive
number such that $d(\gamma(t_{2}),q_{1})<\operatorname{sys}(M)/4$ does not
hold. Define $p_{2}=\gamma(t_{2})$ and thus
$d(p_{2},q_{1})=\operatorname{sys}(M)/4$. Then the path
$\gamma([t_{1},t_{2}])$ is homotopic to $\overline{p_{1}p_{2}}.$

Repeat this process, we obtain a sequence of points
$p_{0}=\gamma(0),\cdots,p_{r}=\gamma(t_{r}),p_{r+1}=p_{0}=\gamma(l)$ on
$\gamma$ and $q_{0},q_{1},\cdots,q_{r},q_{r+1}=q_{0}\in\mathcal{S}$
satisfying:

$d(p_{i},q_{i})<\operatorname{sys}(M)/4,\quad
d(p_{i+1},q_{i})=\operatorname{sys}(M)/4.$

And for $t\in[t_{r},l]$, $d(\gamma(t),p_{r})<\operatorname{sys}(M)/4.$ So the
path $\gamma([t_{r},l])$ is homotopic to $\overline{p_{r}p_{0}}.$

Then the triangle formed by $p_{k},q_{k},p_{k+1}$ has length less than
$2(\operatorname{sys}(M)/4+\operatorname{sys}(M)/4)=\operatorname{sys}(M)$ and
thus is null-homotopic. Also, the triangle formed by $p_{k+1},q_{k},q_{k+1}$
has length less than
$2(\operatorname{sys}(M)/4+\operatorname{sys}(M)/4)=\operatorname{sys}(M)$ and
thus is null-homotopic. Together, we obtain that the circle
$\overline{p_{k}q_{k}}\cup\overline{q_{k}q_{k+1}}\cup\overline{p_{k+1}q_{k+1}}\cup\overline{p_{k}p_{k+1}}$
is homotopically trivial. Thus the curve $\gamma$ can be homotopic to a
piecewise geodesic connected by points
$q_{0},q_{1},\cdots,q_{r},q_{r+1}=q_{0}$.

For each $\gamma_{i}$, we obtain that it is homotopic to a piecewise geodesic
connected by points $p_{0}^{(i)},p_{1}^{(i)},\cdots,p_{r+1}^{(i)}=p_{0}^{(i)}$
lying in $\gamma_{i}$ and also a piecewise geodesic connected by points
$q_{0}^{(i)},q_{1}^{(i)},\cdots,q_{r+1}^{(i)}=q_{0}^{(i)}$ lying in
$\mathcal{S}.$ Note that $p_{0}^{(i)}=q_{0}^{(i)}=q_{0}.$

Therefore, we obtain an abstract connected graph $\Gamma^{\prime}$ with
vertices $q_{k}^{(i)}\in\mathcal{S}$ appeared above, $q,q^{\prime}$ has an
edge if and only if they appear as $q=q_{k}^{(i)},q^{\prime}=q_{k+1}^{(i)}$
for some $i$. There is a (not necessarily injective) map from $\Gamma$ to
$\Gamma^{\prime}$, taking $p_{k}^{(i)}$ to $q_{k}^{(i)}$ and the edge
$\overline{p_{k}^{(i)}p_{k+1}^{(i)}}$ to
$\overline{q_{k}^{(i)}q_{k+1}^{(i)}}.$ Denote the image of $\gamma_{i}$ in
$\Gamma^{\prime}$ by $\sigma_{i}.$ So $\sigma_{i},i=1,\cdots,2g,$ are
different nontrivial loops based at $q_{0}$.

Moreover, there is a natural map $\iota:\Gamma^{\prime}\rightarrow M$ by the
natural map taking the edge between $q_{k},q_{k+1}$ to a fixed geodesic
between $q_{k},q_{k+1}$. Note that by our construction, for $i=1,\cdots,2g$,
$\gamma_{i}$ is homotopic to $\iota(\sigma_{i}).$

Suppose there is a loop $\sigma_{i_{1}}^{n_{1}}\cdots\sigma_{i_{l}}^{n_{l}}$
is homotopically trivial in $\Gamma^{\prime}$. It is clear that
$\iota(\sigma_{i_{1}}^{n_{1}}\cdots\sigma_{i_{l}}^{n_{l}})\in\pi_{1}(M)$ is
also homotopically trivial. Since $\gamma_{i}$ is homotopic to
$\iota(\sigma_{i})$, we obtain
$\gamma_{i_{1}}^{n_{1}}\cdots\gamma_{i_{l}}^{n_{l}}$ is also homotopically
trivial.

Therefore, $\pi_{1}(\Gamma^{\prime})$ has at least $2g$ generators and thus
the Betti number of $\Gamma^{\prime}$, $b_{1}(\Gamma^{\prime})$, is not less
than $2g$.

Let $N$ denote the number of vertices in $\Gamma^{\prime}$. So
$N\leq|\mathcal{S}|$. Denote the number of edges in $\Gamma^{\prime}$ by $K$.
Then $K\leq\frac{N(N-1)}{2}$. So

$\displaystyle 2g\leq b_{1}(\Gamma^{\prime})$ $\displaystyle\leq$
$\displaystyle\frac{N(N-1)}{2}-N+1$ $\displaystyle=$
$\displaystyle\frac{(N-1)(N-2)}{2}$ $\displaystyle\leq$
$\displaystyle\frac{(|\mathcal{S}|-1)(|\mathcal{S}|-2)}{2}.$

Note that

$\operatorname{Area}(B(p,\operatorname{sys}(M)/8))\geq\pi\left(\operatorname{sys}(M)/8\right)^{2},\quad\operatorname{Area}(B(p,3\operatorname{sys}(M)/8))\geq\pi\left(3\operatorname{sys}(M)/8\right)^{2}.$

Since the disjoint balls $B(q,\operatorname{sys}(M)/8)(q\in\mathcal{S})$ lie
in $M\setminus B(x,3\operatorname{sys}(M)/8)$, we have

$\displaystyle|\mathcal{S}|$ $\displaystyle\leq$
$\displaystyle\frac{\operatorname{Area}(M)-\operatorname{Area}(B(x,\frac{3}{8}\operatorname{sys}(M)))}{\pi(\operatorname{sys}(M)/8)^{2}}$
$\displaystyle\leq$
$\displaystyle\frac{\operatorname{Area}(M)-\pi(\frac{3}{8}\operatorname{sys}(M))^{2}}{\pi(\operatorname{sys}(M)/8)^{2}}$
$\displaystyle\leq$
$\displaystyle\frac{64}{\pi}\frac{\operatorname{Area}(M)}{\operatorname{sys}(M)^{2}}-9.$

Assume that $M$ is not Loewner, then
$\frac{\operatorname{Area}(M)}{\operatorname{sys}(M)^{2}}<\frac{\sqrt{3}}{2}.$
The above inquality of $|\mathcal{S}|$ implies that
$|\mathcal{S}|\leq\frac{32\sqrt{3}}{\pi}-9=8.64252$ and thus
$|\mathcal{S}|\leq 8$.

Using $2g\leq(|\mathcal{S}|-1)(|\mathcal{S}|-2)/2$, we obtain

$g\leq\frac{(|\mathcal{S}|-1)(|\mathcal{S}|-2)}{4}=10.25.$

Therefore, if $M$ is not Loewner, then $g\leq 10.$ ∎

######  Remark 4.3.

Gromov [4, Proposition 5.3.A] proved the same inequality

$2g\leq\frac{(N-1)(N-2)}{2},$

where the graph $\Gamma$ is given by a short basis of the homology group. Then
he obtain a lower bound for
$\frac{\operatorname{Area}(M)}{\operatorname{sys}^{\prime}(M)^{2}}$, where
$\operatorname{sys}^{\prime}(M)$ denotes the homological systole.

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