# Direct measurement of the low energy resonances in
${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$ reaction

Shahina  J. Görres  D. Robertson  M. Couder  O. Gomez  A. Gula  Department of
Physics and Astronomy, University of Notre Dame, Notre Dame, Indiana 46556,
USA  The Joint Institution of Nuclear Astrophysics-Center for the Evolution of
the Elements, University of Notre Dame, Notre Dame, Indiana 46556, USA  M.
Hanhardt  Department of Physics, South Dakota School of Mines and Technology,
Rapid City, South Dakota 57701, USA  South Dakota Science and Technology
Authority, Sanford Underground Research Facility, Lead, South Dakota 57754,
USA  T. Kadlecek  Department of Physics, South Dakota School of Mines and
Technology, Rapid City, South Dakota 57701, USA  R. Kelmar  P. Scholz  A.
Simon  E. Stech  Department of Physics and Astronomy, University of Notre
Dame, Notre Dame, Indiana 46556, USA  The Joint Institution of Nuclear
Astrophysics-Center for the Evolution of the Elements, University of Notre
Dame, Notre Dame, Indiana 46556, USA  F. Strieder  Department of Physics,
South Dakota School of Mines and Technology, Rapid City, South Dakota 57701,
USA  M. Wiescher  Department of Physics and Astronomy, University of Notre
Dame, Notre Dame, Indiana 46556, USA  The Joint Institution of Nuclear
Astrophysics-Center for the Evolution of the Elements, University of Notre
Dame, Notre Dame, Indiana 46556, USA

###### Abstract

The ${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$ is an important reaction in
stellar helium burning environments as it competes directly with one of the
main neutron sources for the s-process, the
${}^{22}\rm{Ne}(\alpha,n)^{25}\rm{Mg}$ reaction. The reaction rate of the
${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$ is dominated by the low energy
resonances at $E_{\alpha}^{lab}$ = 650 and 830 keV respectively. The
$E_{\alpha}^{lab}$ = 830 keV resonance has been measured previously, but there
are some uncertainties in the previous measurements. We confirmed the
measurement of the $E_{\alpha}^{lab}$ = 830 keV resonance using implanted 22Ne
targets. We obtained a resonance strength of $\omega\gamma$ = 35 $\pm$ 4 $\mu
eV$, and provide a weighted average of the present and previous measurements
of $\omega\gamma$ = 35 $\pm$ 2 $\mu eV$ with reduced uncertainties compared to
previous studies. We also attempted to measure the strength of the predicted
resonance at $E_{\alpha}^{lab}$ = 650 keV directly for the first time and
found an upper limit of $\omega\gamma$ $\mathrm{<0.15}$ $\mu eV$ for the
strength of this resonance. In addition, we also studied the $E_{P}^{lab}$=
851 keV resonance in ${}^{22}\rm{Ne}(p,\gamma)^{23}\rm{Na}$, and obtained a
resonance strength of $\omega\gamma$ = 9.2 $\pm$ 0.7 eV with significantly
lower uncertainties compared to previous measurements.

###### pacs:

##  I Introduction

The slow neutron capture process (s-process) is responsible for the creation
of half the elements beyond iron. It takes place when the timescale between
successive neutron captures is much larger than their lifetime of
$\beta$-decay Rolfs  _et al._ (1989). Hence, less neutron rich nuclei are
produced along the line of $\beta$ stability. The s-process can be categorized
into two types: the weak s-process responsible for the creation of elements
over the range 60 $<$ A $<$ 90, and the main s-process which creates nuclei
with A $>$ 90 Bisterzo  _et al._ (2015). The temperature range of interest is
0.2 to 0.6 GK for both scenarios. Both processes require a sufficiently large
neutron flux. However, the source of these neutrons is still under debate. In
AGB stars, the main s-process nucleosynthesis occurs during the late stages of
stellar evolution when the star has a degenerate C-O core, a thin radiative
He-intershell and an expanded convective envelope. During the AGB phase, the
star experiences a series of flashes called thermal pulses which are triggered
by the sudden activation of the 3$\alpha$-process at the base of the He-
intershell at highly degenerate gas conditions Bisterzo  _et al._ (2015). In
this intershell environment, two important reactions are activated,
${{}^{13}{\rm C}(\alpha,n)^{16}{\rm O}}$ prior to the thermal pulse and
${}^{22}\rm{Ne}(\alpha,n)^{25}\rm{Mg}$ during the thermal pulse, which act as
the main sources of free neutrons for the s-process. Both
${}^{22}\rm{Ne}(\alpha,n)^{25}\rm{Mg}$ and ${{}^{13}{\rm C}(\alpha,n)^{16}{\rm
O}}$ provide the neutron flux in low mass-AGB stars, ${{}^{13}{\rm
C}(\alpha,n)^{16}{\rm O}}$ being the dominant contributor. The 13C buildup
occurs in a thin zone (the 13C pocket) via the capture of mixed-in hydrogen on
the abundant 12C via 12C(p,$\gamma$)13N($\beta^{+}\nu$)13C. ${{}^{13}{\rm
C}(\alpha,n)^{16}{\rm O}}$ burns radiatively for an extended time releasing a
high neutron flux, when the temperature exceeds 0.8 $\times 10^{8}$ K Bisterzo
_et al._ (2015). The second dominant neutron source,
${}^{22}\rm{Ne}(\alpha,n)^{25}\rm{Mg}$, has a negative Q-value of -478 keV,
and thus requires a sufficiently high temperature. Such conditions are reached
only during the advanced thermal pulses in low mass AGB stars and the core
helium burning phase of massive stars.

${}^{22}\rm{Ne}(\alpha,n)^{25}\rm{Mg}$ also acts as the main neutron source
for the weak s-process in core helium burning of massive stars Käppeler  _et
al._ (1994). The 22Ne is produced from the large amount of 14N left at the end
of the CNO cycle in the H-burning core via the following reaction chain:

${}^{14}{\rm N}(\alpha,\gamma)^{18}{\rm F}(\beta^{+},\nu)^{18}{\rm
O}(\alpha,\gamma)^{22}\rm{Ne}$ (1)

However, because of its negative Q-value the temperature during core helium
burning is not sufficient to trigger the ignition of
${}^{22}\rm{Ne}(\alpha,n)^{25}\rm{Mg}$ at once. Only towards the end of the
helium burning phase with the stellar core gradually contracting, high
temperature and density conditions sufficient for the above reaction are
reached. Hence, the ${}^{22}\rm{Ne}(\alpha,n)^{25}\rm{Mg}$ reaction is
operative only during the peak of the helium flash and at the end of core
helium burning.

At lower temperature the neutron-producing role of
${}^{22}\rm{Ne}(\alpha,n)^{25}\rm{Mg}$ is complicated by the competing
${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$ reaction. The
${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$ reaction has a positive Q-value
$=10614.74\pm 0.03$ keV and, therefore, starts operating at relatively low
temperature, before ${}^{22}\rm{Ne}(\alpha,n)^{25}\rm{Mg}$ can kick in. As a
consequence, it is important to investigate the reaction rate of
${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$ in order to constrain the neutron
production for the weak s-process.

At stellar helium burning temperature, low energy resonances at
$E_{\alpha}^{lab}$ = 650 and 830 keV lie within the Gamow window and dominate
the ${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$ reaction rate.
${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$ proceeds through resonant capture
reaction mechanism, thus populating natural parity states in the compound
nucleus 26Mg.Talwar  _et al._ (2016) Above T=0.3 GK, the rate is dominated by
the $E_{\alpha}^{lab}$ = 830 keV resonance which has been observed in both
$(\alpha,n)$ and $(\alpha,\gamma)$ direct measurement experiments.

The ${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$ reaction was first measured
directly by Wolke et al. Wolke  _et al._ (1989) for $E_{\alpha}^{lab}$ = 0.71
to 2.25 MeV. They found 15 new resonances in the energy range covered, with
the lowest energy resonance at 830 keV. They concluded that the reaction rate
is strongly impacted by the 830 keV resonance, whose resonance strength was
measured to be ${\omega\gamma}$ = $36\pm 4$ $\mu$eV. For this measurement,
they used a differentially pumped gas target system with Neon gas enriched to
$99\%$ in ${}^{22}\rm{Ne}$. The $\gamma$ ray transitions were detected with
Ge(Li) detectors. A recent direct measurement of this resonance was also
performed by Hunt et al. Hunt  _et al._ (2019) who obtained its resonance
strength to be ${\omega\gamma}$ = $46\pm 12$ $\mu$eV. Unlike Wolke et al.
Wolke  _et al._ (1989), they used implanted ${}^{22}\rm{Ne}$ targets to
determine the resonance strength from the thick target yield curve. They
detected the entire cascade of $\gamma$ rays using a
$\gamma\gamma$-coincidence spectrometer, consisting of high-purity germanium
(HPGe) detector surrounded by a NaI(Tl) annulus.

Complementary to the direct measurements, Talwar et al. Talwar  _et al._
(2016) performed indirect measurements using (6Li,d) $\alpha$\- transfer and
$(\alpha,\alpha^{{}^{\prime}})$ inelastic scattering techniques to populate
states that are most likely to appear as resonances in the
${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$ reaction. They observed a strong
transition in both (6Li,d) and $(\alpha,\alpha^{{}^{\prime}})$ measurements at
${\rm E}_{x}=11.167$ MeV, which corresponds to a resonance in the 22Ne +
$\alpha$ system at $E_{\alpha}^{lab}$ = 650 keV. They assigned a spin-parity
of $J^{\pi}=(1^{-},2^{+})$ to this state, and obtained a spectroscopic factor
of $S_{\alpha}=0.36$ (corresponding to $\Gamma_{\alpha}=0.18$ $\mu$eV). Based
on this, they suggested a resonance strength of $\omega\gamma=$ 0.54 $\pm$
0.07 $\mu$eV. They concluded that this state completely dominates the
${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$ reaction rate between T $\simeq
0.2-0.4$ GK and would also potentially dominate the
${}^{22}\rm{Ne}(\alpha,n)^{25}\rm{Mg}$ reaction rate below 0.2 GK. Therefore,
it is essential to measure this low energy resonance in
${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$ in order to accurately predict the
total neutron flux for the weak s-process.

In this work, we present the direct measurements of these two low-energy
resonances at $E_{\alpha}^{lab}$ = 650 and 830 keV which play an important
role in the reaction rate of ${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$.
Several resonances are expected below the 650 keV resonance Talwar  _et al._
(2016). However, the penetrability rapidly drops with beam energy and the
corresponding alpha widths are at least two or three orders of magnitude
lower. Hence, the resulting resonance strength would be well below the
sensitivity of this experiment. For this reason, we concentrated on the search
for the 650 keV resonance which would have the largest impact on the neutron
production for the weak s-process if the predicted strength is confirmed
Talwar  _et al._ (2016).

##  II Experimental setup

###  II.1 Target Preparation and Characterization

The 22Ne targets were made and tested at the University of Notre Dame using
the 5U St.Ana Pelletron accelerator. $\mathrm{Ne}$ gas enriched in
${}^{22}\rm{Ne}$ to 99$\%$ was used to produce a 22Ne beam with an energy of
200 keV, which was implanted onto thick Ta backings, 3.8 cm $\times$ 3.8 cm in
size. Tantalum was used as a backing, as it has a high concentration of 22Ne
at implantation saturation Selin  _et al._ (1967) and beam induced reactions
with the backing are minimized. The 22Ne beam shape was defined by slits in
front of the target, and the beam was wobbled in both x and y directions to
have uniform implantation of 22Ne onto Ta. To measure the saturation curve
several 20Ne targets with increasing implantation dosage of 30, 60, 90, 120,
150, 180, 225, 270 and 300 mC were produced. The saturation was achieved near
300 mC, and hence all the implanted 22Ne targets used during the experiment
had a 300 mC dosage over a target area of 3 cm2.

The implantation profile of the 22Ne targets was investigated using the 1278
keV resonance in the ${}^{22}\rm{Ne}(p,\gamma)^{23}\rm{Na}$ reaction Depalo
_et al._ (2015). The efficiency of the emitted $\gamma$-rays was measured by
using a p-type coaxial germanium detector operating at +4000V. The detector
was placed at an angle of 55∘ with respect to the beam direction to minimize
angular distribution effects. The distance of the detector from the target was
20 cm to reduce summing effects. Moreover, a 1 mm thick lead sheet was
attached to the front face of the detector to reduce the $\gamma$-ray flux
from inelastic scattering from the Ta backing. The absolute full-energy peak
efficiency was measured using calibrated radioactive sources like 56Co and
60Co. The sources were placed exactly at the target position, 20 cm from the
detector to have the same geometry. The efficiency curve was extended to 10.76
MeV using the well known resonance at 992 keV in the
${}^{27}\rm{Al}(p,\gamma)^{28}\rm{Si}$ reaction Antilla  _et al._ (1977). The
absolute efficiency curve for the Ge detector is shown in Fig. 1. It was
obtained by normalizing the efficiency at higher energies, to the efficiency
of the 1779 keV $\gamma$-ray in the ${}^{27}\rm{Al}(p,\gamma)^{28}\rm{Si}$
resonance, taking into account the well-known branching ratios and angular
distributions Antilla  _et al._ (1977). The relative efficiency was converted
into absolute efficiency using the yield value of $(1.08\pm 0.06)\times
10^{-9}$ $\gamma-$ rays (1779 keV) per incident proton Antilla  _et al._
(1977), which corresponds to a resonance strength of 1.93 $\pm$ 0.13 eV Paine
and Sargood (1979).

Figure 1: Absolute $\gamma$-ray detection efficiency as a function of energy
for the Ge detector used for the determination of target stoichiometry.

The stoichiometry of the 22Ne targets was determined using the well-known
resonance at 1278 keV in ${}^{22}\rm{Ne}(p,\gamma)^{23}\rm{Na}$ Depalo  _et
al._ (2015) using the thick target yield formula Iliadis (2007),

$\omega\gamma=\frac{2Y\epsilon_{\mathrm{eff}}}{\lambda^{2}\eta},$ (2)

where $Y$ is the yield, $\eta$ is the efficiency, $\lambda$ is the deBroglie
wavelength of the incident particle and $\epsilon_{\mathrm{eff}}$ is the
effective stopping power of protons given by:

$\epsilon_{\mathrm{eff}}=\frac{M_{Ne}}{M_{P}+M_{Ne}}\left[\epsilon_{Ne}+\left(\frac{N_{Ta}}{N_{Ne}}\right)\epsilon_{Ta}\right],$
(3)

where $N_{Ta}/N_{Ne}$ is the initial target stoichiometry, $M_{P}$ and
$M_{Ne}$ are the masses of the projectile and target nuclei (in amu), while
$\epsilon_{Ne}$ and $\epsilon_{Ta}$ denote the stopping power of the protons
in Ne and Ta respectively, which were obtained using SRIM Ziegler  _et al._
(2010). The quoted uncertainties of these values for the relevant proton
energy are 1.6$\%$ and 3$\%$ for Ne and Ta, respectively.

The yield curve of the 1278 keV resonance was measured with protons from the
St.Ana accelerator utilizing the same target station which has been used for
the Ne implantation. The yield curve is shown in Fig. 2, which is obtained
after taking the average of the absolute yield calculated using the following
$\gamma$-ray transitions: 2540, 2980, 3914, 6102, 7036 and 9576 keV. The yield
from the individual gamma rays was corrected with the branching ratio obtained
from Bakkum and Van Der Leun (1989) and angular distribution obtained from
Viitasalo  _et al._ (1972). The yield curve in Fig. 2 depicts the distribution
of the implanted ${}^{22}\rm{Ne}$ in the Ta backing. In order to obtain the
stoichiometry of the implanted target, Equation. 2 was first used to calculate
the effective stopping power of the incident protons, using the resonance
strength of $\mathrm{\omega\gamma}$ = 10.8 $\pm$ 0.7 eV from Depalo  _et al._
(2015). The effective stopping power thus obtained was then substituted in
Equation. 3 to compute the stoichiometry of the target at maximum yield, which
turned out to be $N_{Ta}/N_{Ne}$= $2.7\pm 0.3$. The total number $n$, of
$\mathrm{Ne}$ nuclei implanted onto the Ta backing was thereafter obtained
using the formula Iliadis (2007),

$n=\frac{2A_{Y}}{\lambda^{2}\omega\gamma},$ (4)

where $\mathrm{A_{Y}}$ is the area under the
${}^{22}\rm{Ne}(p,\gamma)^{23}\rm{Na}$ resonance yield curve. $\mathrm{A_{Y}}$
was obtained by numerical integration of the excitation curve shown in Fig. 2.
The total number of $\mathrm{Ne}$ nuclei implanted, $n$, was found to be (6.21
$\pm$ 0.37) $\times$ $10^{17}$ $\mathrm{atoms/cm^{2}}$. The error of the
integration and the statistical errors are small compared to the uncertainty
of the resonance strength. The value obtained for the total number of
implanted Ne nuclei is consistent with the implantation dose within
uncertainty and indicates an implantation efficiency of nearly 100$\%$.

Figure 2: Yield curve for the $E_{P}^{\mathrm{lab}}$= 1278 keV resonance in
the ${}^{22}\rm{Ne}(p,\gamma)^{23}\rm{Na}$ reaction.

###  II.2 CASPAR accelerator

The measurement of the low energy resonances in
${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$ was performed at CASPAR (Compact
Accelerator System for Performing Astrophysical Research), located on the 4850
ft. level of the Sanford Underground Research Facility in Lead, South Dakota,
at the former site of the Homestake Gold mine Robertson  _et al._ (2016); cas
. The rock above acts as an effective shield from the cosmic-ray induced
background, thus helping to measure the weak resonances which have a low
signal-to-background ratio. The 1 MV JN positive ion accelerator at CASPAR was
used to deliver protons and alpha beams onto the target with intensity ranging
from 10 to 100 $\mu$A. The energy calibration of the beam was performed by
measuring the front edges of the resonances in
${}^{18}\rm{O}(\alpha,\gamma)^{22}\rm{Ne}$ at 750 and 767 keV Vogelaar  _et
al._ (1990) and scanning the well known resonances in
${}^{27}\rm{Al}(p,\gamma)^{28}\rm{Si}$ Endt and van dèr Leun (1973). The beam
energy uncertainty amounted to $\pm$2 keV. The beam was sent through an
analyzing magnet which bent it by 25∘ before transporting it to the
experimental end station. The beam was defocused and delivered to the target.
The size of the beam spot was nearly 1 cm in diameter. The target was mounted
inside the High EffiCiency TOtal absorption spectrometeR (HECTOR) Reingold
_et al._ (2019) at the end of the beamline, as shown in Fig. 3. The setup is
identical to the setup used for the measurement of
${}^{27}\rm{Al}(p,\gamma)^{28}\rm{Si}$ resonances Olivas-Gomez  _et al._
(2022) and the details of the setup are shown in Fig. 2 of Olivas-Gomez  _et
al._ (2022). The electrically isolated target chamber acted as a Faraday cup,
which was used for the integration of the beam current. To limit the buildup
of carbon on the target, a copper tube cooled to liquid nitrogen temperature
was placed inside the beamline, extending from the cold trap up to the target
surface. The cold trap was electrically isolated and biased to -300 V in order
to suppress the secondary electrons. The target was also water cooled in order
to reduce the damage caused by heating due to high beam intensities.

Figure 3: Experimental setup showing HECTOR placed at the end of the beamline
at the CASPAR underground lab. The target was placed at the center of the
detector cube.

###  II.3 $\gamma$-Summing Detector: HECTOR

HECTOR consists of an array of 16 $\mathrm{NaI(Tl)}$ crystals with two photo-
multiplier tubes attached to each segment. Each $\mathrm{NaI(Tl)}$ crystal is
housed in an aluminum casing. The crystals are arranged in a frame to form a
cube 16x16x16 inches in dimension. The center of the detector array consists
of a 60 mm borehole which allows the target to be placed at the center giving
a 4$\pi$ total coverage for the detection of $\gamma$-rays. The advantage of
using HECTOR for the ${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$ measurement
is its high detection efficiency. The signal from each photomultiplier tube is
recorded individually and then gain-matched and summed offline, to improve the
detector resolution. A digital data acquisition system is used to readout the
data from the 16 $\mathrm{NaI(Tl)}$ detectors. A detailed description of the
HECTOR and its data-acquisition can be found in Reingold  _et al._ (2019). In
this work, known level schemes were used to determine the summing efficiency
of HECTOR for the measured resonances. This approach was introduced and tested
for the ${}^{27}\rm{Al}(p,\gamma)^{28}\rm{Si}$ resonances in the recent work
of Olivas-Gomez  _et al._ (2022) and was applied in the analysis of the
${}^{18}\rm{O}(\alpha,\gamma)^{22}\rm{Ne}$ reaction Dombos  _et al._ (2022).

##  III Experimental Procedure and Analysis

The low energy resonances at $E_{\alpha}^{lab}$ = 650 and 830 keV in the
${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$ reaction were measured using the
implanted ${}^{22}\rm{Ne}$ targets (as discussed in section II.1) at CASPAR
Robertson  _et al._ (2016). The stability of the implanted ${}^{22}\rm{Ne}$
targets was monitored throughout the course of the experiment using the well
known resonances at $E_{p}^{lab}$ = 479 keV and 851 keV in the
${}^{22}\rm{Ne}(p,\gamma)^{23}\rm{Na}$ reaction Depalo  _et al._ (2015); the
1278 keV resonance of this reaction is not accessible with the CASPAR
accelerator. Fig. 4 shows the yield curve for the 479 keV resonance of
${}^{22}\rm{Ne}(p,\gamma)^{23}\rm{Na}$ for one of the 22Ne targets. The yield
is obtained using the sum peak in HECTOR Reingold  _et al._ (2019), at
$E_{\Sigma}=E_{CM}+Q$ = 9252.27 keV (where $E_{CM}$ is the center-of-mass
energy of the projectile target system and Q is the reaction Q-value). The
filled-circle (blue) and open-circle (orange) data points show the difference
in the yield at the beginning and the end of the experiment, after
accumulating a charge of nearly 4 C. The targets were fairly stable and the
yield decreased only by 5$\%$ at the end of the experiment, as shown in Fig.
4. No heat damage or blistering was caused by the high intensity of beam
current. As a consequence, we changed each target after it has been exposed to
an accumulated charge of around 4 C.

Figure 4: Yield curve for the $E_{P}^{\mathrm{lab}}$= 479 keV resonance in the
${}^{22}\rm{Ne}(p,\gamma)^{23}\rm{Na}$ reaction. The yield curve is obtained
from the sum peak in HECTOR Reingold  _et al._ (2019). The filled-circle
(blue) and open-circle (orange) data points show the difference in yield at
the beginning and at end of the experiment. The statistical error in the data
is $\simeq 1\%$, which is why the error bars are not visible.

To optimize the experimental yield of the narrow resonance at 830 keV in
${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$, it has to be measured at the
maximum of the 22Ne distribution. For this purpose, the energy loss of protons
from the resonance energy of 479 keV till the maximum of the ${}^{22}\rm{Ne}$
distribution (as shown in Fig. 4) in ${}^{22}\rm{Ne}(p,\gamma)^{23}\rm{Na}$,
was converted to the energy loss of $\alpha$ particles in
${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$ using the relative stopping powers
from SRIM Ziegler  _et al._ (2010). This implies that the maximum of the
${}^{22}\rm{Ne}$ distribution for the $\alpha$-particle beam would correspond
to an $\alpha$ energy of 910 keV for the $E_{\alpha}^{lab}$ = 830 keV
resonance. The ${}^{22}\rm{Ne}$ concentration is constant to 10$\%$ within
$\pm$ 30 keV of this energy. To confirm the position of the top of the thick
target yield curve/plateau, a scan of the $E_{\alpha}^{lab}$ = 830 keV
resonance was performed using large energy steps. Thus data was acquired at
the top of the plateau, i.e., at $E_{\alpha}^{lab}$ = 910 keV, with a total
accumulated charge of Q = 2.94 C on the ${}^{22}\rm{Ne}$ target. The beam-
induced background was measured using an implanted 20Ne target at the same
$\alpha$-particle energy. The sum peak spectrum for the 830 keV resonance is
shown in Fig. 5a. The solid (blue) histogram shows the on-resonance spectrum
taken with a 22Ne target, where the sum peak is located at an energy of
$E_{\Sigma}$ = 11.3 MeV. The dashed (orange) histogram shows the background
spectrum obtained using a 20Ne target. The sum peak from the background
20Ne($\alpha$,$\gamma$)24Mg reaction is located at $E_{\Sigma}$ = 10.02 MeV,
which is well separated from the sum peak of
${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$.

Computation of the strength of the 830 keV resonance requires the experimental
yield (discussed above) as well as the efficiency of the sum peak, as evident
from Equation. 2. The efficiency of the 830 keV resonance sum peak was
obtained using GEANT4 Agostinelli  _et al._ (2003) simulations. The level
scheme and the branching ratio for the reaction were taken from the previous
measurements performed by Wolke  _et al._ (1989); Hunt  _et al._ (2019). The
entire $\gamma$-cascade was simulated and the resultant simulated spectrum is
shown in Fig. 5a, as the dash dotted (brown) histogram. Adding the beam
induced background to the simulation shows excellent agreement with the
${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$ experimental spectrum as shown in
Fig. 5b.

The sum peak was fitted with a Gaussian plus a linear function shown as thick
solid (green) line in Fig. 5, and was integrated within $E_{\Sigma}-3\sigma$
to $E_{\Sigma}+3\sigma$ range to get the total number of events under the
peak. The simulated sum peak was fitted using the same procedure. The
efficiency is given by the ratio of the number of counts in the simulated sum
peak to that of the total simulated cascade events.

Figure 5: (a) Sum spectra for the $E_{\alpha}^{lab}$= 830 keV resonance. The
solid (blue) histogram shows the sum spectrum from the
${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$ reaction and the sum peak is
located at $E_{\Sigma}=E_{CM}+Q$ = 11.3 MeV. The dashed (orange) histogram
shows the beam induced background using the 20Ne target. The dash dotted
(brown) histogram shows the simulated spectrum obtained from GEANT4
Agostinelli  _et al._ (2003). The thick solid (green) line shows the Gaussian
plus a linear function fit to the sum peak. (b) The dashed (orange) histogram
is the sum of beam induced background and simulation. It shows excellent
agreement with the experimental spectrum shown in solid (blue).

The strength of the 830 keV resonance in
${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$ was determined relative to the
well known 479 keV resonance in ${}^{22}\rm{Ne}(p,\gamma)^{23}\rm{Na}$ Depalo
_et al._ (2015) using the expression for the maximum yield of an infinitely
thick target Iliadis (2007),

$\frac{\omega\gamma_{1}}{\omega\gamma_{2}}=\left(\frac{\epsilon_{r,1}}{\epsilon_{r,2}}\right)\left(\frac{\lambda_{r,2}^{2}}{\lambda_{r,1}^{2}}\right)\left(\frac{Y_{1}}{Y_{2}}\right)\left(\frac{\eta_{2}}{\eta_{1}}\right),$
(5)

where the subscripts 1 and 2 correspond to the resonance of interest and the
standard resonance. In this approach, only relative values of the stopping
powers and efficiencies are needed. The efficiency of the 479 keV resonance in
${}^{22}\rm{Ne}(p,\gamma)^{23}\rm{Na}$ ($\eta_{2}$) was obtained in the same
way as 830 keV resonance, using the level scheme and branching ratio from
Longland  _et al._ (2010). The error in the resonance strength
$\omega\gamma_{1}$ depends on the accuracy of $\omega\gamma_{2}$, and the
uncertainties of the ratios of stopping powers, efficiencies and yields. Using
this relative approach, we obtained $\omega\gamma$ = 35 $\pm$ 4 $\mu$eV for
this resonance. The uncertainty in the resonance strength is dominated by the
uncertainty of $\omega\gamma_{2}$ ($E_{P}^{lab}$ = 479 keV resonance) which is
6.4$\%$ and the relative stopping power of alphas and protons in Ta. The
uncertainty in the stopping power for alphas in Ta is 6$\%$ and for protons in
Ta is 3.4$\%$. The statistical uncertainty associated with the sum peak
integral is 3.5$\%$ for the 830 keV resonance and 1.4$\%$ for the 479 keV
resonance. The error of the relative efficiency is 5$\%$ mainly resulting from
the uncertainty in the branching ratio of the 830 keV resonance in
${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$.

Figure 6: Sum spectra for the $E_{\alpha}^{lab}$= 650 keV resonance. The solid
(blue) histogram depicts the sum spectrum from the
${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$ reaction, whereas the dashed
(orange) histogram shows the beam induced background using the 20Ne target.
The two solid vertical lines shows the region of integration for the sum peak.
The background (orange) histogram was normalized to the same accumulated
charge.

The other low energy resonance in ${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$
at $E_{\alpha}^{lab}$= 650 keV was measured in the same way as the
$E_{\alpha}^{lab}$= 830 keV resonance. Data was collected at
$E_{\alpha}^{lab}$= 711 keV, with a total accumulated charge of Q = 22.9 C on
top of the resonance. To determine the beam-induced background, data was taken
with a 20Ne target at the same $\alpha$ particle energy, with a total
accumulated charge of Q = 6.52 C. The sum spectrum for $E_{\alpha}^{lab}$= 650
keV resonance is shown in Fig. 6. The solid (blue) histogram shows the on
resonance spectrum taken with a ${}^{22}\rm{Ne}$ target, where the sum peak is
located at an energy of $E_{\Sigma}$ = 11.16 MeV. The dashed (orange)
histogram shows the background spectrum which was taken with the 20Ne target.
The width of the region of integration was taken to be the same as the 830 keV
resonance sum peak, shifting only by the center of mass energy of the incident
$\alpha$-particle beam. The two vertical lines show the region of integration
for the 650 keV resonance sum peak. The background on the low energy side of
the peak is arising mainly from neutrons, generated by $(\alpha,n)$ reactions
induced by $\alpha$-particle decay from the uranium and thorium present in the
surrounding rock. The other possible source of background of high energy
$\gamma$ rays is from the 11B$(\alpha,\gamma)^{15}$N reaction Wang  _et al._
(1991) which has a Q-value = 10991.18 keV. 11B is usually present in the Ta
backing as a contaminant. A narrow resonance in 11B$(\alpha,\gamma)^{15}$N
reaction Wang  _et al._ (1991) is known at an energy of 606 keV, which
corresponds to an excitation energy of 11.44 MeV.

The resonance strength of the $E_{\alpha}^{lab}$= 650 keV resonance in
${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$ was determined relative to the
$E_{\alpha}^{lab}$= 830 keV resonance using Eq. 5. The efficiency of the sum
peak for the $E_{\alpha}^{lab}$= 650 keV resonance was first calculated taking
the same decay scheme as the $E_{\alpha}^{lab}$= 830 keV resonance, since the
decay scheme is unknown for this resonance. Using this relative approach, we
obtained an upper limit $\omega\gamma$ $\mathrm{<0.11}$ $\mu eV$ for the
$E_{\alpha}^{lab}$= 650 keV resonance. However, to calculate the influence of
the choice of the involved $\gamma$-cascades on the upper limit, we simulated
various cascades with different $\gamma$-ray multiplicities and
$\gamma$-energies. These resulted in an upper limit of 0.15 $\mu$eV
independent of the actual branching ratio of this resonance.

During the course of the experiment, we repeatedly measured the $E_{P}^{lab}$
= 851 keV resonance in ${}^{22}\rm{Ne}(p,\gamma)^{23}\rm{Na}$. This resonance
was used to monitor the target stability during the experiment, primarily
because it is one of the strongest low energy resonances in
${}^{22}\rm{Ne}(p,\gamma)^{23}\rm{Na}$ Endt (1990). However, the strength of
the resonance is poorly known ($\pm$ 36$\%$). For this reason, we determined
its resonance strength relative to the known $E_{P}^{lab}$ = 479 keV resonance
($\omega\gamma$ = 0.594 $\pm$ 0.038 eV) Depalo  _et al._ (2015) using Eq. 5.
The level scheme and branching ratio to calculate the efficiency of the 851
keV resonance sum peak were taken from Bakkum and Van Der Leun (1989). We
obtained a resonance strength of $\omega\gamma$ = 9.2 ± 0.7 eV for this
resonance. This value is in agreement with the previously reported value of
$\omega\gamma$ = 7.0 ± 2.5 eV Endt (1990) but with significantly lower
uncertainty. With this improved accuracy this strong low energy resonance can
now be used to trace Ne implanted in various materials with high sensitivity
without the need for proton energies past 1 MeV.

Table 1: Comparison of the previous literature resonance strength values with the present work for the $E_{\alpha}^{lab}$= 830 keV resonance Work | $\mathrm{\omega\gamma[\mu eV]}$
---|---
Wolke et al. Wolke  _et al._ (1989) | 36 $\pm$ 4
Jaeger (Thesis) Jaeger (2001) | 33 $\pm$ 4
Hunt et al. Hunt  _et al._ (2019) | 46 $\pm$ 12
This work | 35 $\pm$ 4
Weighted average111Common uncertainties of the individual values are negligible (for details see text) | 35 $\pm$ 2

##  IV Discussion

The present resonance strength for the 830 keV resonance is compared with the
results of previous measurements in Table. 1. Wolke et al. Wolke  _et al._
(1989) used one passively shielded Ge detector whereas Jaeger et al. Jaeger
(2001) used one passively and actively shielded Ge detector. Both of these
measurements used windowless gas cell target with enriched ${}^{22}\rm{Ne}$
gas. Hunt et al. Hunt  _et al._ (2019) also used one passively and actively
shielded detector and the target was produced by implantation of
${}^{22}\rm{Ne}$ onto Titanium. In contrast to the previous measurements, our
present experiment utilized a $\gamma$-summing detector which was located on
the 4850 ft. level of the Sanford Underground Research Facility in Lead, South
Dakota Robertson  _et al._ (2016) which reduced cosmic ray induced background
significantly. Here, the targets consisted of ${}^{22}\rm{Ne}$ implanted onto
Ta sheets. A common uncertainty of these experiments is the $\alpha$ stopping
power obtained from SRIM Ziegler  _et al._ (2010). The uncertainty in the
alpha stopping power for pure ${}^{22}\rm{Ne}$ gas is 2$\%$ which is small
compared to the total error of 10$\%$ of the two gas target experiments Wolke
_et al._ (1989); Jaeger (2001). Titanium backings were used by Hunt et al.
which accounted for an uncertainty of 5$\%$ for the alpha stopping power which
is small compared to the total error of 26$\%$. The present experiment uses Ta
backings which have an uncertainty of 5$\%$ for the alpha stopping power. Our
present measurement is relative to the ${}^{22}\rm{Ne}(p,\gamma)^{23}\rm{Na}$
479 keV resonance and we never make use of the absolute value of the stopping
power. Only the ratio of alpha to proton stopping power is required. For these
reasons common uncertainties of the individual experimental values are
negligible. This leads to a weighted average of the previous and present
measurements of $\omega\gamma$ = 35 $\pm$ 2 $\mu eV$.

While we confirm the previous studies of the resonance strength for the 830
keV resonance in ${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$, the strength of
the corresponding resonance in the ${}^{22}\rm{Ne}(\alpha,n)^{25}\rm{Mg}$
still carries large uncertainties. Previous direct studies of this resonance
in ${}^{22}\rm{Ne}(\alpha,n)^{25}\rm{Mg}$ Jaeger (2001); Harms  _et al._
(1991); Giesen  _et al._ (1993) suggests values for its resonance strength
which differ substantially (up to a factor of four) amongst each other. These
measurements also differ from the prediction of the coincidence studies of the
decay of this level using the 22Ne$(^{6}$Li$,d\gamma)$ and the
22Ne$(^{6}$Li$,dn)$ reactions Ota  _et al._ (2020). Unlike the direct
measurements the coincidence studies suggest a comparable strength between the
$(\alpha,n)$ and $(\alpha,\gamma)$ decay modes. Further studies of the
$(\alpha,n)$ reactions are needed to constrain the resonance strengths.

For the lowest energy resonance at $E_{\alpha}^{lab}$ = 650 keV, the observed
yield only provides an upper limit for the resonance strength ($\omega\gamma$
$\mathrm{<0.15}$ $\mu eV$). This limit is a factor of 4 lower than suggested
by Talwar  _et al._ (2016) on the basis of a J${}^{\pi}=1^{-}$ assignment
obtained from the angular distribution analysis of the 22Ne(6Li,d)26Mg and
26Mg($\alpha,\alpha^{\prime})^{26}$Mg studies. Low energy studies of the
22Ne(6Li,d)26Mg and 22Ne(7Li,t)26Mg reactions Jayatissa  _et al._ (2020) on
the other hand observed the $E_{x}$ = 11.3 MeV ($E_{\alpha}^{lab}$ = 830 keV)
state but did not populate the state at $E_{x}$ = 11.17 MeV
($E_{\alpha}^{lab}$ = 650 keV). The measurement performed by Talwar  _et al._
(2016) was at about twice the center of mass energy of Ota  _et al._ (2020),
whereas Jayatissa  _et al._ (2020) did the measurement at sub-Coulomb
energies. For this reason it could be possible that Talwar  _et al._ (2016)
observed a high spin state while this is unlikely for the low energy
experiments performed by Ota  _et al._ (2020) and Jayatissa  _et al._ (2020).
This might suggest that the state at $E_{x}$ = 11.17 MeV ($E_{\alpha}^{lab}$ =
650 keV), is a high spin state, e.g. J${}^{\pi}=4^{+}$, which cannot be
excluded from angular distributions Talwar  _et al._ (2016). The single
particle width ( $\Gamma_{sp}=5\times 10^{-10}$ eV) is three orders of
magnitude lower than that of a J${}^{\pi}=1^{-}$ state ( $\Gamma_{sp}=5\times
10^{-7}$ eV), predicted by Talwar  _et al._ (2016). Several states have been
observed in this excitation range of 26Mg, which cannot be resolved in the
transfer and scattering studies discussed here, so it cannot be excluded that
the $\alpha$ elastic scattering populates a different state than the $\alpha$
transfer reactions. The $\alpha$ capture data discussed here indicate that the
state at $E_{x}$ = 11.17 MeV is a high spin state and for this reason should
have a distinctively smaller resonance strength than predicted by Talwar  _et
al._ (2016) based on a 1- assignment.

If the resonance at $E_{\alpha}^{lab}$ = 650 keV would indeed correspond to a
high spin state, 4+, then its contribution to the
${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$ reaction rate would be reduced by
two orders of magnitude because of the reduction of the penetrability. In this
case its contribution to the total ($\alpha,\gamma$) reaction rate would be
less significant and would only play an important role in the temperature
range around 0.2 GK (see e.g. Figure 8 of Talwar  _et al._ (2016)). At higher
temperatures the reaction rate would be mostly dominated by the resonance at
$E_{\alpha}^{lab}$ = 830 keV and towards lower temperatures by resonances
corresponding to neutron bound states in 26Mg. As a consequence the depletion
of 22Ne during the lower temperature burning phases between helium flashes in
the AGB phase and the cooler phases of core helium burning would be reduced.
This in turn would translate into an enhanced production of s-process elements
which would be increased compared to the network calculation of Talwar  _et
al._ (2016) (see e.g. Figures 14 and 17 of Talwar  _et al._ (2016)).

##  V Summary

In summary, we measured the two low energy resonances in
${}^{22}\rm{Ne}(\alpha,\gamma)^{26}\rm{Mg}$, which dominate the reaction rate
at stellar helium burning temperatures around 0.3 GK. The measurement of the
$E_{\alpha}^{lab}$ = 830 keV resonance confirms the previous results with
considerably smaller uncertainties. Based on our present measurement we
provide a new weighted average value of the resonance strength, which is 35
$\pm$ 2 $\mu eV$. We performed the first direct measurement of the lowest
energy resonance at $E_{\alpha}^{lab}$ = 650 keV. We provide an upper limit
for the resonance strength which is considerably lower than suggested by
Talwar  _et al._ (2016). For future measurements, either an extended gas
target could be used in an underground environment, which would improve the
yield substantially compared to using the implanted ${}^{22}\rm{Ne}$ target or
an experiment could be performed using inverse kinematics and a recoil
separator. However, considering the discussion of the properties of this
resonance a direct measurement might not be able to observe this resonance if
it is indeed a high spin state. In addition, we also measured the
$E_{P}^{lab}$= 851 keV resonance in ${}^{22}\rm{Ne}(p,\gamma)^{23}\rm{Na}$,
and obtained a resonance strength $\omega\gamma$ = 9.2 $\pm$ 0.7 eV, with
significantly lower uncertainties compared to previous measurements.

###### Acknowledgements.

The authors would like to thank the staff of the Sanford Underground Research
Facility (SURF) for their hospitality and technical support given during the
course of the experiment. This research was funded by the National Science
Foundation through Grant No. PHY-2011890 (University of Notre Dame Nuclear
Science Laboratory), Grant No. PHY-1430152 (the Joint Institute for Nuclear
Astrophysics - Center for the Evolution of the Elements), Grant No.
PHY-1913746 (The Compact Accelerator System for Performing Astrophysical
Research at the Sanford Underground Research Facility), the Sanford
Underground Research Facility (SURF, Grant No. DE-SC0020216) and Grant No.
PHY-1614442.

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# Maximum Principles and Consequences for $\gamma$-translators in
${\mathbb{R}}^{n+1}$

José Torres Santaella  Departamento de Geometría y Topología
Universidad de Granada
Granada
España<EMAIL_ADDRESS>
###### Abstract.

In this paper we obtain several properties of translating solitons for a
general class of extrinsic geometric curvature flows given by a homogeneous,
symmetric, smooth non-negative function $\gamma$ defined in an open cone
$\Gamma\subset{\mathbb{R}}^{n}$. The main results are tangential principles,
nonexistence theorems for closed and entire solutions, and a uniqueness result
that says that any strictly convex $\gamma$-translator defined on a ball with
a single end $\mathcal{C}^{2}$-asymptotic to a cylinder is the “bowl”-type
solution found in [Shati] up to vertical translations .

The author was partially supported by the project PID2020-116126GB-I00 funded
by MCIN/ AEI /10.13039/501100011033, by the project PY20-01391 (PAIDI 2020)
funded by Junta de Andalucía FEDER and by the framework of IMAG-Mar ía de
Maeztu grant CEX2020- 001105-M funded by MCIN/AEI/ 10.13039/50110001103. .

##  1\. Introduction

Geometric evolution equations for hypersurfaces have had a remarkable
development over the last decades. This kind of equations lead to interesting
non-linear PDE’s that have been used to solve important open questions in
mathematics and physics.

In this paper we are interested in hypersurfaces in ${\mathbb{R}}^{n+1}$ that
evolve under translations in a fixed unitary direction when an extrinsic
geometric flows is applied to them.

More precisely, let $M_{0}=F_{0}(M)$ be an immersed hypersurface in
${\mathbb{R}}^{n+1}$, we will say that $M_{0}$ evolves under a $\gamma$-flow
if there exist a $1$-parameter family of immersions
$F:M\times[0,T)\to{\mathbb{R}}^{n+1}$ such that

(1) $\displaystyle\begin{cases}\left<{\dfrac{\partial F}{\partial
t}(x,t),\nu(x,t)}\right>=\gamma(\lambda(x,t)),\mbox{ in }M\times(0,T),\\\
F(x,0)=F_{0}(x),\end{cases}$

where $\left<{\cdot,\cdot}\right>$ and $\nu(x,t)$ are the euclidean inner
product and the inward unit normal vector of $M_{t}:=F(M,t)$ in
${\mathbb{R}}^{n+1}$, respectively. In addition, the function $\gamma$ is
evaluated in the principal curvature vector
$\lambda(x,t)=(\lambda_{1}(x,t),\ldots,\lambda_{n}(x,t))$ of $M_{t}$ related
to $\nu(x,t)$ and satisfies the following properties:

  1. a)

$\gamma:\Gamma\to(0,\infty)$ is a smooth symmetric function (i.e.:
$\gamma(\lambda_{\sigma(1)},\ldots,\lambda_{\sigma}(n))=\gamma(\lambda_{1},\ldots,\lambda_{n})$
for every permutation $\sigma\in S_{n}$) and it is defined on a symmetric open
cone $\Gamma$ such that
$\Gamma_{+}:=\left\\{\lambda\in{\mathbb{R}}^{n}:\lambda_{i}>0\right\\}\subset\Gamma\subset{\mathbb{R}}^{n}$.

  2. b)

$\gamma$ is $\alpha$-homogeneous (i.e.: for every $\lambda\in\Gamma$ and
$c>0$, $\gamma(c\lambda)=c^{\alpha}\gamma(\lambda)$) for some $\alpha>0$.

  3. c)

$\gamma$ is increasing in each arguments (i.e.: for every $\lambda\in\Gamma$
and $\forall i=1,\ldots,n$,
$\dfrac{\partial\gamma}{\partial\lambda_{i}}(\lambda)>0$).

  4. d)

$\gamma$ posses a continuous extension to $0$ at the boundary of $\Gamma$
(i.e.: there exist a contunous function
$\tilde{\gamma}:\overline{\Gamma}\to[0,\infty)$ such that
$\left.\tilde{\gamma}\right|_{\Gamma}=\gamma$ and
$\left.\tilde{\gamma}\right|_{\partial\Gamma}=0$).

Recall that the principal curvatures of $M_{t}$ in ${\mathbb{R}}^{n+1}$ are
the eigenvalues of the Weingarten map. In local coordinates, it is given by
$h^{i}_{j}=g^{ik}h_{kj}$, where $g^{ij}$ denotes the coefficients of the
inverse of the metric tensor $g_{ij}$ of $M_{t}$, and $h_{ij}$ denotes the
coefficients of the second fundamental form of $M_{t}$, respectively. The
coefficients of the second fundamental form correspond to the tangent
components of the covariant derivative of the unit normal map (up to
orientation), where in local coordinates is given by
$h_{ij}=\left<{\nabla_{i}\nu,e_{j}}\right>$ where $\left\\{e_{i}\right\\}$
denotes a base of $T_{p}M_{t}$.

######  Example 1.1.

The class of functions $\gamma:\Gamma\to\mathbb{R}$ that satisfies the above
hypotheses with $\alpha=1$ is vast and includes:

  * •

The mean curvature $H=\lambda_{1}+\ldots+\lambda_{n}$, supported in
$\Gamma_{1}=\left\\{\lambda\in\mathbb{R}^{n}:H>0\right\\}$.

  * •

The $k$-th roots of the symmetric elemental polynomial $\sqrt[k]{S_{k}}$,
where

$\displaystyle S_{k}(\lambda)=\sum_{1\leq i_{1}<\ldots<i_{k}\leq
n}\lambda_{i_{1}}\ldots\lambda_{i_{k}},$

supported in the g$\accentset{\circ}{a}$rdin cone
$\Gamma_{k}=\left\\{\lambda\in\mathbb{R}^{n}:S_{l}(\lambda)>0,\>l=1,\ldots,k\right\\}$.

  * •

The inverse of the $k$-th harmonic sum $\left(\sum\limits_{1\leq
i_{1}<\ldots<i_{k}\leq
n}\dfrac{1}{\lambda_{i_{1}}+\ldots+\lambda_{i_{k}}}\right)^{-1}$ supported in
$\Gamma=\left\\{\lambda\in\mathbb{R}^{n}:\lambda_{1}+\ldots+\lambda_{k}>0\right\\}$,
where $\lambda_{1}\leq\ldots\leq\lambda_{n}$ has been imposed.

  * •

Any $1$-homogeneous symmetric combination of the above functions.

We note that by removing hypothesis d, the Hessian quotients functions
$Q_{k,l}=\left(\dfrac{S_{k}}{S_{l}}\right)^{\frac{1}{k-l}}$ supported in
$\Gamma_{k}$ can be included in this class of functions. We refer the reader
to [chow2020extrinsic] for an introductory book in this area of mathematics.

Recall that in this paper we focused in solution to (1) that evolves under
translations in a fixed unitary direction. This means that the $1$-parameter
family of immersion is given (up to tangential diffeomorphism) by

$\displaystyle F(x,t)=F_{0}(x)+vt,$

where $F_{0}$ is the initial immersion and $v\in{\mathbb{S}}^{n}$ is fixed.
This type of solution is know as translating soliton of the $\gamma$-flow, or
$\gamma$-translator for short.

We note that every time slice $M_{t}$ of a $\gamma$-translators satisfies the
equation

(2) $\displaystyle\gamma(\lambda)=\left<{\nu,v}\right>,$

and after a translation and rotation, we may only consider Equation (2) with
$v=e_{n+1}$, since $M_{t}$ is invariant under euclidean isometries.

Finally, $\gamma$-translators has been widely studied when $\gamma=H$. We
refer the reader to [Paco1] for a complete introduction to $H$-translators in
${\mathbb{R}}^{n+1}$ and its relation with singularities with the mean
curvature flow and minimal hypersurface theory.

On the other hand, for general curvature functions there are several works
under different additional assumptions on the curvature function $\gamma$. We
refer the reader to [andrews_2004] for a general behavior under $\gamma$-flows
when $\gamma$ is $1$-homogenous and the initial data is compact, to
[langford2020sharp] and [Lynch2021uniqueness] for the asymptotic analysis
under singularities of these flows.

The first result of this paper is about closed $\gamma$-translator which reads
as follows.

######  Theorem 1.2.

Let $\Gamma=\left\\{\lambda\in{\mathbb{R}}^{n}:\gamma(\lambda)>0\right\\}$ and
assume that $\gamma:\Gamma\to(0,\infty)$ satisfies properties a-c. Then, there
is no closed immersed $\gamma$-translator in ${\mathbb{R}}^{n+1}$ such that
its principal curvatures belong to the cone $\Gamma$.

This result is a well know fact for $H$-translators, since they are minimal
hypersurfaces in $({\mathbb{R}}^{n+1},e^{\left<{x,e_{n+1}}\right>}dx^{2})$.

######  Remark 1.3.

As an easy consequence of Theorem 1.2, it follows that there is no complete
totally umbilical strictly convex $\gamma$-translator in ${\mathbb{R}}^{n+1}$
without boundary. It remains open if it can be removed the completeness
hypothesis on the $\gamma$-translator.

Next, we develop an interior and a boundary tangential principle for
$\gamma$-translators in ${\mathbb{R}}^{n+1}$ which reads as follows.

######  Theorem 1.4.

Let $\Sigma_{1},\Sigma_{2}\subset{\mathbb{R}}^{n+1}$ be two embedded connected
$\gamma$-translators such that

  1. (1)

$\gamma:\Gamma\to[0,\infty)$ satisfies properties a-d in $\Gamma_{+}$ with
$\alpha>0$.

  2. (2)

$\Sigma_{1}$ is strictly convex (i.e.: $\lambda\in\Gamma_{+}$ in
$\Sigma_{1}$).

  3. (3)

$\Sigma_{2}$ is convex (i.e.: $\lambda\in\overline{\Gamma}_{+}$ in
$\Sigma_{2}$).

Then,

  1. a)

(Interior tangential principle) Assume that there exists an interior point
$p\in\Sigma_{1}\cap\Sigma_{2}$ such that the tangent spaces coincide at $p$.
If $\Sigma_{1}$ lies at one side of $\Sigma_{2}$, then both hypersurfaces
coincide.

  2. b)

(Boundary tangential principle) Assume that the boundaries
$\partial\Sigma_{i}$ lie in the same hyperplane $\Pi$ and the intersection of
$\Sigma_{i}$ with $\Pi$ is transversal. If $\Sigma_{1}$ lies at one side of
$\Sigma_{2}$ and there exist $p\in\partial\Sigma_{1}\cap\partial\Sigma_{2}$
such that the tangent spaces to $\Sigma_{i}$ and $\partial\Sigma_{i}$
coincide, then both hypersurfaces coincide.

Next, we consider an important class of $\gamma$-translators in
${\mathbb{R}}^{n+1}$ called “bowl”-type solutions founded in [Shati]. These
hypersurfaces are complete rotationally symmetric strictly convex graphs that
are defined in a ball of radius $r_{0}:=\sqrt[\alpha]{\gamma(1,\ldots,1)}$ or
are entire (i.e.: defined in all ${\mathbb{R}}^{n}$) which are
asymptotic111When $\gamma(0,1,\ldots,1)>0$. to

$x_{n+1}=\frac{1}{(\alpha+1)\gamma(0,1,\ldots,1)}|x|^{\alpha+1}+o(|x|^{\alpha+1}),\mbox{
as }|x|\to\infty$

In fact, in Theorems 1.3 and 1.4 in [Shati], the author characterized this
dichotomy in terms of the curvature function $\gamma$:

  * •

If $\gamma(0,1,\ldots,1)>0$, then the “bowl”-type solution is entire.

  * •

If $\gamma(0,1,\ldots,1)=0$, and, under the constraint
$\gamma(x,y,\ldots,y)=1$, $x\to L>0$ as $y\to\infty$, then the “bowl”-type
solution is defined in a ball.
In addition, if $L=0$, then we have the following extra conditions:

    * –

If $x=O(y^{-(2\alpha-1)})$, then the “bowl”-type solution is entire.

    * –

If there are positive constants $C$ and $k\in(0,2\alpha-1)$ such that $x\geq
Cy^{-k}$ for big enough $y$, then the “bowl”-type solution is defined in a
ball.

As a corollary of the tangential principle, we show a non-existence results
for entire $\gamma$-translators in ${\mathbb{R}}^{n+1}$.

######  Corollary 1.5.

Assume that $\gamma:\Gamma\to[0,\infty)$ satisfies Properties a-d with
$\alpha>\frac{1}{2}$.Then, if the “bowl”-type $\gamma$-translator is defined
in a ball, then there cannot exist a complete convex entire
$\gamma$-translators in ${\mathbb{R}}^{n+1}$.

The main result of this paper is a uniqueness theorem for complete
$\gamma$-translators in ${\mathbb{R}}^{n+1}$ which are strictly convex graphs
defined over ball which reads as follows.

######  Theorem 1.6.

Let $\Sigma\subset{\mathbb{R}}^{n+1}$ be a complete $\gamma$-translator such
that

  1. (1)

$\gamma:\Gamma\to(0,\infty)$ satisfies properties a-d in $\Gamma_{+}$ with
$\alpha>0$.

  2. (2)

$\Sigma$ is strictly convex graph over a ball
$B_{r}^{n}(0)\subset{\mathbb{R}}^{n}$.

  3. (3)

$\Sigma$ posses a single end $\mathcal{C}^{2}$-asymptotic to the cylinder
${\mathbb{S}}^{n-1}(r)\times{\mathbb{R}}$ (i.e.: the principal curvatures of
$\Sigma$ satisfy

$\displaystyle\min\left\\{\lambda_{i}(p):i=1,\ldots,n\right\\}=$
$\displaystyle\lambda_{1}(p)\to 0,$ $\displaystyle\forall
i\in\left\\{2,\ldots,n\right\\},\>$
$\displaystyle\lambda_{i}(p)\to\dfrac{1}{r},$

as $|p|\to\infty$).

Then $\Sigma$ is rotationally symmetric with respect the $x_{n+1}$-axis.

Finally, by the uniqueness result exposed in [Shati], we obtain the following
corollary.

######  Corollary 1.7.

Let $\Sigma$ be a $\gamma$-translator as in Theorem 1.6 with
$\alpha>\dfrac{1}{2}$ such that the “bowl”-type $\gamma$-translator is defined
in a ball of radius $\sqrt[\alpha]{\gamma(1,\ldots,1)}$. Then, $\Sigma$
coincide with the “bowl”-type $\gamma$-translator up to vertical translations.

######  Example 1.8.

We note that Corollary 1.7 can be applied to translators of the curvature
function $\gamma=HS_{n}$. Indeed, when we consider this function evaluated in
the positive cone $\Gamma_{+}$ it satisfies properties (a)-(d).

Moreover, the “bowl”-type solution is defined in the ball of radius
$r=\sqrt[\alpha]{\gamma(1,\ldots,1)}=\sqrt[n+1]{n}$. To check this, we used
the characterization given in [Shati],

$\displaystyle 1=\gamma(x,y,\ldots,y)=x^{2}y^{n-1}+(n-1)xy^{n-1},$

and we see that

$\displaystyle
x=\dfrac{1}{\sqrt{y^{n-1}+\dfrac{(n-1)^{2}y^{2n}}{4}}+\dfrac{(n-1)y^{n}}{2}}.$

Therefore, $x\geq\dfrac{y^{-n}}{n}$ for big enough $y$, and since $0<n<2n+1$,
it follows that the “bowl”-type solution is defined in a ball.

The organization of this article goes as follows: In Section 2, we prove
Theorem 1.2. In Section 3, we show the tangential principles: Theorem 1.4 and
Corollary 1.5. In Section 4, we prove the uniqueness results of Theorem 1.6
and Corollary 1.7. In Appendix we show that the “bowl”-type
$\gamma$-translator which is defined in a ball of radius $r_{0}$ is
$\mathcal{C}^{2}$-asymptotic to the cylinder
${\mathbb{S}}^{n-1}(r_{0})\times{\mathbb{R}}$.

Acknowledgment: The author would like to thank F. Martín and M. Sáez for
bringing this problem to his attention and for all the support they have
provided.

##  2\. Proof of Thoerem 1.2

Now we are going to prove that there is no closed $\gamma$-translator in
${\mathbb{R}}^{n+1}$ such that $\gamma:\Gamma\to(0,\infty)$ satisfies
properties a-c with $\alpha>0$, where
$\Gamma=\left\\{\lambda\in{\mathbb{R}}^{n};\gamma(\lambda)>0\right\\}$, and
the principal curvatures of the $\gamma$-translator belong to $\Gamma$.
The proof is based in the form of the equation that satisfies the height
function of the $\gamma$-translator.

###### Proof of Theorem 1.2.

Let us assume that $\Sigma$ is a compact $\gamma$-translator without boundary
in ${\mathbb{R}}^{n+1}$, and without loss of generality, we will also assume
that $0\in\Sigma$.

Next, we choose normal coordinates centred at $p\in\Sigma$ given by
$\left\\{e_{i}\right\\}_{i=1}^{n}\subset T_{p}\Sigma$. This means that
$\left\\{e_{i}\right\\}$ is an orthornomal basis of eigenvalues of the shape
operator of $\Sigma$ (i.e.:
$h^{i}_{j}(p)=h_{ij}(p)=\lambda_{i}(p)\delta_{j}^{i}$ where $h^{i}_{j}$,
$h_{ij}$ and $\delta_{j}^{i}$ are the coefficients of the shape operator,
second fundamental form and delta Kronecker of $\Sigma$, respectively).

Then, it follows that the height function $u(p)=\left<{p,e_{n+1}}\right>$
satisfies the following equations

$\displaystyle\nabla_{i}u=\left<{e_{i},e_{n+1}}\right>,\>|\nabla
u|=\left|e_{n+1}^{\top}\right|,$
$\displaystyle\nabla_{j}\nabla_{i}u=\left<{\nabla_{j}e_{i},e_{n+1}}\right>=\left<{\nabla_{j}e_{i},\nu}\right>\left<{\nu,e_{n+1}}\right>=-h_{ij}\left<{\nu,e_{n+1}}\right>,$
$\displaystyle\Delta_{\gamma}u:=\dfrac{\partial\gamma}{\partial
h_{ij}}\nabla_{j}\nabla_{i}u=-\alpha\gamma\left<{\nu,e_{n+1}}\right>=-\alpha\left<{\nu,e_{n+1}}\right>^{2}=\alpha(\left|e_{n+1}^{\top}\right|^{2}-1).$

Here $(\cdot)^{\top}$ denotes the orthogonal projection to $T_{p}\Sigma$ in
${\mathbb{R}}^{n+1}$. In addtion, in the third line we used property b to
write the term $\dfrac{\partial\gamma}{\partial h_{ij}}h_{ij}=\alpha\gamma$.
We note that, by property c, it follows that the operator $\Delta_{\gamma}$ is
elliptic when the principal curvatures belong to $\Gamma$.

In consequence, $u$ satisfies the equation

$\displaystyle\Delta_{\gamma}u-\alpha\left|\nabla u\right|^{2}+\alpha=0.$

Finally, by compactness and continuity, $u$ must attains an interior minimum
(recall that $\partial\Sigma=\emptyset$). However, this is impossible since
$\Delta_{\gamma}u\geq 0$ and $\nabla u=0$ at the point where the minimum of
$u$ is reached. Therefore, there cannot exist any closed $\gamma$-translator
such that its principal curvatures belong to $\Gamma$. ∎

##  3\. Proof of Theorem 1.4

The proof of Theorem 1.4 is inspired by the one given in [moller_2014] when
$\gamma=H$.

###### Proof of Thoerem 1.4.

Firstly, we mention that along this proof $D_{i},D_{ij}$ will denote the
derivatives with respect to a local frame for each hypersurface. Moreover, we
will denote by $B^{i}_{r}(x)$ a open ball of radius $r$ centered at
$x\in{\mathbb{R}}^{i}$.

Let $p\in\Sigma_{1}\cap\Sigma_{2}$ be an interior point such that
$T_{p}\Sigma_{1}=T_{p}\Sigma_{2}$ and $\Sigma_{1}$ lies locally at one side of
$\Sigma_{2}$. Then, after a rotation and a translation, there exists $r>0$
such that each $M_{i}\cap B^{n+1}_{r}(p)$ is the graph of a smooth function
$u_{i}:B^{n}_{r}(0)\subset\left\\{x_{n+1}=0\right\\}\to{\mathbb{R}}$, where
$p=(0,u_{i}(0))$. Therefore, since $\Sigma_{1}$ lies at one side of
$\Sigma_{2}$, we may assume that $u_{1}>u_{2}$ in
$B_{r}^{n}(0)\setminus\left\\{0\right\\}$ and $u_{1}(0)=u_{2}(0)$.

Figure 1. Picture of the proof of Theorem 1.2.

Next, we consider a convex combination between $u_{1}$ and $u_{2}$ given by

$\displaystyle u_{s}=(1-s)u_{1}+su_{2},\mbox{ for }s\in[0,1].$

It is not hard to see that for each $s\in(0,1)$, the graph of
$u_{s}:B_{r}(0)\to{\mathbb{R}}$ is strictly convex. In fact, this holds since
the convex combination of a positive definite matrix with a positive semi-
definite matrix is positive definite. Here the involved matrices are the shape
operators of the graphs of $u_{1}$ and $u_{2}$, respectively.

In particular, since for each $s\in(0,1)$, the principal curvatures of the
graph $u_{s}$ lie in the positive cone
$\Gamma_{+}=\left\\{\lambda\in{\mathbb{R}}^{n}:\>\lambda_{i}>0\right\\}$, it
follows that $u_{s}$ can be evaluated in the functional

$\displaystyle E(s)=\gamma(h^{i}_{j}(s))-\left<{\nu(s),e_{n+1}}\right>.$

Recall that the $s$ dependence in $E(s)$ is related to the coefficients of the
shape operator and the unit normal vector of the graph of $u_{s}$.

Furthermore, by the mean value theorem together with the fact that
$E(1)=E(0)=0$, it follows that

$\displaystyle 0=E(1)-E(0)=\dfrac{\partial E}{\partial s}(s_{0}),$

for some $s_{0}\in(0,1)$ (see Fig 1.).

On the other hand, we can explicitly calculate the term $\dfrac{\partial
E}{\partial s}(s)$. Indeed, by denoting $v=u_{2}-u_{1}$ and
$\dot{\gamma}^{\>ij}(s)=\dfrac{\partial\gamma}{\partial
h^{i}_{j}}(h^{i}_{j}(s))$, we have

$\displaystyle\dfrac{\partial E}{\partial s}(s)=$
$\displaystyle\dot{\gamma}^{\>ij}(s)\dfrac{\partial h^{i}_{j}}{\partial
s}(s)-\dfrac{\partial}{\partial s}\dfrac{1}{\sqrt{1+|Du_{s}|^{2}}}$
$\displaystyle=$ $\displaystyle\dot{\gamma}^{\>ij}(s)\dfrac{\partial}{\partial
s}\left[\left(\dfrac{\delta_{ik}}{\sqrt{1+|Du_{s}|^{2}}}-\dfrac{D_{i}u_{s}D_{k}u_{s}}{(1+|Du_{s}|^{2})^{3/2}}\right)D_{kj}u_{s}\right]+\dfrac{\left<{Du_{s},Dv}\right>}{(1+|Du_{s}|^{2})^{\frac{3}{2}}}$
$\displaystyle=$
$\displaystyle\dot{\gamma}^{\>ij}(s)\left[\left(-\dfrac{\delta_{ik}\left<{Du_{s},Dv}\right>}{(1+|Du_{s}|^{2})^{\frac{3}{2}}}-\dfrac{D_{i}vD_{k}u_{s}+D_{k}vD_{i}u_{s}}{(1+|Du_{s}|^{2})^{\frac{3}{2}}}+3\dfrac{D_{i}u_{s}D_{k}u_{s}\left<{Du_{s},Dv}\right>}{(1+|Du_{s}|^{2})^{\frac{5}{2}}}\right)D_{kj}u_{s}\right.$
$\displaystyle\left.+\left(\dfrac{\delta_{ik}}{\sqrt{1+|Du_{s}|^{2}}}+\dfrac{D_{i}u_{s}D_{k}u_{s}}{(1+|Du_{s}|^{2})^{\frac{3}{2}}}\right)D_{kj}v\right]+\dfrac{\left<{Du_{s},Dv}\right>}{(1+|Du_{s}|^{2})^{\frac{3}{2}}}.$

We claim that $v=0$ in $B_{r}(0)$, and we will argue this by contradiction
(i.e.: we will assume that $v$ is not a constant function such that $v(0)=0$
and $v(x)\leq 0$ in $B_{r}^{n}(0)$).

Firstly, we note that by continuity, $v$ reaches a maximum in
$\overline{B_{r}^{n}(0)}$. Let us assume first that the maximum is reached at
$0$. Recall that $v$ satisfies the linear elliptic PDE given by

$\displaystyle 0=$
$\displaystyle\dot{\gamma}^{\>ij}(s_{0})\left(\dfrac{\delta_{ik}}{\sqrt{1+|Du_{s_{0}}|^{2}}}-\dfrac{D_{i}u_{s_{0}}D_{k}u_{s_{0}}}{(1+|Du_{s_{0}}|^{2})^{\frac{3}{2}}}\right)D_{kj}v$
$\displaystyle+\left<{\left[\dot{\gamma}^{\>ij}(s_{0})\left(-\dfrac{\delta_{ik}D_{kj}u_{s_{0}}}{\left(1+|Du_{s_{0}}|^{2}\right)^{\frac{3}{2}}}+3\dfrac{D_{i}u_{s_{0}}D_{k}u_{s_{0}}D_{kj}u_{s_{0}}}{(1+|Du_{s_{0}}|^{2})^{\frac{5}{2}}}\right)+\dfrac{1}{(1+|Du_{s_{0}}|^{2})^{\frac{3}{2}}}\right]Du_{s_{0}},Dv}\right>$
$\displaystyle-\dot{\gamma}^{\>ij}(s_{0})D_{kj}u_{s_{0}}\dfrac{D_{i}u_{s_{0}}D_{k}v+D_{k}u_{s_{0}}D_{i}v}{(1+|Du_{s_{0}}|)^{\frac{3}{2}}}.$

Then, since $\gamma$ is a locally uniformly elliptic operator when the
principal curvatures of the graph of $u_{s_{0}}$ belong to $\Gamma_{+}$, it
follows that the second order term in the above equation

$\displaystyle\dot{\gamma}^{ij}(h^{i}_{j}(s_{0}))\left(\dfrac{\delta_{ik}}{\sqrt{1+|Du_{s_{0}}|^{2}}}-\dfrac{D_{i}u_{s_{0}}D_{k}u_{s_{0}}}{(1+|Du_{s_{0}}|^{2})^{\frac{3}{2}}}\right)D_{kj}v$

is uniformly elliptic in $B_{r}^{n}(0)$.
Consequently, the hypotheses of the strong maximum principle (Theorem 3.5 in
[gilbra_trudinger]) are satisfied, which means that $v$ is constant in
$B_{r}^{n}(0)$, giving a contradiction with our assumption.

On the other hand, let us assume that the maximum of $v$ is reached at some
$x_{0}\in\partial B_{r}^{n}(0)$. Then, since $v$ cannot reach an interior
maximum, it follows that $v(x_{0})>0$.
In particular, the hypotheses of Hopf’s Lemma (Lemma 3.4 in
[gilbra_trudinger]) hold, which implies that $\dfrac{\partial v}{\partial
N}(x_{0})>0$ where $N=\dfrac{x}{r}$ is the outward unit normal of $\partial
B_{r}^{n}(0)$. This means that $\alpha(t)=v\left(\dfrac{x_{0}}{r}t\right)$ is
an increasing function when $t$ is close to $r$.

However, by continuity together with $v\leq 0$ in $B^{n}_{r}(0)$, we can find
a $t_{0}$ close to $r$ such that $\alpha(t_{0})=0$. This contradicts the fact
that $v$ only vanishes at $x=0$. This finishes the proof of the claim $v=0$ in
$B_{r}^{n}(0)$. Therefore, $\Sigma_{1}\cap B^{n+1}(p,r)=\Sigma_{2}\cap
B^{n+1}(p,r)$.

Finally, since both hypersurfaces are connected, it can be showen by a similar
strategy together with the weak uniqueness continuation principle for linear
elliptic equations that $\Sigma_{1}=\Sigma_{2}$.

For the boundary tangency principle, we only need to change $B(p,r)$ with a
hemisphere $B(p,r)\cap\left\\{x_{n+1}\geq 0\right\\}$, here we consider
$\Pi=\left\\{x_{n+1}=0\right\\}$.
Then, since the intersection $\Sigma_{i}\cap\Pi$ is transversal, the function
$v=u_{2}-u_{1}$ satisfies $\dfrac{\partial v}{\partial N}(p)=0$ where
$N=e_{n+1}$ is the normal unit vector of $\partial\Sigma_{i}$ at $p$.
On the other hand, the hypothesis of the Hopf’s Lemma (Lemma 3.4 in
[gilbra_trudinger]) holds, $\dfrac{\partial v}{\partial N}(p)>0$ which is a
contradiction. Therefore, the same argument holds in that case to conclude the
result. ∎

######  Remark 3.1.

We note that generalizing Theorem 1.4 to other type $\gamma$-translators is a
subtle issue. One way to do it without much modification on this proof, would
be by assuming that $\Gamma$ is a convex cone. In fact, under this assumption,
we could take the convex combination of principal curvatures of each
hypersurface instead of the local graphs that represents these hypersurfaces
as in the proof of Theorem 1.4.

We end this section with the proof of Corollary 1.5

###### Proof of Corollary 1.5.

We prove this by contradiction. Assume that there is an entire
$\gamma$-translator $\Sigma$ given by a function
$u:{\mathbb{R}}^{n}\to{\mathbb{R}}$ such that its principal curvatures belong
to the cone $\left\\{\lambda\in{\mathbb{R}}^{n}:\lambda_{i}\geq 0\right\\}$.
This last property is equivalent to $\Sigma$ being convex.

Let $C_{n}$ be the “bowl”-type $\gamma$-translator in ${\mathbb{R}}^{n+1}$.
Recall that $C_{n}$ is the unique222Unique among rotationally symmetric
strictly convex graphs. strictly convex rotationally symmetric graph defined
in $B(0,r_{0})$ which is $\mathcal{C}^{2}$-asymp-totic to the cylinder
${\mathbb{S}}^{n-1}(r_{0})\times{\mathbb{R}}$, where
$r_{0}=\sqrt[\alpha]{\gamma(1,\ldots,1)}$.

Then, translating suitably $C_{n}$ over $\Sigma$, we can find a $t_{0}>0$ such
that $C_{n}+te_{n+1}$ lies strictly above from $\Sigma$ for $t\geq t_{0}$.
Note that this can be done since $C_{n}$ is not an entire graph. Now, we may
translate $C_{n}+t\varepsilon_{n+1}$ downward until it touches $\Sigma$ for
the first time.
Finally, by the interior tangential principle Theorem 1.4, we obtain that
$\Sigma=C_{n}$ which contradicts that $\Sigma$ is entire. ∎

##  4\. Proof of Theorem 1.6

Firstly, since the shape operator of a given hypersurface in
${\mathbb{R}}^{n+1}$ is invariant under isometries of ${\mathbb{R}}^{n+1}$, it
follows that $\gamma(\tilde{\lambda})=\gamma(\lambda)$ where $\tilde{\Sigma}$
is the image of any isometry applied to a hypersurface
$\Sigma\subset{\mathbb{R}}^{n+1}$.
In particular, if $R$ is a rotational field which fixes the $x_{n+1}$-axis in
${\mathbb{R}}^{n+1}$, then

$\displaystyle\left<{\tilde{\nu},e_{n+1}}\right>=\left<{\nu,R^{-1}e_{n+1}}\right>=\left<{\nu,e_{n+1}}\right>=\gamma(\lambda)=\gamma(\tilde{\lambda}).$

This means that $\gamma$-translators remain as $\gamma$-translators after
applying rotational fields which fix the direction of translation under the
$\gamma$-flow.

Consequently, under the hypothesis of Theorem 1.6, it is enough to show that
$\Sigma$ is symmetric along the plane $\left\\{x_{1}=0\right\\}$ to obtain
that $\Sigma$ is rotationally symmetric.

For this purpose we will adopt the following notations and definition used in
[Paco_2014] when $\gamma=H$. Let $A\subset{\mathbb{R}}^{n+1}$ and
$t\in{\mathbb{R}}$, then we set:

  * •

The $1$-parameter family of vertical hyperplanes
$\Pi_{t}=\left\\{x\in{\mathbb{R}}^{n+1}:\textbf{p}(x)=t\right\\}$, where

$\textbf{p}(x_{1},\ldots,x_{n+1})=x_{1}.$

In addition, we denote $\Pi=\Pi_{0}$.

  * •

The $1$-parameter family of horizontal spaces
$Z_{t}=\left\\{x_{n+1}>t\right\\}$.

  * •

The $1$-parameter family of subsets of $A$ given by

$\displaystyle A_{+}(t)$ $\displaystyle=\left\\{x\in A:\textbf{p}(x)\geq
t\right\\},$ $\displaystyle A_{-}(t)$ $\displaystyle=\left\\{x\in
A:\textbf{p}(x)\leq t\right\\},$ $\displaystyle\delta_{t}(A)$
$\displaystyle=A\cap\Pi_{t}.$

Note that $A_{+}(t)$ and $A_{-}(t)$ are the right hand side and the left hand
side, respectively, along $\Pi_{t}$ of $A$ (see Definition 4.1 given below).

  * •

The $1$-parameter families of right and left reflections of $A$, respectively,
along the hyperplane $\Pi_{t}$ given by

$\displaystyle A_{+}^{*}(t)$
$\displaystyle=\left\\{(2t-x_{1},x_{2},\ldots,x_{n+1})\in{\mathbb{R}}^{n+1}:(x_{1},\ldots,x_{n+1})\in
A_{+}(t)\right\\},$ $\displaystyle A_{-}^{*}(t)$
$\displaystyle=\left\\{(2t-x_{1},x_{2},\ldots,x_{n+1})\in{\mathbb{R}}^{n+1}:(x_{1},\ldots,x_{n+1})\in
A_{-}(t)\right\\},$

respectively.

  * •

The orthogonal projection $\pi:{\mathbb{R}}^{n+1}\to{\mathbb{R}}^{n+1}$ to the
hyperplane $\Pi$ given by

$\pi(x_{1},\ldots,x_{n+1})=(0,x_{2},\ldots,x_{n+1}).$

######  Definition 4.1.

Let $A,B$ be two subsets of ${\mathbb{R}}^{n+1}$. We say that $B\leq A$ and it
is read as “$A$ is on the right side of $B$” if, and only if, for every
$x\in\Pi$ such that

$\displaystyle\pi^{-1}(\left\\{x\right\\})\cap A\neq\emptyset\mbox{ and
}\pi^{-1}(\left\\{x\right\\})\cap B\neq\emptyset,$

we have that

(3)
$\displaystyle\sup\left\\{\mathbf{p}(p):p\in\pi^{-1}(\left\\{x\right\\})\cap
B\right\\}\leq\inf\left\\{\mathbf{p}(p):p\in\pi^{-1}(\left\\{x\right\\})\cap
A\right\\}.$

Note that for arbitrary sets, the relation $B\leq A$ is not a partial order,
but for sets given by the graph of an entire function over the plane $\Pi$, it
works as a partial order.

######  Remark 4.2.

The method of moving planes requires specifically two things to be applied:

  * •

The first one is that vertical hyperplanes are to be solutions to Equation
(2). To accomplish this, we add the Property d on $\gamma$.

  * •

The second one is the necessity of tangential interior and boundary
principles, respectively. We will use it to decide what happens when the
hypersurface intersects tangentially the reflection of the same hypersurface
along a moving plane. In our case, Theorem 1.4 gives us these results.

###### Proof of Theorem 1.6..

Recall the hypotheses of this theorem are: $\Sigma\subset{\mathbb{R}}^{n+1}$
is a complete $\gamma$-translator such that

  1. (1)

$\gamma:\Gamma\to[0,\infty)$ satisfies properties a-d.

  2. (2)

$\Sigma$ is strictly convex graph over a ball
$B_{r}^{n}(0)\subset{\mathbb{R}}^{n}$.

  3. (3)

$\Sigma$ has a single end $\mathcal{C}^{2}$-asymptotic to the cylinder
${\mathbb{S}}^{n-1}(r)\times{\mathbb{R}}$.

Firstly, since $\Sigma$ is a strictly convex vertical graph defined in
$B_{r}(0)$, we have

$r=\sup\left\\{t>0:\Sigma\cap\Pi(t)\neq\emptyset\right\\}.$

Then, we consider the set

$\displaystyle\mathcal{A}:=\left\\{t\in[0,r):\Sigma_{+}(t)\mbox{ is a graph
over }\Pi\mbox{ and }\Sigma_{-}(t)\leq\Sigma_{+}^{*}(t)\right\\}.$

Figure 2. The moving plane method acting in a rotationally symmetric strictly
convex surface $M$. Image courtesy of Francisco Martín.

The set $\mathcal{A}$ will act as the parameter of the family of moving
hyperplanes in the application of the moving plane method (see Fig. 2). For
this reason, we want to show that $\mathcal{A}$ is the interval $[0,r)$. In
fact, this will give us that $\Sigma_{-}(0)\leq\Sigma_{+}^{*}(0)$, and by
analogous arguments we will obtain that $\Sigma_{-}^{*}(0)\leq\Sigma_{+}(0)$.
We note that the combination of these two properties imply $\Sigma$ is
symmetric about the hyperplane $\Pi$.

Therefore, in the following claims we will show that $\mathcal{A}=[0,r)$ by
arguing that $\mathcal{A}$ is not empty, and it is an open and closed set of
$[0,r)$.

######  Claim 4.3.

The set $\mathcal{A}\neq\emptyset$. In addition, for every $s\in\mathcal{A}$,
$[s,r)\subset\mathcal{A}$.

###### Proof.

We will show in the following steps that there exists
$\varepsilon\in(0,t_{0})$, such that
$(t_{0}-\varepsilon,t_{0})\subset\mathcal{A}$, giving that
$\mathcal{A}\neq\emptyset$.

######  Step 4.4.

$\Sigma_{+}(t)$ is connected for every $t\in[0,r)$.

###### Proof.

Firstly, lets recall that $\Sigma$ has a single end. This implies that
$\Sigma_{+}(t)$ has only one unbounded component for every $t\in[0,r)$.
Because otherwise, we can choose a compact $\Sigma^{\prime}$ component of
$\Sigma_{+}(t)$. Then, we can translate a hyperplane $\Pi_{t}$ until it
touches $\Sigma^{\prime}$ at a first order contact point333This means that
both hypersurfaces intersect at a point where the first derivatives coincide.
In particular, the tangent planes coincides because the unit normal vectors of
each hypersurface concide at this point.. Note that the tangential principles
Theorem 1.4, applied to $\Sigma^{\prime}$ with $\Pi_{t_{1}}$, for some
$t_{1}>0$, imply that $\Sigma^{\prime}$ is totally geodesic, which contradicts
that $\Sigma$ is strictly convex graph.
Therefore, $\Sigma_{+}(t)$ is connected for every $t\in[0,t_{0})$. ∎

######  Step 4.5.

There exists $\varepsilon_{0}\in(0,t_{0})$ such that $\Sigma_{+}(t)$ is a
graph over $\Pi$ for $t\in(t_{0}-\varepsilon_{0},t_{0})$.

###### Proof.

Firstly, we will show that there exists $\varepsilon_{0}\in(0,r)$ such that

$\displaystyle\left<{\nu,e_{1}}\right>=\dfrac{D_{1}u}{\sqrt{1+|Du|^{2}}}>0$

holds in $\Sigma_{+}(t)$ for all $t\in(r-\varepsilon_{0},r)$. Recall that
$\Sigma$ is a vertical graph of a function
$u:B_{r}^{n}(0)\subset{\mathbb{R}}^{n}\to{\mathbb{R}}$.
For this purpose let us assume the contrary. This means that there exist an
increasing sequence $t_{l}$ converging to $t_{0}$ such that
$D_{1}u(x_{1}^{l},\ldots,x_{n}^{l})\leq 0$ for some
$p_{l}=(x_{1}^{l},\ldots,x_{n}^{l},u(x_{1}^{l},\ldots,x_{n}^{l}))\in\Sigma_{+}(t_{l})$.
Then, since $t_{l}\leq x_{1}^{l}<t_{0}$, it follows that
$\left|p_{l}\right|\to\infty$ . Recall that the end of $\Sigma$ is
$\mathcal{C}^{2}$-asymptotic to the cylinder
${\mathbb{S}}^{n}(r)\times{\mathbb{R}}$, this means that
$\nu(p_{l})\to(1,0\ldots,0)$ as $l\to\infty$. In fact, this occurs because the
intersection of the cylinder ${\mathbb{S}}^{n}(r)$ with the hyperplane
$\Pi_{r}$ occurs at the point with unit normal vector $(1,0,\ldots,0)$. Note
that this contradicts that $D_{1}u(x_{1}^{l},\ldots,x_{n}^{l})=0$ (see Fig 3).

Figure 3. Sequence $\nu(p_{l})\to e_{1}$ as $l\to\infty$ where
$p_{l}\in\Sigma_{+}(t_{l})$. Image courtesy of Francisco Martín.

Consequently, there exists $\varepsilon_{0}\in(0,r)$ such that
$\left<{\nu,e_{1}}\right>>0$ holds at $\Sigma_{+}(t)$ for all
$t\in(r-\varepsilon,r)$. In particular, this fact together with $\Sigma$ being
a complete graph defined in $B^{n}_{r}(0)$444This means that the Gauss map
$\nu:\Sigma\to{\mathbb{S}}^{n}$ is a proper local diffeomorphism which maps
the boundary to the boundary of the image. and $\Sigma_{+}(t)$ being
connected, imply that $\Sigma_{+}(t)$ is a graph over $\Pi$ for all
$t\in(r-\varepsilon_{0},r]$. ∎

######  Step 4.6.

Let $\varepsilon=\dfrac{\varepsilon_{0}}{2}$, then
$\Sigma_{-}(t)\leq\Sigma_{+}^{*}(t)$ for every $t\in[r-\varepsilon,r)$.
Finalizing the first part of the proof of Claim4.3.

###### Proof.

Firstly, since $\Sigma_{+}(t)$ is a graph over $\Pi$ for
$t\in[r-\varepsilon,r)$, the relation “being at the right side” (see
Definition 4.1) is a partial order.
Consequently, it suffices to show that for
$t_{1}=r-\dfrac{\varepsilon_{0}}{2}$, the condition
$\Sigma_{-}(t_{1})\leq\Sigma_{+}^{*}(t_{1})$ (i.e: the reflection of
$\Sigma_{+}(t_{1})$ along $\Pi_{t_{1}}$ is at the right hand side of
$\Sigma_{-}(t_{1})$) is satisfied.
Let us assume that $\Sigma_{-}(t_{1})\not\leq\Sigma_{+}^{*}(t_{1})$, then we
may find a first order contact point $p_{0}$ between $\Sigma_{-}(t_{0})$ and
$\Sigma_{+}^{*}(t_{0})$ for some $0<t_{0}<r-\varepsilon_{0}$. In fact,
$t_{0}\not\in[r-\varepsilon_{0},t_{1}]$ since $\Sigma_{+}(t)$ is a graph over
$\Pi$ for $t\in[r-\varepsilon,r)$, and this would imply that
$\Sigma_{+}(t_{0})$ is not a graph over $\Pi$. Moreover, if $t_{0}=0$, then
the proof finishes, since $0\in\mathcal{A}$ is what we want to prove.
Therefore, the interior and boundary tangential principles in Theorem 1.4
imply that $\Sigma_{+}^{*}(t_{0})=\Sigma_{-}(t_{0})$, which means that
$\Sigma$ is symmetric along $\Pi_{t_{0}}$. But, since the end of $\Sigma$ is
symmetric along the plane $\Pi$, it follows that $t_{0}=0$ a contradiction. ∎

######  Step 4.7.

For every $s\in\mathcal{A}$, $[s,r)\subset\mathcal{A}$.

###### Proof.

Firstly, we note that if $s\in\mathcal{A}$, then $\Sigma_{+}(s)$ is a graph
over $\Pi$. In particular, since $\Sigma_{+}(t)\subset\Sigma_{+}(s)$ for every
$s<t<r$, it follows that $\Sigma_{+}(t)$ is a graph over $\Pi$.

Now, let $t_{1}\in(s,r)$ be such that
$\Sigma_{-}(t_{1})\not\leq\Sigma_{+}^{*}(t_{1})$. Then, we can find
$t_{0}\in(t_{1},r-\varepsilon)$ such that
$\Sigma_{+}^{*}(t_{0})\setminus\delta_{t_{0}}(\Sigma)$ and
$\Sigma_{-}(t_{0})\setminus\delta_{t_{0}}(\Sigma)$. We note that the interior
tangential principle implies that $\Sigma_{+}^{*}(t_{0})=\Sigma_{-}(t_{0})$,
giving that $\Pi_{t_{0}}\neq\Pi$ is hyperplane of symmetry of $\Sigma$, this
contradics that the end of sigma is symmetric along $\Pi$. ∎

∎

######  Claim 4.8.

The set $\mathcal{A}$ is closed in $[0,t_{0})$.

###### Proof.

Let $s_{k}\in\mathcal{A}$ be a sequence of points such that $s_{k}\to s_{0}$.
In the following two steps we are going to show by contradiction that
$s_{0}\in\mathcal{A}$.

######  Step 4.9.

Assume that $\Sigma_{+}(s_{0})$ is not a graph over $\Pi$.

###### Proof.

The assumption of this step implies that we can find $p,q\in\Sigma_{+}(s_{0})$
such that $p=(p_{1},p_{2},\ldots,p_{n+1})$ and
$q=(q_{1},p_{2},\ldots,p_{n+1})$ with $p_{1}<q_{1}$.

Then, by Claim 4.3, we have that $s>s_{0}$ for every $s\in\mathcal{A}$. In
particular, it follows that follows that $p_{1}=s_{0}$. Because otherwise
$q\in\Sigma_{+}(p)$, and $\Sigma_{+}(p)$ would be a graph over $\Pi$ which
does not agree with our assumption.

Moreover, we note that

$\displaystyle q_{1}>s=\dfrac{q_{1}+3s_{0}}{4}>s_{0}=p_{1}.$

Then, since $q\in\Sigma_{+}(s)$, $p\in\Sigma_{-}(s)$ and

$\displaystyle
2s-q_{1}=\dfrac{3s_{0}-q_{1}}{2}=s_{0}+\dfrac{s_{0}-q_{1}}{2}<p_{1},$

it follows that $\Sigma_{-}(s)\not\leq\Sigma_{+}^{*}(s)$, which contradicts
that $s\in\mathcal{A}$. ∎

######  Step 4.10.

Assume that $\Sigma_{+}(s_{0})$ is a graph over $\Pi$ and
$\Sigma_{-}(s_{0})\not\leq\Sigma_{+}^{*}(s_{0})$.

###### Proof.

Firstly, we note that the relation $A\leq B$ is partial order for graphs over
$\Pi$. Then, if $\Sigma_{-}(s_{0})\not\leq\Sigma_{+}^{*}(s_{0})$, by
continuity we may find a first order contact point in the intersection of
these sets which is not in the hyperplane $\Pi_{s_{0}}$, i.e.:

$p_{0}\in(\Sigma_{+}^{*}(s_{0})\cap\Sigma_{-}(s_{0}))-\delta_{s_{0}}(\Sigma).$

In particular, $\textbf{p}(p_{0})=2s_{0}-x_{1}$ for some
$(x_{1},\ldots,x_{n+1})=q_{0}\in\Sigma_{+}(s_{0})$.

Moreover, since $\Sigma_{+}(s_{0})$ is a graph over $\Pi$, we may write

$x_{1}=s_{0}+f_{s_{0}}(0,x_{2},\ldots,x_{n+1})$

for some positive continuous function $f_{s_{0}}:\Pi\to{\mathbb{R}}$. The
reason why $f_{s_{0}}$ is positive is because
$p_{0}\not\in\delta_{s_{0}}(\Sigma)$.

Next, we choose $x_{1}^{k}=s_{k}+f_{s_{0}}(0,x_{2},\ldots,x_{n+1})$. Then,
$q_{k}:=(x_{1}^{k},x_{2},\ldots,x_{n+1})\in\Sigma_{+}(s_{k})$ and

$\displaystyle 2s_{k}-x_{1}^{k}=s_{k}-f_{s_{0}}(0,x_{2},\ldots,x_{n+1}).$

This means that
$p_{k}:=(2s_{k}-x_{1}^{k},x_{2},\ldots,x_{n+1})\in\Sigma_{+}^{*}(s_{k})\cap\Sigma_{-}(s_{k})$.

Then, since $\Sigma_{-}(s_{k})\leq\Sigma_{+}^{*}(s_{k})$, it follows that
$2s_{k}-x_{1}^{k}=s_{k}$ (see a proof in (6) below). In particular,
$f_{s_{0}}(0,x_{2},\ldots,x_{n+1})=0$, which means that
$\textbf{p}(p_{0})=s_{0}=x_{1}$, or equivalently,
$p_{0}\in\delta_{s_{0}}(\Sigma)$ given the desires contradiction. ∎

Therefore, we have proven that $\Sigma_{+}(s_{0})$ is a graph over $\Pi$ and
$\Sigma_{+}^{*}(s_{0})$ is at the right hand side of $\Sigma_{-}(s_{0})$,
finalizing the proof of this claim. ∎

######  Claim 4.11.

The set $\mathcal{A}$ is open in $[0,r)$.

###### Proof.

Let $t_{1}\in\mathcal{A}$, we want to show that there exists $\varepsilon>0$
such that $(t_{1}-\varepsilon,t_{1}]\subset\mathcal{A}$. Then, by the Claim
4.3, the set $(t_{1}-\varepsilon,t_{1}+\varepsilon)$ will be an open
neighborhood of $t_{1}$ contained in $\mathcal{A}$, or equivalently,
$\mathcal{A}$ is an open subset of $[0,r)$.

We divide this proof in the following two steps.

######  Step 4.12.

There exist $\varepsilon_{2}\in(0,t_{1})$ such that
$\Sigma_{+}(t_{1}-\varepsilon_{2})$ is a graph over $\Pi$.

###### Proof.

Since $t_{1}\in\mathcal{A}$, it follows that $\Sigma_{+}(t_{1})$ is a graph
over $\Pi$. Let $h\gg 1$ such that for every $p\in\Sigma_{+}(t_{1})\cap Z_{h}$
the unit normal vector at $p\in\Sigma_{+}(t_{1})\cap Z_{h}$ satisfies
$|\nu(p)-\nu_{{\mathbb{S}}^{n-1}\times{\mathbb{R}}}(p)|<\varepsilon$. Recall
that $Z_{h}=\left\\{x_{n+1}>h\right\\}$. In particular t
Then, if there is $p\in\Sigma_{+}(t_{1})\cap Z_{h}$ such that $\nu(p)\in\Pi$
with $\textbf{p}(p)=\tilde{s}\in[t_{1},r)$, we would have by the boundary
tangential principle that $\Sigma_{+}^{*}(\tilde{s})=\Sigma_{-}(\tilde{s})$
(see Fig 4.). This means that $\Sigma$ is symmetric along
$\Pi_{\tilde{s}}\neq\Pi$, and also the end of $\Sigma$, a contradiction.

Figure 4. View from $x_{n+1}=h\gg 1$ with
$\Sigma_{+}^{*}(t_{1}-\varepsilon_{1})$ is drawn in dashed lines. Image
courtesy of Francisco Martín.

In particular, we have

$\displaystyle\nu(\Sigma_{+}(t_{1}))\cap\Pi=\emptyset.$

Consequently, there exists an $\varepsilon_{0}\in(0,t_{1}]$ such that

$\displaystyle\nu(\Sigma_{+}(t))\cap\Pi=\emptyset,\mbox{ holds for all
}t\in(t_{1}-\varepsilon_{0},t_{1}).$

Then, by combining this with the fact that $\Sigma$ is embedded and
$\Sigma_{+}(t)$ is connected for all $t\in[0,t_{0})$, we obtain that
$\Sigma_{+}(t)\cap Z_{h}$ can be written as a graph over its projection in
$\Pi$ for all $t\in(t_{1}-\varepsilon_{0},t_{1})$.
Furthermore, by taking $\varepsilon_{1}=\dfrac{\varepsilon_{0}}{2}$ as we did
in Claim 4.3, we will get that

(4) $\displaystyle\Sigma_{-}(t_{1}-\varepsilon_{1})\cap
Z_{h}\leq\Sigma_{+}^{*}(t_{1}-\varepsilon_{1})\cap Z_{h}.$

Next, we consider the compact set

$\displaystyle K:=\Sigma\cap\left\\{x_{n+1}\leq h\right\\}.$

We note that

$\displaystyle\left<{\nu,e_{1}}\right>=\dfrac{D_{1}u}{\sqrt{1+|Du|^{2}}}>0$

holds in $K_{+}(t_{1})$ since it is a graph over $\Pi$.
Then, by compactness there exist a $\varepsilon_{2}\in(0,\varepsilon_{1})$
such that $D_{1}u>0$ holds in $K_{+}(t)$ for all
$(t_{1}-\varepsilon_{2},t_{1}]$. Therefore, by combining this with the fact
that $\Sigma$ is embedded and $\Sigma_{+}(t)$ is connected, it follows that
$K_{+}(t)$ can be written as a graph over its projection in $\Pi$ for all
$(t_{1}-\varepsilon_{2},t_{1}]$.

In summary, we have shown that there exists $\varepsilon_{2}\in(0,t_{1})$ such
that $\Sigma_{+}(t)$ can be written as a graph over its projection in $\Pi$
for all $t\in(t_{1}-\varepsilon_{2},t_{1}]$. ∎

######  Step 4.13.

There exists $\varepsilon_{4}\in(0,\varepsilon_{2})$, such that
$\Sigma_{-}(t_{1}-\varepsilon_{4})\leq\Sigma_{+}^{*}(t_{1}-\varepsilon_{4})$.

###### Proof.

Firstly, since $\Sigma_{+}(t)$ is a graph over its projection in $\Pi$ for all
$t\in(t_{1}-\varepsilon_{2},t_{1}]$, we may find a
$\varepsilon_{3}\in(0,\varepsilon_{2})$ such that

(5) $\displaystyle\left(\Sigma_{+}^{*}(t)\cap\Sigma_{-}(t)\cap
K\right)-\delta_{t}(\Sigma)\subset K_{-}(t-\varepsilon_{3}).$

In fact, to see this we note that if

$\displaystyle
x=(x_{1},\ldots,x_{n+1})\in\Sigma_{+}^{*}(t)\cap\Sigma_{-}(t)\cap
K\cap\delta_{t}(\Sigma)^{c},$

we have that $x_{1}<t$ and $x_{1}=2t-y_{1}$ where
$y_{1}=f(0,x_{2},\ldots,x_{n+1})+t$ for some continuous function $f>0$ defined
over a compact set $\Omega_{f}\subset\Pi$.
Therefore, by compactness, we have that $y_{1}-t\geq\varepsilon_{3}$ where

$\varepsilon_{3}=\min\left\\{\varepsilon_{2},\min\limits_{\Omega_{f}}f\right\\}>0.$

This means that for all $t\in(t_{1}-\varepsilon_{3},t_{1}]$ it follows that

$\displaystyle t-x_{1}=y_{1}-t\geq\varepsilon_{3}\Leftrightarrow x_{1}\leq
t-\varepsilon_{3}.$

Moreover, since $\Sigma_{-}(t_{1})\leq\Sigma_{+}^{*}(t_{1})$, it follows that

(6)
$\displaystyle\Sigma_{+}^{*}(t_{1})\cap\Sigma_{-}(t_{1})=\delta_{t_{1}}(\Sigma),$

because otherwise we could find a first order contact point

$p_{0}\in(\Sigma_{+}^{*}(t_{1})\cap\Sigma_{-}(t_{1}))-\delta_{t_{1}}(\Sigma).$

Then, if $p_{0}$ is an interior point, the interior tangential principle
Theorem 1.4 would imply that $\Pi_{\textbf{p}(p_{0})}$ is a hyperplane of
symmetry of $\Sigma$, since
$\Sigma_{-}(\textbf{p}(p_{0}))=\Sigma_{+}^{*}(\textbf{p}(p_{0}))$.
Moreover, if $p_{0}$ is boundary point the same conclusion will hold by using
instead the boundary tangential principle Theorem 1.4. Consequently, either
$\Sigma$ is symmetric about the hyperplane $\Pi$ or
$\Sigma_{-}(t_{1})\cap\Sigma_{+}^{*}(t_{1})=\delta_{t_{1}}(\Sigma)$.

Next, we will show that there exists $\varepsilon_{4}\in(0,\varepsilon_{3})$
such that

$\displaystyle\Sigma_{+}^{*}(t)\cap\Sigma_{-}(t)\cap K=\delta_{t}(\Sigma)\cap
K.$

holds for all $t\in(t_{1}-\varepsilon_{4},t_{1}]$.

Indeed, if it not the case, we may find an increasing sequence $t_{l}$
converging to $t_{1}$ such that

$\displaystyle(\Sigma_{+}^{*}(t_{l})\cap\Sigma_{-}(t_{l})\cap
K)-\delta_{t}(\Sigma)\neq\emptyset,\>\forall l\in{\mathbb{N}}.$

Let
$(x_{1}^{l},\ldots,x_{n+1}^{l})=p_{l}\in(\Sigma_{+}^{*}(t_{l})\cap\Sigma_{-}(t_{l})\cap
K)-\delta_{t}(\Sigma)$. We note that by Equation (5) we have

(7) $\displaystyle x_{1}^{l}\leq t_{l}-\varepsilon_{3}.$

Then, by compactness, we may assume, after taking a subsequence if it is
necessary, that
$p_{l}\to\tilde{p}=(\tilde{x}_{1},\ldots,\tilde{x}_{n+1})\in\Sigma_{+}^{*}(t_{1})\cap\Sigma_{-}(t_{1})\cap
K$.
In particular, Equation(6) gives that $\tilde{x}_{1}=t_{1}$. But, after taking
limits on Equation (7), we see that $\tilde{x}_{1}\leq t_{1}-\varepsilon_{3}$,
given a contradiction with $\tilde{x}_{1}=t_{1}$.

Therefore, there exists $\varepsilon_{4}\in(0,\varepsilon_{3})$ such that

$\displaystyle\Sigma_{+}^{*}(t)\cap\Sigma_{-}(t)\cap K=\delta_{t}(\Sigma)\cap
K$

holds for all $t\in(t_{1}-\varepsilon_{4},t_{1}]$. This means that

$\displaystyle\Sigma_{-}(t)\cap K\leq\Sigma_{+}^{*}(t)\cap K\mbox{ holds for
all }t\in(t_{1}-\varepsilon_{3},t_{1}].$

Thus, combining the above line with Equation (4), we finally obtain

$\displaystyle\Sigma_{-}(t)\leq\Sigma_{+}^{*}(t)\mbox{ holds for all
}t\in(t_{1}-\varepsilon_{4},t_{1}]$

∎

Therefore, by the Steps 4.12 and 4.13, it follows that
$(t_{1}-\varepsilon_{4},t_{1}]\subset\mathcal{A}$, and as we mention in the
beginning of the proof of this claim, the set $\mathcal{A}$ is an open subset
of $[0,t_{0})$. ∎

The proof finishes by noting that $\mathcal{A}$ is a non-empty open and closed
subset of the connected set $[0,t_{0})$. Therefore, $\mathcal{A}=[0,t_{0})$.
In particular, $\Sigma_{-}(0)\leq\Sigma_{+}^{*}(0)$.
An analogous argument will show that $\Sigma_{-}^{*}(0)\leq\Sigma_{+}(0)$.
Therefore, $\Sigma$ is symmetric with respect the hyperplane $\Pi$, finalizing
the proof of Theorem 1.6. ∎

###### Proof of Corollary 1.7 .

Theorem 1.6 implies that $\Sigma$ is a rotationally symmetric graph defined
over the ball of radius $r$. Then, by Theorems 1.3-1.4 in [Shati], it follows
that $r=\sqrt[\alpha]{\gamma(1,\ldots,1)}$ and $\Sigma$ is the “bowl”-type
$\gamma$-translator in ${\mathbb{R}}^{n+1}$ up to vertical translations. ∎

##  5\. Appendix: “Bowl”-type solutions asymptitics to round cylinders

######  Proposition 5.1.

The “bowl”-type solution which is defined in the ball
$B_{r_{0}}(0)\subset{\mathbb{R}}^{n}$ and $\gamma$ satisfies properties a-c
with $\alpha>\dfrac{1}{2}$ is $\mathcal{C}^{2}$-asymptotic to the cylinder
${\mathbb{S}}^{n-1}(r_{0})\times{\mathbb{R}}$ where
$r_{0}=\sqrt[\alpha]{\gamma(1,\ldots,1)}$.

###### Proof.

Firstly, we note that the profile curve of the “bowl”-type $\gamma$-translator
is given by

$\displaystyle\left\\{(r,{\mathbf{u}}(r))\in{\mathbb{R}}^{2}:r\in[0,r_{0})\right\\}.$

Then, by setting ${\mathbf{v}}=\dot{{\mathbf{u}}}$, we have that

(8)
$\displaystyle\dot{{\mathbf{v}}}=(1+{\mathbf{v}}^{2})^{\alpha}f\left(\dfrac{{\mathbf{v}}}{r}\right),$

where $f(x)$ appears in (8) by the implicit function theorem applied on the
level set

$\displaystyle\left\\{(x,y)\in{\mathbb{R}}^{2}:\gamma(x,y,\ldots,y)=1\right\\}.$

Consequently, since the “bowl”-type solution is defined in a ball, it follows
that $\lim\limits_{x\to\infty}f(x)=L\geq 0$.
Next, the principal curvatures of the “bowl”-type solution satisfy

$\displaystyle 0<$
$\displaystyle\lambda_{1}=\dfrac{\dot{{\mathbf{v}}}}{(1+{\mathbf{v}}^{2})^{\frac{3}{2}}}=\dfrac{f\left(\dfrac{{\mathbf{v}}}{r}\right)}{\sqrt{1+{\mathbf{v}}^{2}}}\leq\dfrac{C}{\sqrt{1+{\mathbf{v}}^{2}}},$
$\displaystyle\lambda_{i}=\dfrac{{\mathbf{v}}}{r\sqrt{1+{\mathbf{v}}^{2}}},\mbox{
for }i=2,\ldots,n.$

Therefore, since $\lim\limits_{r\to r_{0}}{\mathbf{v}}(r)=\infty$, it follows
that $\lambda_{1}\to 0$ and $\lambda_{i}=\dfrac{1}{r_{0}}$ as $|p|\to\infty$.
∎

## References