# The 5-dimensional Langevin approach to fission of atomic nuclei

F.A. Ivanyuk<EMAIL_ADDRESS>Institute for Nuclear Research, 03028 Kyiv,
Ukraine  C. Ishizuka<EMAIL_ADDRESS>Tokyo Institute of Technology,
Tokyo, 152-8550 Japan  S. Chiba<EMAIL_ADDRESS>Tokyo Institute
of Technology, Tokyo, 152-8550 Japan  NAT Research Center, NAT Corporation,
3129-45 Hibara Muramatsu, Tokai-mura, Naka-gun, Ibaraki 319-1112 Japan

(today)

###### Abstract

We have generalized the four-dimensional Langevin approach used in our
previous works for the description of fission process to the five-dimensional
by considering the neck parameter $\epsilon$ in the two-center shell model
shape parametrization as an independent dynamical variable. The calculated
results for the mass distribution of fission fragments are in better agreement
with the available experimental data. In particular, the transition from the
mass-symmetric to mass-asymmetric fission via the triple-humped distribution
in fission of thorium isotopes is well reproduced.

nuclear fission, Langevin equations, mass distributions, total kinetic energy

###### pacs:

24.10.-i, 25.85.-w, 25.60.Pj, 25.85.Ca

##  I Introduction

The approach based on Langevin equations langevin  has been successfully
applied in various branches of theoretical physics and chemistry for many
years. In nuclear physics, this approach is used for the description of
fission or fusion processes at excitations above the fission barrier nix76 ;
wada93 ; froebrich ; pomorski1996 ; karpov ; asano ; adeev2005 ; aritomo ;
mazurek ; our17 ; sierk ; scirep ; pasha ; kosenko . In these works the
Langevin equation was solved in 1-5 dimensions with macroscopic werwhe ; wall1
; runswiat  or microscopic hofrep ; hofbook ; ivahof  transport coefficients.
The approach describes quite well the mass distributions and kinetic energies
of fission fragments, the multiplicities of emitted neutrons, and other
observables of fission or fusion processes.

In the Langevin approach one solves the set of differential equations for the
time evolution of collective variables $q_{\mu}$ describing the shape of the
nuclear surface. In our works we used for the shape parametrization that of
the two-center shell model (TCSM) tcsm , see Fig.1. In this model the shape of
the axially symmetric surface is characterized by 5 deformation parameters
$q_{\mu}=z_{0}/R_{0},\delta_{1},\delta_{2},\alpha$ and $\epsilon$. Here
z${}_{0}/R_{0}$ refers to the distance between the centers of left and right
oscillator potentials, $R_{0}$ being the radius of spherical nucleus. The
parameters $\delta_{1}$ and $\delta_{2}$ describe the deformation of the right
and left parts of the nucleus. The fourth parameter $\alpha$ is the mass
asymmetry and the fifth parameter of TCSM shape parametrization $\epsilon$
regulates the neck radius.

In our older Langevin calculations aritomo ; our17  we used a three-
dimensional shape parametrization in which $\delta_{1}$ and $\delta_{2}$ were
assumed to be equal, $\delta_{1}=\delta_{2}$, and the neck parameter was kept
constant, $\epsilon=0.35$. The generalization from three to four dimensions
was carried out in 25 . In that work the deformation parameters $\delta_{1}$
and $\delta_{2}$ were considered as an independent dynamical variables. This
made it possible to include into the consideration shapes with very different
deformations of the left and right parts. Thus, one part could be nearly
spherical while the other is very elongated. This made it possible to
reproduce fermis  the rapid change of mass distributions of fission fragments
from mass-asymmetric (in 256Fm) to mass-symmetric (in 258Fm) fermis , the saw-
tooth structure of neutron multiplicity multy , and the decrease of the total
excitation energy of fission fragments with the increasing of the excitation
energy shimada .

Figure 1: The $z$-dependent part of the TCSM mean-field potential $V(z)$. The
neck parameter $\epsilon$ is defined as the ratio of the potential $V(z)$ at
$z=0$ to the value $V_{0}$ of oscillator potential at $z=0,\epsilon\equiv
V(z=0)/V_{0}$.

At the same time, there are some experimental results on nuclear fission that
are not explained theoretically up to now. In khs  the secondary-beam facility
of GSI Darmstadt was used to study the fission properties of 70 short-lived
radioactive nuclei. Relativistic secondary projectiles were produced by
fragmentation of a 1 A GeV 238U primary beam and identified in nuclear charge
and mass number. These reaction products were excited by electromagnetic
interactions, and fission from excitation energies around 11 MeV was induced.
The elemental yields and the total kinetic energies for a series of neutron-
deficient pre-actinides and actinides from 205At to 234U were determined. The
elemental yields after electromagnetic-induced fission, cover the transition
from symmetric fission at 221Ac to asymmetric fission at 234U. The longest
isotopic sequence, from 217Th to 229Th, was measured for thorium isotopes. In
the transitional region, around 227Th, triple-humped distributions appear,
demonstrating comparable weights for symmetric and asymmetric fission.

The qualitative transition from the mass-symmetric to mass-asymmetric mass
distributions for thorium isotopes was shown in the first paper on the random
walk model random  assuming the overdamped character of nuclear collective
motion. Later (after adding the energy dependence which has some effect at
E*=11 MeV) much better agreement of the calculated fission-fragment charge
yields with GSI data khs  was demonstrated in randrup2013 . The transition
from mass-symmetric to mass-asymmetric in sequence of thorium isotopes was
also found recently by the Polish-Chinese collaboration pomo2021  within the
Born-Oppenheimer approximation.

A more recent experimental study of nuclear fission along the thorium isotopic
sequence was carried out in chatil1 . In that article, the new data on fission
along the thorium isotopic sequence were reported, using the same reaction
mechanism as in khs , but with the new R3B/SOFIA experimental setup which was
conceived to identify the mass and the nuclear charge of the fissioning nuclei
and both fission fragments.

Figure 2: The fission fragment mass distributions for few thorium isotopes.
The experimental data from chatil1  are marked by circles, solid lines show
the 4D Langevin calculations with $\epsilon$=0.35, and the dashed line shows
the 4D Langevin calculation for 222Th with $\epsilon$=0.15.

The experimental data from chatil1  for few thorium isotopes are shown by
circles in Fig.2. Solid lines dhow the 4D Langevin calculations with
$\epsilon$=0.35. One can see that for 230Th the agreement of experimental and
theoretical results is quite reasonable. For lighter isotopes the experimental
mass symmetric peak becomes stronger and stronger, while in calculations the
mass asymmetric peak remains dominant.

As one can see from Fig.2, the mass distribution calculated with the four-
dimensional approach is mainly mass asymmetric. The mass symmetric peak is too
small. All attempts to make it larger varying the starting point or the
scission conditions were not successful. The only noticeable effect on the
height of central peak arose from a variation of the neck parameter
$\epsilon$. With smaller $\epsilon$ the mass symmetric peak became somewhat
larger, see the dash curve in the left part of Fig.2. That was a clear
indication that the parameter $\epsilon$ of TCSM shape parametrization should
not be fixed, but should be considered as an independent dynamical variable,
like the other four deformation parameters.

The results of the five-dimensional Langevin calculations for the sequence of
thorium isotopes are presented below in this preprint.

In Section II we present the main relations of the Langevin approach to the
description of nuclear fission process. In Section III we explain how the
collective potential energy is calculated. Section IV contains the description
of the used mass and friction tensors. The results of numerical 5D calculation
are given in Section V. Section VI contains short summary.

##  II The Langevin approach

The first-order differential equations (Langevin equations) for the time
dependence of the collective variables $q_{\mu}$ and the conjugate momenta
$p_{\mu}$ are:

$\displaystyle\frac{dq_{\mu}}{dt}$ $\displaystyle=$
$\displaystyle\left(m^{-1}\right)_{\mu\nu}p_{\nu},$ (1)
$\displaystyle\frac{dp_{\mu}}{dt}$ $\displaystyle=$
$\displaystyle-\frac{\partial F(q,T)}{\partial
q_{\mu}}-\frac{1}{2}\frac{\partial m^{-1}_{\nu\sigma}}{\partial
q_{\mu}}p_{\nu}p_{\sigma}-\gamma_{\mu\nu}m^{-1}_{\nu\sigma}p_{\sigma}$ (2)
$\displaystyle+$ $\displaystyle g_{\mu\nu}R_{\nu}(t),$

where the sums over the repeated indices are assumed. In Eq. 2 the $F(q,T)$ is
the temperature dependent free energy of the system, $\gamma_{\mu\nu}$ and
($m^{-1})_{\mu\nu}$ are the friction and inverse of mass tensors, and
$g_{\mu\nu}R_{\nu}$(t) is the random force.

The free energy $F(q,T)$ is calculated as the sum of the macroscopic (folded
Yukawa) energy and the temperature dependent shell correction $\delta F(q,T)$.
The shell corrections are calculated from the single-particle energies in the
deformed Woods-Saxon potential pash1 ; pash2  fitted to the mentioned above
TCSM shape parameterizations.

The collective inertia tensor $m_{\mu\nu}$ is calculated within the Werner-
Wheeler approximation werwhe  and for the friction tensor $\gamma_{\mu\nu}$ we
used the wall-and-window formula wall1 ; runswiat .

The random force g${}_{\mu\nu}R_{\nu}$(t) is the product of the normally
distributed white noise $R_{\nu}$(t), $\langle
R_{\mu}(t)R_{\nu}(t^{\prime})\rangle=2\delta_{\mu\nu}\delta(t-t^{\prime})$,
and the temperature-dependent strength factors $g_{\mu\nu}$. The factors gμν
are related to the temperature and friction tensor via the modified Einstein
relation,

$g_{\mu\sigma}g_{\sigma\nu}=T^{\ast}\gamma_{\mu\nu}\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,T^{\ast}=\frac{\hbar\omega_{0}}{2}\,\,\coth\,\,\frac{\hbar\omega_{0}}{2T}\,\,,\,\,\,\,$

where $T^{\ast}$ is the effective temperature pomhof ; hofkid . The parameter
$\omega_{0}$ is the local frequency of collective motion hofkid . The minimum
of $T^{\ast}$ is given by $\hbar\omega_{0}/2$. In the present calculations
$\omega_{0}$ was kept constant, $\omega_{0}$=2 MeV. At large excitations
$T^{\ast}$ is close to $T$.

The temperature $T$ in this context is related to the initial energy
$E^{\ast}_{(in)}$ and the local excitation energy $E^{\ast}$ by,

$E^{\ast}=E_{gs}+E^{\ast}_{(in)}-\frac{1}{2}m^{-1}_{\mu\nu}p_{\mu}p_{\nu}-V_{pot}(q,T=0)=aT^{2},$

where $V_{pot}$ is the potential energy and $a$ is the level density
parameter. More details are given in our earlier publications, see 14 ; 24 ;
25 ; 26 . Usually, the initial values of momenta $p_{\mu}$ are set to zero,
and calculations are started from the ground state deformation.

The calculations are continued until the trajectories reach the ”scission
point”. In our older publications the ”scission point” was defined as the
point in deformation space where the neck radius becomes zero. However, the
zero critical neck radius is not well justified. It was shown in the so called
optimal shapes model strut1963 ; pomiva2  that the nuclear liquid drop loses
stability with respect to elongation at an almost constant elongation
$z_{0}\approx$ 2.3 $R_{0}$ and rather thick neck, $r_{neck}^{crit}$, see
Fig.3. Thus, in all our calculations below we used the finite value of the
critical neck, $r_{neck}^{crit}$ = 1 fm.

Figure 3: The liquid-drop deformation energy of mass-symmetric optimal shapes
as function of the distance $R_{12}$ between the left and right parts of
nucleus, and the shapes at the scission point. The $x_{LD}$ is the fissility
parameter, the ratio of Coulomb and surface energy for spherical shape,
$x_{LD}\equiv E_{Coul}^{(sph}/E_{surf}^{(sph)}$.

##  III The 5D shape parametrization

The potential $V(z)$ in TCSM consists of the two Nilsson potentials with
centers at $z_{1}$ and $z_{2}$, smoothly joined between $z_{1}$ and $z_{2}$ by
a fourth order polynomial in $z$, see Fig.1. The deformation parameters
$\delta_{1}$ and $\delta_{2}$ define the curvature of the parts outside of
$z_{1}$ or $z_{2}$. In the three-dimensional parametrization $\delta_{1}$ and
$\delta_{2}$ are the same. The variation of $\delta_{1}$ at fixed $\delta_{2}$
formally change only the outer part of the potential. But, due to the
continuity conditions at $z_{1}$ and $z_{2}$, requirements of fixed volume and
mass asymmetry, the whole shape, even in the neck region, is modified by
variation of $\delta_{1}$ or $\delta_{2}$, see Fig.4. So, $\delta_{1}$ and
$\delta_{2}$ can be considered as the parameters of deformation of the entire
left and right parts of the shape.

Figure 4: The effect of variation of parameter $\delta_{2}$ on 4D TCSM shapes.

The effect of the variation of the fifth parameter, $\epsilon$, on the nuclear
shape is somewhat different. Formally, parameter $\epsilon$ controls the neck
radius, see Fig.5.

Figure 5: The effect of variation of parameter $\epsilon$ on 5D TCSM shapes.
Figure 6: The 5D TCSM shapes for $\delta_{1}=\delta_{2}$=0.2, $\alpha$=0 at
the scission point for few values of parameter $\epsilon$

However, since the yields and other quantities are calculated at the scission
point with a fixed neck radius, the variation of $\epsilon$ leads to the
incorporation of very elongated or very compact shapes, see Fig.6. Exactly
such shapes contribute to the super-long or super-short fission modes. So, the
inclusion of $\epsilon$ as an additional dynamical variables makes it possible
to describe more accurately the super-long or super-short fission modes.

##  IV The potential energy

As it was mentioned above, we calculate the potential energy within the
macroscopic-microscopic model,

$F(q)=E_{LDM}(q)+\delta F(q,T)$ (3)

The macroscopic part of energy $E_{LDM}(q)$ is calculated within the folded
Yukawa model yamaji1 ; yamaji2 .

The shell correction $\delta F(q,T)$ is calculated by the code of V.
Pashkevich pash1 ; pash2  for Strutinsky’s prescription struti ; brdapa  from
the energies of single-particle states in the deformed Woods-Saxon potential
fitted to the TCSM shapes 25 . At zero temperature the shell correction to the
free energy $\delta F(q,T=0)$ coincides with the shell correction to the
collective potential energy $\delta E(q)$,

$\delta E(q)=\sum_{n,p}\left(\delta E_{shell}^{(n,p)}(q)+\delta
E_{pair}^{(n,p)}(q)\right).$ (4)

The damping of the shell-plus-pairing correction $\delta F(q,T)$ with the
excitation energy was calculated by the method developed in shcot .

An example of the macroscopic deformation energy $E_{LDM}(q)$ is shown in
Fig.7. As one can see, the potential energy is rather flat in $\epsilon$
direction. Consequently, the dynamical trajectories fill all the available
space in $\epsilon$.

Figure 7: The dependence of macroscopic deformation energy of 222Th on the
elongation $z_{0}/R_{0}$ and neck parameter $\epsilon$ for fixed
$\delta_{1}=\delta_{2}=\alpha$=0.2.

The comparison of the total deformation energy of 222Th at the saddle and
above in the ($z_{0},\alpha$) plane is shown in Fig.8. In this plot the energy
was minimized with respect to $\delta_{1},\delta_{2}$ at fixed $\epsilon$ for
$\epsilon$=0.35 and $\epsilon$=0.15. For $\epsilon$=0.35 one clearly sees the
two asymmetric valleys leading to asymmetric fission. For $\epsilon$=0.15 the
mass asymmetric valley is less pronounced. So symmetric divisions may also
occur. These observations are in accord with the fission fragment mass
distribution shown in the left hand part of Fig.2.

Figure 8: The dependence of the potential energy of 222Th, minimized with
respect to $\delta_{1},\delta_{2}$ at fixed $\epsilon$, $\epsilon$=0.35 and
$\epsilon$=0.15, on the elongation $z_{0}/R_{0}$ and asymmetry $\alpha$.

##  V The tensors of friction and inertia

In the present work we use the macroscopic transport coefficients which are
often used for solving the Langevin equations. The macroscopic transport
coefficients depend only on the shape of the system and do not depend on the
excitation energy of the system.

The macroscopic mass tensor $M_{\mu\nu}^{WW}$ is usually defined in Werner-
Wheeler approximation werwhe ,

$M_{\mu\nu}^{WW}=\pi\rho_{{}_{0}}\int\rho^{2}\left[A_{\mu}A_{\nu}+\frac{\rho^{2}}{8}A_{\mu}^{\prime}A_{\nu}^{\prime}\right]dz,$
(5)

where $\rho(z)$ is the profile function of the axially symmetric shape and

$A_{\mu}(z;Q)=\frac{1}{\rho^{2}(z,Q)}\frac{\partial}{\partial
Q_{\mu}}\int_{z}^{z_{{}_{R}}}\rho^{2}(z^{\prime},Q)dz^{\prime}.$ (6)

The macroscopic friction tensor is defined by the so-called wall-and-window
formula wall1 ; runswiat . In the case of axial symmetric shapes, the wall
friction can be written as

$\gamma_{\mu\nu}^{\mathrm{wall}}=\pi\rho_{{}_{0}}v_{F}\int_{z_{{}_{L}}}^{z_{{}_{R}}}dz\,\frac{\partial\rho^{2}}{\partial
Q_{\mu}}\frac{\partial\rho^{2}}{\partial
Q_{\nu}}\left[4\rho^{2}+\left(\frac{\partial\rho^{2}}{\partial
z}\right)^{2}\right]^{-1/2},$ (7)

with $\rho_{0}=3mA/(4\pi R_{0}^{3})$ and $v_{F}$ being the Fermi velocity.

It was also suggested runswiat  that close to scission the wall friction
should be corrected by the ”window term”

$\gamma_{\mu\nu}^{window}=\frac{1}{2}\rho_{0}\bar{v}\left[\Delta\sigma\left(\frac{\partial
R_{12}}{\partial q_{\mu}}\frac{\partial R_{12}}{\partial
q_{\nu}}\right)+\frac{32}{9\Delta\sigma}\frac{\partial V_{L}}{\partial
q_{\mu}}\frac{\partial V_{L}}{\partial q_{\nu}}\right]\ ,$ (8)

where $R_{12}$ is the distance between the centers of mass of left and right
parts of nucleus and $\Delta\sigma$ is the area of the ”‘window”. Then, the
”wall friction” becomes the ”wall-and-window” friction tensor

$\gamma_{\mu\nu}^{w+w}=\gamma_{\mu\nu}^{wall}+\gamma_{\mu\nu}^{window}\ .$ (9)

One expects a smooth transition between the regime in which the wall formula
applies and the part of fission path where wall-and-window friction should be
used. For this Nix and Sierk nixsierk  proposed the phenomenological ansatz

$\gamma_{\mu\nu}^{total}=\sin^{2}(\pi\phi/2)\gamma_{\mu\nu}^{wall}+\cos^{2}(\pi\phi/2)\gamma_{\mu\nu}^{w+w},$
(10)

with $\phi=(r_{neck}/R_{min})^{2}$, where $R_{min}$ is the minimal semi-axes
of the two outer ellipsoidal shapes.

In the old calculations within the Langevin approach it turned that the wall-
and-window friction is to strong. To reproduce the width of fission fragments
mass distribution it was necessary to introduce the reduction factor
$k_{s}=0.27$. Such factor was proposed by Nix and Sierk nix1987  from the
analysis of widths of giant resonances, and we used this shape independent
factor in all our previous works.

The $z_{0}z_{0}$-components of the friction (7) and inertia (5) tensors for
222Th are shown in Fig.9 as function of the elongation $z_{0}/R_{0}$ and mass
asymmetry $\alpha$. The dependence of friction or inertia separately on the
deformation does not has much meaning. This dependence is a consequence of the
definition of the deformation parameters. More meaningful is the ratio of
friction to inertia, the so-called reduced friction coefficient,
$\beta_{\mu\nu}\equiv(\gamma M^{-1})_{\mu\nu}$, or the damping parameter
$\eta_{\mu\nu}\equiv\beta_{\mu\nu}/2\omega_{0}$, where $\omega_{0}$ is the
frequency of local collective vibrations. This is a dimensionless parameter
that defines whether the collective motion is under-damped, $\eta<<$1, or
over-damped $\eta>>$1\. Accordingly, one develops a different models for the
under-damped and over-damped motions. In the model by Kramers kramers  one can
find the expressions for the limits of high viscosity,$\Gamma_{HV}$, and low
viscosity, $\Gamma_{LV}$. These expressions are rather different

$\Gamma_{HV}=\frac{\hbar\omega_{0}}{2\pi}e^{-V_{b}/T}(\sqrt{1+\eta^{2}}-\eta)\,,\quad\Gamma_{LV}=\frac{\hbar\bar{\gamma}}{M}\frac{V_{b}}{T}e^{-V_{b}/T}.$
(11)

The high viscosity limit corresponds to large $\eta$, $\eta>>1$. In the low
viscosity limit $\eta<<1$. So, it is important to know whether the nuclear
fission process is under-damped or over-damped in order to develop the
meaningful models.

In our calculation we use the reduced value of wall and window friction with
the reduction factor $k_{s}$=0.27. The damping factor shown in Fig.10 includes
this factor, $\eta_{\mu\nu}=k_{s}(\gamma M^{-1})_{\mu\nu}/2\omega_{0}$.

Figure 9: The $z_{0}z_{0}$ components of friction (7) and inertia (5) tensors
for 222Th as function of the elongation $z_{0}/R_{0}$ and mass asymmetry
$\alpha$ at fixed $\delta_{1}=\delta_{2}$=0.2 and $\epsilon$=0.35.

The $z_{0}z_{0}$ and $\epsilon\epsilon$ components of damping tensor
$\eta_{\mu\nu}$ for 222Th are shown in Fig.10 for the deformations at the
saddle and above. As one can see, $\eta_{\mu\nu}$ is more stable with respect
to deformation as compared to $\gamma$ or $M$ alone. The value of
$\eta_{z_{0}z_{0}}$ varies in the limits from 2.6 to 0.35. The value of
$\eta_{\epsilon\epsilon}$ varies in the limits from 1.9 to 1.3. Thus, in the
present Langevin treatment with macroscopic transport coefficient the fission
process is neither under-damped nor over-damped. So, the application of the
high viscosity limit $\Gamma_{HV}$ (11) of Kramers’ formula to the nuclear
fission width may not be justified.

Figure 10: The damping parameters $\eta_{z_{0}z_{0}}$ and
$\eta_{\epsilon\epsilon}$ for 222Th as function of the elongation
$z_{0}/R_{0}$ and mass asymmetry $\alpha$ at fixed $\delta_{1}=\delta_{2}$=0.2
and $\epsilon$=0.35.

##  VI Numerical results

In this Section we present the results of the numerical solution of the five-
dimensional Langevin equations (1) with the potential energy and transport
coefficients specified above. In these calculations the damping of shell
correction with the excitation energy was described by the method developed in
shcot . The frequency of local collective vibration was assumed to be
constant, $\hbar\omega_{0}$=2 MeV. In principle, $\omega_{0}$ should depend on
the deformation. But, the account of this dependence makes the computation
time too time consuming.

For solving of Eqs. (1-2), besides the coefficients of equations, one should
fix the initial values and final conditions. The calculations for all thorium
isotopes were started from the same point close to the ground state
deformation, $\\{q_{\mu}\\}$=0.2, 0.2, 0.2, 0.0, 0.35 with zero collective
momenta $\\{p_{\mu}\\}=0$. The integration of equations continued until the
neck radius turned into $r_{neck}^{(crit)}$=1 fm. At this point the solutions
of the equations provide all the information on the shape, collective
velocities and excitation energy of the system. This information makes it
possible to calculate the moments of the density distribution, the mass
distribution of fission fragments, the prescission and total kinetic energies,
excitation energy of nucleus just before the scission.

Unfortunately, the 5D computations are too time consuming. The 5D
configuration space is much larger compared with 4D space. The trajectories
travel in 5D space much longer before they reach the scission point.

Another problem is the lack of the memory space. For the grid in
$z_{0},\delta_{1},\delta_{2},\alpha$ that we use usually, $41\times 31\times
31\times 40$ we can add only 5 points in $\epsilon$. Otherwise the memory of
computer is overfilled. For 10 points in $\epsilon$ we have to reduce twice
the grid in other deformation parameters. In present calculations we used,
$0.1\leq\epsilon\leq 0.5$ with $\Delta\epsilon$=0.1.

The comparison of calculated fission fragments mass distributions (solid
lines) for few thorium isotopes with the experimental data chatil1  (circles)
is shown in Fig.11. One can see that the agreement of the calculated and
experimental results is quite reasonable. The transition from the mass
asymmetric distribution in heavy thorium isotopes to symmetric in light
isotopes via the triple-humped distribution is well reproduced.

Figure 11: The comparison of calculated fission fragments mass distributions
(solid lines) for few thorium isotopes with the experimental data chatil1
(open circles).

The calculated mass distributions of fission fragments for the sequence of
thorium isotopes at the excitation energy $E^{*}$=10 MeV are shown in Fig.12.
Again, the calculated distributions agree qualitatively with the experimental
results of K.-H. Schmidt ${\it at~{}al}$ khs .

Figure 12: The calculated mass distributions of fission fragments for the
sequence of thorium isotopes at the excitation energy $E^{*}$=10 MeV.

##  VII Summary

The calculations within the 5-dimensional dynamical Langevin approach provide
much better agreement with the available experimental data as compared with 4D
calculations. In particular, the transition from the mass-symmetric to mass-
asymmetric fission via the triple-humped distribution in fission of thorium
isotopes is well reproduced.

The use of 5-dimensional Langevin calculations makes theoretical predictions
for the observables of fission process much more reliable.

Acknowledgments. The authors would like to express their gratitude to Profs.
K. Pomorski and J. Randrup for valuable comments and suggestions.

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# Static solutions to symplectic curvature flow in dimension four

Gavin Ball  University of Wisconsin-Madison, Department of Mathematics,
Madison, WI, USA<EMAIL_ADDRESS>https://www.gavincfball.com/

###### Abstract.

This article studies special solutions to symplectic curvature flow in
dimension four. Firstly, we derive a local normal form for static solutions in
terms of holomorphic data and use this normal form to show that every complete
static solution to symplectic curvature flow in dimension four is Kähler-
Einstein. Secondly, we perform an exterior differential systems analysis of
the soliton equation for symplectic curvature flow and use the Cartan-Kähler
theorem to prove a local existence and generality theorem for solitons.

###### Contents

  1. 1 Introduction
  2. 2 Structure equations
  3. 3 Static solutions in dimension four
  4. 4 Local existence of soliton solutions

##  1\. Introduction

An almost-Kähler manifold $(X,\Omega,J)$ consists of a even-dimensional
manifold $X$ endowed with a symplectic form $\Omega$ and a compatible almost
complex structure $J.$ Together, $\Omega$ and $J$ define a Riemannian metric
$g$ on $X.$ One perspective on almost-Kähler geometry is to fix a symplectic
form $\Omega$ on $X,$ choose a compatible $J,$ and think of $J$ and $g$ as
auxiliary tools used to study the symplectic geometry of $(X,\Omega)$. In this
direction, symplectic curvature flow is a degenerate parabolic evolution
equation for almost-Kähler structures introduced by Streets–Tian
[StreetsTian14] given by

(1.1) $\displaystyle\tfrac{\partial}{\partial t}\Omega$
$\displaystyle=-2\rho,$ $\displaystyle\tfrac{\partial}{\partial t}g$
$\displaystyle=-2\rho^{1,1}-2\,\mathrm{Ric}^{2,0+0,2},$

where $\rho$ is the Chern-Ricci form of $(\Omega,J)$ and $\mathrm{Ric}$ is the
Ricci tensor of $g.$ The flow (1.1) preserves the symplectic condition
$d\Omega=0$ and restricts to Kähler-Ricci flow in the case where $J$ is
integrable.

The idea behind symplectic curvature flow is to evolve an initial almost-
Kähler structure on $X$ towards some canonical structure. In their initial
paper, Streets & Tian prove short time existence for (1.1), but it is to be
expected that symplectic curvature flow will encounter singularities in
general. Analogy with other geometric flows, especially Ricci flow, suggests
that singularity formation should be modeled on _soliton_ solutions to (1.1).

######  Definition 1.1.

An almost-Kähler manifold $(X,\Omega,J,g)$ is a _symplectic curvature flow
soliton_ if there exists a constant $\lambda\in\mathbb{R}$ and a vector field
$V$ on $X$ such that

(1.2) $\displaystyle\lambda\Omega+\mathcal{L}_{V}\Omega$
$\displaystyle=-2\rho,$ $\displaystyle\lambda g+\mathcal{L}_{V}g$
$\displaystyle=-2\rho^{1,1}-2\,\mathrm{Ric}^{2,0+0,2}.$

If $\lambda>0,$ $=0,$ or $<0$, we say the soliton is _expanding, steady,_ or
_shrinking_ respectively. If the vector field $V$ vanishes identically, then
we say $(X,\Omega,J,g)$ is a _static solution_ to symplectic curvature flow.

The soliton solutions are precisely the almost-Kähler structures which evolve
by rescaling and diffeomorphisms along (1.1). The static solutions are the
structures which evolve purely by rescaling. For Ricci flow, the static
solutions are the Einstein metrics, and so static solutions to symplectic
curvature flow can be thought of as the almost-Kähler analogues of Einstein
metrics.

###  1.1. Results

In this paper, we study soliton and static solutions to symplectic curvature
flow in dimension four using the techniques of exterior differential systems
and the moving frame. These techniques are well-suited to the non-linear,
overdetermined PDE system (1.2).

Our first main result is Theorem 3.3, which gives a local normal form for
static solutions. We summarize it here as:

######  Theorem 1.2.

Every four-dimensional static solution to symplectic curvature flow is steady
($\lambda=0$). If $(X^{4},\Omega,J,g)$ is a static solution and $p\in X$ is a
point where the Nijenhuis tensor of $J$ is non-vanishing, then there is a
neighbourhood of $p$ with complex coordinates $z_{1},z_{2}$ and a holomorphic
function $h(z_{1})$ such that the almost-Käler structure on $X$ is given by
Equation (3.6).

The geometry of the local normal form (3.6) is constrained enough to preclude
the existence of non-trivial complete static solutions.

######  Corollary 1.3.

A complete static solution to symplectic curvature flow in dimension four is
Kähler-Einstein.

This corollary can be compared to results by Streets–Tian [StreetsTian14] and
Kelleher [Kelleher19] on static solutions. By contrast to their techniques,
our calculations are primarily local in nature. We also note that Pook
[pook2012homogeneous] has constructed compact static solutions to symplectic
curvature flow in dimensions $n(n+1).$

The key ingredient in the proof of Theorem 3.3 is an integrability condition
that must be satisfied by any static solution. The existence of this
integrability condition implies that the overdetermined PDE system describing
static solutions is not involutive in the sense of exterior differential
systems. It is then natural to ask if the more general soliton system is
involutive. Our second main result, Theorem 4.1, shows that the soliton system
is involutive. Hence, the Cartan-Kähler Theorem may be applied to prove the
existence of solutions. We summarize this result here as:

######  Theorem 1.4.

The overdetermined PDE system for four-dimensional symplectic curvature flow
solutions is involutive and symplectic curvature flow solitons exist locally.

In §4.1 we provide an explicit example of homogeneous symplectic curvature
flow soliton. Further examples have appeared in [LauretWill17] and explicit
solutions to symplectic curvature flow have been analyzed in [LauretSymp15,
pook2012homogeneous, FernandezCulma15].

##  2\. Structure equations

Let $X$ be a 4-manifold endowed with an almost Kähler structure
$\left(\Omega,J\right),$ that is to say $\Omega$ is a symplectic form on $X$
and $J$ is a $\Omega$-compatible almost complex structure on $X$. Together,
$\Omega$ and $J$ determine a Riemannian metric $g$ on $X.$

Let $\mathcal{P}$ denote the $\mathrm{U}(2)$-structure on $X$ determined by
the almost Kähler structure $\left(\Omega,J\right)$. Explicitly,
$\pi:\mathcal{P}\to X$ is the principal $\mathrm{U}(2)$-bundle defined by

$\mathcal{P}=\left\\{u:T_{p}X\to\mathbb{C}^{2}\mid\text{$u$ is a complex
linear isomorphism,}\>\>\>u^{*}\Omega_{\mathrm{Std}}=\Omega_{p}\right\\},$

where $\Omega_{\mathrm{Std}}$ denotes the standard Kähler form on
$\mathbb{C}^{2}$,

$\Omega_{\mathrm{Std}}=\tfrac{i}{2}\left(e^{1}\wedge\overline{e^{1}}+e^{2}\wedge\overline{e^{2}}\right).$

Let $\eta$ denote the $\mathbb{C}^{2}$-valued tautological 1-form on
$\mathcal{P},$ defined by $\eta(v)=u(\pi_{*}(v)),$ for $v\in
T_{u}\mathcal{P}.$ Denote the components of $\eta$ with respect to the
standard basis $e_{1},$ $e_{2}$ of $\mathbb{C}^{2}$ by $\eta_{1},\eta_{2}.$
The forms $\eta_{1},\eta_{2}$ are a basis for the semi-basic forms on
$\mathcal{P},$ and they encode the almost complex structure $J$ in the sense
that a 1-form $\theta$ on $X$ is a $(1,0)$-form if and only if the pullback
$\pi^{*}\theta$ lies in $\operatorname{span}(\eta_{1},\eta_{2}).$ We also have
that, on $\mathcal{P},$

$\displaystyle\Omega$
$\displaystyle=\tfrac{i}{2}\eta_{i}\wedge\overline{\eta_{i}},$ $\displaystyle
g$ $\displaystyle=\eta_{i}\cdot\overline{\eta_{i}},$

where $1\leq i\leq 2$ and the unitary summation convention is employed (as it
will be in the remainder of this article).

###  2.1. The first structure equation

On $\mathcal{P},$ Cartan’s first structure equation reads

(2.1)
$d\eta_{i}=-\kappa_{i\overline{j}}\wedge\eta_{j}-\xi_{ij}\wedge\overline{\eta_{j}},$

where $\kappa_{i\overline{j}}=-\overline{\kappa_{j\overline{i}}}$ and
$\xi_{ij}=-\xi_{ji}.$ Here, $\kappa$ is the $\mathfrak{u}(2)$-valued
connection form for the _Chern connection_ on $X.$ The
$\Lambda^{2}_{\mathbb{C}}$-valued 1-form $\xi_{ij}$ is $\pi$-semibasic, so
that there exist functions $A_{ij\overline{k}}$ and $N_{ijk}$ on $X$ such that

$\xi_{ij}=A_{ij\overline{k}}\eta_{k}+N_{ijk}\overline{\eta_{k}}.$

It is easily checked that the symplectic condition $d\Omega=0$ implies
$A_{ij\overline{k}}=0.$ Cartan’s first structure equation (2.1) may therefore
be rewritten as

(2.2)
$d\eta_{i}=-\kappa_{i\overline{j}}\wedge\eta_{j}+N_{ijk}\,\overline{\eta_{j}}\wedge\overline{\eta_{k}}.$

The tensor
$N=N_{ijk}\left(\overline{\eta_{i}\wedge\eta_{j}}\right)\otimes\overline{\eta_{k}}$
descends to $X$ to give a well-defined section of
$\Lambda^{0,2}\otimes\Lambda^{0,1}.$ In fact, $N$ is simply the _Nijenhuis
tensor_ of the almost complex structure $J.$

The structure equations (2.2) are valid in any even dimension. However, they
may be written in a simpler form in complex dimension 2, by exploiting the
fact that $\Lambda^{2}_{\mathbb{C}}$ is 1-dimensional, and this simplification
is worthwhile because it leads to a simplification of the second structure
equation. Let $\varepsilon_{ij}$ denote the totally skew-symmetric symbol with
$\varepsilon_{12}=1/2.$ Let $N_{i}$ denote the functions on $\mathcal{P}$
defined by

$N_{ijk}=\varepsilon_{ij}N_{k}.$

The first structure equation (2.2) may be rewritten as

(2.3)
$d\eta_{i}=-\kappa_{i\overline{j}}\wedge\eta_{j}+\varepsilon_{ij}N_{k}\,\overline{\eta_{j}}\wedge\overline{\eta_{k}}.$

###  2.2. The second structure equations

The identity $d^{2}=0$ applied to equation (2.3) implies the following
equations:

(2.4) $\displaystyle dN_{i}=$
$\displaystyle\overline{\varepsilon_{jk}}A_{ij}\eta_{k}+B\,\eta_{i}+\left(F_{ij}+\varepsilon_{ij}H\right)\overline{\eta_{j}}-\kappa_{i\overline{j}}N_{j}-\kappa_{j\overline{j}}N_{i},$
$\displaystyle d\kappa_{i\overline{j}}=$
$\displaystyle-\kappa_{i\overline{k}}\wedge\kappa_{k\overline{j}}+K_{i\bar{j}k\bar{l}}\overline{\eta_{k}}\wedge\eta_{l}+\left(R+N_{l}\overline{N_{l}}\right)\left(-\tfrac{1}{3}\eta_{i}\wedge\overline{\eta_{j}}-\tfrac{1}{3}\delta_{i\bar{j}}\eta_{k}\wedge\overline{\eta_{k}}\right)$
$\displaystyle+N_{i}\overline{N_{j}}\eta_{k}\wedge\overline{\eta_{k}}+Q_{i\bar{k}}\eta_{k}\wedge\overline{\eta_{j}}+Q_{k\bar{j}}\eta_{i}\wedge\overline{\eta_{k}}-\tfrac{1}{2}A_{ik}\overline{\eta_{j}\wedge\eta_{k}}+\tfrac{1}{2}\overline{A_{jk}}\eta_{i}\wedge\eta_{k}$
$\displaystyle+2B\,\varepsilon_{ik}\overline{\eta_{j}\wedge\eta_{k}}-2\overline{B}\overline{\varepsilon_{jk}}\eta_{i}\wedge\eta_{k},$

for functions $A_{ij},B,F_{ij},H,K_{i\bar{j}k\bar{l}},Q_{i\bar{j}},R$ on
$\mathcal{P}$ having the following symmetries:

$\displaystyle A_{ij}$ $\displaystyle=A_{ji},$ $\displaystyle
K_{i\bar{j}k\bar{l}}$ $\displaystyle=K_{k\bar{j}i\bar{l}}$ $\displaystyle
Q_{i\bar{j}}$ $\displaystyle=-\overline{Q_{j\bar{i}}}$ $\displaystyle F_{ij}$
$\displaystyle=F_{ji},$ $\displaystyle K_{i\bar{j}k\bar{l}}$
$\displaystyle=K_{i\bar{l}k\bar{j}}$ $\displaystyle Q_{i\bar{i}}$
$\displaystyle=0$ $\displaystyle K_{i\bar{i}k\bar{l}}$ $\displaystyle=0$
$\displaystyle K_{i\bar{j}k\bar{l}}$
$\displaystyle=\overline{K_{j\bar{i}l\bar{k}}}$

Each of these functions takes values in an irreducible
$\mathrm{U}(2)$-representation, and the right-hand-side of the equation for
$d\kappa$ in (2.4) represents the irreducible decomposition of the curvature
of the Chern connection of $X.$ The equations (2.4) are called the second
structure equations of the almost Kähler structure $(g,\Omega).$ By a
classical theorem of Cartan, the functions
$A_{ij},B,F_{ij},K_{i\bar{j}k\bar{l}},Q_{i\bar{j}},R$ form a complete set of
second-order invariants of $(g,\Omega).$

The second equation of (2.4) gives the curvature of the Chern connection on
$X.$ The Chern-Ricci form is the trace of this curvature:

$d\kappa_{i\overline{i}}=-R\eta_{i}\wedge\overline{\eta_{i}}-4i\operatorname{Im}\left(\overline{B}\eta_{1}\wedge\eta_{2}\right)-2Q_{i\overline{j}}\overline{\eta_{i}}\wedge\eta_{j}.$

####  2.2.1. The curvature of $g$

The Riemann curvature tensor of a 4-dimensional manifold is a section of a
vector bundle modeled on the $\mathrm{SO}(4)$ representation

$\mathrm{Sym}^{2}\left(\Lambda^{2}\mathbb{R}^{4}\right)\cong\mathbb{R}\oplus\mathrm{Sym}^{2}_{0}\mathbb{R}^{4}\oplus\mathrm{Sym}^{2}_{0}\left(\Lambda^{2}_{+}\mathbb{R}^{4}\right)\oplus\mathrm{Sym}^{2}_{0}\left(\Lambda^{2}_{-}\mathbb{R}^{4}\right).$

Corresponding to each irreducible component, we have the scalar curvature $r$,
the traceless Ricci curvature $\mathrm{Ric}^{0}$, and the self-dual $W^{+}$
and anti-self dual $W^{-}$ Weyl tensors respectively.

The second structure equations (2.4) can be compared with the structure
equations of a Riemannian manifold to write the components of the Riemann
curvature tensor of $g$ in terms of the first and second order invariants of
the almost Kähler structure. The result of this calculation is:

$\displaystyle\mathrm{Scal}(g)$ $\displaystyle=-8N_{i}\overline{N_{i}}-8R,$
$\displaystyle\mathrm{Ric}(g)$
$\displaystyle=A_{ij}\overline{\eta_{i}}\cdot\overline{\eta_{j}}+\overline{A_{ij}}\eta_{i}\cdot\eta_{j}+\left(Q_{i\bar{j}}+N_{i}\overline{N_{j}}\right)\overline{\eta_{i}}\cdot\eta_{j}-\left(\tfrac{1}{2}R+N_{k}\overline{N_{k}}\right)\eta_{i}\cdot\overline{\eta_{i}},$
$\displaystyle W^{+}(g)$
$\displaystyle=\left(4N_{i}\overline{N_{i}}-4R\right)\Omega^{2}-8iB\,\Omega\cdot\left(\eta_{1}\wedge\eta_{2}\right)+8i\overline{B}\,\Omega\cdot\left(\overline{\eta_{1}\wedge\eta_{2}}\right)$
$\displaystyle+2H\,\left(\eta_{1}\wedge\eta_{2}\right)^{2}+2\overline{H}\left(\overline{\eta_{1}\wedge\eta_{2}}\right)^{2}-4N_{i}\overline{N_{i}}\left(\eta_{1}\wedge\eta_{2}\right)\cdot\left(\overline{\eta_{1}\wedge\eta_{2}}\right),$
$\displaystyle W^{-}(g)$
$\displaystyle=-2K_{i\bar{j}k\bar{l}}\left(\overline{\eta}_{i}\wedge\eta_{j}\right)\cdot\left(\overline{\eta_{k}}\wedge\eta_{l}\right),$

where here we are viewing the Ricci tensor as a symmetric 2-tensor, and the
self-dual and anti-self-dual Weyl tensors as sections of
$\mathrm{Sym}^{2}_{0}\left(\Lambda^{2}_{\pm}TX\right).$

####  2.2.2. Symplectic curvature flow

The right hand side of the symplectic curvature flow equation (1.1) may also
be written in terms of second and first order invariants of
$\left(g,\Omega\right).$ We have

(2.5) $\displaystyle\frac{\partial}{\partial t}\Omega$
$\displaystyle=4R\,\Omega+4iQ_{i\bar{j}}\,\overline{\eta_{i}}\wedge\eta_{j}-8\,\operatorname{Im}\left(\overline{B}{\eta_{1}\wedge\eta_{2}}\right),$
$\displaystyle\frac{\partial}{\partial t}g$
$\displaystyle=4R\,g-8Q_{i\overline{j}}\overline{\eta_{i}}\cdot\eta_{j}-4\operatorname{Re}(A_{ij}\overline{\eta_{i}}\cdot\overline{\eta_{j}}),$
$\displaystyle\frac{\partial}{\partial t}J$
$\displaystyle=\left(8\operatorname{Im}\left(\overline{B}{\eta_{1}\wedge\eta_{2}}\right)-4\operatorname{Re}\left(A_{ij}\overline{\eta_{i}}\wedge\overline{\eta_{j}}\right)\right)g^{-1},$

##  3\. Static solutions in dimension four

We now study static solutions to symplectic curvature flow in dimension 4.
These are solutions which evolve strictly by rescaling under the flow, so for
such a solution we must have

(3.1) $\displaystyle\frac{\partial}{\partial t}\Omega$
$\displaystyle=\lambda\Omega,$ $\displaystyle\frac{\partial}{\partial t}g$
$\displaystyle=\lambda g,$ $\displaystyle\frac{\partial}{\partial t}J$
$\displaystyle=0.$

Comparing with (2.5), we see that the static almost Kähler structures are
characterized by the following conditions on their second order invariants:

(3.2) $\displaystyle
R=\frac{\lambda}{4},\>\>\>B=0,\>\>\>Q_{i\bar{j}}=0,\>\>\>A_{ij}=0.$

Therefore, the first and second structures equations for a static solution
reduce to

$\displaystyle d\eta_{i}$
$\displaystyle=-\kappa_{i\overline{j}}\wedge\eta_{j}+\varepsilon_{ij}N_{k}\,\overline{\eta_{j}}\wedge\overline{\eta_{k}},$
$\displaystyle dN_{i}=$
$\displaystyle\left(F_{ij}+\varepsilon_{ij}H\right)\overline{\eta_{j}}-\kappa_{i\overline{j}}N_{j}-\kappa_{j\overline{j}}N_{i},$
$\displaystyle d\kappa_{i\overline{j}}=$
$\displaystyle-\kappa_{i\overline{k}}\wedge\kappa_{k\overline{j}}+K_{i\bar{j}k\bar{l}}\overline{\eta_{k}}\wedge\eta_{l}+\left(\tfrac{\lambda}{4}+N_{l}\overline{N_{l}}\right)\left(-\tfrac{1}{3}\eta_{i}\wedge\overline{\eta_{j}}-\tfrac{1}{3}\delta_{i\bar{j}}\eta_{k}\wedge\overline{\eta_{k}}\right)$
$\displaystyle+N_{i}\overline{N_{j}}\eta_{k}\wedge\overline{\eta_{k}}+Q_{i\bar{k}}\eta_{k}\wedge\overline{\eta_{j}}.$

Differentiating the second equation, we find

$0=d^{2}N_{i}\wedge\overline{\eta_{1}\wedge\eta_{2}}=\left(F_{ij}\overline{N_{j}}+\frac{1}{2}\varepsilon_{ij}H\overline{N_{j}}\right)\eta_{1}\wedge\eta_{2}\wedge\overline{\eta_{1}\wedge\eta_{2}}.$

The equations

(3.3)
$F_{ij}\overline{N_{j}}+\frac{1}{2}\varepsilon_{ij}H\overline{N_{j}}=0,\>\>\>i=1,2$

must therefore be satisfied by any static almost Kähler structure. These
equations give a restriction on the second-order invariants of a static
solution which is not an algebraic consequence of (3.2), so the static
equations (3.2) are not involutive in the sense of exterior differential
systems.

The equation (3.3) may be simplified by adapting coframes. Suppose $N$ is non-
zero at a point $p\in X$. Say a coframe $u\in\mathcal{P}_{p}$ is $N$-adapted
if $N_{2}=0$ at $u\in\mathcal{P}.$ The group $\mathrm{U}(2)$ acts transitively
on $\Lambda^{0,2}\otimes\Lambda^{0,1}$ with stabilizer
$\mathrm{U}(1)\times\mathrm{U}(1),$ so the collection of all $N$-adapted
coframes is a $\mathrm{U}(1)\times\mathrm{U}(1)$-bundle over the locus in $X$
where $N\neq 0.$ For simplicity, we shall assume from now on $N$ is nowhere
vanishing on $X$ (otherwise restrict to the open dense locus where $N\neq 0$).
Denote the bundle of $N$-adapted coframes by $\mathcal{P^{\prime}}\to X.$

Equations (3.3) imply that, on $\mathcal{P}^{\prime},$

$F_{11}=0,\>\>\>F_{12}=\tfrac{1}{2}H.$

Differentiating the identity $N_{2}=0,$ we find that on $\mathcal{P}^{\prime}$

$0=F_{22}\overline{\eta_{2}}-N_{1}\kappa_{2\bar{1}}.$

Define $G=N_{1}F_{22},$ so we have $\kappa_{2\bar{1}}=G\,\overline{\eta_{2}}.$
Let us also define $\mathbb{R}$-valued 1-forms $\alpha=-i\kappa_{1\bar{1}}$
and $\beta=-i\kappa_{2\bar{2}}.$ The forms $\alpha$ and $\beta$ together
define a connection on the $\mathrm{U}(1)\times\mathrm{U}(1)$-bundle
$\mathcal{P}^{\prime}.$ The structure equations restricted to
$\mathcal{P}^{\prime}$ now read

$\displaystyle d\eta_{1}=$ $\displaystyle
i\alpha\wedge\eta_{1}+N_{1}\overline{\eta_{1}\wedge\eta_{2}},$ $\displaystyle
d\eta_{2}=$ $\displaystyle
i\beta\wedge\eta_{2}+G\eta_{1}\wedge\overline{\eta_{2}},$ $\displaystyle
dN_{1}=$ $\displaystyle H\overline{\eta_{2}}+2iN_{1}\alpha+iN_{1}\beta,$
$\displaystyle d\alpha=$ $\displaystyle i\left(\tfrac{1}{3}\lvert
N_{1}\rvert^{2}-\tfrac{1}{6}\lambda-
K_{1\bar{1}1\bar{1}}\right)\eta_{1}\wedge\overline{\eta_{1}}+iK_{2\bar{2}2\bar{1}}\eta_{1}\wedge\overline{\eta_{2}}-iK_{1\bar{1}1\bar{2}}\eta_{2}\wedge\overline{\eta_{1}}$
$\displaystyle+i\left(\tfrac{2}{3}\lvert
N_{1}\rvert^{2}-\tfrac{1}{12}\lambda+\lvert
G\rvert^{2}+K_{1\bar{1}1\bar{1}}\right)\eta_{2}\wedge\overline{\eta_{2}},$
$\displaystyle d\beta=$ $\displaystyle i\left(-\tfrac{1}{3}\lvert
N_{1}\rvert^{2}-\tfrac{1}{12}\lambda+K_{1\bar{1}1\bar{1}}\right)\eta_{1}\wedge\overline{\eta_{1}}-iK_{2\bar{2}2\bar{1}}\eta_{1}\wedge\overline{\eta_{2}}+iK_{1\bar{1}1\bar{2}}\eta_{2}\wedge\overline{\eta_{1}}$
$\displaystyle+i\left(-\tfrac{2}{3}\lvert
N_{1}\rvert^{2}-\tfrac{1}{6}\lambda-\lvert
G\rvert^{2}-K_{1\bar{1}1\bar{1}}\right)\eta_{2}\wedge\overline{\eta_{2}}$

The identities $d^{2}\eta_{i}=0$ imply

$\displaystyle K_{1\bar{1}1\bar{2}}$
$\displaystyle=0,\>\>\>\>K_{2\bar{2}2\bar{1}}=0,$ $\displaystyle
K_{1\bar{1}1\bar{1}}$ $\displaystyle=\tfrac{1}{3}\lvert
N_{1}\rvert^{2}+\tfrac{1}{12}\lambda-\lvert G\rvert^{2},$ $\displaystyle d\,G$
$\displaystyle=G_{\bar{1}}\eta_{1}+G_{2}\overline{\eta_{2}}-iG\alpha+2iG\beta,$

for some $\mathbb{C}$-valued functions $G_{\bar{1}}$ and $G_{2}$ on
$\mathcal{P}^{\prime}.$ The identity $d^{2}\alpha=0$ implies $GG_{2}=0,$ so
$G_{2}$ must vanish identically on $\mathcal{P}^{\prime}$ (since the vanishing
of $G$ implies the vanishing of $G_{2}$).

Next, differentiating the identity $\kappa_{2\bar{1}}=G\,\overline{\eta_{2}}$
implies

$K_{1\bar{2}1\bar{2}}=-\overline{G_{\bar{1}}},\>\>\>\>\>K_{2\bar{1}2\bar{1}}=-G_{\bar{1}}.$

The identity $d^{2}G=0$ yields

$N_{1}G_{\bar{1}}\eta_{1}\wedge\overline{\eta_{1}\wedge\eta_{2}}+G\left(3\lvert
N_{1}\rvert^{2}-\tfrac{1}{2}\lambda\right)\overline{\eta_{2}}\wedge\eta_{1}\wedge\eta_{2}=0,$

leading to two possibilities:

  1. (1)

$G$ vanishes identically on $\mathcal{P}$; or

  2. (2)

$\lvert N_{1}\rvert^{2}=\tfrac{1}{6}\lambda$ and $G_{\bar{1}}=0$ on
$\mathcal{P}^{\prime}.$

Let us suppose case (2) holds. Differentiating $\lvert
N_{1}\rvert^{2}=\tfrac{1}{6}\lambda$ implies $H=0$ on $\mathcal{P},$ so we
have

$dN_{1}=2iN_{1}\alpha+iN_{1}\beta.$

The identity $d^{2}N_{1}$ then implies $\lvert N_{1}\rvert^{2}=0,$
contradicting our assumption that $N$ is non-zero. Therefore case (2) is not
possible and we must instead have case (1) holding: $G=0$ on
$\mathcal{P}^{\prime}.$

Finally, the identity $d^{2}N_{1}\wedge\overline{\eta_{2}}=0$ implies that
$\lambda N_{1}=0,$ so we must have $\lambda=0$ for any non-trivial static
solution to symplectic curvature flow.

At this stage, the structure equations have simplified to

(3.4) $\displaystyle d\eta_{1}=$ $\displaystyle
i\alpha\wedge\eta_{1}+N_{1}\overline{\eta_{1}\wedge\eta_{2}},$ $\displaystyle
d\eta_{2}=$ $\displaystyle i\beta\wedge\eta_{2},$ $\displaystyle dN_{1}=$
$\displaystyle H\overline{\eta_{2}}+2iN_{1}\alpha+iN_{1}\beta,$ $\displaystyle
d\alpha=$ $\displaystyle i\lvert
N_{1}\rvert^{2}\eta_{2}\wedge\overline{\eta_{2}}$ $\displaystyle d\beta=$
$\displaystyle-i\lvert N_{1}\rvert^{2}\eta_{2}\wedge\overline{\eta_{2}}.$

It is easy to check that equations (3.4) represent an involutive prescribed
coframing problem in the sense of [BryEDSNotes]. The primary invariants are
the real and imaginary parts of the function $N_{1},$ the free derivatives are
the real and imaginary parts of $H$ and the tableau of free derivatives is
equivalent to the standard Cauchy-Riemann tableau

$\begin{bmatrix}x&y\\\ -y&x\end{bmatrix}.$

Therefore, Theorem 3 of [BryEDSNotes] may be applied to show that non-trivial
static solutions to symplectic curvature flow exist locally and depend on two
functions of one variable.

We summarize the conclusions of this section in the following theorem.

######  Theorem 3.1.

A non-trivial static solution to symplectic curvature flow in dimension 4 must
have $\lambda=0.$ Real analytic non-trivial static solutions exist locally and
depend on two functions of one variable in the sense of exterior differential
systems.

######  Remark 3.2.

We will show in the following subsection that any static solution must be real
analytic.

####  3.0.1. Integrating the structure equations

In this subsection we shall integrate the structure equations (3.4) to obtain
a local normal form for four-dimensional static solutions to symplectic
curvature flow. This local normal form will allow us to draw conclusions on
the global structure of such solutions.

We begin by noting that $d\left(\alpha+\beta\right)=0,$ so we have locally
$\beta=-\alpha+dg$ for some function $g.$ The function $g$ may be integrated
away in the structure equations, so we may assume we are working on the
$\mathrm{U}(1)$-subbundle $\mathcal{P}^{\prime\prime}\to X$ where
$\beta=-\alpha.$

The distribution $\eta_{1}=0$ descends from $\mathcal{P}^{\prime\prime}$ to
$X$ to give a well-defined real codimension-two distribution on $X.$ Each leaf
of the resulting foliation has a metric given by the restriction of
$\left\lvert N_{1}\right\rvert^{2}\overline{\eta_{2}}\cdot\eta_{2},$ and the
equations

$\displaystyle d\left(N_{1}\overline{\eta_{2}}\right)$
$\displaystyle=2i\alpha\wedge\left(N_{1}\overline{\eta_{2}}\right),$
$\displaystyle d\alpha$
$\displaystyle=-i\left(N_{1}\overline{\eta_{2}}\right)\wedge\overline{\left(N_{1}\overline{\eta_{2}}\right)}$

imply that this metric has constant curvature $-4.$ It follows that if
$U\subset X$ is simply-connected, then there exists a $\mathbb{C}$-valued
function $z_{1}$ and an $\mathbb{R}$-valued function $s$ on
$\mathcal{P^{\prime\prime}}|_{U}$ such that

$N_{1}\,\overline{\eta_{2}}=\frac{e^{is}\,dz_{1}}{1-\left\lvert
z_{1}\right\rvert^{2}},\>\>\>\>\alpha=-\frac{i}{2}\frac{\overline{z_{1}}dz_{1}-z_{1}d\overline{z_{1}}}{1-\left\lvert
z_{1}\right\rvert^{2}}+ds.$

We restrict to the locus $s=0.$ This amounts to restricting the
$\mathrm{U}(1)$-structure $\mathcal{P}^{\prime\prime}$ to an
$\left\\{e\right\\}$-structure over $U$.

The equation

$d\left(N_{1}dz_{1}\right)=\frac{1}{2}\frac{N_{1}z_{z}}{1-\lvert
z_{1}\rvert^{2}}d\overline{z_{1}}\wedge dz_{1}$

implies

(3.5) $N_{1}=\frac{h(z_{1})}{\sqrt{1-\lvert z_{1}\rvert^{2}}}$

for some holomorphic function $h$ of a single complex variable. The
$d\eta_{1}$ equation in (3.4) implies

$d\begin{bmatrix}\eta_{1}\\\
\overline{\eta_{1}}\end{bmatrix}=-\frac{1}{2}\frac{1}{1-\lvert
z_{1}\rvert^{2}}\begin{bmatrix}\overline{z_{1}}dz_{1}-z_{1}d\overline{z_{1}}&2\,dz_{1}\\\
2\,d\overline{z_{1}}&z_{1}d\overline{z_{1}}-\overline{z_{1}}dz_{1}\end{bmatrix}\wedge\begin{bmatrix}\eta_{1}\\\
\overline{\eta_{1}}\end{bmatrix}.$

It follows that

$d\left(\frac{\eta_{1}+z_{1}\overline{\eta_{1}}}{\sqrt{1-\lvert
z_{1}\rvert^{2}}}\right)=0,$

so there exists a coordinate $z_{2}$ on $U$ with

$\eta_{1}=\frac{dz_{2}-z_{1}d\overline{z_{2}}}{\sqrt{1-\lvert
z_{1}\rvert^{2}}}.$

The coordinate $z_{2}$ is unique up to addition of a constant.

We have now proven the first part of the following theorem. The second part
follows by reversing the steps above.

######  Theorem 3.3.

Let $\left(X,\Omega,g\right)$ be a non-trivial 4-dimensional static solution
to symplectic curvature flow and suppose $p\in X$ is a point where the
Nijenhuis tensor is non-vanishing. Then there is a local neighbourhood of $p$
with complex coordinates $z_{1}$ and $z_{2}$ and a holomorphic function
$h(z_{1})$ such that the symplectic form $\Omega$ and metric $g$ on $X$ are
given by

(3.6) $\displaystyle\Omega$ $\displaystyle=\frac{i}{2}\left(\frac{dz_{1}\wedge
d\overline{z_{1}}}{\lvert h(z_{1})\rvert^{2}\left(1-\lvert
z_{1}\rvert^{2}\right)}+dz_{2}\wedge d\overline{z_{2}}\right),$ $\displaystyle
g$ $\displaystyle=\frac{dz_{1}\cdot d\overline{z_{1}}}{\lvert
h(z_{1})\rvert^{2}\left(1-\lvert z_{1}\rvert^{2}\right)}+\frac{1+\lvert
z_{1}\rvert^{2}}{1-\lvert z_{1}\rvert^{2}}dz_{2}\cdot
d\overline{z_{2}}-\operatorname{Re}\left(\frac{\overline{z_{1}}dz_{2}^{2}}{1-\lvert
z_{1}\rvert^{2}}\right).$

Conversely, let $h(z_{1})$ be a meromorphic function on the unit disk and let
$\Sigma\subset\mathbb{C}$ be the subset of the unit disk where $h(z_{1})$ has
no zeros or poles. Then equation (3.6) defines a static solution to symplectic
curvature flow on the 4-manifold $\Sigma\times\mathbb{C}.$

The local normal form of Theorem 3.3 is strong enough to derive global
consequences.

######  Corollary 3.4.

There are no non-trivial complete static solutions to symplectic curvature
flow in dimension 4.

###### Proof.

Let $L$ be a leaf of the foliation $\eta_{1}=0$ on $X$ and let $\widehat{L}$
denote the universal cover of $L.$ The function
$z_{1}:\widehat{L}\to\mathbb{C}$ and holomorphic function $h(z_{1})$ exist
globally on $\widehat{L},$ and $z_{1}$ identifies $\widehat{L}$ with the unit
disk $\mathbb{D}\subset\mathbb{C}.$ Formula (3.6) implies that the boundary
$\left\lvert z_{1}\right\rvert=1$ is at finite distance, and the function
$\left\lvert N_{1}\right\rvert^{2}$ blows up there. Therefore the metric
induced on $\widehat{L}$ is incomplete, hence the metric $g|_{L}$ is
incomplete, hence $g$ is incomplete. ∎

##  4\. Local existence of soliton solutions

In the previous section we showed that the static system was non-involutive in
the sense of exterior differential systems. This non-involutivity lead to
several compatibility conditions which placed severe restrictions on the local
geometry of a static solution. By contrast, the soliton equation is involutive
as shall be explained in this section. The upshot is a local existence and
uniqueness theorem for soliton solutions.

The soliton equations are

(4.1) $\displaystyle\frac{\partial}{\partial t}\Omega$
$\displaystyle=\lambda\Omega+\mathcal{L}_{V}\Omega,$
$\displaystyle\frac{\partial}{\partial t}g$ $\displaystyle=\lambda
g+\mathcal{L}_{V}g,$

where $\lambda\in\mathbb{R}$ is a constant and $V$ is a vector field on $X.$

Let $V$ be a vector field on an almost-Kähler 4-manifold $X,$ thought of as a
$\mathrm{U}(2)$-equivariant map $\mathcal{B}\to\mathbb{C}^{2}$ and let $V_{i}$
be the components of this map with respect to the standard basis of
$\mathbb{C}^{2}.$ There exist functions $U,$ $S_{ij},$ $W_{i\bar{j}},$ and $Y$
with symmetries $W_{i\bar{i}}=0,$ $S_{ij}=S_{ji}$ such that, on $\mathcal{B},$

(4.2)
$dV_{i}+\kappa_{i\overline{j}}V_{j}=\left(W_{i\bar{j}}+Y\delta_{i\bar{j}}\right)\eta_{j}+\left(S_{ij}+U\epsilon_{ij}\right)\overline{\eta_{j}}.$

These functions appear in the Lie derivatives of $\Omega$ and $g$ with respect
to $V$:

$\displaystyle\mathcal{L}_{V}\Omega$
$\displaystyle=-\tfrac{i}{2}\left(W_{i\bar{j}}+\overline{W_{j\bar{i}}}\right)\overline{\eta_{i}}\wedge\eta_{j}+2\,\operatorname{Re}\left(U\right)\Omega+\operatorname{Im}\left(\left(Y-V_{i}\overline{N_{i}}\right)\eta_{1}\wedge\eta_{2}\right)$
$\displaystyle\mathcal{L}_{V}g$
$\displaystyle=2\,\operatorname{Re}\left(\overline{S_{ij}}\eta_{i}\cdot\eta_{j}\right)+4\operatorname{Re}\left(\overline{\epsilon_{kj}}V_{k}\overline{N}_{i}\eta_{i}\cdot\eta_{j}\right)+\left(W_{i\bar{j}}+\overline{W}_{j\bar{i}}\right)+2\,\operatorname{Re}(U)g\overline{\eta}_{i}\cdot\eta_{j}.$

Comparing with the symplectic curvature flow equations (2.5), we see that the
second-order invariants of a soliton solution satisfy

(4.3) $\displaystyle Q_{i\bar{j}}$
$\displaystyle=-\tfrac{1}{4}\operatorname{Re}(W_{i\bar{j}}),$ $\displaystyle
R$ $\displaystyle=\tfrac{1}{2}\operatorname{Re}(Y)+\tfrac{1}{4}\lambda,$
$\displaystyle A_{ij}$
$\displaystyle=-\tfrac{1}{2}S_{ij}+\tfrac{1}{2}\varepsilon_{ik}\overline{V_{k}}N_{j}+\tfrac{1}{2}\varepsilon_{jk}\overline{V_{k}}N_{i},$
$\displaystyle B$
$\displaystyle=\tfrac{1}{8}U+\tfrac{1}{8}\overline{V_{i}}N_{i}.$

Conversely, if $(X,\Omega,J)$ is an almost-Kähler 4-manifold with a vector
field $V$ so that equations (2.3), (2.4), (4.2), (4.3) are satisfied on the
$\mathrm{U}(2)$-bundle $\mathcal{B},$ then $(X,\Omega,J)$ is a soliton for
symplectic curvature flow.

######  Theorem 4.1.

Solitons for the symplectic curvature flow exist locally and depend on 10
functions of 3 variables in the sense of exterior differential systems.

###### Proof.

We first reduce the problem on constructing symplectic curvature flow solitons
to a prescribed coframing problem in the style of [BryEDSNotes].

By the above paragraph, the $\mathrm{U}(2)$-coframe bundle $\mathcal{B}\to X$
of a symplectic curvature soliton carries 1-forms $\eta$ and $\kappa$ and
functions $N_{i},$ $V_{i},$ $W_{i\bar{j}}$, $Y$, $S_{ij},$ $U,$ $F_{ij}$ and
$K_{i\bar{j}k\bar{l}}$ satisfying equations (2.3), (2.4), (4.2), (4.3).
Conversely, if $M$ is an 8-manifold together with 1-forms $\eta_{i}$ and
$\kappa_{i\bar{j}}$ and functions $N_{i},$ $V_{i},$ $W_{i\bar{j}}$, $Y$,
$S_{ij},$ $U,$ $F_{ij}$ and $K_{i\bar{j}k\bar{l}}$ satisfying equations (2.3),
(2.4), (4.2), (4.3), then an argument similar to Theorem 1 of
[BryantBochKah01] implies that (after possibly shrinking $M$) the
4-dimensional leaf space $X$ of the integrable plane field $\eta=0$ carries an
almost-Kähler structure which is a soliton for the symplectic curvature flow
and $M$ may be identified an an open set in the $\mathrm{U}(2)$-bundle
$\mathcal{B}\to X$ associated to this almost-Kähler structure.

The prescribed coframing problem (2.3), (2.4), (4.2), (4.3) is written in a
form where it is natural to attempt to apply Theorem 3 of [BryEDSNotes]. The
‘primary invariants’ are the functions $N_{i},$ $V_{i},$
$\operatorname{Re}(Y),$ $\operatorname{Re}(W_{i\bar{j}}),$ $U,$ $S_{ij},$ and
$K_{i\bar{j}k\bar{l}}$ while the derived invariants consist of the covariant
derivatives of these functions with respect to $\kappa_{i\bar{j}},$ taking the
identity $d^{2}\eta=0$ into account. However, the Cartan’s existence theorem
cannot be applied directly because the tableau of free derivatives is not
involutive. One may compute that it has Cartan characters $(24,22,13,1),$
while the dimension of the prolongation is $106,$ so Cartan’s test fails.
Nevertheless, the system does become involutive after one prolongation. We
omit the details here due to length, but after prolongation the tableau of
free derivatives has Cartan characters $(56,31,10,0)$ and the dimension of the
prolongation is $148=(56)+2(31)+3(10)+4(0),$ so Cartan’s test is passed and
the system is involutive. Existence and generality in the real analytic
category then follows from Theorem 3 of [BryEDSNotes]. ∎

######  Remark 4.2 (Real analyticity).

Streets–Tian [StreetsTian14] prove that symplectic curvature flow is parabolic
modulo diffeomorphism. Similarly, it can be shown that the soliton system is
elliptic modulo diffeomorphism. Therefore, symplectic curvature flow solitons
which are $C^{1,\alpha}$ in harmonic coordinates for some $\alpha>0$ must be
real analytic in those coordinates.

######  Remark 4.3 (The gradient case).

The equation $dV^{\flat}=0$ implies

$\displaystyle U$ $\displaystyle=\overline{V_{i}}N_{i},$
$\displaystyle\operatorname{Im}(Y)$ $\displaystyle=0,$
$\displaystyle\operatorname{Im}(W_{i\bar{j}})$ $\displaystyle=0.$

Adjoining these equations to (2.3), (2.4), (4.2), (4.3) gives a prescribed
coframing problem whose local solutions correspond to gradient solitons.
However, in contrast to the general case, this system is not involutive even
after prolongation because the equation $d^{2}Y=0$ yields a restriction on the
3-jet of a solution. In the language of exterior differential systems, this
problem has _intrinsic torsion_. Unfortunately, the restriction on the 3-jet
is more algebraically complicated than the equations encountered in §3 and the
existence and local generality of gradient solitons remains unknown.

######  Remark 4.4 (Comparison to Laplacian flow).

Symplectic curvature flow has a formal similarity to the Laplacian flow of
closed $\mathrm{G}_{2}$-structures in that both are flows of geometric
structures with torsion defined by closed differential forms. This formal
similarity extends to the local existence theory for static and soliton
solutions. The static solutions to Laplacian flow are the _eigenforms_ ,
analyzed in [BallQuad23] where it was shown that the relevant EDS is not
involutive. By contrast, the general Laplace soliton system is well-behaved.
In a recent paper Bryant [bryant2023generality] has shown that the EDS
describing Laplace solitons is involutive.

###  4.1. An example

Let $\mathrm{G}=\mathrm{SL}_{2}\mathbb{R}\ltimes\mathbb{R}^{2}$ denote the
group of volume preserving affine transformations of $\mathbb{R}^{2}$ and
write the left-invariant Maurer-Cartan form $\mu$ of $\mathrm{G}$ as

$\mu=\begin{bmatrix}0&0&0\\\ \alpha_{1}&\alpha_{3}&\beta-\alpha_{4}\\\
\alpha_{2}&-\beta-\alpha_{4}&-\alpha_{3}\end{bmatrix}.$

Let $\mathrm{S}^{1}$ denote the circle subgroup of $\mathrm{G}$ generated by
the action of the vector field dual to $\beta.$ For each pair of non-zero
numbers $(a,b)\in\mathbb{R}^{2}$ the 2-form $\Omega_{a,b}$ and metric
$g_{a,b}$ defined by

$\displaystyle\Omega_{a,b}$
$\displaystyle=a^{2}\,\alpha_{1}\wedge\alpha_{2}+b^{2}\,\alpha_{3}\wedge\alpha_{4},$
$\displaystyle g_{a,b}$
$\displaystyle=a^{2}\left(\alpha_{1}^{2}+\alpha_{2}^{2}\right)+b^{2}\left(\alpha_{3}^{2}+\alpha_{4}^{2}\right),$

descend to the 4-dimensional quotient $X=\mathrm{G}/\mathrm{S}^{1}$ to define
an almost Kähler structure on $X.$ The $U(2)$-invariants of this structure may
be computed via the Maurer-Cartan equation $d\mu=-\mu\wedge\mu.$ For this
structure, we find that the right hand side of the symplectic curvature flow
equations are given by

$\displaystyle
4R\,\Omega+4iQ_{i\bar{j}}\,\overline{\eta_{i}}\wedge\eta_{j}-8\,\operatorname{Im}\left(\overline{B}{\eta_{1}\wedge\eta_{2}}\right)$
$\displaystyle=4\,\alpha_{3}\wedge\alpha_{4},$ $\displaystyle
4R\,g-8Q_{i\overline{j}}\overline{\eta_{i}}\cdot\eta_{j}-4\operatorname{Re}(A_{ij}\overline{\eta_{i}}\cdot\overline{\eta_{j}})$
$\displaystyle=4\left(\alpha_{3}^{2}+\alpha_{4}^{2}\right).$

Therefore, the 1-parameter family of almost-Kähler structures on $X$ defined
by

$\displaystyle\Omega(t)$
$\displaystyle=a^{2}\,\alpha_{1}\wedge\alpha_{2}+\sqrt{4t+b^{4}}\,\alpha_{3}\wedge\alpha_{4},$
$\displaystyle g(t)$
$\displaystyle=a^{2}\left(\alpha_{1}^{2}+\alpha_{2}^{2}\right)+\sqrt{4t+b^{4}}\left(\alpha_{3}^{2}+\alpha_{4}^{2}\right).$

gives a solution to symplectic curvature flow with initial condition
$\Omega(0)=\Omega_{a,b},$ $g(0)=g_{a,b}.$ For any $a,$ $a^{\prime}$ the
structures $(\Omega_{a,b},g_{a,b})$ and
$(\Omega_{a^{\prime},b},g_{a^{\prime},b})$ are diffeomorphism equivalent, so
in fact each $(\Omega(t),g(t))$ defines a soliton solution to symplectic
curvature flow.

######  Remark 4.5.

The manifold $X$ defined above is non-compact and $\mathrm{G}$ does not admit
a uniform lattice, so it has no compact quotients. However, $\mathrm{G}$ does
admit cocompact lattices $\Gamma$ (for example
$\Gamma=\mathrm{SL}_{2}\mathbb{Z}\ltimes\mathbb{Z}^{2}$) and these give
quotients $\Gamma\backslash X$ with finite volume. The 1-parameter family of
almost-Kähler structures $(\Omega(t),g(t))$ descends to each $\Gamma\backslash
X$ to give a solution of symplectic curvature flow, however the diffeomorphism
between the structures $(\Omega_{a,b},g_{a,b})$ and
$(\Omega_{a^{\prime},b},g_{a^{\prime},b})$ does not descend to
$\Gamma\backslash X,$ so $(\Omega(t),g(t))$ does not give a soliton solution
on $\Gamma\backslash X.$

## References