# First Frequency-Time-Resolved Imaging Spectroscopy Observations of Solar
Radio Spikes

Daniel L. Clarkson School of Physics & Astronomy, University of Glasgow,
Glasgow, G12 8QQ, UK  Eduard P. Kontar School of Physics & Astronomy,
University of Glasgow, Glasgow, G12 8QQ, UK  Mykola Gordovskyy Department of
Physics & Astronomy, University of Manchester, Manchester M13 9PL, UK
Nicolina Chrysaphi LESIA, Observatoire de Paris, Université PSL, CNRS,
Sorbonne Université, Université de Paris, 5 place Jules Janssen, 92195 Meudon,
France  School of Physics & Astronomy, University of Glasgow, Glasgow, G12
8QQ, UK  Nicole Vilmer LESIA, Observatoire de Paris, Université PSL, CNRS,
Sorbonne Université, Université de Paris, 5 place Jules Janssen, 92195 Meudon,
France  Station de Radioastronomie de Nançay, Observatoire de Paris, CNRS,
PSL, Université d’Orléans, Nançay, France

(Received 2021 June 30; Revised 2021 July 26; Accepted 2021 Aug 4)

###### Abstract

Solar radio spikes are short duration and narrow bandwidth fine structures in
dynamic spectra observed from GHz to tens of MHz range. Their very short
duration and narrow frequency bandwidth are indicative of sub-second small-
scale energy release in the solar corona, yet their origin is not understood.
Using the LOw Frequency ARray (LOFAR), we present spatially, frequency and
time resolved observations of individual radio spikes associated with a
coronal mass ejection (CME). Individual radio spike imaging demonstrates that
the observed area is increasing in time and the centroid positions of the
individual spikes move superluminally parallel to the solar limb. Comparison
of spike characteristics with that of individual Type IIIb striae observed in
the same event show similarities in duration, bandwidth, drift rate,
polarization and observed area, as well the spike and striae motion in the
image plane suggesting fundamental plasma emission with the spike emission
region on the order of ${\sim}\>10^{8}$ cm, with brightness temperature as
high as $10^{13}$ K. The observed spatial, spectral, and temporal properties
of the individual spike bursts are also suggesting the radiation responsible
for spikes escapes through anisotropic density turbulence in closed loop
structures with scattering preferentially along the guiding magnetic field
oriented parallel to the limb in the scattering region. The dominance of
scattering on the observed time profile suggests the energy release time is
likely to be shorter than what is often assumed. The observations also imply
that the density turbulence anisotropy along closed magnetic field lines is
higher than along open field lines.

Sun: corona – Sun: turbulence – Sun: radio radiation

††journal: ApJL

##  1 Introduction

Solar activity sporadically releases magnetic energy via solar flares and
coronal mass ejections (CMEs) that brightly manifest via electromagnetic
radiation from X-rays to radio waves (e.g. Holman et al., 2011, as a review).
Solar radio bursts are a signature of electrons accelerated in flares and
CMEs. Solar radio spikes are short duration ($10-1000$ ms) bursts with narrow
spectral widths from $\Delta{f}/f\simeq 0.002-0.01$, observed from $7-8$ GHz
(Staehli & Magun, 1986; Benz et al., 1992) down to decametric frequencies
(Melnik et al., 2014). Spike durations are observed to decrease with
increasing frequency to below $10$ ms at gigahertz frequencies (Benz, 1986;
Staehli & Magun, 1986). Their short duration and narrow frequency range are
indicative of processes that occur on millisecond timescales and hence provide
a unique avenue to study the fastest processes in the solar corona (e.g.
Aschwanden, 2002; Karlický et al., 2021). Indeed, electron acceleration due to
magnetic energy release leads to formation of electron beams that subsequently
excite Langmuir waves that produce the observed radio emission. Therefore,
spike durations represent an upper limit for the energy release time. Spikes
are observed in dynamic spectra either chaotically or grouped in clusters and
in connection with either Type III (Tarnstrom & Philip, 1972a; Guedel & Benz,
1990; Shevchuk et al., 2016; Melnik et al., 2017), Type II (Karlicky, 1984;
Armatas et al., 2019), or Type IV (McKim Malville et al., 1967; Guedel & Benz,
1990; Shevchuk et al., 2016; Bouratzis et al., 2016) solar radio bursts.
Spikes are most abundant between ${\sim}\>300-3000$ MHz (Benz, 1986). Their
frequency drift rates are reported to vary from zero to $>100$ MHz s-1
(Tarnstrom & Philip, 1972b), while their fluxes tend not to exceed a few
hundred solar flux units (sfu; 1 sfu = $10^{-22}$ W m-2 Hz-1). At decametric
wavelengths, spikes show time profiles similar to that of Type III radio
bursts but with shorter durations (Melnik et al., 2014).

It was suggested that the plasma emission mechanism thought to produce Type
III bursts (Ginzburg & Zhelezniakov, 1958) is responsible for spike emission
(Zhelezniakov & Zaitsev, 1975). A comparison of spike emission with Type III
bursts has been made by Tarnstrom & Philip (1972a) who notes that the exciter
spatial extent determines whether a spike or Type III burst is produced.
Tarnstrom & Philip (1972b) proposed that weak electron beams with lower
densities and smaller spatial sizes than those that produce Type III bursts
are responsible for spike bursts. Another proposed mechanism is electron
cyclotron maser (ECM) emission (Holman et al., 1980) due to spikes being
observed in conjunction with Type IV radio bursts, and could serve to explain
spike emission in regions of strong magnetic fields and/or low densities.
Cliver et al. (2011) suggest ECM emission as the source of a spike burst
related to a powerful X-class flare, but notes that below 130 MHz, the
densities required would be too high in a post-eruption loop system for ECM to
operate. Moreover, if one of the brightest events recorded cannot produce
sufficient conditions for ECM emission high in the corona, then it is
infeasible for less powerful events.

Imaging of spike clusters has mainly been conducted at higher frequencies:
Khan & Aurass (2006) image radio spikes at $327$ and $410.5$ MHz using the
Nançay Radio Heliograph (NRH) (Kerdraon & Delouis, 1997) finding the spike
emission region above the soft and hard X-ray sources of the associated flare,
but find no significant motion of the spike sources. The radio images show the
spike bursts were temporally and spatially associated with compressed magnetic
field structures due to a CME. Benz et al. (2002) also show spike locations
away from the flare site with two cases near flare loop tops. Interestingly,
the spikes are observed during the flare decay phase, indicating a possible
link to post-flare acceleration sites. Imaging by Krucker et al. (1995) shows
the spike emission at altitudes where it is proposed that the energy release
causing both spike emission and Type III emission occurs. More recent VLA
imaging in the $1.0-1.6\;\mathrm{GHz}$ range show the spike source to be
located above the flare arcade (Luo et al., 2021). However, the detailed
spatial evolution of individual spikes in time, space and frequency have not
been reported in the literature before.

In this paper, we report for the first time, the frequency and time-resolved
evolution of individual radio spikes produced before and in the wake of a CME
using the LOw Frequency ARray (LOFAR; van Haarlem et al., 2013) tracking of
the spike source motion, as well as the spike characteristics in dynamic
spectra between $30-70$ MHz. The spike observations reveal superluminal source
motion and source size expansion at 100 ms scales consistent with strong
anisotropic scattering of radio-waves in a turbulent corona. The spike
locations before and after the CME are shifted upwards, presumably perturbed
by the CME. The results also confirm not only the similarity of the observed
spike properties with Type IIIb striae, but the co-spatial character of the
Type IIIb and spike sources.

##  2 Spike characteristics

The active region AR12665 on the western solar limb produced a C1.4 class
solar flare between 10:50 to 10:55, with the ejection of a bifurcated jet
(Chrysaphi et al., 2020). Solar soft X-ray flux, radio flux and polarization
dynamic spectra during the eruptive event are shown in Figure 1. The
polarization measurements are provided by the Nançay Decameter Array (NDA)
MEFISTO receiver (Boischot et al., 1980; Lecacheux, 2000; Lecacheux et al.,
2013; Lamy et al., 2018), designed to automatically filter noise. Numerous
short duration and low frequency bandwidth spikes are seen. Some spikes are
chaotically distributed in the spectrum, whilst others form chains similar to
Type IIIb bursts. A Type II burst observed by LOFAR near 11:03 UT is
associated with the CME, as reported by Chrysaphi et al. (2020). In addition,
a cluster of bright Type III bursts occur close to the start of the jet
eruption near 10:52 UT, along with two Type IIIb bursts at 10:42 and 11:21 UT
shown in Figure 2(b,e).

Figure 1: X-ray and radio emissions between 10:40 and 11:40 UT. (a) X-ray
lightcurves from the GOES spacecraft (Garcia, 1994). (b) Dynamic spectrum of
the LOFAR observation. A series of bright Type III bursts can be seen near
10:52, with a Type II burst near 11:03. Spikes are distributed across the
dynamic spectrum with a cluster between 11:10 and 11:21. The white box
represents the region shown in Figure 2(a). (c) Dynamic spectrum of the
circular polarization from the NDA/MEFISTO receiver. (d-f) Polarization from
the NDA/MEFISTO receiver of the Type III and Type II bursts, respectively, at
34.5 MHz indicated by the solid black lines in panel (c), and polarization of
two spike bursts within the region shown in Figure 2(c) at 32.50 and 33.88
MHz. The spikes are highlighted by the arrows.

43 isolated solar radio spikes between 10:40 to 11:36 UT are analysed using
LOFAR tied-array beam-forming mode (van Haarlem et al., 2013) using 24 core
Low Band Antenna stations in the outer LBA configuration with a maximum
baseline of $3.6\;\mathrm{km}$ in the frequency interval $30-70$ MHz. 216
interferometrically synthesized beams image the solar corona up to $\sim 3$ R⊙
with a temporal and spectral resolution of $10$ ms and $12.2$ kHz,
respectively. This enabled spikes with durations $<1$ s and spectral widths
greater than ${\sim}\>24$ kHz to be individually resolved. The spike
observations were temporally decreased in resolution to 20 ms to reduce noise.
The flux was calibrated using observations of Taurus A (see Kontar et al.,
2017, for details). Figure 2 shows zoomed-in versions of the fine structures.

Figure 2: Zoomed in dynamic spectra of numerous spike and Type IIIb features
within Figure 1. (a) Numerous spikes between 11:19:32 and 11:21:06 UT from the
region bounded by the white box in Figure 1. (b) Type IIIb burst near 11:21
UT. (c) Spike cluster between $30-40$ MHz. (d) An individual radio spike. (e)
Type IIIb burst observed near 10:42 UT.

Figure 3: (a) Dynamic spectrum of the single spike shown in Figure 2(d). The
black points show the peak of the Gaussian fit to the flux profile at each
time bin. The green line shows the linear fit to these points to derive the
frequency drift rate. (b) Spike flux profile at $34.5$ MHz. The dashed
vertical line shows the burst peak time, with the horizontal red and blue
lines showing the rise and decay time, respectively. (c) Flux profile at the
time of the burst peak (black) with a Gaussian fit (blue). The horizontal line
marks the FWHM spectral width. (d) LOFAR image at the peak intensity of the
spike at $34.5$ MHz near 11:19:59.8 UT. The green contour marks the 2D
Gaussian fit at the FWHM level with the centroid marked by the green cross.
The white dots represent the phased-array beam pointing, and the white oval
shows the half-maximum synthesized LOFAR beam. (e) Spike centroid motion in
time at a fixed central spike frequency of $34.5$ MHz represented by the
colored triangles overlaid on an SDO/AIA 171 Å image at 11:20:57 UT. The blue
plus symbols show the peak centroid position of the spikes between $30-45$ MHz
before the CME, whilst the white plus symbols show the spike peak centroid
positions after the CME. The grey lines with diamond markers represent the
centroid motion of a single Type IIIb striae near 10:42 UT at $32.95$ MHz,
whereas the open grey triangles represent a striae near 11:21 UT at $31.57$
MHz. (f) Motion of 10 individual spikes. The open triangles represent
frequencies between $43-45$ MHz and closed triangles between $33-35$ MHz. The
color gradients represent time increasing from dark to light. The red diamond
shows the approximate location of the active region.

Figure 4: Spike characteristics from LOFAR data between $30-70$ MHz. The black
diamonds represent striae from Type IIIb bursts during the same event. (a)
Spike FWHM durations (red). The thin-line curves show the collision time,
$\tau_{\mathrm{coll}}\approx T^{3/2}/110n$, (McKim Malville et al., 1967) for
a plasma of the temperatures $T$, where $n$ is the plasma density. The red
line shows the linear fit to the spike durations. (b) Spike FWHM rise times
with linear fit. (c) Spike FWHM decay times with linear fit. The errors in
(a-c) represent the uncertainty due to the temporal resolution. (d) Spike
spectral widths with a linear fit (blue). The uncertainty represents the
1-sigma error provided by the Gaussian fitting procedure. The black solid line
shows the average fit to spike spectral widths from Melnik et al. (2014) with
the grey region showing the upper and lower bounds. The two grey lines show
fits to striae widths from Sharykin et al. (2018). (e) Frequency drift rates
of spikes between $30-70$ MHz with a linear fit (green). The uncertainty
represents the 1-sigma error provided by the Gaussian fitting procedure. The
light and dark grey squares show the striae drift rates from Sharykin et al.
(2018) with corresponding linear fits. (f) Observed FWHM area of spikes at
their peak intensity (purple). The coefficients $\alpha$ and $\beta$ of the
powerlaw fits are $(7.09\pm 3.0)\times 10^{5}$ and $(1.75\pm 0.085)\times
10^{5}$, respectively. The light-grey dashed line shows the LOFAR beam area.
The errors are calculated as in Kontar et al. (2017).

The spike peak fluxes range from $4-66$ sfu, averaging at ${\sim}\>18$ sfu.
The peak flux of the Type IIIb striae are an order of magnitude brighter than
the average spikes, peaking up to ${\sim}\>200$ sfu. The time profiles of
solar radio spikes at a given frequency resembles that of Type III bursts; a
prompt rise time followed by a longer decay. However, the duration of spikes
near $30$ MHz are shorter up to a factor of ${\sim}\>20$. Figure 3(b) shows
the time series at the central spike frequency of $34.5$ MHz with a FWHM
duration of $0.65$ s. Typically, the rise time is shorter than the decay time,
averaging at $0.22$ and $0.26$ s, respectively. The spike durations fall
between $0.2-1.1$ s, average at $0.48$ s (Figure 4(a)), with a rate of change
of $-5\pm 1.8$ ms per MHz. The durations are comparable with nearby Type IIIb
striae as well as the collision time for a plasma of temperatures between
$(0.5-2)\times 10^{6}$ K shown by the curves in Figure 4(a).

The instantaneous frequency flux profile at the burst peak time is symmetrical
(Figure 3(c)). The Gaussian fit gives a FWHM bandwidth of $\Delta{f}=50.34\pm
1.32$ kHz (Figure 3(c)). Average bandwidth is $76.1$ kHz between $30-70$ MHz
and tends to increase with frequency (Figure 4(d)), with some spike widths
reaching up to $250$ kHz near $70$ MHz. Spike spectral widths overlap with
that of Type IIIb striae between $30-46$ MHz from the same event, with striae
widths also observed up to $174$ kHz.

The spike in Figure 3(a) shows a near-zero drift rate of
$\mathrm{d}f/\mathrm{d}t=3.58\pm 1.1$ kHz/s. The majority of spikes have
negative drift rates between zero and $-70$ kHz/s with an outlier near $-123$
kHz/s and eight spikes that have a positive drift rate (Figure 4(e)). The
linear fit indicates no significant change in frequency drift with frequency.
The spike drift rates are comparable with striae drift rates ranging between
$50$ to $-70$ kHz/s. In comparison, the bulk Type IIIb structures shown in
Figure 2(b,e) have drift rates of $-3.14\pm 0.21$ and $-2.31\pm 0.26$ MHz/s,
respectively, measured by a linear fit to the peak flux position at the
central frequency of each striae.

The observed spike FWHM area at the peak of the central frequency is
$202.19\pm 16.3$ arcmin2 at $34.5$ MHz (Figure 3(d)). The observed spike areas
decrease with increasing frequency from $297$ to $122$ arcmin2 between $30$ to
$45$ MHz (Figure 4(f)) in a similar manner to drift-pair bursts (Kuznetsov &
Kontar, 2019), and is approximated with a power law $A\sim f^{-\gamma}$ where
$\gamma=2.3$ and $1.9$ for spikes and striae, respectively. The large
uncertainties are due to their low intensities. The synthesised LOFAR beam
area over this frequency range is $A_{\mathrm{beam}}\simeq 50-113$ arcmin2, so
the LOFAR-beam-corrected source areas are up to $A\simeq 72-184$ arcmin2.
Importantly, the spike areas at a fixed frequency increase over time at tens
of ms scales (Figure 5(d)) with expansion rates between $18-108$ arcmin2/s
that are most pronounced during the decay phase, similar to Type IIIb
observations (Kontar et al., 2017).

One of the intriguing observations is the variability of the radio spike
sources (positions and areas) with time at tens of millisecond scales. The
spike centroid position moves vertically in the plane-of-sky image over $0.65$
s across the solar equator (Figure 3(d)), covering $262$ arcsec $\simeq 0.28$
R⊙ in the image plane, corresponding to the speed of light over the FWHM
duration. The frequency drift rate inferred velocity, assuming a Newkirk
density model (Newkirk, 1961), is ${\sim}\>45.1$ km/s. Spikes with higher
frequency drifts such as $60$ kHz/s at $35.5$ MHz show speeds of up to $680$
km/s. The spikes observed ${\sim}\>10$ mins before the flare-CME appear closer
to the disk centre, while the spikes in the wake of the CME are further away
(Figure 3(e)). The spike source motion follows a trajectory parallel to the
Type IIIb striae observed prior to the CME. Post-CME Type IIIb striae show
motion that originates within the same region as the post-CME spikes (Figure
3(f)), suggesting a common exciter. It is interesting to note that the FWHM
spike source areas are comparable to Type IIIb areas in this event, but
smaller than observed before (e.g. Kontar et al., 2017).

Figure 5(a-c) shows the spike centroid motion in the image plane for the
radial, x, and y positions. The radial distance changes weakly, with the bulk
motion in the y-direction with a superluminal plane-of-sky speed of $10.4$
arcmin/s $\simeq 1.5c$ during the decay phase where the velocity is most
pronounced. This vertical motion is typical for all observed spikes during
this period (see Figure 3(f)), with x and y drift rates between $-0.44$ to
$-2.27$ arcmin/s and $5.25$ to $11.18$ arcmin/s, respectively. In the
y-direction, this corresponds to apparent speeds between $0.76c-1.8c$.

Figure 5: Spike source motion and expansion over the duration of the dynamic
spectrum shown in Figure 2(d) at $34.5$ MHz. The red curve shows normalised
radio flux at the same frequency, with the vertical grey dashed lines at the
peak and FWHM decay time. The blue lines show linear fits over the decay
period. The errors are given as in Kontar et al. (2017). (a) Heliocentric
radial position. (b) Spike centroid velocity in the X direction. (c) Spike
centroid velocity in the Y direction. (d) Observed area and (e,f) widths in
the X and Y direction over time.

##  3 Summary

The spectral and temporal characteristics of the observed spikes are
consistent with those previously reported at similar frequencies (Melnik et
al., 2014; Shevchuk et al., 2016). There is a tendency for shorter durations
and higher bandwidths towards higher frequencies; a similar trend is observed
near GHz frequencies (Guedel & Benz, 1990; Bouratzis et al., 2016; Csillaghy &
Benz, 1993; Benz et al., 1992). The magnetic field strength at this coronal
height ${\sim}\>1.78$ R⊙ by Dulk & McLean (1978),
$B(r)=0.5(r/R_{\odot}-1)^{-1.5}$ G, is $0.73$ G, which is close to estimates
using LOFAR Type IIIb observations by Kolotkov et al. (2018). The electron
cyclotron frequency for this field strength is $f_{ce}\simeq 2.0$ MHz $\ll
f_{pe}\sim 30$ MHz, so the criteria for ECM emission is not satisfied.
Moreover, NDA/MEFISTO measurements indicate that the Type III and Type II
bursts both show left-handed circular polarization up to $-0.4$ and $-0.3$,
respectively (Figure 1). The spikes within the region shown in Figure 2(c) are
also left-hand polarized at $-0.15$ and $-0.1$, indicating that the spikes are
produced by the same emission mechanism. Therefore, we suggest plasma emission
as the source of the radio spikes similar to Type IIIb bursts, but likely from
weaker/slower electron beams. The plasma turbulence also changes the spatial
distribution of Langmuir waves (seen in the numerical simulations by Kontar
(2001)), creating regions of enhanced Langmuir waves and hence electromagnetic
emission seen as striae or spikes.

The radio spike positions are observed before and after the onset of the solar
eruptive event at 10:50 UT. The location of radio spikes and Type IIIb striae
after the CME are $1250-1450$ arcsec away from the Sun centre, compared to
${\sim}\>1100$ arcsec prior to the CME eruption showing a shift away from the
Sun. For the first time, the time-resolved observations of individual spikes
reveal source motions and source size changes at 100 ms scales. The spike
sources (both before and after the CME) follow trajectories approximately
parallel to the solar limb, which contrasts the previously observed radial
motions of Type IIIb bursts (Kontar et al., 2017; Zhang et al., 2019), and the
corresponding spherically symmetric coronal simulations (Chen et al., 2020).

The motion of the spike sources is superluminal ($0.76c-1.8c$) and accompanied
by the superluminal FWHM source size expansion of $7.4$ arcmin/s
${\sim}\>1.1c$. For scatter-dominated sources, the source velocity depends on
the angle between the line-of-sight and the direction from the Sun centre to
the source location, as well as the anisotropy of plasma turbulence (Kontar et
al., 2019; Chen et al., 2020; Kuznetsov et al., 2020). Large heliocentric
angles such as that observed by spikes in this observation are subject to
larger displacements along the direction of the guiding magnetic field and
increased apparent velocities. Superluminal motions are, in fact, observed in
radio-wave propagation simulations (see Figure 5 by Kuznetsov et al., 2020),
suggesting $1.0c-1.1c$ speeds for sources located at heliocentric angles of 30
and 50 degrees.

The previously unobserved non-radial superluminal motion of spike and Type
IIIb sources in this event suggests a different magnetic configuration to what
was simulated by Chen et al. (2020). In anisotropic density turbulence that is
aligned with the magnetic field, radio waves propagate preferentially along
the magnetic field direction (Kontar et al., 2019). The observed spike and
Type IIIb sources are located within the region where the magnetic field is
likely forming loop-like structures (Chrysaphi et al., 2020), so radio-wave
scattering in the region with the magnetic field lines parallel to the limb
could induce the observed direction of the source motion. The extended post-
reconnection closed loops are likely formed within the CME wake and is the
location of weak electron beam acceleration, resulting in Langmuir wave
generation and subsequent spike emission. The simulations of radio-wave
transport by Kuznetsov et al. (2020) show that stronger anisotropy leads to
smaller observed peak source sizes and superluminal velocities. Thus, the
spikes and Type IIIb striae source properties are consistent with the
simulations with anisotropy $\alpha=0.1-0.2$ (Kuznetsov et al., 2020), which
is higher than the anisotropy $\alpha=0.25-0.3$ required in open
configurations to explain Type III burst properties (e.g. Kontar et al., 2019;
Chen et al., 2020). Consequently, the anisotropy of density turbulence in
closed loop configurations should be higher than that along open field lines,
typical for Type III bursts.

Many similarities between spike and Type IIIb striae characteristics such as
duration, spectral width, velocities and observed area suggest a common
physical mechanism. The spike durations decrease on average with increasing
frequency, with bandwidths ranging between $20-100\;\mathrm{kHz}$, covering a
similar range reported in Sharykin et al. (2018) for striae. The spike and
striae drift rates show little dependence on frequency, and indicate
velocities of $10-50$ km/s. Spike drift rates close to $30$ MHz overlap with
the striae drift rates presented Sharykin et al. (2018), however the
comparison diverges above $40$ MHz. The drift rate inferred velocities of our
observed spikes and striae are much less than the bulk speeds of Type III
bursts that propagate at characteristic speeds of ${\sim}\>10^{5}$ km/s
$\simeq c/3$ (Suzuki & Dulk, 1985). The motion of Langmuir waves (Reid &
Kontar, 2021), which is believed to be responsible for Type IIIb striae drift,
could be the explanation for the individual spike drift. However, since the
spikes have on average shorter duration, such drift is likely to be diluted by
scattering effects.

As was noted before (e.g. Melnik et al., 2014), the spike duration is
comparable to the plasma collision time. However, a spread of temperatures
(from $0.5-2$ MK) for the same cluster of spikes are required to explain the
characteristic decay time (Figure 4). Radio spikes above $100$ MHz (Guedel &
Benz, 1990) would need even higher plasma temperatures—$2-4$ MK within the
collisional damping hypothesis. While damping of plasma oscillations should be
present, the large source sizes, the superluminal motion of spike sources, and
the aforementioned morphological similarity to Type IIIb striae implies that
scattering is an important factor in determining spike profiles. Therefore, we
suggest that the scattering of radio waves rather than collisional damping
alone determines the time profile. Spike durations are often used to constrain
the shortest energy release time in flares, with the bandwidth equating to the
size of the acceleration region itself (Benz, 1985). Following the Type IIIb
approach (Kontar et al., 2017; Sharykin et al., 2018), the size of the
emitting source spike region can be estimated as $\Delta{r}\simeq
2L(\Delta{f}/f)\simeq 10^{8}$ cm that corresponds to a subtended solid angle
of ${\sim}\>10^{-2}$ arcmin2. Therefore, the effect of radio-wave propagation
has increased the observed source area by four orders of magnitude. This means
that the brightness temperatures of the spike sources corrected for scattering
could be up to $10^{12}-10^{13}$ K, well above the values $10^{8}-10^{9}$ K
when radio-wave scattering is ignored. With scattering as the determining
factor of the duration and higher resulting brightness temperature, the energy
release responsible for electron acceleration would be much shorter and more
intense than previously assumed in the literature. The characteristic emission
timescale is approximately reduced by the ratio of the observed size to the
emitting region, and could be two orders of magnitude shorter, i.e. tens of
milliseconds instead of ${\sim}\>1$ s as observed.

DLC, EPK, and NV are thankful to Dstl for the funding through the UK-France
PhD Scheme (contract DSTLX-1000106007). EPK was supported by STFC consolidated
grant ST/P000533/1. MG was supported by the STFC grant ST/T00035X/1. NC thanks
CNES for its financial support. The authors acknowledge the support by the
international team grant (http://www.issibern.ch/teams/lofar/) from ISSI Bern,
Switzerland. This paper is based (in part) on data obtained from facilities of
the International LOFAR Telescope (ILT) under project code LC8_027. LOFAR (van
Haarlem et al., 2013) is the Low-Frequency Array designed and constructed by
ASTRON. It has observing, data processing, and data storage facilities in
several countries, that are owned by various parties (each with their own
funding sources), and that are collectively operated by the ILT Foundation
under a joint scientific policy. The ILT resources have benefited from the
following recent major funding sources: CNRS-INSU, Observatoire de Paris and
Université d’Orléans, France; BMBF, MIWF-NRW, MPG, Germany; Science Foundation
Ireland (SFI), Department of Business, Enterprise and Innovation (DBEI),
Ireland; NWO, The Netherlands; The Science and Technology Facilities Council,
UK; Ministry of Science and Higher Education, Poland. The authors thank the
radio astronomy station of Nançay / Scientific Unit of Nançay of the Paris
Observatory (USR 704-CNRS, supported by the University of Orleans, the OSUC
and the Center Region in France) for providing access to NDA observations
accessible online at https://www.obs-nancay.fr.

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# Implications of some mass-capacity inequalities

Pengzi Miao  Department of Mathematics, University of Miami, Coral Gables, FL
33146, USA<EMAIL_ADDRESS>
###### Abstract.

Applying a family of mass-capacity related inequalities proved in [16], we
obtain sufficient conditions that imply the nonnegativity as well as positive
lower bounds of the mass, on a class of manifolds with nonnegative scalar
curvature, with or without a singularity.

##  1\. Introduction

A smooth Riemannian $3$-manifold $(M,g)$ is called asymptotically flat (AF) if
$M$, outside a compact set, is diffeomorphic to $\mathbb{R}^{3}$ minus a ball;
the associated metric coefficients satisfy

$g_{ij}=\delta_{ij}+O(|x|^{-\tau}),\ \partial g_{ij}=O(|x|^{-\tau-1}),\ \
\partial\partial g_{ij}=O(|x|^{-\tau-2}),$

for some $\tau>\frac{1}{2}$; and the scalar curvature of $g$ is integrable.
Under these AF conditions, the limit, near $\infty$,

$\mathfrak{m}=\lim_{r\to\infty}\frac{1}{16\pi}\int_{|x|=r}\sum_{j,k}(g_{jk,j}-g_{jj,k})\frac{x^{k}}{|x|}$

exists and is called the ADM mass [2] of $(M,g)$. It is a result of Bartnik
[3], and of Chruściel [9], that $\mathfrak{m}$ is a geometric invariant,
independent on the choice of the coordinates $\\{x_{i}\\}$.

A fundamental result on the mass and the scalar curvature is the Riemannian
positive mass theorem (PMT):

######  Theorem 1.1 ([20, 22]).

Let $(M,g)$ be a complete, asymptotically flat $3$-manifold with nonnegative
scalar curvature without boundary. Then

$\mathfrak{m}\geq 0,$

and equality holds if and only if $(M,g)$ is isometric to the Euclidean space
$\mathbb{R}^{3}$.

On an asymptotically flat $3$-manifold $(M,g)$ with boundary $\Sigma=\partial
M$, the capacity (or $L^{2}$-capacity) of $\Sigma$ is defined by

$\mathfrak{c}_{{}_{\Sigma}}=\inf_{f}\,\left\\{\frac{1}{4\pi}\int_{M}|\nabla
f|^{2}\right\\},$

where the infimum is taken over all locally Lipschitz functions $f$ that
vanishes on $\Sigma$ and tend to $1$ at infinity. Equivalently, if $\phi$
denotes the function with

$\Delta\phi=0,\ \ \phi|_{\Sigma}=1,\ \text{and}\ \phi\to 0\ \ \text{at}\ \
\infty,$

then,
$\displaystyle\mathfrak{c}_{{}_{\Sigma}}=\frac{1}{4\pi}\int_{M}|\nabla\phi|^{2}=\frac{1}{4\pi}\int_{\Sigma}|\nabla\phi|$,
and

$\phi=\mathfrak{c}_{{}_{\Sigma}}|x|^{-1}+o(|x|^{-1}),\ \ \text{as}\
x\to\infty.$

Regarding the mass and the capacity, if $\Sigma$ is a minimal surface, Bray
showed

######  Theorem 1.2 ([4]).

Let $(M,g)$ be a complete, asymptotically flat $3$-manifold with nonnegative
scalar curvature, with minimal surface boundary $\Sigma=\partial M$. Then

$\mathfrak{m}\geq\mathfrak{c}_{{}_{\Sigma}},$

and equality holds iff $(M,g)$ is isometric to a spatial Schwarzschild
manifold outside the horizon.

In [16, Theorem 7.4], an inequality relating the mass-to-capacity ratio to the
Willmore functional of the boundary was obtained:

######  Theorem 1.3 ([16]).

Let $(M,g)$ be a complete, orientable, asymptotically flat $3$-manifold with
one end, with boundary $\Sigma$. Suppose $\Sigma$ is connected and
$H_{2}(M,\Sigma)=0$. If $g$ has nonnegative scalar curvature, then

(1.1) $\frac{\mathfrak{m}}{\mathfrak{c}_{{}_{\Sigma}}}\geq
1-\left(\frac{1}{16\pi}\int_{\Sigma}H^{2}\right)^{\frac{1}{2}}.$

Here $\mathfrak{m}$ is the mass of $(M,g)$, $\mathfrak{c}_{{}_{\Sigma}}$ is
the capacity of $\Sigma$ in $(M,g)$, and $H$ is the mean curvature of
$\Sigma$. Moreover, equality in (1.1) holds if and only if $(M,g)$ is
isometric to a spatial Schwarzschild manifold outside a rotationally symmetric
sphere with nonnegative mean curvature.

As shown in [16], (1.1) implies the $3$-dimensional PMT. For instance,
assuming $M$ is topologically $\mathbb{R}^{3}$, applying (1.1) to the exterior
of a geodesic sphere $S_{r}$ with radius $r$ centered at any point $p\in M$,
one has

(1.2) $\frac{\mathfrak{m}}{\mathfrak{c}_{{}_{S_{r}}}}\geq
1-\left(\frac{1}{16\pi}\int_{S_{r}}H^{2}\right)^{\frac{1}{2}}.$

Letting $r\to 0$, one obtains $\mathfrak{m}\geq 0$. Earlier proofs of $3$-d
PMT via harmonic functions were given by Bray-Kazaras-Khuri-Stern [6] and
Agostiniani-Mazzieri-Oronzio [1].

Theorem 1.3 follows from two other results (Corollary 7.1 and Theorem 7.3) in
[16]:

######  Theorem 1.4 ([16]).

Let $(M,g)$ be a complete, orientable, asymptotically flat $3$-manifold with
one end, with connected boundary $\Sigma$, satisfying $H_{2}(M,\Sigma)=0$. If
$g$ has nonnegative scalar curvature, then

(1.3) $\left(\frac{1}{\pi}\int_{\Sigma}|\nabla
u|^{2}\right)^{\frac{1}{2}}\leq\left(\frac{1}{16\pi}\int_{\Sigma}H^{2}\right)^{\frac{1}{2}}+1,$

and

(1.4) $\frac{\mathfrak{m}}{2\mathfrak{c}_{{}_{\Sigma}}}\geq
1-\left(\frac{1}{4\pi}\int_{\Sigma}|\nabla u|^{2}\right)^{\frac{1}{2}}.$

Here $u$ is the harmonic function with $u=0$ at $\Sigma$ and $u\to 1$ near
$\infty$. Moreover,

  * •

equality in (1.3) holds if and only if $(M,g)$ is isometric to a spatial
Schwarzschild manifold outside a rotationally symmetric sphere with
nonnegative mean curvature;

  * •

equality in (1.4) holds if and only if $(M,g)$ is isometric to a spatial
Schwarzschild manifold outside a rotationally symmetric sphere.

A corollary of (1.3) (see [16, Theorem 7.2]) is an upper bound on the
capacity-to-area-radius ratio, first derived by Bray and the author [7].

######  Theorem 1.5 ([7]).

Let $(M,g)$ be a complete, orientable, asymptotically flat $3$-manifold with
one end, with boundary $\Sigma$. Suppose $\Sigma$ is connected and
$H_{2}(M,\Sigma)=0$. If $g$ has nonnegative scalar curvature, then

(1.5)
$\frac{2\mathfrak{c}_{{}_{\Sigma}}}{r_{{}_{\Sigma}}}\leq\left(\frac{1}{16\pi}\int_{\Sigma}H^{2}\right)^{\frac{1}{2}}+1.$

Here $\mathfrak{c}_{{}_{\Sigma}}$ is the capacity of $\Sigma$ in $(M,g)$ and
$r_{{}_{\Sigma}}=\left(\frac{|\Sigma|}{4\pi}\right)^{\frac{1}{2}}$ is the
area-radius of $\Sigma$. Moreover, equality holds if and only if $(M,g)$ is
isometric to a spatial Schwarzschild manifold outside a rotationally symmetric
sphere with nonnegative mean curvature.

In this paper, we give some other applications of (1.1), (1.3) and (1.4).

First, for later purposes, we remark on the topological assumption
“$H_{2}(M,\Sigma)=0$” in Theorems 1.3 – 1.5 above: the assumption is imposed
only to ensure each regular level set of the harmonic function $u$, vanishing
at the boundary and tending to $1$ near $\infty$, to be connected in the
interior of $M$ (see the paragraph preceding the proof of Theorem 3.1 in
[16]); indeed, (1.1), (1.3) and (1.4) (and all other results from [16]) hold
if “$H_{2}(M,\Sigma)=0$” is replaced by assuming

  * ($\ast$)

each closed, connected, orientable surface in the interior of $M$ either is
the boundary of a bounded domain, or together with $\Sigma$ forms the boundary
of a bounded domain.

Now we motivate the main tasks in this paper. Let us first return to the
setting of (1.2), in which the surface $S_{r}$ “closes up nicely” (to bound a
geodesic ball). In this setting, by a result of Mondino and Templeton-Browne
[18], $\\{S_{r}\\}$ can be perturbed to yield another family of surfaces
$\\{\Sigma_{r}\\}$ so that, as $r\to 0$,

(1.6)
$\int_{\Sigma_{r}}H^{2}=16\pi-\frac{8\pi}{3}R(p)r^{2}+\frac{4\pi}{3}\left[\frac{1}{9}R(p)^{2}-\frac{4}{15}|\mathring{\mbox{Ric}}(p)|^{2}-\frac{1}{5}\Delta
R(p)\right]r^{4}+O(r^{5}).$

Here $R$ denotes the scalar curvature and
$\mathring{\mbox{Ric}}=\mbox{Ric}-\frac{1}{3}Rg$ is the traceless part of Ric,
the Ricci tensor. Applying (1.1) to the exterior of these $\Sigma_{r}$ in
$(M,g)$, one obtains

(1.7)
$\frac{\mathfrak{m}}{\mathfrak{c}_{{}_{\Sigma_{r}}}}\geq\frac{1}{12}R(p)r^{2}+\left[\frac{1}{90}|\mathring{\mbox{Ric}}(p)|^{2}-\frac{1}{864}R(p)^{2}+\frac{1}{120}\Delta
R(p)\right]r^{4}+O(r^{5}).$

If $R\geq 0$, (1.7) shows the inequality $\mathfrak{m}\geq 0$ as well as the
rigidity of $\mathfrak{m}=0$.

In general, (1.1) suggests that, if it is applied to obtain $\mathfrak{m}\geq
0$ on an $(M,g)$, the manifold boundary $\Sigma$ does not need to admit a
“nice fill-in”. Rewriting (1.1) as

$\mathfrak{m}\geq\mathfrak{c}_{{}_{\Sigma}}\left[1-\left(\frac{1}{16\pi}\int_{S_{r}}H^{2}\right)^{\frac{1}{2}}\right],$

one may seek conditions on metrics $g$ with a “singularity” so that
$\mathfrak{m}\geq 0$ while $g$ is allowed to be incomplete.

Similarly, on an $(M,g)$ with two ends, one of which is asymptotically flat
(AF), assuming it admits a harmonic function $u$ that tends to $1$ at the AF
end and tends to $0$ at the other end, one may aim to apply (1.4), i.e.

$\mathfrak{m}\geq
2\mathfrak{c}_{{}_{\Sigma}}\left[1-\left(\frac{1}{4\pi}\int_{\Sigma}|\nabla
u|^{2}\right)^{\frac{1}{2}}\right],$

to bound $\mathfrak{m}$ via the energy of $u$ on the entire $(M,g)$.

Below we formulate a class of manifolds to carry out the above mentioned
tasks. Throughout the paper, let $N$ be a noncompact, connected, orientable
$3$-manifold. We assume $N$ admits an increasing exhaustion sequence of
bounded domains with connected boundary. Precisely, this means there exists a
sequence of closed, orientable surfaces $\\{\Sigma_{k}\\}_{k=1}^{\infty}$ in
$N$ such that

  * •

$\Sigma_{k}$ is connected;

  * •

$\Sigma_{k}=\partial D_{k}$ for a precompact domain $D_{k}\subset N$; and

  * •

$\bar{D}_{k}\subset D_{k+1}$ and $N=\cup_{k=1}^{\infty}D_{k}$. Here
$\bar{D}_{k}=D_{k}\cup\Sigma_{k}$ is the closure of $D_{k}$ in $N$.

Fix a point $p\in N$, let $M=N\setminus\\{p\\}$. On $M$, let $g$ be a smooth
metric that is asymptotically flat near $p$. We refer $p$ as the
asymptotically flat (AF) $\infty$ of $(M,g)$. Unless otherwise specified, we
do not impose assumptions on the behavior of $g$ near $\Sigma_{k}$ as
$k\to\infty$. In particular, $(M,g)$ does not need to be complete,

Given any closed, connected surface $S\subset M$, we say $S$ encloses $p$ if
$S=\partial D_{{}_{S}}$ for some precompact domain $D_{{}_{S}}\subset N$ such
that $p\in D_{{}_{S}}$. Let $\mathcal{S}$ denote the set of all such surfaces
$S\subset M$ enclosing $p$. Clearly, $\Sigma_{k}\in\mathcal{S}$ for large $k$.
Define

(1.8) $\mathfrak{c}(M,g)=\inf_{S\in\mathcal{S}}\,\mathfrak{c}_{{}_{S}}.$

Here $\mathfrak{c}_{{}_{S}}$ is the capacity of $S$ in the asymptotically flat
$(E_{{}_{S}},g)$, where

$E_{{}_{S}}=(D_{{}_{S}}\setminus\\{p\\})\cup S.$

As a functional on $\mathcal{S}$, the capacity $\mathfrak{c}_{{}_{S}}$ has a
monotone property, that is if $S_{1},S_{2}\in\mathcal{S}$ and
$D_{{}_{S_{1}}}\subset D_{{}_{S_{2}}}$, then
$\mathfrak{c}_{{}_{S_{1}}}\geq\mathfrak{c}_{{}_{S_{2}}}$. Such a property
readily implies $\\{\mathfrak{c}_{{}_{\Sigma_{k}}}\\}$ is monotone non-
increasing and

(1.9) $\mathfrak{c}(M,g)=\lim_{k\to\infty}\mathfrak{c}_{{}_{\Sigma_{k}}}.$

Standard arguments show $\mathfrak{c}(M,g)>0$ if and only if there exists a
harmonic function $w$ on $(M,g)$ such that $0<w<1$ on $M$ and $w(x)\to 1$ at
$\infty$ (i.e. as $x\to p$). (See Proposition 3.1 in Section 3.)

Figure 1. On the left is an examples of $(M,g)$ with $\mathfrak{c}(M,g)=0$;
the arrow denotes the AF end; $\\{\Sigma_{k}\\}$ may approach a “singularity”
as $k\to\infty$. On the right is an example of $(M,g)$ with
$\mathfrak{c}(M,g)>0$; besides the AF end, $(M,g)$ has another end with
suitable growth.

For manifolds $(M,g)$ with $\mathfrak{c}(M,g)=0$, we seek conditions that
imply the AF end of $(M,g)$ has mass $\mathfrak{m}\geq 0$, see Theorem 2.1 and
Remark 2.2. For $(M,g)$ with $\mathfrak{c}(M,g)>0$, we explore for sufficient
conditions that bound $\mathfrak{m}$ from below via $\mathfrak{c}(M,g)$, see
Theorem 3.1 and Corollary 3.1.

##  2\. Singular metrics with $\mathfrak{m}\geq 0$

Let $N$, $M$ and $g$ be given in the definition of $\mathfrak{c}(M,g)$ in
(1.8). Given $S\in\mathcal{S}$, let

$W(S)=\int_{S}H^{2}.$

We want to apply (1.1) to $(E_{{}_{S}},g)$. For this purpose, we assume the
background manifold $N$ satisfies $H_{2}(N)=0$. Under this assumption, any
closed, connected surface $S^{\prime}$ in $M=N\setminus\\{p\\}$ is the
boundary of a bounded domain $D\subset N$. If $p\not\in D$, then $D\subset M$;
if $p\in D$, then $S^{\prime}$ is homologous to $S\in\mathcal{S}$. Therefore,
condition $(\ast)$ holds on $E_{{}_{S}}$.

The following is a direct corollary of (1.1).

######  Proposition 2.1.

Suppose $H_{2}(N)=0$ and $(M,g)$ has nonnegative scalar curvature. Then

(2.1) $\mathfrak{c}_{{}_{S_{k}}}W(S_{k})^{\frac{1}{2}}\to 0\ \text{along a
sequence }\\{S_{k}\\}\subset\mathcal{S}\ \ \Rightarrow\ \mathfrak{m}\geq 0.$

###### Proof.

If $W(S_{k})\leq 16\pi$ for some $k$, then (1.1) implies $\mathfrak{m}\geq 0$.

Suppose $W(S_{k})>16\pi$ for every $k$, then
“$\mathfrak{c}_{{}_{S_{k}}}W(S_{k})^{\frac{1}{2}}\to 0$” implies
“$\mathfrak{c}_{{}_{S_{k}}}\to 0$”. Rewriting (1.1) as

(2.2)
$\mathfrak{m}\geq\mathfrak{c}_{{}_{S_{k}}}\left[1-\left(\frac{1}{16\pi}W(S_{k})\right)^{\frac{1}{2}}\right]$

and letting $k\to\infty$, we have $\mathfrak{m}\geq 0$. ∎

Given $S\in\mathcal{S}$, let
$\displaystyle\mathfrak{m}_{{}_{H}}(S)=\frac{r_{{}_{S}}}{2}\left(1-\frac{1}{16\pi}W(S)\right)$
denote the Hawking mass of $S$ ([11]). Inequality (2.3) in the next
Proposition is comparable to the result of Huisken and Ilmanen [13] on the
relation between $\mathfrak{m}$ and $\mathfrak{m}_{{}_{H}}(S)$.

######  Proposition 2.2.

Suppose $H_{2}(N)=0$ and $(M,g)$ has nonnegative scalar curvature. If a
surface $S\in\mathcal{S}$ satisfies $W(S)\geq 16\pi$, then

(2.3)
$\mathfrak{m}\geq\mathfrak{c}_{{}_{S}}\left[1-\left(\frac{1}{16\pi}W(S)\right)^{\frac{1}{2}}\right]\geq\mathfrak{m}_{{}_{H}}(S).$

###### Proof.

If $W(S)\geq 16\pi$, then (1.5) implies

(2.4)
$\begin{split}\mathfrak{c}_{{}_{S}}\left[1-\left(\frac{1}{16\pi}\int_{S}H^{2}\right)^{\frac{1}{2}}\right]\geq&\
\frac{r_{{}_{S}}}{2}\left[1-\frac{1}{16\pi}\int_{S}H^{2}\right]=\mathfrak{m}_{{}_{H}}(S).\end{split}$

This combined with (1.1) proves (2.3). ∎

######  Remark 2.1.

Similar to (2.1), a condition of “$r_{{}_{S_{k}}}W(S_{k})\to 0$” along
$\\{S_{k}\\}\subset\mathcal{S}$ also implies “$\mathfrak{m}\geq 0$”. However,
if $\inf_{k}W(S_{k})\geq 16\pi$, then

$``r_{{}_{S_{k}}}W(S_{k})\to 0"\ \Rightarrow\ ``r_{{}_{S_{k}}}\to 0\
\text{and}\ r_{{}_{S_{k}}}W(S_{k})^{\frac{1}{2}}\to 0"\ \Rightarrow\
``\mathfrak{c}_{{}_{S_{k}}}\to 0",$

where the last step is by (1.5). Combined with (2.4), this implies the
assumption of “$\mathfrak{c}_{{}_{S_{k}}}W(S_{k})^{\frac{1}{2}}\to 0$” in
Proposition 2.1.

In what follows, let $\\{\Sigma_{k}\\}\subset\mathcal{S}$ be the sequence of
surfaces given in the introduction. The numerical value of
$\mathfrak{c}_{{}_{\Sigma_{k}}}$ depends on $g$ near the AF end. However, a
property of “$\mathfrak{c}_{{}_{\Sigma_{k}}}\to 0$” does not. This was shown
by Bray and Jauregui [5] in the context of $(M,g)$ having a zero area
singularity. Their argument applies to
“$\mathfrak{c}_{{}_{\Sigma_{k}}}W(\Sigma_{k})^{\frac{1}{2}}\to 0$”. To
illustrate this, it is convenient to adopt a notion of relative capacity (see
[14] for instance). Given two surfaces $S,\tilde{S}\in\mathcal{S}$, suppose
$S\cap\tilde{S}=\emptyset$ and $D_{{}_{\tilde{S}}}\subset D_{{}_{S}}$. The
capacity of $S$ relative to $\tilde{S}$ is

(2.5)
$\mathfrak{c}_{{}_{(S,\tilde{S})}}=\frac{1}{4\pi}\int_{D_{{}_{S}}\setminus
D_{{}_{\tilde{S}}}}|\nabla v|^{2},$

where $v$ is the harmonic function on $D_{{}_{S}}\setminus D_{{}_{\tilde{S}}}$
with $v=0$ at $S$ and $v=1$ at $\tilde{S}$.

######  Proposition 2.3.

Let $\tilde{S}\in\mathcal{S}$ be a fixed surface. Then, as $k\to\infty$,

$\mathfrak{c}_{{}_{\Sigma_{k}}}W(\Sigma_{k})^{\frac{1}{2}}\to 0\
\Longleftrightarrow\
\mathfrak{c}_{{}_{(\Sigma_{k},\tilde{S})}}W(\Sigma_{k})^{\frac{1}{2}}\to 0.$

###### Proof.

For large $k$, let $u_{k}$, $v_{k}$ be the harmonic function on
$D_{k}\setminus\\{p\\}$, $D_{k}\setminus D_{{}_{\tilde{S}}}$, with boundary
values $u_{k}=0$ at $\Sigma_{k}$, $u_{k}\to 1$ at the AF $\infty$, $v_{k}=0$
at $\Sigma_{k}$, $v_{k}=1$ at $\tilde{S}$, respectively. Let
$\beta_{k}=\min_{\tilde{S}}u_{k}$. By the maximum principle, $v_{k}\geq
u_{k}\geq\beta_{k}v_{k}$ on $D_{k}\setminus D_{{}_{\tilde{S}}}$, which implies
$\partial_{\nu}v_{k}\geq\partial_{\nu}u_{k}\geq\beta_{k}\partial_{\nu}v_{k}$
at $\Sigma_{k}$. Here $\nu$ denotes the unit normal to $\Sigma_{k}$ pointing
to $\infty$. Since
$4\pi\mathfrak{c}_{{}_{\Sigma_{k}}}=\int_{\Sigma_{k}}\partial_{\nu}u_{k}$ and
$4\pi\mathfrak{c}_{{}_{(\Sigma_{k},\tilde{S})}}=\int_{\Sigma_{k}}\partial_{\nu}v_{k}$,
one has

$\beta_{k}^{-1}\mathfrak{c}_{{}_{\Sigma_{k}}}\geq\mathfrak{c}_{{}_{(\Sigma_{k},\tilde{S})}}\geq\mathfrak{c}_{{}_{\Sigma_{k}}}.$

The claim follows by noting that $\beta_{k}$ has a uniform positive lower
bound as $k\to\infty$. ∎

As an application of Propositions 2.1 and 2.3, we have

######  Theorem 2.1.

Let $N$ be a noncompact, connected, orientable $3$-manifold. Suppose
$H_{2}(N)=0$. Let $M=N\setminus\\{p\\}$ where $p$ is a point in $N$. Let $g$
be a smooth metric with nonnegative scalar curvature on $M$ such that $g$ is
asymptotically flat near $p$. Assume there is a precompact domain $D\subset N$
such that $p\in D$ and $(N\setminus D,g)$ is isometric to

$((0,\delta]\times\Sigma,\bar{g}+h),$

where

  * •

$\delta>0$ is a constant, $\Sigma$ is a closed, connected, orientable surface;

  * •

$\bar{g}=dr^{2}+a(r)^{2}\sigma$, in which $\sigma$ is a given metric on
$\Sigma$ and $a(r)$ is a positive function on $(0,\delta]$; and

  * •

$\lambda^{-1}\leq|\bar{g}+h|_{\bar{g}}\leq\lambda$ for some constant
$\lambda>0$.

Then

(2.6) $\lim_{r\to
0}\left(\int_{r}^{\delta}\frac{1}{a(x)^{2}}\,dx\right)^{-1}\left[|a^{\prime}(r)|+a(r)|\bar{\nabla}h|_{\bar{g}}\right]=0\
\Longrightarrow\ \mathfrak{m}\geq 0.$

######  Remark 2.2.

If $a(r)=r^{b}$ for a constant $b>0$, then (2.6) translates to

$\lim_{r\to 0}\,r^{3b-2}\left(1+r|\bar{\nabla}h|_{\bar{g}}\right)=0\
\Rightarrow\ \mathfrak{m}\geq 0.$

This in particular implies, if $g$ has a conical or $r^{b}$-horn type
singularity modeled on $\bar{g}=dr^{2}+r^{2b}\sigma$ near $r=0$, then, under a
mild asymptotic assumption of

$\lambda^{-1}\leq|\bar{g}+h|_{\bar{g}}\leq\lambda\ \ \text{and}\ \
r|\bar{\nabla}h|_{\bar{g}}=O(1),$

one has “$b>\frac{2}{3}\Rightarrow\mathfrak{m}\geq 0$”. (Related results on
PMT with isolated singularities can be found in [21, 15, 10]).

###### Proof of Theorem 2.1.

Let $\Sigma_{r}=\\{r\\}\times\Sigma$, $r\in(0,\delta]$. For $s\in(0,\delta)$,
let $\bar{\mathfrak{c}}_{{}_{(\Sigma_{s},\Sigma_{\delta})}}$,
$\bar{W}(\Sigma_{s})$ denote the capacity of $\Sigma_{s}$ relative to
$\Sigma_{\delta}$, the Willmore functional of $\Sigma_{s}$, respectively, with
respect to $\bar{g}$.

The function
$u(r)=\left(\int_{s}^{\delta}a(x)^{-2}\,dx\right)^{-1}\,\int_{s}^{r}a(x)^{-2}\,dx$
is $\bar{g}$-harmonic on $[s,\delta]\times\Sigma$ with $u=0$ at $\Sigma_{s}$
and $u=1$ at $\Sigma_{\delta}$. This implies

(2.7)
$\bar{\mathfrak{c}}_{{}_{(\Sigma_{s},\Sigma_{\delta})}}=\frac{|\Sigma|_{\sigma}}{4\pi}\left(\int_{s}^{\delta}a(x)^{-2}\,dx\right)^{-1},$

where $|\Sigma|_{\sigma}$ is the area of $(\Sigma,\sigma)$. The mean curvature
$\bar{H}$ of $\Sigma_{s}$ with respect to $\bar{g}$ is
$\bar{H}=2a^{-1}a^{\prime}$. Hence,

$\bar{W}(\Sigma_{s})=4a^{\prime}(s)^{2}|\Sigma|_{\sigma}.$

We compare $\bar{\mathfrak{c}}_{{}_{(\Sigma_{s},\Sigma_{\delta})}}$ and
$\mathfrak{c}_{{}_{(\Sigma_{s},\Sigma_{\delta})}}$. Let $\bar{\nabla}$,
$\nabla$ and $dV_{\bar{g}}$, $dV_{g}$ denote the gradient, the volume form
with respect to $\bar{g}$, $g$, respectively. Since
$\mathfrak{c}_{{}_{(\Sigma_{s},\Sigma_{\delta})}}$ equals the infimum of the
$g$-Dirichlet energy of functions that vanish at $\Sigma_{s}$ and equal $1$ at
$\Sigma_{\delta}$, we have

(2.8) $\begin{split}\mathfrak{c}_{{}_{(\Sigma_{s},\Sigma_{\delta})}}\leq&\
\int_{[s,\delta]\times\Sigma}|\nabla u|_{g}^{2}\,dV_{g}\\\ \leq&\
C\int_{[s,\delta]\times\Sigma}|\bar{\nabla}u|_{\bar{g}}^{2}\,dV_{\bar{g}}=C\,\bar{\mathfrak{c}}_{{}_{(\Sigma_{s},\Sigma_{\delta})}}.\end{split}$

Here $C>0$ denotes a constant independent on $s$ and we have used the
assumption $\lambda^{-1}\leq|g|_{\bar{g}}\leq\lambda$.

We also compare $\bar{W}(\Sigma_{s})$ and $W(\Sigma_{s})$. Let
$\bar{\displaystyle}{\mathbb{II}}$ denote the second fundamental form of
$\Sigma_{s}$ with respect to $\bar{g}$. Direct calculation shows

(2.9)
$H-\bar{H}=|\bar{\displaystyle}{\mathbb{II}}|_{\bar{g}}\,O(|h|_{\bar{g}})+O(|\bar{\nabla}h|_{\bar{g}}).$

(For instance, see formula (2.33) in [17] and the proof therein.) Therefore,

(2.10) $\begin{split}H^{2}=&\
\bar{H}^{2}+\left[|\bar{\displaystyle}{\mathbb{II}}|_{\bar{g}}\,O(|h|_{\bar{g}})+O(|\bar{\nabla}h|_{\bar{g}})\right]^{2}+\bar{H}\left[|\bar{\displaystyle}{\mathbb{II}}|_{\bar{g}}\,O(|h|_{\bar{g}})+O(|\bar{\nabla}h|_{\bar{g}})\right]\\\
=&\
\bar{H}^{2}+|\bar{\displaystyle}{\mathbb{II}}|_{\bar{g}}^{2}\,O(|h|_{\bar{g}})+\bar{H}\,O(|\bar{\nabla}h|_{\bar{g}})+O(|\bar{\nabla}h|_{\bar{g}}^{2}).\end{split}$

Let $d\sigma_{g}$, $d\sigma_{\bar{g}}$ denote the area form on $\Sigma_{s}$
with respect to $g$, $\bar{g}$, respectively. Then

(2.11) $\begin{split}\int_{\Sigma_{s}}H^{2}\,d\sigma_{g}\leq&\
C\int_{\Sigma_{s}}H^{2}\,d\sigma_{\bar{g}}\\\ =&\
C\,\bar{W}(\Sigma_{s})+\left[|\bar{\displaystyle}{\mathbb{II}}|_{\bar{g}}^{2}\,O(|h|_{\bar{g}})+\bar{H}\,O(|\bar{\nabla}h|_{\bar{g}})+O(|\bar{\nabla}h|_{\bar{g}}^{2})\right]\,|\Sigma_{s}|_{\bar{g}}.\end{split}$

Plugging in $\bar{W}(\Sigma_{s})=4{a^{\prime}}^{2}|\Sigma|_{\sigma}$,
$|\bar{\displaystyle}{\mathbb{II}}|^{2}_{\bar{g}}=2a^{-2}{a^{\prime}}^{2}$,
and $\bar{H}^{2}=4a^{-2}{a^{\prime}}^{2}$, we have

(2.12) $\begin{split}W(\Sigma_{s})\leq&\
C\,|\Sigma|_{\sigma}\left[{a^{\prime}}^{2}+{a^{\prime}}^{2}\,|h|_{\bar{g}}+aa^{\prime}|\bar{\nabla}h|_{\bar{g}}+a^{2}|\bar{\nabla}h|_{\bar{g}}^{2}\right]\\\
\leq&\
C\,|\Sigma|_{\sigma}\left[{a^{\prime}}^{2}(1+|h|_{\bar{g}})+a^{2}|\bar{\nabla}h|_{\bar{g}}^{2}\right].\end{split}$

As $|h|_{\bar{g}}$ is bounded by assumption, it follows from (2.7), (2.8) and
(2.12) that

(2.13)
$\mathfrak{c}_{{}_{(\Sigma_{s},\Sigma_{\delta})}}W(\Sigma_{s})^{\frac{1}{2}}\leq
C\left(\int_{s}^{\delta}a(x)^{-2}\,dx\right)^{-1}\left[|a^{\prime}|+a|\bar{\nabla}h|_{\bar{g}}\right].$

(2.6) now follows from Propositions 2.1 and 2.3. ∎

######  Remark 2.3.

The negative mass Schwarzschild manifolds are known to have an $r^{b}$-horn
type singularity with $b=\frac{2}{3}$ (see [5, 21] for instance). In [5], Bray
and Jauregui developed a theory of “zero area singularities” (ZAS) modeled on
the singularity of these manifolds. Among other things, they introduced a
notion of the mass of ZAS. In [19, Theorem 4.8], Robbins showed the ADM mass
of an asymptotically flat $3$-manifold with a single ZAS is at least the ZAS
mass. The conclusion on the $r^{b}$-horn type singularity in Remark 2.2 can
also be reached via the results on ZAS in [5, 19].

We end this section by applying (1.1) to obtain information of $W(\cdot)$ in
the negative mass Schwarzschild manifolds.

######  Proposition 2.4.

Consider a spatial Schwarzschild manifold with negative mass, i.e.

$(M_{\mathfrak{m}},g_{\mathfrak{m}})=\left((0,\infty)\times
S^{2},\frac{1}{1+\frac{2m}{r}}dr^{2}+r^{2}\sigma_{o}\right),$

where $(S^{2},\sigma_{o})$ denotes the standard unit sphere and the mass
$\mathfrak{m}=-m$ is negative. Let $\Sigma\subset M_{\mathfrak{m}}$ be any
connected, closed surface that is homologous to a slice $\\{r\\}\times S^{2}$.
Let $r_{max}(\Sigma)=\max_{{}_{x\in\Sigma}}\,r(x).$ Then

(2.14) $W(\Sigma)\geq 16\pi\left(1+\frac{2m}{r_{max}(\Sigma)}\right).$

In particular, $W(\Sigma)\to\infty$ as $r_{max}(\Sigma)\to 0$.

###### Proof.

(1.1) implies

(2.15) $\left(\frac{1}{16\pi}W(\Sigma)\right)^{\frac{1}{2}}\geq
1+\frac{m}{\mathfrak{c}_{{}_{\Sigma}}}.$

Let $\Sigma_{*}=\\{r_{max}(\Sigma)\\}\times\mathbb{S}^{2}$. As $\Sigma_{*}$
encloses $\Sigma$,

(2.16) $\mathfrak{c}_{{}_{\Sigma_{*}}}\geq\mathfrak{c}_{{}_{\Sigma}}.$

Since $\mathfrak{m}=-m<0$, the above implies

(2.17) $\left(\frac{1}{16\pi}W(\Sigma)\right)^{\frac{1}{2}}\geq
1+\frac{m}{\mathfrak{c}_{{}_{\Sigma_{*}}}}.$

Direct calculation gives

(2.18)
$\mathfrak{c}_{{}_{\Sigma_{*}}}=\frac{m}{\left(1+\frac{2m}{r_{max}(\Sigma)}\right)^{\frac{1}{2}}-1}.$

(2.14) follows from (2.17) and (2.18). ∎

Proposition 2.4 gives another perspective of the singularity of
$(M_{\mathfrak{m}},g_{\mathfrak{m}})$ via the Willmore functional $W(\cdot)$.

##  3\. Bounding $\mathfrak{m}$ via $\mathfrak{c}(M,g)$

Let $(M,g)$ be given in the definition of $\mathfrak{c}(M,g)$ in (1.8). In
this section, we relate $\mathfrak{m}$ and $2\mathfrak{c}(M,g)$ assuming
$\mathfrak{c}(M,g)>0$. We begin with a characterization of
$\mathfrak{c}(M,g)>0$ which follows from standard arguments on harmonic
functions.

######  Proposition 3.1.

Let $\mathfrak{c}(M,g)$ be defined in (1.8). Then $\mathfrak{c}(M,g)>0$ if and
only if there exists a harmonic function $w$ on $(M,g)$ such that $0<w<1$ on
$M$ and $w(x)\to 1$ at $\infty$ (i.e. as $x\to p$).

###### Proof.

For each $k$, let $u_{k}$ be the harmonic function on
$(E_{{}_{\Sigma_{k}}},g)$ with $u_{k}\to 1$ as $x\to\infty$ and $u_{k}=0$ at
$\Sigma_{k}$. Given any surface $S\in\mathcal{S}$, by the maximum principle,
$\\{u_{k}\\}$ forms an increasing sequence in the exterior of $S$ relative to
$\infty$ (i.e. in $D_{{}_{S}}\setminus\\{p\\}$). Interior elliptic estimates
imply $\\{u_{k}\\}$ converges to a harmonic function $u_{\infty}$ on $M$
uniformly on compact sets in any $C^{i}$-norm. The limit $u_{\infty}$
satisfies $0<u_{\infty}\leq 1$ and $u_{\infty}\to 1$ as $x\to\infty$. By the
strong maximum principle, either $u_{\infty}\equiv 1$ or $0<u_{\infty}<1$.

Suppose $(M,g)$ admits a harmonic $w$ with $0<w<1$ and $w\to 1$ at $\infty$.
Then $w$ is an upper barrier for $\\{u_{k}\\}$, which implies $u_{\infty}\leq
w$, and hence $0<u_{\infty}<1$. In this case, $\mathfrak{c}(M,g)$ must be
positive. Otherwise, if $\mathfrak{c}(M,g)=0$, then
$\lim_{k\to\infty}\int_{E_{{}_{\Sigma_{k}}}}|\nabla u_{k}|^{2}=0$, which would
imply $\int_{K}|\nabla u_{\infty}|^{2}=0$ on any compact set $K$ in $M$, and
hence $u_{\infty}\equiv 1$, a contradiction.

Next suppose $\mathfrak{c}(M,g)>0$. We want to show $0<u_{\infty}<1$. If not,
$u_{\infty}\equiv 1$ on $M$. Pick any surface $S\in\mathcal{S}$, then
$\lim_{k\to\infty}u_{k}=1$ at $S$. Let $\beta_{k}=\min_{S}u_{k}$ for large
$k$. Let $\tilde{u}_{k}$ be the harmonic function on $E_{{}_{S}}$ with
$\tilde{u}_{k}\to 1$ at $\infty$ and $\tilde{u}_{k}=\beta_{k}$ at $S$. By the
maximum principle, $u_{k}\geq\tilde{u}_{k}$. Therefore, $\tilde{c}_{k}\geq
c_{k}$, where $\tilde{c}_{k}$, $c_{k}$ are the coefficients in the expansions

$\tilde{u}_{k}=1-\tilde{c}_{k}|x|^{-1}+o(|x|^{-1}),$
$u_{k}=1-c_{k}|x|^{-1}+o(|x|^{-1}),$

as $x\to\infty$. Here we have $c_{k}=\mathfrak{c}_{{}_{S_{k}}}$ and

$4\pi\tilde{c}_{k}=\lim_{r\to\infty}\int_{|x|=r}\frac{\partial\tilde{u}_{k}}{\partial\nu}=\int_{S}\frac{\partial\tilde{u}_{k}}{\partial\nu},$

where $\nu$ denotes the corresponding unit normal pointing to $\infty$.
Elliptic boundary estimates applied to $w_{k}=\tilde{u}_{k}-\beta_{k}$ shows

$\lim_{k\to\infty}\max_{S}|\nabla w_{k}|=0.$

Consequently, $\tilde{c}_{k}\to 0$ as $k\to\infty$. Combined with
$\tilde{c}_{k}\geq\mathfrak{c}_{k}$, this shows

$\mathfrak{c}(M,g)=\lim_{k\to\infty}\mathfrak{c}_{{}_{S_{k}}}=0,$

which is a contradiction. Therefore, $0<u_{\infty}<1$. This completes the
proof. ∎

######  Remark 3.1.

One may further require $w$ satisfies $\inf_{M}w=0$ in Proposition 3.1. To see
this, it suffices to examine the proof beginning with assuming
$\mathfrak{c}(M,g)>0$. In this case, we have shown $0<u_{\infty}<1$ on $M$.
Suppose $\inf_{M}u_{\infty}>0$, consider
$v=(1-\inf_{M}u_{\infty})^{-1}(u_{\infty}-\inf_{M}u_{\infty})$. Then
$v<u_{\infty}$ and $v$ also acts as a barrier for $\\{u_{k}\\}$, which implies
$u_{\infty}\leq v$, a contradiction. Hence, $\inf_{M}u_{\infty}=0$.

Next, we focus on the case in which the function $u$ tends to zero at “the
other end”.

######  Proposition 3.2.

Suppose there is a harmonic function $u$ on $(M,g)$ with $0<u<1$, $u(x)\to 1$
at $\infty$ (i.e. as $x\to p$), and $\lim_{k\to\infty}\max_{\Sigma_{k}}u=0$.
Then

(3.1) $\mathfrak{c}(M,g)=\mathcal{C},$

where $\mathcal{C}>0$ is the coefficient in the expansion of

$u=1-\mathcal{C}|x|^{-1}+o(|x|^{-1})$

in the AF end.

###### Proof.

Let $u_{k}$ be the harmonic function on $(E_{{}_{\Sigma_{k}}},g)$ with
$u_{k}\to 1$ at $\infty$ and $u_{k}=0$ at $\Sigma_{k}$. Then $u_{k}\leq u$,
which implies $\mathcal{C}\leq c_{k}$, where
$c_{k}=\mathfrak{c}_{{}_{\Sigma_{k}}}$ is the coeffiicent in

$u_{k}=1-c_{k}|x|^{-1}+o(|x|^{-1})$

as $x\to\infty$. This shows

$\mathcal{C}\leq\lim_{k\to\infty}\mathfrak{c}_{{}_{\Sigma_{k}}}=\mathfrak{c}(M,g).$

To show the other direction, consider $\alpha_{k}=\max_{\Sigma_{k}}u$. On
$E_{{}_{\Sigma_{k}}}$, by the maximum principle,
$\frac{1}{1-\alpha_{k}}(u-\alpha_{k})\leq u_{k}$, which implies
$\frac{1}{1-\alpha_{k}}C\geq c_{k}$. As $\alpha_{k}\to 0$, this gives

$\mathcal{C}\geq\lim_{k\to\infty}(1-\alpha_{k})\mathfrak{c}_{{}_{\Sigma_{k}}}=\mathfrak{c}(M,g).$

Therefore, $\mathcal{C}=\mathfrak{c}(M,g)$. ∎

We are now in a position to derive applications of (1.4).

######  Theorem 3.1.

Let $N$ be a noncompact, connected, orientable $3$-manifold admitting an
exhaustion sequence of precompact domains $D_{k}$ with connected boundary
$\partial D_{k}$, $k=1,2,\ldots$. Suppose $H_{2}(N)=0$. Let
$M=N\setminus\\{p\\}$ where $p$ is a point in $N$. Let $g$ be a smooth metric
with nonnegative scalar curvature on $M$ such that $g$ is asymptotically flat
near $p$. Assume there is a harmonic function $u$ on $(M,g)$ with $0<u<1$,
$u(x)\to 1$ as $x\to p$, and $\displaystyle\lim_{k\to\infty}\max_{{\partial
D_{k}}}u=0$. Then

  1. (i)

The limit $\displaystyle\lim_{t\to 0}\int_{u^{-1}(t)}|\nabla u|^{2}$ exists
(finite or $\infty$), where $t\in(0,1)$ is a regular value of $u$; and

  2. (ii)

$\displaystyle\mathfrak{m}\geq 2\mathfrak{c}(M,g)\left[1-\lim_{t\to
0}\left(\frac{1}{4\pi}\int_{u^{-1}(t)}|\nabla
u|^{2}\right)^{\frac{1}{2}}\right]$.

###### Proof.

Given a regular value $t\in(0,1)$ of $u$, let $\Sigma_{t}=u^{-1}(t)$.
$\Sigma_{t}$ is a closed, orientable surface in $M=N\setminus\\{p\\}$. Let
$\Sigma_{t}^{(1)}$ denote any connected component of $\Sigma_{t}$. Since
$H_{2}(N)=0$, $\Sigma_{t}^{(1)}$ is the boundary of a bounded domain
$\Omega_{1}$ in $N$. If $p\notin\Omega_{1}$, then $u$ is identically a
constant by the maximum principle. Hence, $\Sigma_{t}^{(1)}$ encloses $p$. As
a result, if there are two connected components of $\Sigma_{t}$, then both of
them enclose $p$, and thus form the boundary of a bounded domain in $M$. By
the maximum principle, $u$ is a constant, which is a contradiction. Therefore,
$\Sigma_{t}$ is connected. Since $t$ is arbitrary, this in particular shows
(1.4) is applicable to $(E_{t},g)$, where $E_{t}=\\{u\geq t\\}\subset M$ is
the exterior of $\Sigma_{t}$ with respect to $\infty$.

Applying (1.4) to $(E_{t},g)$, we have

(3.2) $\frac{\mathfrak{m}}{2\mathfrak{c}_{{}_{\Sigma_{t}}}}\geq
1-\left(\frac{1}{4\pi}\int_{\Sigma_{t}}|\nabla
u_{t}|^{2}\right)^{\frac{1}{2}}.$

Here $u_{t}=\frac{1}{1-t}(u-t)$ is the harmonic function on $(E_{t},g)$ that
tends to $1$ at $\infty$ and equals $0$ at $\Sigma_{t}$,
$\mathfrak{c}_{{}_{\Sigma_{t}}}=\frac{1}{1-t}\mathcal{C}$, and $\mathcal{C}$
is the coefficient in the expansion of

$u=1-\mathcal{C}|x|^{-1}+o(|x|^{-1}).$

It follows from (3.2) that

(3.3)
$\frac{\mathfrak{m}}{2\mathcal{C}}\geq\frac{1}{1-t}-\frac{1}{(1-t)^{2}}\left(\frac{1}{4\pi}\int_{\Sigma_{t}}|\nabla
u|^{2}\right)^{\frac{1}{2}}.$

Consider the function

$\mathcal{B}(t)=\frac{1}{(1-t)}\left[4\pi-\frac{1}{(1-t)^{2}}\int_{\Sigma_{t}}|\nabla
u|^{2}\right].$

In [16, Theorem 3.2 (ii)], we showed $\mathcal{B}(t)$ is monotone
nondecreasing in $t$ if $g$ has nonnegative scalar curvature. As a result,

$\lim_{t\to 0}\mathcal{B}(t)\ \text{exists}.$

Consequently,

$\lim_{t\to 0}\int_{\Sigma_{t}}|\nabla u|^{2}\ \text{exists}.$

This proves (i). (ii) follows from (3.3), (i) and Proposition 3.2. ∎

We have not assumed $g$ to be complete on $M$ so far. In particular, $(M,g)$
in Theorem 3.1 could just be the interior of an AF manifold with boundary
$\Sigma$ and the function $u$ may simply be the restriction, to the interior,
of the harmonic function that tends to $1$ at $\infty$ and vanishes at
$\Sigma$. In that extreme case, $\lim_{t\to 0}\int_{\Sigma_{t}}|\nabla
u|^{2}=\int_{\Sigma}|\nabla u|^{2}$ and (ii) reduces to (1.4).

If $g$ is complete on $M$, we have the following corollary.

######  Corollary 3.1.

Let $N$, $p$, $M$, $g$ and $u$ be given as in Theorem 3.1. Suppose $(M,g)$ is
complete and has Ricci curvature bounded from below. Then

(3.4) $\mathfrak{m}\geq 2\mathcal{C},$

where $\mathcal{C}=\mathfrak{c}(M,g)$ is the coefficient in the expansion of

$u=1-\mathcal{C}|x|^{-1}+o(|x|^{-1})$

as $x\to p$.

Corollary 3.1 relates to a result of Bray [4]. In [4, Theorem 8], Bray proved
that, if $(M,g)$ is a complete asymptotically flat $3$-manifold with
nonnegative scalar curvature which has multiple AF ends and mass
$\mathfrak{m}$ in a chosen end, then

$\mathfrak{m}\geq 2\mathcal{C},$

where $\mathcal{C}$ is the coefficient in the expansion
$u=1-\mathcal{C}|x|^{-1}+o(|x|^{-1})$ at the chose end, and $u$ is the
harmonic function that tends to $1$ at the chosen end and approaches $0$ at
all other AF ends.

Bray’s theorem allows $M$ to have more general topology and more than two
ends. Its proof made use of the $3$-d PMT. Complete manifolds whose ends are
all asymptotically flat necessarily have bounded Ricci curvature. In this
sense, Corollary 3.1 provides a partial generalization of Bray’s result.

###### Proof of Corollary 3.1.

Let $\Sigma_{t}$ be given in the proof of Theorem 3.1. Since $(M,g)$ is
complete and has Ricci curvature bounded from below, by the gradient estimate
of Cheng and Yau [8], $\max_{\Sigma_{t}}\,u^{-1}|\nabla u|\leq\Lambda$ where
$\Lambda$ is a constant independent on $t$. Combined with
$\int_{\Sigma_{t}}|\nabla u|=4\pi\mathcal{C}$, this shows

$\frac{1}{4\pi}\int_{\Sigma_{t}}|\nabla u|^{2}\leq\mathcal{C}\,\Lambda t,$

which implies

(3.5) $\lim_{t\to 0}\int_{\Sigma_{t}}|\nabla u|^{2}=0.$

It follows from (3.5) and Theorem 3.1 (ii) that $\mathfrak{m}\geq
2\mathcal{C}$. ∎

######  Remark 3.2.

Let $R$, Ric denote the scalar curvature, Ricci curvature of $g$. Since

$R\geq 0\ \text{and}\ \mbox{Ric}\ \text{bounded from above}\ \Rightarrow\
\mbox{Ric}\ \text{bounded from below},$

Corollary 3.1 also holds if the assumption of “Ric bounded from below” is
replaced by “Ric bounded from above”.

######  Remark 3.3.

As used in Bray’s work [4], the inequality $\mathfrak{m}\geq 2\mathcal{C}$ has
a geometric interpretation that asserts the mass of the conformally deformed
metric $u^{4}g$, which might not be complete, is nonnegative. Instead of
$\mathfrak{m}\geq 2\mathcal{C}$, a weaker inequality
$\mathfrak{m}\geq\mathcal{C}$ was obtained by Hirsch, Tam and the author in
[12].

Acknowledgement. I thank Christos Mantoulidis who introduced me to the work
[18]. I also thank Professor Xianzhe Dai for several helpful communications.

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