Technical report

Bragg-Edge Elastic Strain Tomography for in situ Systems from Energy-Resolved
Neutron Transmission Imaging

J.N. Hendriks, A.W.T. Gregg, C.M. Wensrich, A.S. Tremsin, T. Shinohara, M.
Meylan, E.H. Kisi, V. Luzin, and O. Kirsten

  * $\bullet$

Please cite this version:
J.N. Hendriks, A.W.T. Gregg, C.M. Wensrich, A.S. Tremsin, T. Shinohara, M.
Meylan, E.H. Kisi, V. Luzin, and O. Kirsten. Bragg-Edge Elastic Strain
Tomography for in situ Systems from Energy-Resolved Neutron Transmission
Imaging. Physical Review Materials, 1.5 (2017): 053802

###### Abstract

Technological developments in high resolution time-of-flight neutron detectors
have raised the prospect of tomographic reconstruction of elastic strain
fields from Bragg-edge strain images. This approach holds the potential to
provide a unique window into the full triaxial stress field within solid
samples. While general tomographic reconstruction from these images has been
shown to be ill-posed, an injective link between measurements and boundary
deformations exists for systems subject to in situ applied loads in the
absence of residual stress. Recent work has provided an algorithm to achieve
tomographic reconstruction for this class of mechanical system. This paper
details an experiment focused on providing a proof-of-concept for this
algorithm by carrying out a full tomographic reconstruction of a biaxial
strain field within a non-trivial steel sample. This work was carried out on
the RADEN energy-resolved neutron imaging instrument at the Japan Proton
Accelerator Research Complex with validation through Digital Image Correlation
and constant wavelength neutron strain scans from the KOWARI diffractometer
within the Australian Centre for Neutron Scattering. Aside from a small
systematic error present within individual measurements, the experiment was a
success, and now serves as a practical demonstration Bragg-edge transmission
strain tomography for in situ systems.

##  1 INTRODUCTION

Strain tomography has the potential to dramatically transform the way in which
experimental mechanics is carried out. In much the same way that regular
computed tomography has transformed medicine and other sciences, a full-field
approach to measuring triaxial strain within solid materials will have an
impact across a range of areas. For example, Additive Manufacturing (AM)
processes allow tremendous flexibility in component design. AM components are
now finding their way into critical aerospace applications, such as fuel
nozzles [1] and turbine blades [2]. Residual strain locked in by the printing
process has a critical impact on the strength of the resulting components and
must be understood to advance this technology. In contrast to current
destructive and semi-destructive techniques [3, 4], 2D Digital Image
Correlation (DIC) [5] and point-wise X-ray and neutron diffraction methods [6,
7, 8], Bragg-edge neutron strain tomography promises to reconstruct the full
three-dimensional strain field and hence stress within printed engineering
parts on the scale of centimetres.

The rapid development of pixelated ‘time-of-flight’ detectors has made Bragg-
edge analysis a prominent method for strain imaging. Bragg-edges are sudden
increases in the relative transmission through polycrystalline solids as a
function of wavelength (see Figure 1b).

These edges are formed through diffraction; a neutron of wavelength $\lambda$
can be coherently scattered by crystal planes with spacing $d$, provided that
the scattering angle, $\theta$, satisfies Bragg’s law; $\lambda=2d\sin\theta$.
An abrupt increase in transmission occurs once $\lambda=2d$ is exceeded as a
neutron cannot be scattered by more than $2\theta=180^{\circ}$ (i.e.
backscattered). Like many diffraction techniques Bragg-edge location and
geometry can provide a wealth of information on the internal structure of
materials [9].

While other approaches are available (e.g. [10]), Bragg-edge analysis
typically involves time-of-flight energy resolved measurements at pulsed
neutron sources such as ISIS (UK), J-PARC (Japan) or SNS (USA). This technique
relies upon the direct relationship between the velocity of a neutron and its
wavelength. At a pulsed source, discrete bursts of polychromatic neutrons are
created through spallation processes. As each pulse travels down a flight tube
(typically 15-$\mathrm{2}\mathrm{0}\mathrm{m}$) it becomes elongated as the
fast neutrons (short wavelengths) outpace slower neutrons (long wavelengths)
as illustrated in Figure 1a. The time of arrival of each neutron is then a
proxy for wavelength, allowing the relative transmission spectra through
samples to be directly determined.

(a)

(b)

(c)

(d)

Figure 1: (Colour online) (a) Energy-resolved Time-of-Flight (TOF)
transmission geometry at a pulsed neutron source. (b) Bragg-edges in iron
powder from a TOF transmission measurement [9]. (c) and (d) A high resolution
strain image ($\times 10^{6}$) from a two-dimensional steel compact-tension
crack sample using a Micro-Channel Plate (MCP) detector by [11] (Finite
Element Model on the left, measurement on the right).

The position of a Bragg-edge can be obtained by fitting the integral form of
the Kropff model [12] to the measured spectra. The relative shift in these
positions can then provide a measure of strain of the form [9];

$\langle\epsilon\rangle=\frac{\lambda-\lambda_{0}}{\lambda_{0}},$ (1)

where $\langle\epsilon\rangle$ is the measured normal strain in the beam
direction, $\lambda$ is the position of the Bragg-edge, and $\lambda_{0}$ is
the Bragg-edge position in an unstrained sample. Strains measured in this way
are the average normal component in the direction of the transmitted ray. As
with all diffraction techniques, these measurements refer only to the elastic
component and hence are directly related to stress through Hooke’s law.

(a)

(b)

Figure 2: (a) The coordinate system for a single strain measurement modelled
by the Longitudinal Ray Transform (2). (b) The boundary nodes and 2D Finite
Element plate model used in the reconstruction process.

Initially this work revolved around helium-based detectors [13], however the
current state-of-the-art 10B-based Micro-Channel Plate (MCP) technology due to
Tremsin et al. [14] [15] is now capable of spatial resolutions down to
$\mathrm{5}\mathrm{5}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ [16, 11, 17] (e.g.
see Figure 1d).

These developments have raised the tantalising prospect of strain tomography
which has formed a focus for research over the past decade. As opposed to
conventional tomographic imaging, this problem focuses on the reconstruction
of a tensor field; a significantly more complex problem.

The measurement from a single ray observed by such a detector can be modelled
by the Longitudinal Ray Transform (LRT);

$\langle\epsilon\rangle=\mathbf{I}_{\boldsymbol{\epsilon}}=\frac{1}{L_{0}}\int_{0}^{L_{0}}\epsilon_{ij}\left(\boldsymbol{x}_{0}+s\hat{\boldsymbol{n}}\right)\hat{n}_{i}\hat{n}_{j}\,\mathrm{d}s,$
(2)

where a ray in the direction $\hat{\boldsymbol{n}}$ enters the sample at
$\boldsymbol{x}_{0}$, and $L_{0}$ is the irradiated sample length (see Figure
2(a)). Lionheart and Withers[18], and prior work by Sharafutdinov [19],
clearly demonstrates the general strain tomography problem from this transform
is ill-posed. For conservative strain fields the LRT is only sensitive to
boundary deformations, implying multiple strain fields can project to the same
set of strain images (solutions to the problem are not unique). However,
reconstruction can still be achieved if additional information is included
through assumptions (special cases), or constraints imposed by the physics of
the system. For example, most of the initial work in this area is limited to
special cases (e.g. axisymmetry [20, 21, 22] and granular systems [23]).

A recent breakthrough in this area has provided a reconstruction technique for
strain fields in samples subjected to in situ loadings [24]; i.e. boundary
tractions in the absence of eigen-strain (e.g. plasticity), or body forces. In
this case, the lack of eigen-strain and body forces provides a direct link
between the strain field and the boundary deformation [24]. This direct
relationship provides a method for reconstruction. Strain fields resulting
from in situ loadings represent a large class of mechanical systems; their
reconstruction represents an important step towards general strain tomography.

This paper presents an experimental proof-of-concept for this algorithm;
representing the first practical tomographic reconstruction of a non-
axisymmetric strain field.

##  2 EXPERIMENTAL SET-UP

This experiment was conducted over a 4 day period in November 2016 at the
Japan Proton Accelerator Research Complex (J-PARC) on the RADEN energy
resolved neutron imaging instrument. The system under examination consisted of
a small, two-dimensional, C-shaped, EN-26 steel sample subject to a
compressive load of approximately $\mathrm{7}\mathrm{k}\mathrm{N}$ (Figure 3).
The sample had an inner diameter of $\mathrm{7}\mathrm{m}\mathrm{m}$, outer
diameter of $\mathrm{2}\mathrm{0}\mathrm{m}\mathrm{m}$ and thickness
$\mathrm{1}\mathrm{6}\mathrm{m}\mathrm{m}$. Prior to loading, the sample was
quenched and tempered at $560^{\circ}$C for one hour to relieve residual
strains. The resulting hardness of the material was around 250HV. Load was
applied by two M6 aluminium bolts (7075-T6) in a small die constructed from a
high strength aluminium alloy (7075-T6). Aluminium was chosen to provide
relative transparency to neutrons. The two aluminium bolts were each torqued
to $1.$\mathrm{7}\mathrm{N}\mathrm{m}$$ to apply the
$\mathrm{7}\mathrm{k}\mathrm{N}$ compressive load.

The sample geometry was carefully chosen to produce regions of high strain at
some distance from the applied loads. This creates a strain field that is
trivial to model numerically; the effect of boundary conditions is limited to
small regions around the contact points.

Figure 3: The experimental system and measurement geometry; a small, two-
dimensional, C-shaped steel sample contained within an aluminium loading
system positioned between the source and the MCP detector. The beam incident
angle shown is for a stage rotation of $\theta=90^{\circ}$.

The sample and loading system were placed on a rotation stage between the beam
source and an MCP detector [11], as illustrated in Figure 3. This detector
consists of a $512\times 512$ array of pixels with a spatial resolution of
$\mathrm{5}\mathrm{5}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ and nano second
temporal resolution. The actual spatial resolution during the experiment was
assessed using a ‘Siemens star’ radial grating and found to be around
$\mathrm{1}\mathrm{0}\mathrm{0}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$.

After reference open-beam and $\lambda_{0}$ measurements, a series of 86 in-
plane Bragg-edge strain projections were measured. Each of these projections
represents a one-dimensional strain profile through the sample in the
transverse direction. Figure 4 shows two examples at different angles. The
projection angles, $\theta$, were based on golden angle increments to evenly
distribute directions over $360^{\circ}$ irrespective of beam time
interruptions.

Strain measurements were based on the relative shift of the $(110)$ Bragg-
edge. Each projection was sampled for one hour at a beam power of
$\mathrm{1}\mathrm{5}\mathrm{0}\mathrm{k}\mathrm{W}$ and an
$\mathrm{1}\mathrm{8}\mathrm{m}$ source distance. A Gaussian image filter
($\sigma=4\text{ pixels}$) was applied to each time bin before grouping pixels
into columns two pixels wide spanning the full sample length. Given the
constraints of available beam time, individual projections tended to be noisy.
The grey band in Figure 4 represents one standard deviation either side of the
mean; this is a lower bound of the true uncertainty.

(a)

(b)

Figure 4: A comparison between typical experimental strain profiles, and
profiles generated from simulated FEA data. The grey band represents one
standard deviation either side of the mean. (a) Profiles for
$\theta=0^{\circ}$. (b) Profiles for $\theta=90^{\circ}$

Reconstruction from these profiles was achieved using the method presented by
Wensrich et al. [24]. A summary of this approach is as follows;

Where the strains are a due to a continuous displacement field
$\boldsymbol{\phi}$, the LRT (2) can be written as;

$\mathbf{I}_{\boldsymbol{\epsilon}}=\frac{1}{L_{0}}(\phi_{i}\left(\boldsymbol{x}_{0}+L_{0}\hat{\boldsymbol{n}}\right)-\phi_{i}\left(\boldsymbol{x}_{0}\right))\hat{n}_{i},$
(3)

where $\phi_{i}\left(\boldsymbol{x}_{0}+L_{0}\hat{\boldsymbol{n}}\right)$ and
$\phi_{i}\left(\boldsymbol{x}_{0}\right)$ are the boundary displacements at
exit and entry locations of the ray. With the aid of linear shape functions, a
system of equations relating $M$ measurements,
$\langle\boldsymbol{\epsilon}\rangle$, to the displacements of $N$ discrete
points along the boundary of the sample, $\boldsymbol{\Phi}$ can be written in
the form;

$\mathbf{\langle}\boldsymbol{\epsilon}\mathbf{\rangle}_{[M\times
1]}=\mathbf{A}_{[M\times 2N]}\boldsymbol{\Phi}_{[2N\times 1]}.$ (4)

A least squares approach can then be used to solve for $\boldsymbol{\Phi}$.

From $\boldsymbol{\Phi}$, the internal strain field can then be found by
solving the resulting Dirichlet boundary problem using Finite Element Analysis
(FEA). This method was slightly modified to account for rays passing through
multiple segments in a non-convex sample, and was shown to converge to the
true solution in simulation.

This algorithm requires detailed knowledge of the sample position, center of
rotation, and orientation. These were determined by matching the known sample
geometry to edge height profile over all projections.

##  3 RESULTS, VALIDATION AND ERROR ANALYSIS

The least-squares solution for 47 boundary nodes was found from the 86
projections, which then formed the boundary conditions for the two-
dimensional, plane-stress, finite element model shown in Figure 2(b)). Figure
5 shows the reconstructed field in terms of the three cartesian components of
strain; $\epsilon_{xx}$, $\epsilon_{xy}$, and $\epsilon_{yy}$. For
$\epsilon_{yy}$, a band of compression is observed along the vertical axis in
line with the applied load, with a maximum corresponding to
$\sigma_{yy}=$\mathrm{-}\mathrm{3}\mathrm{9}\mathrm{0}\mathrm{M}\mathrm{Pa}$$
concentrated on the inside surface. A corresponding region of tensile strain
exists on the left hand side as a result of the bending load (maximum of
$\sigma_{yy}=$\mathrm{1}\mathrm{3}\mathrm{6}\mathrm{M}\mathrm{Pa}$$).
$\epsilon_{xx}$ and $\epsilon_{xy}$ have a less-direct relationship to the
applied load.

Figure 5: The reconstructed full-field two-dimensional strain components
($\epsilon_{xx}$, $\epsilon_{xy}$, and $\epsilon_{yy}$) compared to an FEA
simulation and DIC results.

Figure 5 also shows a corresponding strain field predicted from an FEA model.
Several assumptions are made by this model (e.g. symmetric uniform edge load,
and plane stress), however it was validated was against constant wavelength
neutron strain scanning on the KOWARI diffractometer at the Australian Center
for Neutron Scattering within the Australian Nuclear Science and Technology
Organisation (ANSTO). This instrument provided measurements of $\epsilon_{xx}$
and $\epsilon_{yy}$ along the sample centreline based on the relative shift of
the (211) $\alpha-\text{iron}$ diffraction peak using a gauge volume of
$$\mathrm{1}\times\mathrm{1}\times\mathrm{1}\mathrm{4}\mathrm{m}\mathrm{m}$^{3}$,
and a wavelength of $1.67$Å(i.e. $90^{\circ}$ geometry). Figure 6 shows these
measurements directly compared to the tomographic reconstruction and FEA
results at corresponding locations. The similarity between FEA predictions and
the strain scan demonstrates the models suitability as a reference by which
the quality of the reconstruction can be assessed.

Figure 6: Comparison of the reconstructed and FEA strain fields to a constant
wavelength strain scan from the KOWARI diffractometer.

DIC was also carried out during sample loading, the results of which are also
shown in Figure 5. DIC provides an estimate of full-field two-dimensional
strain at the surface, by comparing digital images of the sample before and
after loading [25]. These strains are computed as the gradient of displacement
field calculated through cross-correlation. Strain within most of the sample
was at the lower end of what we could resolve using this technique, however
orders of magnitude and structure were similar to the FEA model.

A comparison of the reconstruction to the FEA model, DIC results, and the
KOWARI strain scan shows close agreement, however, the reconstruction tends to
have a lower magnitude – by approximately 15% overall. Outside of this
reduction in magnitude there are small regions of significant discrepancy near
the inside surface of the ‘C’. These discrepancies were greater than expected.
This expected error was based on a set of simulations, that provide an
estimate of the uncertainty from Gaussian measurement noise with standard
deviation $1\times 10^{-4}$. This suggested convergence to the true solution
with a standard deviation less than $5\times 10^{-5}$.

The reduction in magnitude was also observed in individual strain profiles
(e.g Figure 4). Similarly, discrepancies were observed in the profiles in
areas corresponding to the inside surface of the ‘C’; for example the region
from $5-7.$\mathrm{5}\mathrm{m}\mathrm{m}$$ in Figure 4a. This would suggests
a systematic error is present in the strain measurement.

This systematic error may arise from a variety of sources. Of particular note,
the regions of worst discrepancy tended to coincide with regions of high
strain gradient. The effect of strain gradients on the geometry of Bragg-edges
is not well known. The Kropff model [12, 9] defines the geometry via five
parameters, one of which specifies location. This is assumed to have an
independent and direct relationship with average strain along the path; this
may not be true when strain gradients exist.

Another potential source of error is related to the spatial smoothing of
required to reduce measurement uncertainty; this has the effect of tempering
variation in the resulting profiles. Longer sampling times or brighter sources
may reduce the need for smoothing in future. Note that beam divergence may
play a similar role. It is important that work continues on resolving these
issues.

##  4 CONCLUSION

Aside from these systematic errors, this approach has performed well at
reconstructing this strain field from experimental data. This represents the
first tomographic reconstruction of a non-axisymmetric strain field from
Bragg-edge measurements. This field was developed by an in situ loading and
the reconstruction hinged upon this assumption, however this special case is a
significant step towards achieving general strain field tomography.

This demonstration provides a basis for future efforts focussed on extending
to three-dimensional strain fields, as well as other approaches that may allow
for general reconstruction of a broader class of strain field (e.g. residual
strain). A central component of this future work revolves around the
development of algorithms suitable for the general case where plastic strains
may present. In this case, the link between internal strain and boundary
deformation is severed, violating the key assumption of the algorithm
demonstrated in this paper. In place of this, future algorithms will be
developed through a consideration of the physics of equilibrium or
minimisation of strain energy to constrain the reconstructions to the true
solution. While these future algorithms continue to be a work-in-progress, the
present work serves to provide a demonstration that the Bragg-edge
transmission strain tomography has practical potential.

##  5 ACKNOWLEDGEMENTS

This work is supported by the Australian Research Council through the
Discovery Project scheme (ARCDP170102324). Access to RADEN instrument (J-PARC)
and KOWARI (ANSTO) was made possible through their respective user programs
(proposals 2016A0032 and P5293). Additional support during the ANSTO
experiment was provided by the Australian Institute for Nuclear Science and
Engineering (AINSE).

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# LOFAR search for radio emission from galaxies hosting tidal disruption
events

Kamakshi Kaushik  Interdisciplinary School of Sciences
Savitribai Phule Pune University
Pune, Maharashtra, 411007, India  Amitesh Omar  Aryabhatta Research Institute
of Observational Sciences,
Manora Peak, Nainital,
Uttrakhand, 263001, India  Brijesh Kumar  Aryabhatta Research Institute of
Observational Sciences,
Manora Peak, Nainital,
Uttrakhand, 263001, India  Kuntal Mishra  Aryabhatta Research Institute of
Observational Sciences,
Manora Peak, Nainital,
Uttrakhand, 263001, India  Jagdish Joshi  Aryabhatta Research Institute of
Observational Sciences,
Manora Peak, Nainital,
Uttrakhand, 263001, India

###### Abstract

Radio emission from 23 tidal disruption event host galaxies were searched in
the 144 MHz LOFAR-LoTSS2 images. Three host galaxies are detected with diffuse
radio emission, which can be interpreted as either galactic synchrotron radio
emission or diffuse radio halo in dense galactic environments. Non-detection
of (transient) radio emission in majority of galaxies could be due to self-
absorption of radio emission associated with the tidal disruption event.

##  1 Introduction

A total of 56 tidal disruption events (TDEs) are catalogued in Gezari (2021).
A star of mass $M_{*}$ and radius $R_{*}$ upon passing within the tidal radius
$R_{*}(M_{BH}/M_{*})^{1/3}$ of a black hole (BH) with mass $M_{BH}$, is
tidally ripped apart and the event is known as a TDE (Rees, 1988). A TDE can
result in to a slowly varying transient electromagnetic signal from X-rays to
radio bands on time-scales of years. The radio follow-ups of TDEs are
discussed in Alexander et al. (2020). Due to self-absorption, radio spectrum
shows a turnover around a few GHz, where the emission peaks. The emission
decreases at both higher and lower frequencies from the peak frequency. The
radio spectrum evolves over several years and turnover frequency is expected
to shift at lower frequencies although with a decrement in the total radio
flux (Cendes et al., 2021). This makes detection of radio emission from TDEs
at low radio frequencies extremely difficult with the present generation of
radio telescopes, and may only be possible at very late stages of evolution in
some very bright TDEs.

Studying the properties of host galaxies is important in understanding TDEs.
It was found that TDEs are hosted dominantly in post starburst galaxies, where
the current star formation rate is very low (French et al., 2017). This makes
host galaxies weak radio sources. In a study of a sample of 513 PSB galaxies,
only 12 galaxies were detected in radio emission above 1 mJy using the 1.4 GHz
VLA FIRST images (Shin et al., 2011). At lower radio frequencies, diffuse
galactic synchrotron radio emission is expected to be enhanced. Radio
detection from TDE host galaxies at low radio frequency (e.g., 144 MHz) can be
important for several reasons, e.g., persistent radio continuum from host
galaxy can be detected and subtracted to get radio light curves of TDEs, radio
morphology and properties of the TDE host galaxies can be studied etc.

##  2 LOFAR detections

The LoTSS2 survey using the LOFAR provides 144 MHz images covering about 27%
of the northern sky at a median rms of 75 $\mu$Jy beam-1 (Shimwell et al.,
2022). The rms in different regions can vary due to variations in de-
convolution errors and the sky noise. The LOFAR observations were carried out
during 2014 to 2022. In the present release of the LoTSS2, 23 TDEs are covered
in the survey images. The timings of the LOFAR detections for these TDEs are
between 9029 days and 57 days from the reported day of the TDE transient,
wherever known. As some X-ray sources discovered in the X-ray surveys were
later identified as TDEs, in those cases the actual epochs of the TDEs remain
unknown. We carried out a visual inspection of all 23 fields in the LoTSS2
images. The radio emission is detected from three TDE host galaxies. These
images are presented in Fig. 1. The individual objects are discussed below.

###  2.1 AT2018iih

AT2018iih was discovered as a transient by the ATLAS facility on November 9,
2018 (Tonry et al., 2018) and later classified as a TDE using the the Zwicky
Transient Facility (van Velzen et al., 2021). It is hosted in a dwarf galaxy
at $z\sim 0.212$ having a blue magnitude of $\sim 18$. The 144 MHz LOFAR radio
emission is diffuse and covers several galaxies in the vicinity (see Fig.1a).
The epoch of the LOFAR radio observation is 97 days after the discovery of the
TDE. The host galaxy appears to be in a multi-galaxy environment and most
likely interacting with other galaxies. The total radio flux at 144 MHz is
estimated to be $\sim 6$ mJy. The radio morphology appears similar to the
radio emission in a combined halo formed in a compact group of interacting
galaxies.

###  2.2 RBS 1032

The RBS 1032 was first detected as a soft and luminous X-ray source in the
ROSAT Bright Survey (RBS) in 1990. Later, it was suggested to be a TDE
candidate based on multiple X-ray observations with ROSAT and XMM-Newton, and
constraining light-curve decay models (Maksym et al., 2014). It is hosted in a
dwarf galaxy at $z\sim 0.026$. The epoch of the LOFAR radio observation is at
least 23 years after the TDE event. The radio morphology appears similar to
the radio halo emission in galaxy clusters and groups (see Fig.1b). However,
surprisingly no other bright galaxy is seen within the radio extent. There is
also no associated diffuse X-ray emission in the ROSAT and XMM-Newton images.
The radio emission is connected to the TDE host galaxy but not centered on it.
This radio morphology at the moment remains unexplained and is highly
intriguing in absence of bright optical sources and diffuse X-ray emission.
The total radio flux is estimated to be $\sim 4$ mJy.

###  2.3  NGC 5905 (RBS 1475)

RBS 1475 was also first detected as a soft and luminous X-ray source by ROSAT
and later identified as a TDE candidate based on the X-ray light-curve from
multiple X-ray observations at different epochs (Bade et al., 1996). It is
hosted in a nearby ($z\sim 0.011$) spiral galaxy NGC 5905. The epoch of the
LOFAR radio observation is at least 24 years after the TDE event. The 144 MHz
morphology is typical of a spiral galaxy showing diffuse emission associated
with the the disk and the spiral arms (see Fig. 1c). It has a central bright
radio source indicating that the nucleus is active. The radio flux of the
central source is estimated to be $\sim 39$ mJy and that of the diffuse
emission is estimated to be $\sim 100$ mJy. In a previous attempt with the
GMRT, total 1.4 GHz flux associated with the central source was estimated as
$\sim 9.5$ mJy with no significant disc radio emission (Raichur et al., 2015).

###  2.4 Sw 1644+57 (non-detection)

We also report here an important non-detection at 144 MHz with 3$\sigma$ limit
on the flux as $\sim 0.3$ mJy for a well-studied TDE Sw 1644+57. The epoch of
the LOFAR observation is 2806 days after its discovery. It was first
discovered by Swift and classified as a Gamma Ray Burst GRB110328A (Cummings
et al., 2011), but later identified as a TDE (Bloom et al., 2011). Fading
radio emission associated with the TDE has been detected previously. Using the
radio analysis based on the self-absorption modeling of radio flux at GHz
frequencies (van Velzen et al., 2021), a radio flux much below 0.1 mJy at the
LOFAR frequency is predicted. Therefore a non-detection at 144 MHz is
consistent here with the previous models.

(a) AT2018iih (0.1 mJy beam${}^{-1}\times(1,2,3,4)$)

(b) RBS 1032 (0.04 mJy beam${}^{-1}\times(1,2,3,...,9)$)

(c) NGC 5905 (thin 0.15 mJy beam${}^{-1}\times(1,2,3,4,5)$, thick 1 mJy
beam${}^{-1}\times(1,2,3,4)$)

Figure 1: 144 MHz radio contours of TDE host galaxies overlaid upon the
optical r-band SDSS images. The contour levels are provided in parentheses
after the TDE name. The green box indicates location of TDE host galaxy.

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# On minimal higher genus fillings

Gregory R. Chambers  Department of Mathematics, Rice University, Houston, TX
<EMAIL_ADDRESS>
###### Abstract.

In this article, we prove that if $(M,g)$ is a genus $G$ orientable surface
with a single boundary component $S^{1}$, and if $(D,g_{0})$ is a disc such
that interior points are connected by unique geodesics and

$d_{(D,g_{0})}(x,y)\geq d_{(M,g)}(x,y)$

for all $x,y\in\partial M=\partial D$, then

$(1+\frac{2G}{\pi})\textrm{Area}(M,g)\geq\textrm{Area}(D,g_{0}).$

##  1\. Introduction

In “Filling Riemannian manifolds” [6], M. Gromov conjectured the following:

######  Conjecture 1.1 (Gromov).

Suppose that $(M,g)$ is a Riemannian orientable surface with a single boundary
component $S^{1}$ of length $2\pi$. If, for every $x,y\in S^{1}$

$d_{S^{1}}(x,y)=d_{(M,g)}(x,y),$

then the area of $(M,g)$ is no less than the area of the round hemisphere of
intrinsic diameter $\pi$.

Gromov also observed that if $M$ has genus $0$, this result can be obtained
from Pu’s inequality [9], which is a sharp inequality relating the length of
the systole (shortest noncontractible loop) to the area of $\mathbb{R}P^{2}$
with a Riemannian metric.

In 2002, S. Ivanov [8] gave a different proof of this result, in fact proving
something more general:

######  Theorem 1.1 (S. Ivanov).

If $(D,g)$ and $(D,g_{0})$ are two Riemannian discs with

$d_{g}(x,y)\geq d_{g_{0}}(x,y)$

for all $x,y\in\partial D$, and if interior points of $(D,g_{0})$ are
connected by unique geodesics, then

$\textrm{Area}(D,g)\geq\textrm{Area}(D,g_{0}).$

Conjecture 1.1 is also known to be true if $(M,g)$ is a genus $1$ surface;
this was proved using different methods in 2003 by V. Bangert, C. Croke, S.
Ivanov, and M. Katz (see [1]).

Our main result generalizes Theorem 1.1:

######  Theorem 1.2.

Suppose that $(D,g_{0})$ is a Riemannian $2$-disc $D$, and that $(M,g)$ is an
oriented surface with boundary $\partial M=\partial D=S^{1}$ and genus $G$. If

  1. (1)

For every two points $x$ and $y$ in the interior of $D$, there is a unique
geodesic from $x$ to $y$.

  2. (2)

For every $p$ and $q$ on $\partial D=\partial M$,

$d_{g_{0}}(p,q)\leq d_{g}(p,q).$

then

$Area(D,g_{0})\leq(1+\frac{2G}{\pi})Area(M,g).$

The problem of comparing the geometry of manifolds based on their boundary
data has been studied in other contexts as well. For example, see [2], [4],
[5], [3], and [7].

Acknowledgments The author would like to thank Michael Wolf and for many
useful conversations surrounding the ideas in this article, as well for making
numerous suggestions to earlier drafts. He would also like to thank Yevgeny
Liokumovich for several discussions at the beginning of this project. The
author was supported in part by NSF grant DMS-1906543.

##  2\. Overview of proof

As mentioned in the introduction, the proof will build upon the methods
developed by S. Ivanov in [8]. Following this article, we begin by defining
“special distance functions” as follows. Selecting a point $p$ on the boundary
of $M$ (which is equal to the boundary of $D$), we define the function
$f_{p}:M\rightarrow\mathbb{R}$ by

$f_{p}(x)=\max_{q\in\partial M}(|pq|_{0}-|xq|).$

Here, we are using the same notation as in [8]; $|\cdot|_{0}$ denotes the
distance between points in $(D,g_{0})$, and $|\cdot|$ denotes the distance
between points in $(M,g)$. Note that the function $f_{p}$ is well-defined
since $\partial M$ is compact. Similarly, we define
$\tilde{f}_{p}:D\rightarrow\mathbb{R}$ by

$\tilde{f}_{p}(x)=\max_{q\in\partial D}(|pq|_{0}-|xq|_{0}).$

The proof now works as follows. We argue that, for a point $p\in\partial
M=\partial D$, $f_{p}$ and $\tilde{f}_{p}$ agree on the boundary. We then
choose a sequence of distinct points $p_{1},\dots,p_{n}$ on $\partial
M=\partial D$, and consider the $2$-forms $\sum_{i=1}^{n}f_{p_{i}}\wedge
f_{p_{i+1}}$ on $M$ and
$\sum_{i=1}^{n}\tilde{f}_{p_{1}}\wedge\tilde{f}_{p_{i+1}}$ on $D$. We show
that both are exact, and so by Stokes’ Theorem their integrals can be written
as integrals of a $1$-form over $\partial M$ and $\partial D$. Since these
boundaries agree and the functions agree on them (for all
$p_{1},\dots,p_{n}$), these forms also are equal on the boundaries and so the
integrals are equal as well.

We then argue that, as this sequence of points becomes dense in the boundary,
the integral over the disc goes to $\pi\textrm{Area}(D,g_{0})$, and the
integral over the genus $1$ surface is bounded by
$\pi(1+\frac{2G}{\pi})\textrm{Area}(M,g)$. Since they agree (for any sequence
of points) up to an arbitrarily small perturbation, we obtain the desired
result.

##  3\. Properties of $f_{p}$ and $\tilde{f}_{p}$

Throughout this section, the majority of the statements and their proofs
follow their counterparts in [8]. We begin with the following lemma:

######  Lemma 3.1.

Suppose that $p\in\partial M$, and $f_{p}$ is defined as above. Then the
following are true:

  1. (1)

$f_{p}$ is a nonexpanding function with respect to the metric $g$, that is,
$|f_{p}(x)-f_{p}(y)|\leq|xy|$ for all $x,y\in M$.

  2. (2)

$f_{p}$ is differentiable almost everywhere.

  3. (3)

If $x\in\partial M$, then $f_{p}(x)=|px|_{0}$.

###### Proof.

We will prove the three parts of the lemma in order:

  1. (1)

Fix $p\in\partial M$; $f_{p}$ is the supremum of functions
$f_{p,q}(x)=-|xq|+|pq|_{0}$ over all $q\in\partial M$. We observe that

$|f_{p,q}(x)-f_{p,q}(y)|=||xq|-|yq||\leq|xy|,$

where the last inequality is due to the reverse triangle inequality. Thus,
each $f_{p,q}$ is nonexpanding, and so the supremum over $q\in\partial M$ is
also nonexpanding.

  2. (2)

If $x\in\partial M$, then $|xq|\geq|xq|_{0}$ for all $q\in\partial M$ from the
assumption about the boundary distances in $(D,g_{0})$ and $(M,g)$. Then
$|pq|_{0}-|xq|\leq|pq|_{0}-|xq|_{0}\leq|px|_{0}$ from the triangle inequality.
Furthermore, choosing $q=x$ we have $|px|_{0}-|xx|=|px|_{0}$, completing the
proof.

  3. (3)

This follows from the fact that $f_{p}$ is nonexpanding, and so is Lipschitz
(with Lipschitz constant $1$).

∎

We also have the following lemma:

######  Lemma 3.2.

Suppose that we fix a point $p\in\partial D$ (which is also equal to $\partial
M$). Defining $\tilde{f}_{p}$ as above, we have $\tilde{f}_{p}(x)=|xp|_{0}$
for all $x\in D$.

###### Proof.

Fix $x$ in the interior of $D$, and let $q\in\partial D$ be the unique point
on $\partial D$ where the geodesic from $p$ to $x$ intersects $\partial D$.
Note that $q$ is unique by the assumption on the uniqueness of geodesics
connecting interior points in $(D,g_{0})$.

Since this geodesic is unique, it is minimal, and
$|pq|_{0}=|px|_{0}+|xq|_{0}$, so
$\tilde{f}_{p}(x)\geq|pq|_{0}-|qx|_{0}=|px|_{0}$. In addition,
$\tilde{f}_{p}(x)\leq\tilde{f}_{p}(p)+|px|_{0}=|px|_{0}$ from Part 1 of Lemma
3.1 (note that $(D,g_{0})$ satisfies the hypotheses of that lemma). ∎

######  Lemma 3.3.

Suppose that $p\in\partial M=\partial D$, and $f_{p}$ and $\tilde{f}_{p}$ are
defined as above. Then $f_{p}(x)=\tilde{f}_{p}(x)$ for every $x\in\partial
D=\partial M$.

###### Proof.

This lemma follows immediately from Lemma 3.1 and Lemma 3.2. ∎

######  Definition 3.1.

Suppose that $x$ is a point in the interior of $M$, and $p$ is a point on
$\partial M$. A point $q\in\partial M$ is called a point of maximum if
$f_{p}(x)=|pq|_{0}-|xq|$. It is called a _nearest_ point of maximum if, for
every point of maximum $q^{\prime}$, $|xq|\leq|xq^{\prime}|.$

######  Lemma 3.4.

Suppose that $p\in\partial M$, and $f_{p}$ is as above. If $x$ is an interior
point of $M$ and $f_{p}$ is differentiable at $x$, then $|df_{p}(x)|=1$.
Moreover, if $q\in\partial M$ is a point of maximum, and if
$\gamma:[0,|xq|]\rightarrow M$ is the unit-speed shortest geodesic from $x$ to
$q$, then $\textrm{grad}f_{p}(x)=\gamma^{\prime}(0)$. Furthermore, there is a
unique point of maximum to $x$.

###### Proof.

This proof will follow that of Lemma 1.2 in [8]. As in that proof, we will
prove the statement for gradients instead of derivatives.

Fix a point $x$ in the interior of $M$. Suppose that $p\in\partial M$, and
$f_{p}$ is differentiable at $x$, $q$ is a point of maximum for $p$, and the
curve $\gamma:[0,|xq|]\rightarrow M$ is a unit-speed shortest curve connecting
$x$ to $q$ in $(M,g)$; $\gamma(0)=x$, $\gamma(|xq|)=q$, and
$|\gamma(t)\gamma(t^{\prime})|=|t-t^{\prime}|$ for all
$t,t^{\prime}\in[0,|xq|]$. Since $x$ is in the interior of $M$, an initial arc
of $\gamma$ is contained in the interior of $M$ and so is a geodesic. As a
result, $\gamma$ is differentiable at $0$, and $|\gamma^{\prime}(0)|=1$. Since
$f_{p}$ is nonexpanding and $f_{p}(q)=|pq|_{0}=f_{p}(x)+|xq|$,
$f_{p}(\gamma(t))=f_{p}(x)+t$ for all $t\in[0,|xq|]$.

From this observation that $f_{p}$ grows at unit rate along $\gamma$, and
since this is the maximal growth rate (since $f_{p}$ is nonexpanding),
$\textrm{grad}f_{p}(x)=\gamma^{\prime}(0)$, and so
$|\textrm{grad}f_{p}(x)|=1$.

To prove that there is a unique point of maximum to $x$, suppose that $q_{1}$
and $q_{2}$ are both nearest points of maximum to $x$. Then the shortest
geodesic from $x$ to $q_{1}$ and from $x$ to $q_{2}$ must have the same
gradient at $x$. This is because the gradient along each curve has magnitude
$1$, which is the maximal rate of growth of the function. Hence, both shortest
curves start at $x$ with the same tangent vector, and so due to the uniqueness
of geodesics, that geodesic must hit the boundary of $M$ at $q_{1}$ first, or
at $q_{2}$ first. Thus, since the distances from $q_{1}$ to $x$ and $q_{2}$ to
$x$ are equal, $q_{1}=q_{2}$. ∎

We now investigate what happens if we have a finite set of points on the
boundary of $M$. For the remainder of the article, whenever we consider a
sequence of points $\\{p_{i}\\}$ on $\partial M$ or $\partial D$, we will
assume that they are all distinct. Furthermore, whenever we consider a
sequence of points $\\{q_{i}\\}$ on $\partial M$ or $\partial D$ formed by
taking the nearest points of maximum of unique $\\{p_{i}\\}$ with respect to
an interior point $x$, we will assume that $\\{q_{i}\\}$ are all unique as
well. This may require a small perturbation of the points $\\{p_{i}\\}$; this
does not affect the validity of the proofs throughout this article.

######  Lemma 3.5.

Suppose that $p_{1},\dots,p_{n}$ are points on $\partial M$ in
counterclockwise order. In addition, suppose that $x$ is a point in the
interior of $M$, and that $f_{p_{1}},\dots,f_{p_{n}}$ are all differentiable
at $x$. Let $q_{1},\dots,q_{n}$ be the respective nearest points of maximum of
$p_{1},\dots,p_{n}$ with respect to $x$. If $p_{1},\dots,p_{n}$ and
$q_{1},\dots,q_{n}$ are all distinct, then $q_{1},\dots,q_{n}$ are also in
counterclockwise order on $\partial M$.

To prove this lemma, we may assume that $n=3$ without a loss in generality.
This is because the cyclic ordering of a collection of points on $S^{1}$ is
determined by the cyclic ordering of all triplets. For the next lemma, assume
that $p_{1},p_{2},$ and $p_{3}$ and $q_{1},q_{2}$, and $q_{3}$ be as in the
hypotheses of Lemma 3.5.

######  Lemma 3.6.

Fix $x$ in the interior of $M$, and suppose that $p_{1},p_{2},$ and $p_{3}$
and $q_{1},q_{2},$ and $q_{3}$ are points on $\partial M=\partial D$ so that
$q_{i}$ satisfies $f_{p_{i}}(x)=|xq_{i}|+|p_{i}q_{i}|_{0}$, $f_{p_{i}}$ is
differentiable at $x$ for all $i$, and $q_{i}\neq p_{i}$ for all $i$. If
$i\neq j$, the pair $\\{p_{i},q_{j}\\}$ does not separate the pair
$\\{p_{j},q_{i}\\}$. By this we mean that both $p_{j}$ and $q_{i}$ lie in the
same component of $S^{1}\setminus\\{p_{i},p_{j}\\}$.

###### Proof.

Suppose that two pairs did separate; without loss of generality we may assume
that $i=1$ and $j=2$. Let $\alpha$ be a minimizing geodesic from $p_{1}$ to
$q_{2}$ in $(D,g_{0})$, and let $\beta$ be a minimizing geodesic from $p_{2}$
to $q_{1}$ in $(D,g_{0})$. Since $D$ is a disc, and the two pairs separate,
$\alpha$ and $\beta$ must intersect at a point $z\in D$. Thus, we have

$|p_{1}q_{2}|_{0}+|p_{2}q_{1}|_{0}=|p_{1}z|_{0}+|zq_{2}|_{0}+|p_{2}z|_{0}+|zq_{1}|_{0}\geq|p_{1}q_{1}|_{0}+|p_{2}q_{2}|_{0},$

where the last inequality is due to the triangle inequality. As a result,

$\displaystyle|p_{1}q_{2}|_{0}-|xq_{2}|+|p_{2}q_{1}|_{0}-|xq_{1}|$
$\displaystyle\geq|p_{1}q_{1}|_{0}-|xq_{1}|+|p_{2}q_{2}|_{0}-|xq_{2}|$
$\displaystyle=f_{p_{1}}(x)+f_{p_{2}}(x).$

In addition, we observe that $|p_{1}q_{2}|_{0}-|xq_{2}|\leq f_{p_{1}}(x)$ and
$|p_{2}q_{1}|_{0}-|xq_{1}|\leq f_{p_{2}}(x)$ from the definition of
$f_{p_{1}}$ and $f_{p_{2}}$. Thus, all of the above inequalities must be
equalities, and so $q_{1}$ and $q_{2}$ are points of maximum for both $p_{1}$
and $p_{2}$. ∎

We use Lemma 3.6 to prove Lemma 3.5; this proof is identical to the second
half of the proof of Lemma 1.2 from [8] on page 5.

##  4\. Oriented triangles and Proof of Theorem 1.2

######  Definition 4.1.

Suppose that $a$ and $b$ are points on $S^{1}\subset\mathbb{R}^{2}$. The
triangle $\Delta_{ab}$ is the one formed by the segment from $a$ to $b$, the
segment from $a$ to $(0,0)$, and from $b$ to $(0,0)$. Furthermore,
$\Delta_{ab}$ carries an orientation. If $a$ and $b$ are antipodal or equal,
then we say that $\Delta_{ab}$ is neutrally oriented. If not, then $a$ and $b$
separate $S^{1}$ into two closed arcs $\alpha$ and $\beta$ with nonempty
disjoint interiors, and such that the length of $\alpha$ is less than the
length of $\beta$ (with respect to the standard measure on $S^{1}$).

Case 1: If we parametrize $\alpha$ in a counterclockwise fashion, and if
$\alpha$ starts at $a$ and ends at $b$, then the triangle is positively
oriented.

Case 2: If we parametrize $\alpha$ in a counterclockwise fashion, and if
$\alpha$ starts at $b$ and ends at $a$, then the triangle is negatively
oriented.

The _oriented area_ of $\Delta_{ab}$ is the area of the triangle if the
orientation is positive, and the negative of the area of the triangle if the
orientation is negative. If $\Delta_{ab}$ is neutrally oriented, then its area
is $0$ and so we define its oriented area to be $0$. With a slight abuse of
notation, we will denote this as $\textrm{Area}(\Delta_{ab})$.

######  Lemma 4.1.

Suppose that we have a sequence of ordered collections of points
$\mathcal{P}_{1},\dots,\mathcal{P}_{k},\dots$ from $S^{1}$, and define
$|\mathcal{P}_{k}|$ to be

$\sum_{i=1}^{n}\textrm{Area}(\Delta_{p_{i}p_{i+1}})$

where $p_{1},\dots,p_{n}$ are the points (in counterclockwise order) that
comprise $\mathcal{P}_{k}$ (and $p_{n+1}=p_{n}$). In addition, for every $k$,
let $\ell(k)$ be the length of the _longest_ segment of
$S^{1}\setminus\mathcal{P}_{k}$ (using the standard measure on $S^{1}$). Note
that $\ell(k)>0$ for all $k$ (since each collection of points is finite).

If $\lim_{k\rightarrow\infty}\ell(k)=0$, then

$\lim_{k\rightarrow\infty}|\mathcal{P}_{k}|=\pi.$

###### Proof.

The proof of this fact is contained in the proof of Lemma 2.1 from [8]; the
limit is the area of a unit disc, which is $\pi$. Note that for $k$
sufficiently large, each triangle formed from the collection $\mathcal{P}_{k}$
is positively oriented. ∎

######  Proposition 4.2.

Suppose that $p_{1},\dots,p_{n}$ are distinct counterclockwise points on
$\partial M$, and let $x$ in the interior of $M$ such that
$f_{p_{1}},\dots,f_{p_{n}}$ are all differentiable at $x$. In addition, let
$q_{1},\dots,q_{n}$ be points on $\partial M$ which are nearest points of
minima to $x$ with respect to $p_{1},\dots,p_{n}$; suppose that they are
distinct - by Lemma 3.5 they are oriented counterclockwise. In addition, we
have that there are unique length minimizing geodesics
$\gamma_{1},\dots,\gamma_{n}$ from $x$ to $q_{1},\dots,q_{n}$ respectively.

If $\mathcal{P}=\\{v_{1},\dots,v_{n}\\}$ are the tangent vectors, then

$\sum_{i=1}^{n}\textrm{Area}(\Delta_{v_{i}v_{i+1}})\leq\pi(1+\frac{2G}{\pi}).$

######  Lemma 4.3.

Suppose that $q_{1},\dots,q_{n}$ are distinct points on $\partial D$ which are
oriented in counterclockwise order, and suppose that $x$ is a point on the
interior of $D$. If $\gamma_{1},\dots,\gamma_{n}$ are disjoint simple curves
such that $\gamma_{i}$ starts at $x$ and ends at $q_{i}$. Then
$q_{1},\dots,q_{n}$ also are oriented in a counterclockwise fashion, as are
the tangent vectors $\\{v_{1},\dots,v_{n}\\}$.

###### Proof.

This follows from the topology of the disc; in particular, the Jordan Curve
Theorem directly leads to the conclusion of this lemma. ∎

For now, we will leave the statement of the Proposition 4.2 as is, and will
postpone the proof to the next section. We now prove Theorem 1.2 following the
same technique as [8].

######  Definition 4.2.

We define the $2$-form $\omega_{n}$ on $\mathbb{R}^{n}$ by

$\omega_{n}=\sum_{i=1}^{n}dx_{i}\wedge dx_{i+1}.$

######  Lemma 4.4.

For every $n\geq 2$, define

$F_{M}:M\rightarrow\mathbb{R}^{n}$

by

$F(x)=(f_{p_{1}}(x),\dots,f_{p_{n}}(x))$

. We then have

$\int_{M}F_{M}^{*}\omega_{n}\leq 2\pi(1+\frac{2G}{\pi})\textrm{Area}(M,g).$

###### Proof.

Let $\eta$ be the $2$-form $F_{M}^{*}\omega_{n}$ on $M$; $\eta$ is measurable
and so $\eta(x)=A(\eta)dA$, where $A$ is a measurable real-valued function
which is differentiable almost everywhere, and $dA$ is the standard area form
(which exists since $M$ is orientable). Observe that
$\eta=F_{M}^{*}\omega_{n}=\sum_{i=1}^{n}df_{p_{i}}\wedge f_{p_{i+1}}$. Observe
further that $1/2A(df_{p_{i}}\wedge df_{p_{i+1}}$ is equal to the oriented
area of the triangle $\Delta v_{i}v_{i+1}$ where $v_{i}=df_{p_{i}}(x)$ and
$v_{i+1}=df_{p_{i+1}}(x)$ (here, $v_{i}$ and $v_{i+1}$ are in in
$T^{*}_{x}M$). As such, $1/2A(\eta)$ is equal to the sum of the oriented areas
of the triangles $\Delta v_{i}v_{i+1}$ with $i\in\\{1,\dots,n\\}$.

Note that all $v_{i}$ are unit vectors of $T^{*}_{x}M$, and so are in
$UT^{*}_{x}M$. Then by Proposition 4.2 and Lemma 4.3 together imply that

$1/2A(\eta)\leq\pi(1+\frac{2G}{\pi}).$

As a result,

$\int_{M}\eta\leq\int_{M}A(\eta)dA\leq\pi(1+\frac{2G}{\pi})\textrm{Area}(M,g).$

∎

######  Lemma 4.5.

Suppose that $\\{\mathcal{P}_{1},\dots\\}$ is a sequence of collections of
points on $\partial D$. For each sequence
$\mathcal{P}_{k}=\\{p_{n},\dots,p_{n}\\}$, define

$F_{D}:D\rightarrow\mathbb{R}^{n}$

by

$F(x)=(\tilde{f}_{p_{1}}(x),\dots,\tilde{f}_{p_{n}}(x)).$

We then have

$\lim_{n\rightarrow\infty}\int_{D}F_{D}^{*}\omega_{n}=2\pi\textrm{Area}(D,g_{0}).$

###### Proof.

In this case, by Lemma 3.1 we have $\tilde{f}_{p}(x)=|px|_{0}$,
$\tilde{f}_{p}$ is differentiable everywhere on the interior of $D$ and
geodesics are unique, since $\textrm{grad}f_{p}(x)$ is the unit vector
opposite to the tangent of the unique geodesic from $x$ to $p$. Thus, the map
that sends $p$ to $df_{p}(x)$ is then a homeomorphism from $S^{1}$ to
$UT^{*}_{x}D$. Let $\mathcal{P}=\\{p_{i}\\}$ is a sequence of counterclockwise
points on $\partial D$, let $\delta(\mathcal{P})$ be the length of the longest
segment of $S^{1}\setminus\\{p_{i}\\}$.

Let us choose a collection of points on $\partial D$
$\\{\mathcal{P}_{1},\mathcal{P}_{2},\dots\\}$ so that
$\delta(\mathcal{P}_{i})\rightarrow 0$ as $i\rightarrow\infty$. As in the
proof of Lemma 4.4, for a given $\mathcal{P}_{n}$ the corresponding points
$\\{v_{i}\\}$ on $UT^{*}_{x}D$ are the images of the points under the
homeomorphism from $\partial D=S^{1}\rightarrow UT^{*}_{x}D$.

Since it is a homeomorphism, if $\mathcal{V}_{n}$ is the set of points from
$UT^{*}_{x}D$ that correspond to $\mathcal{P}_{k}$, the longest segment of
$UT^{*}_{x}D\setminus\mathcal{V}_{n}$ also goes to $0$ as $k$ goes to
$\infty$. As per the proof of Lemma 4.4, we observe that if
$\eta=F_{D}^{*}\omega_{n}$, then $\eta=A(\eta)dA$ and $1/2A(\eta)$ is equal to
the sum of the oriented areas of the triangles $\Delta v_{i}v_{i+1}$. Lemma
4.1 implies that all of these triangles are positively oriented for
sufficiently large $n$, and the fact that the longest segment of
$UT^{*}_{x}D\setminus\mathcal{V}_{n}$ goes to $0$ as $k$ goes to $\infty$
implies that, as $k$ goes to $\infty$, $1/2A(\eta)$ goes to $\pi$, so
$A(\eta)$ goes to $2\pi$. ∎

###### Proof of Theorem 1.2.

Let $F_{M}$, $F_{D}$, and $\omega_{n}$ be defined as above. From Lemma 3.3, if
$x$ is a point on $\partial D=\partial M$, and if $p$ is another point on
$\partial D=\partial M$, then $f_{p}(x)=\tilde{f}_{p}(x)=|px|_{0}$. Thus,
$F_{M}$ and $F_{D}$ agree on $\partial M=\partial D$. Since $\omega_{n}$ is a
closed $2$-form on $\mathbb{R}^{n}$, it is exact, and so $\omega_{n}=d\mu_{n}$
for some $1$-form $\mu_{n}$. Then
$F_{M}^{*}\omega_{n}=F_{M}^{*}d\mu_{n}=dF_{M}^{*}\mu_{n}$ and
$F_{D}^{*}\omega_{n}=F_{D}^{*}d\mu_{n}=dF_{D}^{*}\mu_{n}$. Using Stokes’
Theorem and the fact that $F_{M}^{*}\mu_{n}=F_{D}^{*}\mu_{n}$ on $\partial
M=\partial D$, we have

$\displaystyle\int_{M}A(\omega_{n})dA$
$\displaystyle=\int_{M}F_{M}^{*}\omega_{n}$ $\displaystyle=\int_{\partial
M}F_{M}^{*}\mu_{n}$ $\displaystyle=\int_{\partial D}F_{D}^{*}\mu_{n}$
$\displaystyle=\int_{D}F_{D}^{*}\omega_{n}$
$\displaystyle=\int_{D}A(\omega_{n})dA$

This is true for any collection $\mathcal{P}$ of points on $S^{1}=\partial
M=\partial D$. For any such collection, by Lemma 4.4,
$2\pi\textrm{Area}(M,g)\geq\int_{M}A(\omega_{n})dA$. By Lemma 4.5, there is a
sequence of collections $\\{\mathcal{P}_{k}\\}$ of points on the boundary so
that $\int_{D}A(\omega_{n})dA\rightarrow 2\pi\textrm{Area}(D,g_{0})$ as
$k\rightarrow\infty$.

Combining these estimates and taking the limit as $k\rightarrow\infty$, we
have

$2\pi(1+\frac{2G}{\pi})\textrm{Area}(M,g)\geq 2\pi\textrm{Area}(D,g_{0}).$

Dividing both sides by $2\pi$ completes the proof. ∎

##  5\. Proof of Proposition 4.2

We begin this section with defining and proving the existence of special
sequences of curves with respect to an interior point $x$ of $(M,g)$.

######  Definition 5.1.

Suppose that $(M,g)$ is a genus $G$ surface, and $x$ is an interior point of
$M$. We say that a sequence of $2G$ curves $\eta_{1},\dots,\eta_{2G}$ which
start and end at $x$ form a _bouquet_ with respect to $x$ if the following are
true:

  1. (1)

If we remove $\eta_{1},\dots,\eta_{2G}$ from $(M,g)$, then we obtain a half-
open annulus. One of the boundary components is the original $S^{1}$ (which we
will call $C_{1}$), and one is an open $S^{1}$ that comes from removing these
curves (which we will call $C_{2}$).

  2. (2)

All $\\{\eta_{i}\\}$ are disjoint except for their initial and final points.

  3. (3)

Suppose that $q$ is a point on $\partial M$, and $\nu$ is a curve from $q$ to
$x$ which is length minimizing. Then $\nu$ only intersects each $\eta_{i}$ at
$x$, the initial point of $\nu$.

This is depicted in Figure 1.

Figure 1. A bouquet for the genus-1 surface

######  Lemma 5.1.

Suppose that $(M,g)$ is a genus $G$ surface, and $x$ is an interior point of
$M$. Then there exists a bouquet $\eta_{1},\dots\eta_{2G}$ with respect to
$x$.

###### Proof.

We will prove the existence of such a sequence of curves by performing $G$
pairs of operations. We will define the first pair of operations in detail,
and then the others will be defined analogously. Along the way, we will show
that Properties 2 and 3 above (the no intersection properties) must be true
for every intermediate sequence of curves. After $i$ operations, we will
obtain $2i$ curves, which when removed from $M$ will result in a genus $G-i$
surface with two holes, one equal to the original $S^{1}$, and one equal to
the open curve resulting from removing the $2i$ curves. After $G$ operations,
the result is an annulus with two boundary components, as desired.

Let us now describe the process, starting from the first step. To form
$\eta_{1}$, we choose the shortest loop with endpoints equal to $x$ which is
noncontractible and has minimal length. Clearly, minimality guarantees that it
is simple. Suppose that $q$ is a point on the boundary of $M$, and $\nu$ is a
shortest curve from $x$ to $q$. If the tangent vector of $\nu$ is parallel to
the initial or final tangent vector of $\eta_{1}$, then the first segment of
$\nu$ will be equal to $\eta_{1}$, which means that $\nu$ is not the shortest
curve.

Suppose that $y$ is another point of intersection; by the same argument as
above, the tangent vector of $\nu$ and $\eta_{1}$ at the point $y$ must be
linearly independent. Then we can consider the two arcs from $x$ along
$\eta_{1}$ to $y$, as well as the segment of $\nu$ from $x$ to $y$. If either
of the first two arcs are shorter than the arc along $\nu$, then $\nu$ is not
the shortest curve. If the arc along $\nu$ is shorter or equal in length to
both of the original arcs, then we can replace one of the two arcs of
$\eta_{1}$ with the arc of $\nu$ from $x$ to $y$ so that the result is
noncontractible. In addition, after smoothing out the resulting singularity,
we can conclude that this new curve is both noncontractible and shorter than
$\eta_{1}$, which is a contradiction.

If we cut along $\eta_{1}$, we obtain a genus $G-1$ surface with three
boundary components. One of these components corresponds to the original
boundary component of the surface, and the other two come from cutting the
original surface along $\eta_{1}$. There are two points, $x_{1}$ and $x_{2}$,
one on each of these components, which correspond to the original point $x$
before the unfolding.

We define $\eta_{2}$ be the shortest curve that goes from the point $x_{1}$ on
one of the boundary components to the point $x_{2}$ on the other boundary
component; this curve does not touch either of the components with $x_{1}$ or
$x_{2}$ since these components, as well as $\eta_{2}$, are all geodesics (and
are uniquely determined by initial point and initial tangent vector). It is
also simple, as it must be length minimizing. Suppose that $\nu$ is the
shortest curve from $x$ to $q$, a point on the boundary of $M$. Suppose that
$\nu$ intersects $\eta_{2}$ at a point $y$ other than $x$. The tangents of
these curves at $y$ are linearly independent (otherwise $\nu$ would contain
$\eta_{2}$ as a component, which would make it not the shortest curve). Then
we can follow the same procedure as before using $\eta_{2}$ instead of
$\eta_{1}$; we can replace a segment of $\nu$ with a segment of $\eta_{2}$ to
form a shorter competitor to $\nu$ (which is impossible), or replace a segment
of $\eta_{2}$ with a segment of $\nu$ to form a better competitor to
$\eta_{2}$, which is also impossible. This step is shown in Figure 2.

Once we have found $\eta_{1}$ and $\eta_{2}$, if $G>1$ we move on to form
$\eta_{3}$ and $\eta_{4}$. To form $\eta_{3}$, we do the following. First, we
cut along $\eta_{1}$ and $\eta_{2}$, the result being a surface with two
boundary components, one ($B_{1}$) which corresponds to the original boundary
of the surface, and one ($B_{2}$) which corresponds to cutting along
$\eta_{1}$ and $\eta_{2}$. $B_{2}$ contains $4$ points which correspond to
$x$; $x_{1},\dots,x_{4}$. We then choose $\eta_{3}$ to be the shortest curve
which starts at some $x_{i}$ and ends at some $x_{j}$ (potentially with
$i=j$), and which is noncontractible relative to $B_{2}$. $\eta_{3}$ can then
be interpreted as a curve on the original surface $M$ starting and ending at
$x$.

Due to the uniqueness of geodesics, $\eta_{3}$ intersects $B_{2}$ only at its
endpoints. If $\nu$ is a length minimizing curve from $x$ to $y$, a point on
$\partial M$, then it does not intersect $\eta_{3}$. This argument is the same
as that employed to show that this does not happen for $\eta_{1}$.

As before, to form $\eta_{4}$, we cut along $\eta_{1},\eta_{2},$ and
$\eta_{3}$. The result is a genus $G-2$ surface with three boundary
components, one which corresponds to the original boundary component, and two
which come from cutting along $\eta_{1},\eta_{2},$ and $\eta_{3}$. Each of
these two components, $X$ and $Y$, contain points which correspond to $x$.

We then define $\eta_{4}$ as the shortest curve which starts at a point
corresponding to $x$ on $X$ to a point which corresponds to $x$ on $Y$. Again,
the uniqueness of geodesics implies that $\eta_{4}$ only intersects
$\eta_{1},\eta_{2},$ and $\eta_{3}$ at its endpoints, and is simple (as it is
length minimizing). Furthermore, $\eta_{4}$ corresponds to a loop which starts
and ends at $x$ on the original surface $M$. If $\nu$ is a length minimizing
curve which start at $x$ and ends at $y$ on $\partial M$, then it does not
intersect $\eta_{4}$ except at $x$ by the same argument that we employed above
to show that the same property holds for $\eta_{2}$.

If $G>2$, then we execute the same procedure to form $\eta_{5}$ and
$\eta_{6}$. We continue to define $\eta_{2i+1}$ and $\eta_{2i+2}$ until we
obtain $\eta_{1},\dots,\eta_{2G}$. These curves satisfy all of the desired
properties, completing the proof. ∎

Figure 2. Building a bouquet for the genus-1 surface

###  5.1. Proof of Proposition 4.2

To complete the proof of Proposition 4.2, we will require a combinatorial
result which is proved in the next section. We will proceed assuming this
result.

To begin, let $x$ be a point in the interior of $M$. Suppose that
$q_{1},\dots,q_{n}$ are distinct points on $\partial M=S^{1}$ which are
ordered counterclockwise on $\partial M$. Suppose further that there are
unique minimizing geodesics $\alpha_{1},\dots,\alpha_{n}$ which start at $x$
and end at $q_{1},\dots,q_{n}$ (respectively). We select a bouquet
$\\{\beta_{1},\beta_{2},\dots,\beta_{2G-1},\beta_{2G}\\}$ with respect to $x$;
such a bouquet exists by Lemma 5.1.

We first observe that $\alpha_{i}$ and $\alpha_{j}$ with $i\neq j$ cannot be
equal. This follows from the fact that $q_{i}\neq q_{j}$. Next, $\alpha_{i}$
and $\alpha_{j}$ only intersect at $x$. If there is another point of
intersection $y$ of $\alpha_{i}$ and $\alpha_{j}$, then since they are not
equal this would imply that there were multiple shortest curves from $x$ to
$q_{i}$ and $q_{j}$, which is impossible.

Combining these observations with the definition of a bouquet,
$\\{\alpha_{i}\\}$ do not intersect $\\{\beta_{i}\\}$ or each other (except at
$x$). As such, if we remove $\\{\beta_{i}\\}$ from $M$, we then obtain an
annulus with two boundary components. Let $C_{1}$ be the component which is
equal to $\partial M$, and $C_{2}$ is the new boundary component formed by
removing $\\{\beta_{1},\dots,\beta_{2G}\\}$. In this annulus $\mathcal{A}$,
$\alpha_{1},\dots,\alpha_{n}$ each starts at one of the $4G$ vertices of the
polygon $C_{2}$ (corresponding to the initial tangent vector of $\alpha_{i}$
at $x$), and ends at $q_{1},\dots,q_{n}$ (respectively). This is shown in
Figure 3.

Figure 3. The objects from the proof of Proposition 4.2

Let us be more specific. The tangent vectors of both ends of each curve in
$\\{\beta_{i}\\}$ divide $UT_{x}$ up into segments; each segment corresponds
to one of the $4G$ vertices of the polygon with geodesic edges formed by
cutting $M$ along $\\{\beta_{i}\\}$ and unfolding the result. Let $v_{i}$ be
the unit vector in $UT_{x}$ given by $\alpha_{i}^{\prime}(0))$ (where
$\alpha_{i}$ is unit-speed parametrization). Then $\alpha_{i}$ starts at the
point on the polygon which corresponds to the interval of $UT_{x}$ in which
$v_{i}$ lies (after the tangent vectors
$\\{\beta_{i}^{\prime}(0),-\beta_{i}^{\prime}(1)\\}$ are all removed).
Furthermore, the order of all $\\{v_{i}\\}$ lying in the same segment
corresponds to the order of the tangent vectors of $\\{\alpha_{i}\\}$ at their
mutual starting point.

This setup satisfies the hypotheses of Proposition 6.1 (stated and proved in
the next section), and so the total area of all of the triangles
$\Delta_{v_{1}v_{2}},\Delta_{v_{2}v_{3}},\dots,\Delta_{v_{n-1}v_{n}},\Delta_{v_{n}v_{1}}$
is at most $\pi$; this completes the proof.

##  6\. Combinatorial Results

We will begin with the setup for Proposition 6.1, the main proposition of this
section.

Fix a counterclockwise sequence of distinct vectors $v_{1},\dots,v_{n}$ on
$S^{1}$, and fix a positive integer $G\geq 1$. Suppose that
$\tau_{1},\dots,\tau_{4G}$ are segments of $S^{1}$ whose union is $S^{1}$,
have nonempty and disjoint interiors, and $v_{i}$ is in the interior of some
$\tau_{j}$.

Let $\mathcal{A}$ be the annulus $\\{(x,y):1\leq||(x,y)||\leq
2\\}\subset\mathbb{R}^{2}$, let $C_{1}$ be the inner boundary with radius $1$,
and $C_{2}$ be the outer boundary with radius $2$ (see Figure 3). Suppose that
$p_{1},\dots,p_{4G}$ are points on $C_{2}$ oriented counterclockwise.

Lastly, suppose that, for every $\tau_{i}$, we associate it with one of the
points $p_{j}$ by a bijective function
$f:\\{1,\dots,4G\\}\rightarrow\\{1,\dots,4G\\}$. We then consider curves
$\alpha_{1},\dots,\alpha_{n}$ with the following properties:

  1. (1)

Each $\alpha_{i}$ is simple.

  2. (2)

Each $\alpha_{i}$ starts at $C_{2}$, and ends at $C_{1}$.

  3. (3)

The initial point of $\alpha_{i}$ is one of the points $p_{j}$. In particular,
the initial point of $\alpha_{i}$ is equal to $p_{f(i)}$.

  4. (4)

For each $p_{i}$, then curves which start at $p_{i}$ are exactly
$\alpha_{a_{i}},\dots,\alpha_{b_{i}-1}$ for some $b_{i}\geq a_{i}\geq 1$. Note
that this can be empty.

  5. (5)

For every $\alpha_{i}$ and $\alpha_{j}$ with $i\neq j$, the endpoints of
$\alpha_{i}$ and $\alpha_{j}$ are distinct, and $\alpha_{i}$ and $\alpha_{j}$
are disjoint except for possibly their initial points.

  6. (6)

All $\\{\alpha)\\}$ are length minimizing geodesics.

For each $i$, let $v_{i}=\alpha_{i}^{\prime}(0)$.

The main result concerns the order of the endpoints of $\\{\alpha_{i}\\}$ as
defined above. We express this in terms of the total area of the triangles
formed by points $\\{v_{i}\\}$ as follows:

######  Proposition 6.1.

Suppose that both $f$ and $\\{\alpha_{i}\\}$ are defined as above. Let
$\\{q_{i}\\}$ be the endpoints of $\\{\alpha_{i}\\}$, respectively. Then

$\sum_{i=1}^{n}\textrm{Area}(\Delta_{v_{i}v_{i+1}})=\sum_{i=1}^{n}\textrm{Area}(\Delta_{\alpha_{i}^{\prime}(0)\alpha_{i+1}^{\prime}(0)})\leq\pi(1+\frac{2G}{\pi}).$

Before we prove Proposition 6.1, we will need an additional lemma.

######  Lemma 6.2.

Suppose that $a$ and $b$ are points on $S^{1}$ (with unit radius). Then

$-\frac{1}{2}\leq\textrm{Area}(\Delta_{ab})\leq\frac{1}{2}.$

###### Proof.

We need only prove the upper bound; the proof of the lower bound is analogous.
Without loss of generality, we may assume that $b=(1,0)$ and $a$ lies on the
upper half-circle. Then the area of $\Delta_{ab}$ is equal to
$\frac{\sin\theta}{2}$, where $\theta$ is the counterclockwise angle from $b$
to $a$ with $\theta\in[0,\pi]$. On this interval, $\frac{\sin\theta}{2}$ is
maximized at $\theta=\pi/2$, and attains a maximum of $1/2$.

∎

We now move on to the proof of Proposition 6.1:

###### Proof.

Let us first consider each sequence $v_{a_{i}},\dots,v_{b_{i}-1}$ (with
$v_{j}=\alpha_{j}^{\prime}(0)$). Since the curves are disjoint and are
geodesics, the $\\{v_{j}\\}$ are all distinct, and
$v_{a_{i}},\dots,v_{b_{i}-1}$ are either all counterclockwise or all
clockwise. Thus,

$\sum_{j=a_{i}}^{b_{i}-1}\textrm{Area}(\Delta
v_{j}v_{j+1})\leq\frac{\textrm{Length}(\tau_{i})}{2},$

since all $v_{a_{i}},\dots,v_{b_{i}-1}$ lie in an interval of $S^{1}$ of
length equal to that of $\tau_{i}$ (using the notation defined above).

From this, we can conclude that

$\sum_{k=1}^{4G}\sum_{j=a_{k}}^{b_{k}-1}\textrm{Area}(\Delta v_{j}v_{j+1})\leq
1/2\sum_{k=1}^{4G}\textrm{Length}(\tau_{k})=\pi.$

The series

$\sum_{i=1}^{n}\textrm{Area}(\Delta_{v_{i}v_{i+1}})$

has exactly $4G$ more terms than

$\sum_{k=1}^{4G}\sum_{j=a_{k}}^{b_{k}-1}\textrm{Area}(\Delta v_{j}v_{j+1}).$

Each one of these terms, from Lemma 6.2, is at most 1/2, and so we get the
bound

$\sum_{i=1}^{n}\textrm{Area}(\Delta_{v_{i}v_{i+1}})\geq\pi+\frac{4G}{2}=\pi(1+\frac{2G}{\pi}),$

completing the proof. ∎

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