# A direct unified wave-particle method for simulating non-equilibrium flows

Junzhe Caoa<EMAIL_ADDRESS>Sha Liuab<EMAIL_ADDRESS>Sirui
Yanga<EMAIL_ADDRESS>Chengwen Zhongab<EMAIL_ADDRESS>aSchool
of Aeronautics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072,
China
bInstitute of Extreme Mechanics, Northwestern Polytechnical University, Xi’an,
Shaanxi 710072, China

###### Abstract

In this work, the Navier-Stokes (NS) solver is combined with the Direct
simulation Monte Carlo (DSMC) solver in a direct way, under the wave-particle
formulation [J. Comput. Phys. 401, 108977 (2020)]. Different from the
classical domain decomposition method with buffer zone for overlap, in the
proposed direct unified wave-particle (DUWP) method, the NS solver is coupled
with DSMC solver on the level of algorithm. Automatically, in the rarefied
flow regime, the DSMC solver leads the simulation, while the NS solver leads
the continuum flow simulation. Thus advantages of accuracy and efficiency are
both taken. At internal flow regimes, like the transition flow regime, the
method is accurate as well because a kind of mesoscopic modeling is proposed
in this work, which gives the DUWP method the multi-scale property.
Specifically, as to the collision process, at $t<\tau$, it is supposed that
only single collision happens, and the collision term of DSMC is just used. At
$t>\tau$, it is derived that $1-\tau/\Delta t$ of particles should experience
multiple collisions, which will be absorbed into the wave part and calculated
by the NS solver. Then the DSMC and NS solver can be coupled in a direct and
simple way, bringing about multi-scale property. Different from the original
wave-particle method, in the proposed DUWP method, the wave-particle
formulation is no more restricted by the Boltzmann-BGK type model and the
enormous research findings of DSMC and NS solvers can be utilized into much
more complicated flows, like the thermochemical non-equilibrium flow. In this
work, one-dimensional cases in monatomic argon gas are preliminarily tested,
such as shock structures and Sod shock tubes.

###### keywords:

Multi-scale flow; Wave-particle formulation; Unified numerical scheme

††journal: ???

##  1 Introduction

With the development of hypersonic vehicle, spacecraft and micro-
electromechanical system, the multi-scale non-equilibrium flow receives more
attention and it is a challenging topic to develop corresponding multi-scale
numerical methods, which can accurately simulate flows from the continuum flow
regime to the free molecular flow regime. In recent years, both the stochastic
particle method [1, 2] and discrete velocity method (DVM) [3, 4, 5, 6, 7, 8]
are successfully developed. Nevertheless, one of the most important problems
faced in these numerical methods is the efficiency problem. Though a lot of
methods are proposed for acceleration or reducing memory consumption [9, 10,
11, 12, 13], there is still need for further improvement. The root cause of
efficiency problem is that when the non-equilibrium feature becomes stronger,
there are rather more freedom degrees to be described. For example, the
dimensionality curse of DVM makes the discrete velocity space quiet expensive
for simulating hypersonic flows.

When the flow is rarefied enough, the Monte Carlo method is very efficiency to
describe the velocity distribution function, which makes stochastic particle
methods suitable for simulating rarefied flows, and the most famous method of
them is the Direct simulation Monte Carlo (DSMC) method[14]. It has been
developed to maturity and used to simulate complex flows like thermochemical
non-equilibrium flows in flow mechanism studies[15, 16] and engineering
applications[17, 18]. However, the efficiency of stochastic particle methods
reduces rapidly when the flow becomes near-continuum or low-speed. Generally,
the number of particles to be numerically simulated should be prohibitively
large when using stochastic particle methods to simulate near-continuum flows
and low-speed flows, and the time step in DSMC is seriously restricted in the
near-continuum flow regime. On the other hand, macroscopic methods for
simulating continuum flows are adequately developed as well, based on Euler
equations and Navier-Stokes (NS) equations[19, 20, 21]. These methods are
efficient and accurate in the continuum flow regime, but invalid on rarefied
flows. As a result, domain decomposition methods are developed to implement
the two categories methods (macroscopic methods and stochastic particle
methods, mainly the DSMC method) in their corresponding suitable domains, and
set a buffet zone for overlap[22, 23, 24, 25, 26]. These methods achieve
success. However, the accuracy is not sufficient in transitional flows and
buffer zones of unsteady multi-scale flows.

Recently, a unified gas-kinetic wave-particle (UGKWP) method is proposed [27,
28] on the framework of unified gas-kinetic scheme (UGKS)[29]. It is a multi-
scale multi-efficiency preserving (MMP) method, and is rapidly developed [30,
31, 32]. Benefitting from the analytical solution of Boltzmann-BGK type model,
the stochastic particles are coupled with macroscopic methods (wave part, like
the gas-kinetic scheme (GKS)[33]) in the level of algorithm, which is
different from the empirically artificial divide of different methods on the
mesh in domain decomposition methods. Because the UGKWP method is designed
fully in accordance with the Boltzmann-BGK type model, it is accurate enough
in all flow regimes, naturally better than domain decomposition methods,
especially when simulating complex multi-scale flows, so the UGKWP achieves
success on the study of complex multi-scale flow mechanism[34, 35]. However,
aiming at more complicated multi-physical field multi-scale flows, for
example, thermochemical non-equilibrium flows, it is a hard task to extend the
Boltzmann-BGK type model. As to monatomic gas mixtures, the key of modeling is
the recovery of three coefficients, affecting the diffusion, viscosity and
heat flux. In the Andries-Aoki-Perthame (AAP) model[36, 37] and Ellipsoidal
Statistical-BGK(ES-BGK) type model[38, 39], there are a maximum of two correct
transport coefficients. In the model proposed in Ref.[40, 41], the three
transport coefficients are recovered, but it is to be improved by
incorporating the correct Soret and Dufour coefficients. As to polyatomic gas
mixtures[42, 43] or even reactive mixtures[44, 45, 46], much more relaxation
rates of macroscopic variable need to be recovered, like the relaxation rates
of internal energy (rotational and vibrational energy). Some work has been
done to deal with these problems. In Ref.[47], a direct relaxation (DR)
process is proposed to decrease the difficulty in building the Boltzmann-BGK
type model. The only parameters used in the DR are relaxation rates of
macroscopic variable. In Ref.[31], the multi-scale mechanism of UGKWP method
is simplified into the quantified model-competition (QMC) mechanism. Though
the Boltzmann-BGK type model or DR is still needed, the algorithm is much
simplified with acceptable loss of accuracy. In Ref.[48], instead of the hard
work on improving the Boltzmann-BGK type model, the relaxation time $\tau$ of
each particle is modified. And the early-rising problem of shock wave
simulation is solved. The direct modeling theory[49] implied in it is
desirable to be developed.

When the hypersonic vehicle or spacecraft flies at high Mach number and high
attitude, the rarefication effect is strongly associated with thermochemical
non-equilibrium effect, and it is of vital importance to develop a physical
model or numerical method which can simulate this kind of flow efficiently and
accurately. The wave-particle formulation is perfect but existing methods are
all restricted by the Boltzmann-BGK type model. In this work, the stochastic
unified method for multi-scale flows is reflected, and a methods called direct
unified wave-particle (DUWP) method is proposed to get rid of the restriction.
The procedure of DUWP method can be listed as: particle transfer part,
macroscopic flux transfer part, particle collision part, and particle
absorption or separation part. In the particle transfer part, particles are
freely transported without collision. The macroscopic flux is directly
calculated by NS solvers. In the particle collision part, the collision term
of DSMC is just used. And in the particle absorption or separation part, a
simple criteria is set for sampling. As a consequence, the DSMC and NS solver
are coupled in a direct and simple way, bringing about multi-scale property.
And the enormous research findings of DSMC and CFD can be utilized.

The remainder of this paper is organized as follows: the basis of gas kinetic
theory is introduced in Sec.2. Sec.3 is the construction of DUWP method. The
numerical test cases are conducted in Sec.4. The conclusions are in Sec.5.

##  2 The gas kinetic theory

In the gas-kinetic theory, the physical system is described by the
distribution function $f\left(\bm{x},\bm{\xi},\bm{\eta},t\right)$ which means
the probability density of particles that arrive at position $\mathbf{x}$ at
time $t$ with velocity $\bm{\xi}$ and internal energy represented by the
equivalent velocity $\bm{\eta}$. As to the equilibrium state, the distribution
function is totally determined by macroscopic values, known as Maxwellian
distribution function:

$f=g={\left({\frac{1}{{2\pi
RT}}}\right)^{\frac{{D+3}}{2}}}\exp\left({-\frac{{{\mathbf{c}}\cdot{\mathbf{c}}+{\bm{\eta}}\cdot{\bm{\eta}}}}{{2RT}}}\right),$
(1)

where $R$ is specific gas constant, $T$ is temperature, $D$ is the degree of
freedom about the rotational and the vibrational motion of molecules,
$\mathbf{c}=\bm{\xi}-\mathbf{U}$ is the peculiar velocity (the macroscopic
velocity $\mathbf{U}$ is also known as the mean velocity of all particles).
The macroscopic conservative variables
$\bm{\Psi}=\left(\rho,\rho\mathbf{U},\rho E\right)$(density, momentum and
energy) can be directly obtained from the distribution function $f$ by the
following relation: ($\bm{\varphi}=\left({1,\bm{\xi},{1\over
2}\left({\bm{\xi}\cdot\bm{\xi}+\bm{\eta}\cdot\bm{\eta}}\right)}\right)$)

$\bm{\Psi}=\int_{R^{3}}\rho\bm{\varphi}fd\bm{\xi}\int_{R^{D}}d\bm{\eta}.$ (2)

The integral is called the “moment” in the gas-kinetic theory. The stress
$\mathbf{S}$ and heat flux $\mathbf{Q}$ can also be obtained by the following
moments:

$\displaystyle{\mathbf{S}}=\int_{R^{3}}\rho\mathbf{cc}(f-g)d\bm{\xi}\int_{R^{D}}d\bm{\eta},$
(3) $\displaystyle{\mathbf{Q}}=\int_{R^{3}}\rho\mathbf{c}\left[{{1\over
2}m\left({\mathbf{c}\cdot\mathbf{c}+\bm{\eta}\cdot\bm{\eta}}\right)f}\right]d\bm{\xi}\int_{R^{D}}d\bm{\eta}.$

##  3 A direct unified wave-particle method

###  3.1 Modeling of collision source term

The collision source term is described as Fig.1. The coordinate is the
evolution of time, divided into three parts, and $\tau=\frac{\mu}{p}$ is the
molecular mean collision time ($\mu$ is the viscosity coefficient and $p$ is
pressure). At $\left[{0,\tau}\right]$, it is supposed that only single
collision happens. Microscopic mechanism leads this time interval and the DSMC
method is used. At $\left[{n\tau,\infty}\right]$ ($n$ is large enough), the
macroscopic mechanism leads the flow and the NS solver is used. At
$\left[{\tau,n\tau}\right]$, it is supposed that multiple collision happens,
and the mesoscopic model should be conducted.

Fig. 1:  Modeling of collision source term.

The BGK model equation is taken for qualitative guide. Separating the
transport term out, the BGK model equation becomes:

$\frac{\partial f}{\partial t}=\frac{g-f}{\tau}.$ (4)

It is generally known that the Prantl number of BGK model is fixed to $1$,
which means its relaxation rate of heat flux is always wrong, and the Shakhov-
BGK model is a popular improved model, as:

$\displaystyle\frac{\partial f}{\partial t}=\frac{g^{S}-f}{\tau},$ (5)
$\displaystyle
g^{S}=g\left[{1+(1-\rm{Pr})\frac{\mathbf{c}\cdot\mathbf{Q}}{5pRT}\left({\frac{|c|^{2}+|\eta|^{2}}{RT}-5}\right)}\right].$

However, as discussed in the introduction, as to thermochemical non-
equilibrium flows, much more relaxation rates should be matched, thus the
model will be prohibitively complicated or even unknown. In the DUWP method,
the model is written as:

$\frac{\partial f}{\partial t}=\frac{\mathscr{G}-f}{\tau}.$ (6)

It is assumed that $\mathscr{G}$ exists but it is not needed to get its
specific expression, because the NS solver will be directly used to calculate
this part. Its implicit differential solution is:

$f(t)=\frac{\tau}{\tau+t}f(0)+\frac{t}{\tau+t}\mathscr{G}.$ (7)

From Eq.7, it can be derived that at $\left[{\tau,\Delta t}\right]$, $P$ of
particles should be absorbed into the wave part ($\mathscr{G}$). $\Delta t$ is
the time step. The expression of $P$ is:

$P=\left\\{\begin{array}[]{lr}0,&\Delta t\leq\tau,\\\
\frac{t}{\tau+t}|_{t=\Delta t-\tau}=\frac{\Delta t-\tau}{\Delta
t-\tau+\tau}=1-\frac{\tau}{\Delta t},&\Delta t>\tau.\end{array}\right.$ (8)

###  3.2 Multi-scale flux

An important experience in designing multi-scale numerical methods in the FVM
framework, like the UGKS and UGKWP method, is that the flux must be multi-
scale. The core of multi-scale flux is utilizing the integral solution of
model equation. Taking the BGK-type model as an example:

$\frac{\partial f}{\partial t}+\bm{\xi}\cdot\frac{\partial
f}{\partial\mathbf{x}}=\frac{\mathscr{G}-f}{\tau}.$ (9)

The integral solution of Eq.9 is:

$f\left({\mathbf{0},t}\right)=\frac{1}{\tau}\int^{t}_{0}\mathscr{G}\left[{-\bm{\xi}\left({t-\tilde{t}}\right),\tilde{t}}\right]e^{\frac{\tilde{t}-t}{\tau}}d\tilde{t}+e^{-t/\tau}f\left({-\bm{\xi}t,0}\right),$
(10)

and its flux can be calculated as: ($\mathbf{n}$ denotes the unit normal
vector of interface, and $S$ is the area of interface. $\Delta t$ is the
length of time step.)

$\displaystyle\mathbf{F}=\frac{1}{\Delta t}\int_{0}^{\Delta
t}\int_{R^{D}}\int_{R^{3}}\rho\bm{\varphi}f\left({\mathbf{0},t}\right)\bm{\xi}\cdot\mathbf{n}Sd\bm{\xi}d\bm{\eta}dt$
(11) $\displaystyle=$ $\displaystyle\frac{1}{\Delta t}\int_{0}^{\Delta
t}\int_{R^{D}}\int_{R^{3}}\rho\bm{\varphi}\left\\{{\frac{1}{\tau}\int^{t}_{0}\mathscr{G}\left[{-\bm{\xi}\left({t-\tilde{t}}\right),\tilde{t}}\right]e^{\frac{\tilde{t}-t}{\tau}}d\tilde{t}}+e^{-t/\tau}f\left({-\bm{\xi}t,0}\right)\right\\}\bm{\xi}\cdot\mathbf{n}Sd\bm{\xi}d\bm{\eta}dt.$

Substitute second-order Taylor expansion of
$\mathscr{G}\left[{-\bm{\xi}\left({t-\tilde{t}}\right),\tilde{t}}\right]$,

$\mathscr{G}\left({\mathbf{x},t}\right)=\mathscr{G}|_{0}+\frac{\partial\mathscr{G}}{\partial\mathbf{x}}|_{0}\cdot\mathbf{x}+\frac{\partial\mathscr{G}}{\partial
t}|_{0}t,$ (12)

into Eq.10, it can be derived into:

$f\left({\mathbf{0},t}\right)=\frac{1}{\tau}\int^{t}_{0}\left[{\mathscr{G}|_{0}+\frac{\partial\mathscr{G}}{\partial\mathbf{x}}|_{0}\cdot\left({-\bm{\xi}}\right)\left({t-\tilde{t}}\right)+\frac{\partial\mathscr{G}}{\partial
t}|_{0}\tilde{t}}\right]e^{\frac{\tilde{t}-t}{\tau}}d\tilde{t}+e^{-t/\tau}f\left({-\bm{\xi}t,0}\right).$
(13)

Substituting Eq.13 into Eq.11, it is derived as:

$\displaystyle\mathbf{F}=\frac{1}{\Delta
t}\int_{R^{D}}\int_{R^{3}}\rho\bm{\varphi}\left({\delta_{1}\mathscr{G}|_{0}+\delta_{2}\frac{\partial\mathscr{G}}{\partial\mathbf{x}}|_{0}\cdot\bm{\xi}+\delta_{3}\frac{\partial\mathscr{G}}{\partial
t}|_{0}}\right)\bm{\xi}\cdot\mathbf{n}Sd\bm{\xi}d\bm{\eta}$ (14)
$\displaystyle+$ $\displaystyle\frac{1}{\Delta t}\int_{0}^{\Delta
t}\int_{R^{D}}\int_{R^{3}}\rho\bm{\varphi}e^{-t/\tau}f\left({-\bm{\xi}t,0}\right)\bm{\xi}\cdot\mathbf{n}Sd\bm{\xi}d\bm{\eta}dt,$

where,

$\displaystyle\delta_{1}$ $\displaystyle=\Delta t-\tau\left({1-e^{-\Delta
t/\tau}}\right),$ (15) $\displaystyle\delta_{2}$
$\displaystyle=2\tau^{2}\left({1-e^{-\Delta t/\tau}}\right)-\tau\Delta
t-\tau\Delta te^{-\Delta t/\tau},$ $\displaystyle\delta_{3}$
$\displaystyle=\frac{\Delta t^{2}}{2}-\tau\Delta t+\tau^{2}\left({1-e^{-\Delta
t/\tau}}\right).$

Eq.14 is the core of success of UGKS and UGKWP method. The former term denotes
the macroscopic affect on the flux. Only macroscopic variables and their
gradients are needed to calculate it. The latter term denotes the free
transport part. In the UGKWP method, it is described by particles. As
Ref.[31], the former term is further rewritten as:

$\displaystyle\frac{1}{\Delta
t}\int_{R^{D}}\int_{R^{3}}\rho\bm{\varphi}\left({\delta_{1}\mathscr{G}|_{0}+\delta_{2}\frac{\partial\mathscr{G}}{\partial\mathbf{x}}|_{0}\cdot\bm{\xi}+\delta_{3}\frac{\partial\mathscr{G}}{\partial
t}|_{0}}\right)\bm{\xi}\cdot\mathbf{n}Sd\bm{\xi}d\bm{\eta}$ (16)
$\displaystyle=$ $\displaystyle\frac{1}{\Delta
t}\int_{R^{D}}\int_{R^{3}}\rho\bm{\varphi}\left\\{{\delta_{1}\left[{\mathscr{G}|_{0}-\varepsilon\tau\left({\frac{\partial\mathscr{G}}{\partial\mathbf{x}}|_{0}\cdot\bm{\xi}+\frac{\partial\mathscr{G}}{\partial
t}|_{0}}\right)}\right]+\left[{\frac{\Delta
t^{2}}{2}-\tau^{2}+\left({\tau^{2}+\tau\Delta t}\right)e^{-\Delta
t/\tau}}\right]\frac{\partial\mathscr{G}}{\partial
t}|_{0}}\right\\}\bm{\xi}\cdot\mathbf{n}Sd\bm{\xi}d\bm{\eta}.$

If the high order temporal term $\left[{\frac{\Delta
t^{2}}{2}-\tau^{2}+\left({\tau^{2}+\tau\Delta t}\right)e^{-\Delta
t/\tau}}\right]\frac{\partial\mathscr{G}}{\partial t}|_{0}$ is omitted, the
formula becomes,

$\frac{1}{\Delta
t}\int_{R^{D}}\int_{R^{3}}\rho\bm{\varphi}\left\\{{\delta_{1}\left[{\mathscr{G}|_{0}-\varepsilon\tau\left({\frac{\partial\mathscr{G}}{\partial\mathbf{x}}|_{0}\cdot\bm{\xi}+\frac{\partial\mathscr{G}}{\partial
t}|_{0}}\right)}\right]}\right\\}\bm{\xi}\cdot\mathbf{n}Sd\bm{\xi}d\bm{\eta}.$
(17)

It is found that the distribution function here is in line with the Chapmann-
Enskog expansion, which is consistent with the N-S equations. In the view of
BGK model or Shakhov model, the complicated integration in Eq.14 can be
replaced by simple N-S solver as Ref.[31]. Furthermore, extending the flow to
multi-component flow or even thermochemical non-equilibrium flow, this term
can also be calculated by corresponding N-S solver, while the BGK-type model
equation for such kind of complicated flow is unavailable at this moment. The
parameter,

$\varepsilon=\frac{-2\left({1-e^{-\Delta t/\tau}}\right)+\frac{\Delta
t}{\tau}\left({1+e^{-\Delta t/\tau}}\right)}{\frac{\Delta
t}{\tau}-\left({1-e^{-\Delta t/\tau}}\right)},$ (18)

is named as non-equilibrium correction factor for N-S solver’s viscosity. When
$\Delta t\gg\tau$, $\varepsilon\rightarrow 1$. As a consequence, in the
process of calculating multi-scale flux, the multi-scale model is not relied
on. Only N-S solver for calculating the former term in Eq.14 is needed (as
Eq.17), and free-transport particles for calculating the latter term in Eq.14
is needed.

In the DUWP method, the implicit differential method is used instead of
integral formula with $exp()$ function. To match this, as Ref.[31], the time
integration in Eq.11 is not used. Instead, the solution at $t=\Delta t$ is
used, and it is derived that:

$\displaystyle\mathbf{F}=\int_{R^{D}}\int_{R^{3}}\rho\bm{\varphi}f\left({\mathbf{0},\Delta
t}\right)\bm{\xi}\cdot\mathbf{n}Sd\bm{\xi}d\bm{\eta}$ (19) $\displaystyle=$
$\displaystyle\int_{R^{D}}\int_{R^{3}}\rho\bm{\varphi}\left({1-e^{-\frac{\Delta
t}{\tau}}}\right)\left[{\mathscr{G}|_{0}-\varepsilon_{1}\tau\left({\frac{\partial\mathscr{G}}{\partial\mathbf{x}}|_{0}\cdot\bm{\xi}+\frac{\partial\mathscr{G}}{\partial
t}|_{0}}\right)}\right]\bm{\xi}\cdot\mathbf{n}Sd\bm{\xi}d\bm{\eta},$

where,

$\varepsilon_{1}=1-\frac{\Delta t}{\tau}\frac{e^{-\frac{\Delta
t}{\tau}}}{1-e^{-\frac{\Delta t}{\tau}}}.$ (20)

If a second-order approximation is used to $e^{-\frac{\Delta
t}{\tau}}=\frac{1}{1+\frac{\Delta t}{\tau}+\frac{\Delta t^{2}}{2\tau^{2}}}$, a
more simple formula can be derived from $\varepsilon_{1}$:
$\varepsilon_{12}=\frac{\Delta t}{2\tau+\Delta t}$. The most simple parameter
is that $\varepsilon_{0}=1$. The profile of different kinds of $\varepsilon$
is shown in Fig.2.

Fig. 2:  Profile of $\varepsilon$.

###  3.3 Algorithm of DUWP

The algorithm of DUWP is summed as follows:

Step (1)

Initial state. Give an initial state with macroscopic variables and
microscopic particles, and,

$\bm{\Psi}_{t}=\bm{\Psi}_{w}+\bm{\Psi}_{p},$ (21)

where subscript “t” denotes the total conservative macroscopic value in this
cell, “w” denotes the conservative macroscopic value recorded by the
macroscopic part (wave part), and “p” denotes the conservative macroscopic
value counted by the stochastic particle part.

Step (2)

Collision. According to Eq.8, $P$ of particles will be absorbed into wave part
(deleted) and $1-P$ of wave part will be converted into particles as:

$\displaystyle\xi_{1}$ $\displaystyle=U_{t,1}+\sqrt{2RT_{t}}cos(2\pi
rand)\sqrt{-ln(rand)},$ (22) $\displaystyle\xi_{2}$
$\displaystyle=U_{t,2}+\sqrt{2RT_{t}}cos(2\pi rand)\sqrt{-ln(rand)},$
$\displaystyle\xi_{3}$ $\displaystyle=U_{t,3}+\sqrt{2RT_{t}}cos(2\pi
rand)\sqrt{-ln(rand)},$ $\displaystyle\xi_{j}$
$\displaystyle=\sqrt{2RT_{t}}cos(2\pi rand)\sqrt{-ln(rand)}.$

Indeed, when sampling particles from the wave part, an assumption is
implicated that $\mathscr{G}\approx g$. Then, the collision term of DSMC
method is used to the particles part, which will be introduced in Appendix A.

Step (3)

Flux. Particles are marked and transported as:

$\displaystyle m^{n+1}=$ $\displaystyle m^{n},$ (23)
$\displaystyle\bm{\xi}^{n+1}=$ $\displaystyle\bm{\xi}^{n},$
$\displaystyle\mathbf{x}^{n+1}=$
$\displaystyle\mathbf{x}^{n}+\bm{\xi}^{n}\Delta t.$

In this equation, superscript denotes step number, $m$ and $\mathbf{x}$ denote
mass and position vector of a particle, respectively. The contribution to flux
of these particles is calculated as:

$\mathbf{F}^{p}_{i}=\sum\limits_{j\in M(i)}\left\\{{-\sum\limits_{k\in
N(ij)}m_{k}\bm{\varphi}_{k}+\sum\limits_{k\in
N(ji)}m_{k}\bm{\varphi}_{k}}\right\\}/V_{i},$ (24)

where $M(i)$ is the gather of adjacent cells of cell $i$, and $N(ij)$ is the
gather of particles passing through the interface from cell $i$ to cell $j$.
Then, the flux of macroscopic part is calculated by N-S solvers. After that,
the macroscopic variables at next time step can be renewed as:

$\bm{\Psi}^{n+1}_{i}=\bm{\Psi}^{n}_{i}+\mathbf{F}^{p,n}_{i}-\frac{\Delta
t_{i}}{V_{i}}\sum\limits_{j\in M(i)}\mathbf{F}^{w,n}_{ij}S_{ij},$ (25)

where $\mathbf{F}^{w}$ is the flux of wave part (NS solver).

Step (4)

If the simulation is not finished, go back to step $1$.

A diagrammatic drawing is shown as Fig.3. The gray block denotes a cell. Red
color and band denote wave, while blue color and spheres denote particles.

Fig. 3:  A diagrammatic drawing for DUWP.

##  4 Test cases

One-dimensional cases in monatomic argon gas are firstly tested, like shock
structure and Sod shock tube. Sod shock tube is taken as a typical unsteady
case and different Knudsen numbers are considered.

###  4.1 Shock structure

Shock structure is a typical non-equilibrium flow. When Mach number is high,
its nonequilibrium feature is strong. It is a classical case to test the
ability of models or numerical methods to describe non-equilibrium flows. In
this case, Mach numbers are taken as $3.0$ and $8.0$, and parameters are:
$\gamma=5/3$, $\rm{Pr}=2/3$. The dynamic viscosity is calculated through
$\mu\sim T^{\omega}$. The size of mesh is set to be a quarter of inflow
molecular mean free path $\lambda$, which is calculated through:

$\displaystyle\lambda$
$\displaystyle=\frac{1}{\beta}\frac{\mu}{p}\sqrt{\frac{RT}{2\pi}},$ (26)
$\displaystyle\beta$
$\displaystyle=\frac{{5(\alpha+1)(\alpha+2)}}{{4\alpha(5-2\omega)(7-2\omega)}}.$

Other parameters are set referring to Ref.[50, 51], as follows. When Mach
number is $3.0$, $\alpha=1.0$ and $\omega=0.5$. The reference data are results
of Boltzmann equation[52] and UGKS[53]. Fig.4 shows the comparison results.
The x coordinate is nondimensionalized through $0.5\sqrt{\pi}\lambda$. And the
vertical coordinates are nondimensionalized as follows:

$\displaystyle\hat{\rho}$ $\displaystyle=\frac{\rho}{\rho_{\rm{up}}},\hat{T}$
$\displaystyle=\frac{T}{T_{\rm{up}}},$ (27) $\displaystyle\hat{\Pi}$
$\displaystyle=\frac{\Pi}{p_{\rm{up}}},\hat{Q}$
$\displaystyle=\frac{Q}{p_{\rm{up}}\sqrt{2RT_{\rm{up}}}},$

where subscript “up” denotes inflow parameter. When Mach number is $8.0$,
$\alpha=1.0$ and $\omega=0.68$. The reference data are results of DSMC[54] and
UGKS[53]. Fig.5 shows the comparison results. The x coordinate is
nondimensionalized through $\beta\pi\lambda$. And the vertical coordinates are
nondimensionalized as follows:

$\displaystyle\hat{\rho}$
$\displaystyle=\frac{\rho-\rho_{\rm{up}}}{\rho_{\rm{down}}-\rho_{\rm{up}}},\hat{T}$
$\displaystyle=\frac{T-T_{\rm{up}}}{T_{\rm{down}}-T_{\rm{up}}},$ (28)
$\displaystyle\hat{\Pi}$
$\displaystyle=\frac{\Pi}{\rho_{\rm{up}}(2RT_{\rm{up}})},\hat{Q}$
$\displaystyle=\frac{Q}{\rho_{\rm{up}}(2RT_{\rm{up}})^{1.5}},$

where subscript “down” denotes outflow parameter. The mass of a particle is
$\frac{\rho_{up}\Delta x}{500}$, and $600$ steps of time average are taken.
Indeed, in this case the DUWP method is equivalent to the DSMC method, because
the mesh size is much smaller than $\lambda$ and $\Delta t<\tau$.

Fig. 4:  Results of $\rm{Ma}=3.0$ argon gas shock structure: (a) Density and
temperature, (b) stress and heat flux.

Fig. 5:  Results of $\rm{Ma}=8.0$ argon gas shock structure: (a) Density and
temperature, (b) stress and heat flux.

###  4.2 Sod shock tube

Sod shock tube is a typical non-equilibrium flow. It can test the ability of
method to simulate unsteady flow, and with different inflow viscosity
parameters taken, the flow regime can be set from rarefied to continuum. In
this case, Knudsen numbers are taken from $1E-5$, $1E-4$, $3E-4$, $1E-3$ to
$1E-2$, and parameters are: $\gamma=5/3$, $\rm{Pr}=2/3$, $\alpha=1.0$,
$\omega=0.81$. The dynamic viscosity is calculated through $\mu\sim
T^{\omega}$. $100$ cells are divided in the length of tube, which is the
reference length $1.0$. The upstream and downstream parameters are taken as:

Table I: Upstream and downstream parameters of Sod shock tube $\rho_{up}$ | $u_{up}$ | $T_{up}$ | $P_{up}$ | $\rho_{down}$ | $u_{down}$ | $T_{down}$ | $P_{down}$
---|---|---|---|---|---|---|---
$1.0$ | $0.0$ | $1.0$ | $1.0$ | $0.125$ | $0.0$ | $0.8$ | $0.1$

The mass of a particle is $\frac{\rho_{up}\Delta x}{2000}$, and $50$ steps of
ensemble average are taken. Results at $t=0.15$ are shown in Fig.6. Fig.6
shows the ratio of wave at different Knudsen numbers. At $Kn=1E-5$, it is
almost pure NS solver, and at $Kn=1E-2$ it is pure DSMC solver. Reference
results are calculated by the UGKS method, and results of DUWP are in line
well at each state.

Fig. 6:  Results of Sod shock tube: (a) $Kn=1E-5$ , (b) $Kn=1E-4$ , (c)
$Kn=3E-4$ , (d) $Kn=1E-3$ , (e) $Kn=1E-2$ , (f) ratio of wave.

##  5 Conclusions

The proposed DUWP method works well in one-dimensional cases. The method will
be extended to multi-dimension next. The essential point is how to deal with
the great change of mesh size in the whole domain. And it is desirable to
verify the accuracy of DUWP to predict stress and heat flux at the wall.

## Appendix A: DSMC collision term and its nondimensionalization

As to the collision term of DSMC, the no-time counter (NTC) method is used.
For each time step, we need to sample
$\frac{1}{2}N_{p}(N_{p}-1)F_{N}(\sigma_{T}\xi_{r})_{max}\Delta t/V_{c}$
couples and decide whether they collide. As to each couple sampled, the
probability of collision is
$\frac{\sigma_{T}\xi_{r}}{(\sigma_{T}\xi_{r})_{max}}$. Among the parameters,
$N_{p}$ is the number of numerical particles; $F_{N}$ means that each
numerical particle denotes $F_{N}$ real particles; $V_{c}$ is the volume of
the cell; $\sigma_{T}=\pi d^{2}$ is the total collision cross-section;
$\bm{\xi}_{r}=\bm{\xi}_{a}-\bm{\xi}_{b}$ is the relative velocity between the
two numerical particles $a$ and $b$; $\bm{\xi}_{r,max}$ is always estimated.
$d$ for variable hard sphere (VHS) model is calculated as follows, and for
hard sphere (HS) model, $d$ doesn’t change:

$\frac{d}{d_{ref}}=\left({\frac{\xi_{r}}{\xi_{r,ref}}}\right)^{-\upsilon},$
(29)

where $d_{ref}$ is the particle reference diameter,
$\xi_{r,ref}=\sqrt{2kT_{ref}/m_{r}}/\Gamma(2.5-\omega)^{0.5/\upsilon}$ ($k=mR$
is the Boltzmann const,$m_{r}=\frac{m_{a}m_{b}}{m_{a}+m_{b}}=0.5m$ is the
reduced mass,$\Gamma()$ is the Gamma function), $\upsilon=\frac{2}{\eta-1}$ is
the exponent of intermolecular law, and
$\omega=\frac{1}{2}\frac{\eta+3}{\eta-1}$ is the viscosity index. The
collision process of two particles is as follows:(superscript “*” denotes
parameters after collision)

$\displaystyle cos\theta=2rand-1,\phi=2\pi rand,$ (30)
$\displaystyle\bm{\xi}^{*}_{r}=(\xi_{r}sin\theta cos\phi,\xi_{r}sin\theta
sin\phi,\xi_{r}cos\theta),$
$\displaystyle\bm{\xi}_{1}^{*}=\bm{\xi}_{m}+\frac{m_{b}}{m_{a}+m_{b}}\bm{\xi}_{r}^{*},\bm{\xi}_{2}^{*}=\bm{\xi}_{m}-\frac{m_{a}}{m_{a}+m_{b}}\bm{\xi}_{r}^{*},$
$\displaystyle\bm{\xi}_{m}=\frac{m_{a}\bm{\xi}_{a}+m_{b}\bm{\xi}_{b}}{m_{a}+m_{b}}.$

More introductions about the NTC, collision model and corresponding parameters
can be referred to monographs for DSMC[14].

A convenient nondimensionalization method is suggested as follows. Referring
to the definition of mean free path:

$\lambda=\frac{1}{\sqrt{2}\pi d^{2}n},$ (31)

where $n=F_{N}N/V_{c}$ means the number of real physical particles per volume,
it is convenient to take the inflow parameters as reference:

$\lambda_{ref}=\frac{1}{\sqrt{2}\pi
d_{ref}^{2}F_{N,ref}N_{ref}/V_{c}}\Rightarrow
d_{ref}^{2}=\frac{1}{\sqrt{2}\pi\lambda_{ref}F_{N,ref}N_{ref}/V_{c}},$ (32)

Then $F_{N,ref}N_{ref}/V_{c}$ can be counteracted in
$\frac{1}{2}N(N-1)F_{N}(\sigma_{T}c_{r})_{max}\Delta t/V_{c}$, which is the
number of sampled numerical particles per time step. So it is as follows for
the VHS model:

$\displaystyle\frac{1}{2}N(N-1)F_{N}(\sigma_{T}c_{r})_{max}\Delta
t/V_{c}=\frac{1}{2\sqrt{2}\lambda_{ref}}\frac{\rho}{\rho_{ref}}(N-1)\Delta
t\left({\frac{c_{r,max}}{c_{r,ref}}}\right)^{-2\upsilon}c_{r,max},$ (33)
$\displaystyle\frac{\sigma_{T}c_{r}}{(\sigma_{T}c_{r})_{max}}=\frac{c_{r}}{(c_{r})_{max}}^{1-2\upsilon}.$

## Acknowledgements

This work is supported by National Natural Science Foundation of China
(11902264, 11902266, 12072283, 12172301) and 111 project of China (B17037).
Junzhe Cao thanks Mr. Hao Jin at School of Aeronautics, Northwestern
Polytechnical University for details of DSMC method.

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# The Demailly systems with the vortex ansatz

Arindam Mandal  Department of Mathematics, Indian Institute of Science,
Bangalore 560012, India<EMAIL_ADDRESS>
###### Abstract.

For an arbitrary-rank vector bundle over a projective manifold, J.-P. Demailly
proposed several systems of equations of Hermitian-Yang-Mills type for the
curvature tensor to settle a conjecture of Griffiths on the equivalence of
Hartshorne ampleness and Griffiths positivity. In this article, we have
studied two proposed systems and proved that these equations have smooth
solutions for the Vortex bundle using the continuity method.

†† Keywords: Holomorphic vector bundle, Ample vector bundle, Hermitian metric,
curvature tensor, Griffiths positivity, Nakano positivity, dual Nakano
positivity, elliptic operator.

##  1\. Introduction

Let $X$ be an $n$-dimensional projective manifold. A rank-$r$ holomorphic
vector bundle $E$ over $X$ is said to be ample in the Hartshorne sense
[Har1966] if and only if the line bundle $\mathcal{O}_{\mathbb{P}(E)}(1)$ is
ample over $\mathbb{P}(E)$. The Chern curvature tensor $\Theta_{E,h}$ of a
Hermitian metric $h$ is said to be Griffiths positive if
$\langle\sqrt{-1}\Theta_{E,h}(\zeta,\bar{\zeta}).v,v\rangle_{h}$ is positive
for all decomposable nonzero elements $\zeta\otimes v\in T_{X}\otimes E$, and
Nakano positive if the bilinear form on $T_{X}\otimes E$ defined by
$\sqrt{-1}\Theta_{E,h}$ is positive. Nakano positivity and dual Nakano
positivity (the bundle $(E^{*},h^{*})$ is Nakano negative) imply Griffiths
positivity, which is equivalent to dual Griffiths positivity (the bundle
$(E^{*},h^{*})$ is Griffiths negative). Griffiths positivity implies
ampleness. B. Berndtsson [Bo2009] has proved that for every positive integer
$m$, $S^{m}E\otimes detE$ is Nakano positive if $E$ is ample. The tangent
bundle $T{\mathbb{P}^{n}}$ of the complex projective space $\mathbb{P}^{n}$ is
ample but not Nakano positive, and ampleness does not imply Nakano positivity
(see [Dem2021] for details). A conjecture of Griffiths [Gri1970] asks if
Hartshorne ampleness implies Griffiths positivity. This conjecture is still
open in its full generality. However, the conjecture holds for vector bundles
on smooth curves, i.e., for $n=1$, see [CaFl1990] and [Um1973] for more
details. Much work has been done in this direction (see [LSY2010, MoTa2007,
Nau2017] and the references therein). J. P. Demailly [Dem2021] introduced
systems of PDE of Hermitian-Yang-Mills type for the curvature tensor to prove
the equivalence between ampleness and Griffiths positivity. Let $(E,H_{0})$ be
smooth Hermitian holomorphic vector bundle of rank $r$ over $X$ such that $E$
is ample and $\omega_{0}=\sqrt{-1}\Theta_{detE,detH_{0}}>0$. Then one of
Demailly’s systems for time-dependent metrics $h_{t},t\in[0,1]$ is as follows

$\begin{split}&\omega_{0}^{-n}det_{T_{X}\otimes
E^{*}}\Big{(}\sqrt{-1}\Theta_{E,h_{t}}+(1-t)\alpha\omega_{0}\otimes
Id_{E^{*}}\Big{)}^{\frac{1}{r}}=\Big{(}\frac{detH_{0}}{deth_{t}}\Big{)}^{\lambda}a_{0},\\\
&\omega_{0}^{-n}\Big{(}\omega_{0}^{n-1}\wedge\sqrt{-1}(\Theta_{E,h_{t}}-\frac{1}{r}\Theta_{detE,deth_{t}}\otimes
Id_{E})\Big{)}\\\
&\qquad\qquad=-\varepsilon\Big{(}\frac{detH_{0}}{deth_{t}}\Big{)}^{\mu}ln\Bigg{(}\frac{h_{t}H_{0}^{-1}}{det(h_{t}H_{0}^{-1})^{\frac{1}{r}}}\Bigg{)},\end{split}$
(1)

where $\lambda\geq 0,\mu\in\mathbb{R}$, $\varepsilon>0$ and
$a_{0}=\omega_{0}^{-n}det_{T_{X}\otimes
E^{*}}\Big{(}\sqrt{-1}\Theta_{E,h_{0}}+\alpha\omega_{0}\otimes
Id_{E^{*}}\Big{)}^{\frac{1}{r}}>0$. The metric $h_{0}$ is a solution of the
second equation (cushioned Hermite-Einstein equation) of system (1) at $t=0$
with the condition $deth_{0}=detH_{0}$. With the notations same as above, one
more variant of the above system is

$\begin{split}&\omega_{0}^{-n}det_{T_{X}\otimes
E^{*}}\Big{(}\sqrt{-1}\Theta_{E,h_{t}}+(1-t)\alpha\omega_{0}\otimes
Id_{E^{*}}\Big{)}^{\frac{1}{r}}=\Big{(}\frac{detH_{0}}{deth_{t}}\Big{)}^{\lambda}a_{0},\\\
&\omega_{0}^{-n}\Big{(}\omega_{t}^{n-1}\wedge\sqrt{-1}(\Theta_{E,h_{t}}-\frac{1}{r}\Theta_{detE,deth_{t}}\otimes
Id_{E})\Big{)}\\\
&\qquad\qquad=-\varepsilon\Big{(}\frac{detH_{0}}{deth_{t}}\Big{)}^{\mu}ln\Bigg{(}\frac{h_{t}H_{0}^{-1}}{det(h_{t}H_{0}^{-1})^{\frac{1}{r}}}\Bigg{)},\end{split}$
(2)

where
$\omega_{t}=\frac{1}{1+r\alpha}(\sqrt{-1}\Theta_{detE,deth_{t}}+r(1-t)\alpha\omega_{0})$
and the metric $h_{0}$ is a solution of second equation of the system (2)
satisfying $deth_{0}=detH_{0}$. Even though for both of the above systems, the
existence of the metric $h_{0}$ is clear from [UhYa1986], for our purpose, in
the case of Vortex bundle, we shall discuss the existence separately
(subsections 2.1 and 3.1). If one can prove the existence of $h_{t}$ for all
time $t\in[0,1]$, for one of the above systems, then $h_{t}$ will be dual
Nakano positive for $t=1$. Thus a stronger result than Griffiths conjecture
will have been proven, which will certainly settle the Griffiths conjecture.
It turns out that there exists ample bundle which is not dual Nakano positive;
see [Dem2021] for a detailed example. Therefore one should not expect the
existence of solutions of Demailly’s systems for all time $t\in[0,1]$ in
general. V. P. Pingali [Pin2021] has studied the system (1) for the direct sum
of ample line bundles on Riemann surfaces using Leray-Schauder degree theory.
In this article, using the continuity method, we have studied systems (1) and
2 on the Vortex bundle.

We follow the construction of Vortex bundles as in [Gar1993] and [Pin2020].
Let $\Sigma$ be a compact Riemann surface with a background Hermitian metric
$k$ on an ample holomorphic line bundle $L$ such that
$\omega_{\Sigma}=\sqrt{-1}\Theta_{k}$ is the Kähler metric, where $\Theta_{k}$
is the curvature of the metric $k$. Consider $\mathbb{CP}^{1}$ with the metric
$h_{FS}$ on the line bundle $\mathcal{O}(1)$, who’s curvature is the Fubini-
study metric $\omega_{FS}=\frac{\sqrt{-1}dz\wedge d\bar{z}}{(1+|z|^{2})^{2}}$.
Define the rank-2 vector bundle $E$ on the projective manifold
$X=\Sigma\times\mathbb{CP}^{1}$ by
$E=\pi_{1}^{*}((r_{1}+1)L)\otimes\pi_{2}^{*}(r_{2}\mathcal{O}(2))\oplus\pi_{1}^{*}(r_{1}L)\otimes\pi_{2}^{*}((r_{2}+1)\mathcal{O}(2))$,
where $\pi_{1}:X\rightarrow\Sigma$ and $\pi_{2}:X\rightarrow\mathbb{CP}^{1}$
are the projection maps and $r_{1},r_{2}$ are positive integers. Let $\phi\in
H^{0}(\Sigma,L)$ be a global holomorphic section. We see that multiplication
of the metric $k$ by a nonzero constant does not change the curvature
$\Theta_{k}$. So we assume $k$ has been rescaled so that
$|\phi|_{k}^{2}\leq\frac{1}{2}$. Define a holomorphic structure on $E$ by the
second fundamental form
$\beta=\pi_{1}^{*}\phi\otimes\pi_{2}^{*}\Big{(}\frac{\sqrt{8\pi}dz}{(1+|z|^{2})^{2}}\otimes
d\bar{z}\Big{)}$. Let $E$ be equipped with metric
$h_{t}=\pi_{1}^{*}(e^{-(f_{t}+\psi_{t})}k^{r_{1}+1})\otimes\pi_{2}^{*}(h_{FS}^{2r_{2}})\oplus\pi_{1}^{*}(e^{-f_{t}}k^{r_{1}})\otimes\pi_{2}^{*}(h_{FS}^{2r_{2}+2})$,
where $f_{t}$ and $\psi_{t}$ are smooth function on $\Sigma$. Suppose
$\tilde{h}_{t}=\pi_{1}^{*}(e^{-(f_{t}+\psi_{t})}k^{r_{1}+1})\otimes\pi_{2}^{*}(h_{FS}^{2r_{2}})$,
$\tilde{g}_{t}=\pi_{1}^{*}(e^{-f_{t}}k^{r_{1}})\otimes\pi_{2}^{*}(h_{FS}^{2r_{2}+2})$
and $g_{t}=e^{-\psi_{t}}k$. Then the Chern connection of $(E,h_{t})$ for the
holomorphic structure given by $\beta$ is given by the following connection
matrix

$A_{h_{t}}=\begin{bmatrix}A_{\tilde{h}_{t}}&\beta\\\
-\beta^{{\dagger}_{g_{t}}}&A_{\tilde{g}_{t}}\end{bmatrix}.$

Its curvature matrix is

$\Theta_{h_{t}}=\begin{bmatrix}\Theta_{\tilde{h}_{t}}-\beta\wedge\beta^{{\dagger}_{g_{t}}}&\nabla^{1,0}\beta\\\
-\nabla^{0,1}\beta^{{\dagger}_{g_{t}}}&\Theta_{\tilde{g}_{t}}-\beta^{{\dagger}_{g_{t}}}\wedge\beta\end{bmatrix}.$

Let
$H_{0}=\pi_{1}^{*}(k^{r_{1}+1})\otimes\pi_{2}^{*}(h_{FS}^{2r_{2}})\oplus\pi_{1}^{*}(k^{r_{1}})\otimes\pi_{2}^{*}(h_{FS}^{2r_{2}+2})$
be the background metric on the Vortex bundle $E$. Then
$\omega_{0}=(2r_{1}+1)\omega_{\Sigma}+(4r_{2}+2)\omega_{FS}$. Choosing
$\lambda=0,\mu=0$, and $\varepsilon=1$, the system (1) for the Vortex bundle
will be the following decoupled system of equations.

$\begin{split}&\Big{\\{}\Big{(}\Delta
f_{t}+\Delta{\psi}_{t}+(r_{1}+1)+\alpha(1-t)(2r_{1}+1)\Big{)}\Big{(}2r_{2}+|\phi|_{g_{t}}^{2}+\\\
&\alpha(1-t)(4r_{2}+2)\Big{)}\Big{(}\Delta
f_{t}+r_{1}+\alpha(1-t)(2r_{1}+1)\Big{)}\\\
&\Big{(}(2r_{2}+2)-|\phi|_{g_{t}}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\Big{\\}}+\sqrt{-1}\frac{\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g_{t}}}}{\omega_{\Sigma}}\\\
&\Big{(}2r_{2}+|\phi|_{g_{t}}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\Big{(}\Delta
f_{t}+r_{1}+\alpha(1-t)(2r_{1}+1)\Big{)}\\\ =&\Big{\\{}\Big{(}\Delta
f_{0}+\Delta\psi_{0}+(r_{1}+1)+\alpha(2r_{1}+1)\Big{)}\Big{(}2r_{2}+|\phi|_{g_{0}}^{2}+\alpha(4r_{2}+2)\Big{)}\\\
&\Big{(}(2r_{2}+2)-|\phi|_{g_{0}}^{2}+\alpha(4r_{2}+2)\Big{)}\Big{(}\Delta
f_{0}+r_{1}+\alpha(2r_{1}+1)\Big{)}\Big{\\}}\\\
&+\sqrt{-1}\frac{\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g_{0}}}}{\omega_{\Sigma}}\Big{(}2r_{2}+|\phi|_{g_{0}}^{2}+\alpha(4r_{2}+2)\Big{)}\\\
&\Big{(}\Delta f_{0}+r_{1}+\alpha(2r_{1}+1)\Big{)},\end{split}$ (3)

$(2r_{1}+1)\big{(}|\phi|_{g_{t}}^{2}-1\big{)}+\big{(}\Delta\psi_{t}+1\big{)}(2r_{2}+1)=(2r_{1}+1)(4r_{2}+2)\psi_{t},$
(4)

where
$h_{0}=\pi_{1}^{*}(e^{-(f_{0}+\psi_{0})}k^{r_{1}+1})\otimes\pi_{2}^{*}(h_{FS}^{2r_{2}})\oplus\pi_{1}^{*}(e^{-f_{0}}k^{r_{1}})\otimes\pi_{2}^{*}(h_{FS}^{2r_{2}+2})$
is the solution of the equation (4) at $t=0$, satisfying $deth_{0}=detH_{0}$.
The existence of such $h_{0}$ is discussed in the subsection 2.1 and
$\alpha>0$ is a large enough constant so that
$\sqrt{-1}\Theta_{h_{0}}+\alpha\omega_{0}\otimes Id_{E^{*}}>0$ in the sense of
Nakano and $\Delta f_{0}+r_{1}+\alpha(2r_{1}+1)>0$. We now state one of our
results.

######  Theorem 1.1.

The system defined by equations (3) and (4) has smooth solution
$(f_{t},\psi_{t})$ such that $\Delta f_{t}+r_{1}+\alpha(1-t)(2r_{1}+1)>0$ for
all $t\in[0,1].$

If we choose $\lambda=0$ and $\mu=0$, then the system (2) for the Vortex
bundle will be the following coupled system of equations.

$\begin{split}&\Big{\\{}\Big{(}\Delta
f_{t}+\Delta{\psi}_{t}+(r_{1}+1)+\alpha(1-t)(2r_{1}+1)\Big{)}\Big{(}2r_{2}+|\phi|_{g_{t}}^{2}+\\\
&\alpha(1-t)(4r_{2}+2)\Big{)}\Big{(}\Delta
f_{t}+r_{1}+\alpha(1-t)(2r_{1}+1)\Big{)}\\\
&\Big{(}(2r_{2}+2)-|\phi|_{g_{t}}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\Big{\\}}\\\
&+\sqrt{-1}\frac{\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g_{t}}}}{\omega_{\Sigma}}\Big{(}2r_{2}+|\phi|_{g_{t}}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\\\
&\Big{(}\Delta f_{t}+r_{1}+\alpha(1-t)(2r_{1}+1)\Big{)}\\\
=&\Big{\\{}\Big{(}\Delta
f_{0}+\Delta\psi_{0}+(r_{1}+1)+\alpha(2r_{1}+1)\Big{)}\Big{(}2r_{2}+|\phi|_{g_{0}}^{2}+\alpha(4r_{2}+2)\Big{)}\\\
&\Big{(}(2r_{2}+2)-|\phi|_{g_{0}}^{2}+\alpha(4r_{2}+2)\Big{)}\Big{(}\Delta
f_{0}+r_{1}+\alpha(2r_{1}+1)\Big{)}\Big{\\}}\\\
&+\sqrt{-1}\frac{\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g_{0}}}}{\omega_{\Sigma}}\Big{(}2r_{2}+|\phi|_{g_{0}}^{2}+\alpha(4r_{2}+2)\Big{)}\\\
&\Big{(}\Delta f_{0}+r_{1}+\alpha(2r_{1}+1)\Big{)},\end{split}$ (5)
$\begin{split}&\Big{\\{}2\Big{(}2\Delta
f_{t}+\Delta\psi_{t}+(1+2\alpha(1-t))(2r_{1}+1)\Big{)}\big{(}|\phi|_{g_{t}}^{2}-1\big{)}\\\
&+\big{(}\Delta\psi_{t}+1\big{)}\big{(}1+2\alpha(1-t)\big{)}(4r_{2}+2)\Big{\\}}\\\
&\qquad\qquad=2(2r_{1}+1)(4r_{2}+2)(2\alpha+1)\varepsilon\psi_{t},\end{split}$
(6)

where
$h_{0}=\pi_{1}^{*}(e^{-(f_{0}+\psi_{0})}k^{r_{1}+1})\otimes\pi_{2}^{*}(h_{FS}^{2r_{2}})\oplus\pi_{1}^{*}(e^{-f_{0}}k^{r_{1}})\otimes\pi_{2}^{*}(h_{FS}^{2r_{2}+2})$
is the solution of the equation (6) at $t=0$, satisfying $deth_{0}=detH_{0}$.
The existence of such $h_{0}$ is discussed in subsection 3.1 and $\alpha>0$ is
a large enough constant so that
$\sqrt{-1}\Theta_{h_{0}}+\alpha\omega_{0}\otimes Id_{E^{*}}>0$ in the sense of
Nakano and $\Delta f_{0}+r_{1}+\alpha(2r_{1}+1)>0$. Finally, we have the
following result.

######  Theorem 1.2.

For large enough $\varepsilon>0$, the system defined by equations (5) and (6)
has smooth solution $(f_{t},\psi_{t})$ such that $\Delta
f_{t}+r_{1}+\alpha(1-t)(2r_{1}+1)>0$ for all $t\in[0,1].$

Demailly’s original approach involved the method of continuity. However,
proving openness for Demailly’s systems is the most challenging part because
the required positivity properties for openness may not be preserved along the
continuity path. Even in the case of a direct sum of ample line bundles on a
Riemann surface, it appears hard to prove, and therefore the Leray-Schauder
degree method was used in [Pin2021]. The main point of this article is to
provide a proof-of-concept for Demailly’s approaches. We hope that the
techniques used for the vortex bundle generalize to more complicated
situations.

We briefly describe the strategy of the proofs. System (1) is decoupled and
hence is relatively easier to handle (Section 2). On the other hand, unlike
System (1), System (2) is truly a coupled system. To demonstrate openness, the
key point is to prove the lower bound for $\Delta\psi_{t}$, independent of $t$
and $\varepsilon$ so that
$\Delta\psi_{t}+1+2(2r_{1}+1)(4r_{2}+2)(2\alpha+1)\varepsilon$ can be made
positive for large $\varepsilon$. However, as one will see, to get such
estimates, it is crucial to observe that the lower bound of $\Delta\psi_{0}$
itself is independent of $\varepsilon$. These calculations are rather delicate
and carried out in Section 3.

##  2\. Proof of Theorem 1.1

For the remainder of the paper, we drop the parameter $t$ for notational
convenience. We denote constants by $C$ that may vary from line to line and
are independent of $t$ unless specified.

###  2.1.  Existence of solution at $t=0$ for the first system

Recall that
$h_{0}=\pi_{1}^{*}(e^{-(f_{0}+\psi_{0})}k^{r_{1}+1})\otimes\pi_{2}^{*}(h_{FS}^{2r_{2}})\oplus\pi_{1}^{*}(e^{-f_{0}}k^{r_{1}})\otimes\pi_{2}^{*}(h_{FS}^{2r_{2}+2})$
is the solution of the equation (4) at $t=0$, and $deth_{0}=detH_{0}$.
Therefore, $2f_{0}+\psi_{0}=0$ and $\psi_{0}$ satisfying
$(2r_{1}+1)\big{(}|\phi|_{g_{0}}^{2}-1\big{)}+\big{(}\Delta\psi_{0}+1\big{)}(2r_{2}+1)=(2r_{1}+1)(4r_{2}+2)\psi_{0}$.
We shall show that $\psi_{0}$ exists by the method of continuity. Now define
$L_{s}(\psi_{0})=\Delta\psi_{0}+1-s(1-|\phi|_{g_{0}}^{2})\frac{2r_{1}+1}{2r_{2}+1}-2(2r_{1}+1)\psi_{0}$,
where $s\in[0,1]$. Let $S:=\\{s\in[0,1]|L_{s}=0\text{ has a smooth solution at
}s\\}$. Clearly $\psi_{0}=\frac{1}{2(2r_{1}+1)}$ is a solution of $L_{0}=0$.
Thus $0\in S$. Now
$DL_{s}(\psi_{0})[\delta\psi]=\Delta\delta\psi-s|\phi|^{2}_{g_{0}}\frac{2r_{1}+1}{2r_{1}+1}\delta\psi-2(2r_{1}+1)\delta\psi$.
By maximum principle we get $Ker(DL_{s})=\\{0\\}$. Since $DL_{s}$ is formally
self-adjoint, we get $DL_{s}$ is an isomorphism. Hence by implicit function
theorem for Banach manifolds, we get $S$ is open. Now we must prove that $S$
is closed. Let us first prove a prior bound on the term $|\phi|_{g_{0}}^{2}$.

######  Lemma 2.1.

$|\phi|_{g_{0}}^{2}<1$.

###### Proof.

We know

$\partial\bar{\partial}|\phi|_{g_{0}}^{2}=-\Theta_{g_{0}}|\phi|^{2}_{g_{0}}+\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g_{0}}}.$

At the point $p$ of maximum of $|\phi|_{g_{0}}^{2}$,
$\sqrt{-1}\partial\bar{\partial}|\phi|_{g_{0}}^{2}(p)\leq 0$. Thus we see that
$\sqrt{-1}\Theta_{g_{0}}(p)\geq 0$, which implies
$\big{(}1+\Delta\psi_{0}\big{)}(p)\geq 0$. If $\psi_{0}(p)\geq 0$, we have
$|\phi|^{2}_{g_{0}}(p)=|\phi|^{2}_{k}(p)e^{-\psi_{0}(p)}\leq|\phi|^{2}_{k}(p)\leq\frac{1}{2}$.
Otherwise $\psi_{0}(p)<0$, and equation (4) implies
$(2r_{1}+1)\big{(}|\phi|_{g_{0}}^{2}-1\big{)}<0$. Hence, We get
$|\phi|^{2}_{g_{0}}<1$. ∎

Let $s\in S$, then
$\Delta\psi_{0}+1=s(1-|\phi|_{g_{0}}^{2})\frac{2r_{1}+1}{2r_{2}+1}-2(2r_{1}+1)\psi_{0}$.
Applying maximum principle and Lemma 2.1 we get $||\psi_{0}||_{C^{0}}<C$ and
hence $||\Delta\psi_{0}||_{C^{0}}<C$. Now by the Arzela-Ascoli theorem, we see
that $S$ is closed. This completes the proof of the existence of $h_{0}$.

###### Theorem 1.1.

Since the system is decoupled, we shall solve the equation (4) for $\psi$
using the continuity method. Then we shall solve equation (3) using the same
method. Let

$\displaystyle I^{\prime}_{1}:=\Big{\\{}t\in[0,1]\Big{|}\text{ equation
\eqref{02} has smooth solution $\psi_{t}$ at $t$ }\Big{\\}}.$

Since $\psi_{0}$ solves the equation at $t=0$, so $0\in I_{1}^{\prime}$. Hence
$I^{\prime}_{1}$ is non-empty. We want to show $I^{\prime}_{1}$ is closed and
open. Then by connectedness we can conclude that $I^{\prime}_{1}=[0,1]$.

###  2.2.  Closedness of $I^{\prime}_{1}$

If we prove suitable uniform estimates for $\psi_{t}$ whenever $t\in
I^{\prime}_{1}$, then using the Arzela-Ascoli theorem we can prove that
$I^{\prime}_{1}$ is closed.

A similar calculation as in Lemma 2.1 shows the following estimates.

######  Lemma 2.2.

$|\phi|^{2}_{g}<1$ for all $t\in[0,1].$

Now let us prove $C^{0}$-estimates for $\psi_{t}$.

######  Lemma 2.3.

If $t\in I_{1}^{\prime}$, then there exists a constant $C$ such that
$||\psi_{t}||_{C^{0}}\leq C$.

###### Proof.

Suppose $\psi$ attains its maximum at a point $p$, then $\Delta\psi(p)\leq 0$.
Using this along with equation (4), we conclude the following inequality

$\displaystyle(2r_{1}+1)(4r_{2}+2)\psi(p)\leq(2r_{2}+1).$ (7)

At a minimum point $q$ for $\psi$ we have $\Delta\psi(q)\geq 0.$ Equation
$\eqref{02}$ gives

$\displaystyle-(2r_{1}+1)\leq(2r_{1}+1)(4r_{2}+2)\psi(q).$ (8)

∎

Applying Lemma 2.3 in equation (4), we have the following estimate.

######  Lemma 2.4.

Any smooth $\psi$ that solves the equation (4) satisfies
$||\Delta\psi||_{C^{0}}\leq C$ for some positive constant $C$.

###  2.3. Openness of $I^{\prime}_{1}$

Let us define the map $T_{2}:C^{2,\beta}\times[0,1]\rightarrow C^{0,\beta}$ by
$T_{2}\Big{(}\psi,t\Big{)}=(2r_{1}+1)\big{(}|\phi|_{g}^{2}-1\big{)}+\big{(}\Delta\psi+1\big{)}(2r_{2}+1)-(2r_{1}+1)(4r_{2}+2)\psi$.
Its linearization is

$\begin{split}&DT_{2}(\psi,t)[\delta\psi]\\\
&=(2r_{2}+1)\Delta\delta\psi-(2r_{1}+1)|\phi|^{2}_{g}\delta\psi-(2r_{1}+1)(4r_{2}+2)\delta\psi.\end{split}$
(9)

If $\delta\psi\in Ker(DT_{2})$, then $DT_{2}(\psi)[\delta\psi]=0$. Now maximum
principle implies that $\delta\psi=0$. Since $T_{2}$ is self-adjoint and
$Ker(DT_{2})=0$, it is an isomorphism.
From Lemma 2.3, Lemma 2.4, and by bootstrapping, we conclude that solutions
are smooth. Therefore, we have uniform estimates of solutions of the equation
(4) and its derivative of all orders.

Now we can solve the equation (3) for the variable $f$. As we mentioned
earlier, we shall do so by the continuity method. We let

$\displaystyle I_{1}^{\prime\prime}:=\Big{\\{}t\in[0,1]\text{ \Big{|} equation
(\ref{01}) has smooth solution $f_{t}$ at $t$ and }$ $\displaystyle\textbf{
$\Delta f_{t}+r_{1}+\alpha(1-t)(2r_{1}+1)>0$ }\Big{\\}}.$

At $t=0$, $f_{0}$ is a solution of equation (3) and $\Delta
f_{0}+r_{1}+\alpha(1-t)(2r_{1}+1)>0$, so $0\in I_{1}$. Hence $I_{1}$ is non-
empty. We must show that $I_{1}^{\prime\prime}$ is both open and closed.

###  2.4.  Closedness of $I^{\prime\prime}_{1}$

Let us prove the closedness of $I_{1}^{\prime\prime}$ by proving estimates for
$f$ and its derivatives.

######  Lemma 2.5.

There exists a constant $C$ such that whenever $t\in I^{\prime\prime}_{1}$, we
have $||\Delta f_{t}||_{C^{0}}\leq C$.

###### Proof.

Since $t\in I^{\prime\prime}_{1}$, we have $\Delta
f>-\big{(}r_{1}+\alpha(2r_{1}+1)\big{)}$.
Now from equation (3) we compute,

$\begin{split}&\Big{\\{}\Big{(}\Delta
f+\Delta\psi+(r_{1}+1)+\alpha(1-t)(2r_{1}+1)\Big{)}\Big{(}2r_{2}+|\phi|_{g}^{2}\\\
&+\alpha(1-t)(4r_{2}+2)\Big{)}\Big{(}\Delta
f+r_{1}+\alpha(1-t)(2r_{1}+1)\Big{)}\\\
&\Big{(}(2r_{2}+2)-|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\Big{\\}}\\\
&\leq\Big{(}\Delta
f_{0}+\Delta\psi_{0}+(r_{1}+1)+\alpha(2r_{1}+1)\Big{)}\Big{(}2r_{2}+|\phi|_{g_{0}}^{2}+\alpha(4r_{2}+2)\Big{)}\\\
&\Big{(}(2r_{2}+2)-|\phi|_{g_{0}}^{2}+\alpha(4r_{2}+2)\Big{)}\Big{(}\Delta
f_{0}+r_{1}+\alpha(2r_{1}+1)\Big{)}\\\
&+\sqrt{-1}\frac{\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g_{0}}}}{\omega_{\Sigma}}\Big{(}2r_{2}+|\phi|_{g_{0}}^{2}+\alpha(4r_{2}+2)\Big{)}\\\
&\Big{(}\Delta f_{0}+r_{1}+\alpha(2r_{1}+1)\Big{)}.\end{split}$ (10)

Now appealing to Lemma 2.4, we get $\Delta f\leq C$. ∎

Let $G$ be the Green’s function of the metric $\omega_{\Sigma}$ such that
$-C\\{1+|ln(d_{\omega_{\Sigma}}(x,y))|\\}\leq G(x,y)\leq 0$. Then for any
continuous function $f$, we have the following Green representation formula:

$\displaystyle
f(x)=\displaystyle\frac{\int_{\Sigma}f(y)\omega_{\Sigma}(y)}{\int_{\Sigma}\omega_{\Sigma}(y)}+\int_{\Sigma}G(x,y)\Delta
f(y)\omega_{\Sigma}(y)$ (11)

Using the formula (11) and Lemma 2.5 we have the following:

######  Lemma 2.6.

If $t\in I^{\prime\prime}_{1}$ then $||f_{t}||_{C^{0}}\leq C$, for some
positive constant $C$.

Let $t_{n}\in I^{\prime\prime}_{1}$ be such that $t_{n}\rightarrow t$. To
prove $I^{\prime\prime}_{1}$ is closed, we must show $t\in
I^{\prime\prime}_{1}$. As $t_{n}\in I^{\prime\prime}_{1}$, we have $\Delta
f_{t_{n}}+r_{1}+\alpha(1-t)(2r_{1}+1)>0$. Arzela-Ascoli theorem, together with
the above estimates, we get a subsequence of $t_{n}$ again call the
subsequence by $t_{n}$ such that $f_{t_{n}}\rightarrow f$ in $C^{2,\alpha}$.
Then by usual bootstrapping argument we have $f$ is smooth and $,\Delta
f+r_{1}+\alpha(1-t)(2r_{1}+1)\geq 0$. If $\Delta
f+r_{1}+\alpha(1-t)(2r_{1}+1)=0$ at some point, then equation (3) gives a
contradiction. Hence $\Delta f+r_{1}+\alpha(1-t)(2r_{1}+1)>0$. Therefore $t\in
I_{1}^{\prime\prime}$.

We now proceed to prove the openness of $I_{1}^{\prime\prime}$.

###  2.5. Openness of $I^{\prime\prime}_{1}$

Let $\mathcal{B}$ be the subset in $C^{2,\beta}$ defined by
$\mathcal{B}:=\Big{\\{}f\in C^{2,\beta}\Big{|}\Delta
f+r_{1}+\alpha(1-t)(2r_{1}+1)>0,\displaystyle\int_{\Sigma}f\omega_{\Sigma}=0\Big{\\}}$.
Now let us define the map $T_{1}:\mathcal{B}\times[0,1]\rightarrow
C^{0,\beta}$ by

$\begin{split}&T_{1}(f,t)\\\ =&\bigg{\\{}\Big{(}\Delta
f+\Delta{\psi}+(r_{1}+1)+\alpha(1-t)(2r_{1}+1)\Big{)}\Big{(}2r_{2}+|\phi|_{g}^{2}+\\\
&\alpha(1-t)(4r_{2}+2)\Big{)}\Big{(}\Delta
f+r_{1}+\alpha(1-t)(2r_{1}+1)\Big{)}\\\
&\Big{(}(2r_{2}+2)-|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\\\
&+\sqrt{-1}\frac{\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g}}}{\omega_{\Sigma}}\Big{(}2r_{2}+|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\\\
&\Big{(}\Delta
f+r_{1}+\alpha(1-t)(2r_{1}+1)\Big{)}\bigg{\\}}-\bigg{\\{}\Big{(}\Delta
f_{0}+\Delta\psi_{0}+(r_{1}+1)+\alpha(2r_{1}+1)\Big{)}\\\
&\Big{(}2r_{2}+|\phi|_{g_{0}}^{2}+\alpha(4r_{2}+2)\Big{)}\Big{(}(2r_{2}+2)-|\phi|_{g_{0}}^{2}+\alpha(4r_{2}+2)\Big{)}\\\
&\Big{(}\Delta
f_{0}+r_{1}+\alpha(2r_{1}+1)\Big{)}+\sqrt{-1}\frac{\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g_{0}}}}{\omega_{\Sigma}}\Big{(}\Delta
f_{0}+r_{1}+\alpha(2r_{1}+1)\Big{)}\\\
&\Big{(}2r_{2}+|\phi|_{g_{0}}^{2}+\alpha(4r_{2}+2)\Big{)}\bigg{\\}}.\end{split}$
(12)

Then its linearization at a point $(f,t)\in\mathcal{B}\times[0,1]$ will be

$\begin{split}&DT_{1}(f,t)[\delta f]\\\
=&\Big{(}2r_{2}+|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\bigg{[}\Big{(}(2r_{2}+2)-|\phi|_{g}^{2}+\\\
&\alpha(1-t)(4r_{2}+2)\Big{)}\Big{(}2\Delta
f+\Delta\psi+\big{(}1+2\alpha(1-t)\big{)}(2r_{1}+1)\Big{)}\\\
&+\sqrt{-1}\frac{\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g}}}{\omega_{\Sigma}}\bigg{]}\Delta\delta
f.\end{split}$ (13)

If $t\in I^{\prime\prime}_{1}$, then clearly $DT_{1}(f,t)$ is an isomorphism.
Therefore by the implicit function theorem on Banach manifolds, we get
$I^{\prime\prime}_{1}$ is open. This completes the proof of the Theorem (1.1).
∎

##  3\. Proof of theorem 1.2

Our next concern is to prove Theorem (1.2). First, let us discuss the solution
of the second system at $t=0$.

###  3.1.  Exixtence of solution at $t=0$ for the second system

Recall that
$h_{0}=\pi_{1}^{*}(e^{-(f_{0}+\psi_{0})}k^{r_{1}+1})\otimes\pi_{2}^{*}(h_{FS}^{2r_{2}})\oplus\pi_{1}^{*}(e^{-f_{0}}k^{r_{1}})\otimes\pi_{2}^{*}(h_{FS}^{2r_{2}+2})$
is the solution of the equation (6) at $t=0$, and $deth_{0}=detH_{0}$.
Therefore $(f_{0},\psi_{0})$ satisfying $2f_{0}+\psi_{0}=0$ and

$\begin{split}&\Big{\\{}2\Big{(}2\Delta
f_{0}+\Delta\psi_{0}+(1+2\alpha)(2r_{1}+1)\Big{)}\big{(}|\phi|_{g_{0}}^{2}-1\big{)}+\big{(}\Delta\psi_{0}+1\big{)}\big{(}1+2\alpha\big{)}(4r_{2}+2)\Big{\\}}\\\
&=2(2r_{1}+1)(4r_{2}+2)(2\alpha+1)\varepsilon\psi_{0}.\end{split}$

Equivalently, we shall solve the following equation for $\psi_{0}$,

$\displaystyle\Delta\psi_{0}+1=(1-|\phi|_{g_{0}}^{2})\frac{2r_{1}+1}{2r_{2}+1}+2(2r_{1}+1)\varepsilon\psi_{0}.$

Similar arguments in subsection 2.1 prove the existence of required
$\psi_{0}$. Moreover, we get the following estimates independent of
$\varepsilon$.

######  Lemma 3.1.

There exists positive constant $C$ independent of $\varepsilon$ such that
$||\varepsilon\psi_{0}||_{C^{0}}+||\Delta\psi_{0}||_{C^{0}}<C$.

Let

$\displaystyle I_{2}:=\Big{\\{}t\in[0,1]\Big{|}\text{ the system defined by
equations \eqref{1} and \eqref{2} has smooth}$ $\displaystyle\text{ solution
$(f_{t},\psi_{t})$ at $t$ },\Delta
f_{t}+r_{1}+\alpha(1-t)(2r_{1}+1)>0\Big{\\}}.$

At $t=0$, $(f_{0},\psi_{0})$ solves the system and $\Delta
f_{0}+r_{1}+\alpha(2r_{1}+1)>0$. So $I_{2}$ is non-empty. Now let us prove
$I_{2}$ is closed by proving some a priori estimates.

###  3.2.  Closedness of $I_{2}$

A similar proof as Lemma 2.1 gives the following result.

######  Lemma 3.2.

$|\phi|^{2}_{g_{t}}<1$ for $t\in I_{2}.$

###### Proof.

A simple computation using normal coordinates gives the following identity

$\partial\bar{\partial}|\phi|_{g_{t}}^{2}=-\Theta_{g_{t}}|\phi|^{2}_{g_{t}}+\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g_{t}}}.$
(14)

Let $|\phi|^{2}_{g_{t}}$ attains its maximum at a point $p$, then
$\sqrt{-1}\partial\bar{\partial}|\phi|_{g_{t}}^{2}(p)\leq 0$. So we must have
$\sqrt{-1}\Theta_{g_{t}}(p)\geq 0$ i.e., $(1+\Delta\psi_{t})(p)\geq 0$. If
$\psi_{t}(p)\geq 0$, then
$|\phi|^{2}_{g_{t}}(p)=|\phi|^{2}_{k}(p)e^{-\psi_{t}(p)}\leq|\phi|^{2}_{k}(p)\leq\frac{1}{2}$.
Otherwise, we have $\psi_{t}(p)<0$. Then from equation (6), we get
$\Big{(}2\Delta
f_{t}+\Delta\psi_{t}+(1+2\alpha(1-t))(2r_{1}+1)\Big{)}(|\phi|^{2}_{g_{t}}-1)(p)<0$.
Since $t\in I_{2}$, it follows that $|\phi|^{2}_{g_{t}}<1$. ∎

Next, we will prove $C^{0}$ estimates of $\psi_{t}$.

######  Lemma 3.3.

If $t\in I_{2}$, then $||\varepsilon\psi_{t}||_{C^{0}}\leq C$, where $C$ is
independent of $\varepsilon$.

###### Proof.

Suppose $\psi$ attains its maximum at a point $p$, then $\Delta\psi(p)\leq 0$.
From equation (6) and $\Delta f+r_{1}+\alpha(1-t)(2r_{1}+1)>0$ we get

$2(2r_{1}+1)(4r_{2}+2)(2\alpha+1)\varepsilon\psi(p)\leq(1+2\alpha)(4r_{2}+2)$

which yields

$\displaystyle\varepsilon\psi(p)\leq\frac{1}{2(2r_{1}+1)}.$ (15)

At a minimum point $q$ for $\psi$ we have $\Delta\psi(q)\geq 0$. Now in view
of equation (5) this gives

$\begin{split}&\Big{\\{}\Big{(}\Delta
f+r_{1}+\alpha(1-t)(2r_{1}+1)\Big{)}\Big{(}(2r_{2}+2)-|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\\\
&\Big{(}2r_{2}+|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\Big{\\}}(q)\leq\Big{\\{}\Big{(}\Delta
f_{0}+\Delta\psi_{0}+(r_{1}+1)+\alpha(2r_{1}+1)\Big{)}\\\
&\Big{(}2r_{2}+|\phi|_{g_{0}}^{2}+\alpha(4r_{2}+2)\Big{)}\Big{(}(2r_{2}+2)-|\phi|_{g_{0}}^{2}+\alpha(4r_{2}+2)\Big{)}\\\
&\Big{(}\Delta
f_{0}+r_{1}+\alpha(2r_{1}+1)\Big{)}+\sqrt{-1}\frac{\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g_{0}}}}{\omega_{\Sigma}}\\\
&\Big{(}2r_{2}+|\phi|_{h_{0}}^{2}+\alpha(4r_{2}+2)\Big{)}\Big{(}\Delta
f_{0}+r_{1}+\alpha(2r_{1}+1)\Big{)}\Big{\\}}(q),\end{split}$ (16)

Applying Lemma 3.1 to the right-hand side term of the equation (16), the
following inequality holds for some $C>0$, independent of $\varepsilon$

$\begin{split}&\Big{\\{}\Big{(}\Delta
f+r_{1}+\alpha(1-t)(2r_{1}+1)\Big{)}\Big{(}(2r_{2}+2)-|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\\\
&\Big{(}2r_{2}+|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\Big{\\}}(q)\leq
C.\end{split}$ (17)

Equation (6) implies

$\Big{\\{}4(|\phi|_{g}^{2}-1)\Big{(}\Delta
f+r_{1}+\alpha(1-t)(2r_{1}+1)\Big{)}\Big{\\}}(q)\leq
2(2r_{1}+1)(4r_{2}+2)(2\alpha+1)\varepsilon\psi(q)$ (18)

Using Lemma3.2, equations (17) and (18) we can see

$\displaystyle 2(2r_{1}+1)(4r_{2}+2)(2\alpha+1)\varepsilon\psi(q)$
$\displaystyle\geq\frac{2C\big{(}|\phi|^{2}_{g}-1\big{)}}{\Big{(}(2r_{2}+2)-|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\Big{(}2r_{2}+|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}}(q)$
$\displaystyle\geq-C.$ (19)

Therefore, equations (15) and (3.2) establish the result. ∎

Now we are in a position to prove an essential estimate, which will be used to
show closedness as well as openness.

######  Lemma 3.4.

For $t\in I_{2}$, $-\tilde{C}\leq\Delta\psi_{t}\leq C$ for some positive
constant $\tilde{C}$ independent of $\varepsilon$.

###### Proof.

Since $t\in I_{2}$, we have $\Delta f+r_{1}+\alpha(1-t)(2r_{1}+1)>0$. So from
equations (6) we get

$\begin{split}&\big{(}1+\Delta\psi\big{)}\Big{(}2(|\phi|^{2}_{g}-1)+(1+2\alpha(1-t))(4r_{2}+2)\Big{)}\\\
&\geq 2(2r_{1}+1)(4r_{2}+2)(2\alpha+1)\varepsilon\psi_{t}.\end{split}$

Thus using Lemma3.3 we see $\Delta\psi\geq-\tilde{C}$, for some positive
$\tilde{C}$ independent of $\varepsilon$. Now equation (5) gives

$\begin{split}&\big{(}1+\Delta\psi\big{)}\Big{(}\Delta
f+r_{1}+\alpha(1-t)(2r_{1}+1)\Big{)}\Big{(}(2r_{2}+2)-|\phi|_{g}^{2}\\\
&+\alpha(1-t)(4r_{2}+2)\Big{)}\Big{(}2r_{2}+|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\leq
C.\end{split}$

Using equation (6) we have

$\begin{split}&\big{(}1+\Delta\psi\big{)}\bigg{\\{}\big{(}1+\Delta\psi\big{)}\Big{(}2(|\phi|^{2}_{g}-1)+\Big{(}1+2\alpha(1-t)\Big{)}(4r_{2}+2)\Big{)}\\\
&\qquad\qquad-2(2r_{1}+1)(4r_{2}+2)(2\alpha+1))\varepsilon\psi\bigg{\\}}\\\
&\leq\displaystyle\frac{C}{\left((2r_{2}+2)-|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\right)\left(2r_{2}+|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\right)},\end{split}$

which implies $\Delta\psi\leq C$. ∎

With these above estimates of $\psi_{t}$ in hand, we now estimate $f_{t}$ as
follows.

######  Lemma 3.5.

If $t\in I_{2}$, then $||\Delta f_{t}||\leq C$.

###### Proof.

Since $t\in I_{2}$, we have $\Delta f+r_{1}+\alpha(1-t)(2r_{1}+1)>0$.
Therefore from the equation (5) we get

$\Big{(}\Delta
f+\Delta\psi+(r_{1}+1)+\alpha(1-t)(2r_{1}+1)\Big{)}\Big{(}\Delta
f+r_{1}+\alpha(1-t)(2r_{1}+1)\Big{)}\leq C$

Therefore $\Delta f\leq C$, otherwise it contradicts Lemma 3.4. ∎

Using the formula (11) and Lemma 3.5, we have the following.

######  Lemma 3.6.

There exists positive constant $C$ such that $||f_{t}||_{C^{0}}\leq C$,
whenever $t\in I_{2}$.

Now one can easily conclude that $I_{2}$ is closed as follows.

Let $t_{n}\in I_{2}$ be such that $t_{n}\rightarrow t$. To prove $I_{2}$ is
closed, we must show $t\in I_{2}$. As $t_{n}\in I_{2}$, we have $\Delta
f_{t_{n}}+r_{1}+\alpha(1-t)(2r_{1}+1)>0$. Arzela-Ascoli theorem, together with
the above estimates, we get a subsequence of $t_{n}$ again call the
subsequence by $t_{n}$ such that $f_{t_{n}}\rightarrow f$ and
$\psi_{t_{n}}\rightarrow\psi$ in $C^{2,\beta}$. Then $\Delta
f+r_{1}+\alpha(1-t)(2r_{1}+1)\geq 0$ and by usual bootstrapping argument we
have $f$ and $\psi$ are smooth. Now if $\Delta
f+r_{1}+\alpha(1-t)(2r_{1}+1)=0$ at some point, then (5) gives a
contradiction. Hence $\Delta f+r_{1}+\alpha(1-t)(2r_{1}+1)>0$. Therefore $t\in
I_{2}$.

The only thing we are left with is proving the openness of $I_{2}$.

###  3.3. Openness of $I_{2}$

For $0<\beta<1$, let $\mathcal{C}$ be the subset of $C^{2,\beta}\times
C^{2,\beta}$ defined by $\mathcal{C}:=\Big{\\{}(f,\psi)\in C^{2,\beta}\times
C^{2,\beta}\Big{|}\Delta
f+r_{1}+\alpha(1-t)(2r_{1}+1)>0,\displaystyle\int_{\Sigma}f\omega_{\Sigma}=0\Big{\\}}$.
Now let us define the map $T:\mathcal{C}\times[0,1]\rightarrow
C^{0,\beta}\times C^{0,\beta}$ by
$T(f,\psi,t)=\Big{(}T_{1}(f,\psi,t),T_{2}(f,\psi,t)\Big{)}$, where

$\begin{split}&T_{1}(f,\psi,t)\\\ =&\bigg{\\{}\Big{(}\Delta
f+\Delta\psi+(r_{1}+1)+\alpha(1-t)(2r_{1}+1)\Big{)}\Big{(}2r_{2}+|\phi|_{g}^{2}+\\\
&\alpha(1-t)(4r_{2}+2)\Big{)}\Big{(}\Delta
f+r_{1}+\alpha(1-t)(2r_{1}+1)\Big{)}\\\
&\Big{(}(2r_{2}+2)-|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}+\sqrt{-1}\frac{\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g}}}{\omega_{\Sigma}}\\\
&\Big{(}2r_{2}+|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\Big{(}\Delta
f+r_{1}+\alpha(1-t)(2r_{1}+1)\Big{)}\bigg{\\}}\\\ &-\bigg{\\{}\Big{(}\Delta
f_{0}+\Delta\psi_{0}+(r_{1}+1)+\alpha(2r_{1}+1)\Big{)}\Big{(}2r_{2}+|\phi|_{g_{0}}^{2}+\alpha(4r_{2}+2)\Big{)}\\\
&\Big{(}(2r_{2}+2)-|\phi|_{g_{0}}^{2}+\alpha(4r_{2}+2)\Big{)}\Big{(}\Delta
f_{0}+r_{1}+\alpha(2r_{1}+1)\Big{)}\\\
&+\sqrt{-1}\frac{\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g_{0}}}}{\omega_{\Sigma}}\Big{(}2r_{2}+|\phi|_{g_{0}}^{2}+\alpha(4r_{2}+2)\Big{)}\\\
&\Big{(}\Delta f_{0}+r_{1}+\alpha(2r_{1}+1)\Big{)}\bigg{\\}},\end{split}$ (20)

and

$\begin{split}&T_{2}(f,\psi,t)\\\ =&2\bigg{(}2\Delta
f+\Delta\psi+\big{(}1+2\alpha(1-t)\big{)}(2r_{1}+1)\bigg{)}(|\phi|_{g}^{2}-1)\\\
&+\big{(}\Delta\psi+1\big{)}\big{(}1+2\alpha(1-t)\big{)}(4r_{2}+2)\\\
&-2(2r_{1}+1)(4r_{2}+2)(2\alpha+1)\varepsilon\psi.\end{split}$ (21)

Then the linearization of $T_{1}$ at a point $(f,\psi)$ will be

$\begin{split}&DT_{1}(f,\psi,t)[\delta f,\delta\psi]\\\
=&\Big{(}2r_{2}+|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\bigg{[}\Big{(}(2r_{2}+2)-|\phi|_{g}^{2}+\\\
&\alpha(1-t)(4r_{2}+2)\Big{)}\Big{(}2\Delta
f+\Delta\psi+\big{(}1+2\alpha(1-t)\big{)}(2r_{1}+1)\Big{)}\\\
&+\sqrt{-1}\frac{\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g}}}{\omega_{\Sigma}}\bigg{]}\Delta\delta
f+\Big{[}\Big{(}2r_{2}+|\phi|_{g}^{2}+\\\
&\alpha(1-t)(4r_{2}+2)\Big{)}\Big{(}\Delta
f+r_{1}+\alpha(1-t)(2r_{1}+1)\Big{)}\\\
&\Big{(}(2r_{2}+2)-|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\Big{]}\Delta\delta\psi\\\
&-\Big{(}\Delta
f+r_{1}+\alpha(1-t)(2r_{1}+1)\Big{)}\bigg{[}|\phi|_{g}^{2}\Big{\\{}2(1-|\phi|_{g}^{2})\\\
&\Big{(}\Delta f+\Delta\psi+(r_{1}+1)+\alpha(1-t)(2r_{1}+1)\Big{)}+\\\
&\sqrt{-1}\frac{\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g}}}{\omega_{\Sigma}}\Big{\\}}+\sqrt{-1}\frac{\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g}}}{\omega_{\Sigma}}\\\
&\Big{(}2r_{2}+|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\bigg{]}\delta\psi\\\
&+\Big{(}\Delta f+r_{1}+\alpha(1-t)(2r_{1}+1)\Big{)}\\\
&\bigg{[}\sqrt{-1}\frac{\partial(\delta\psi)\wedge\nabla^{0,1}\phi^{{\dagger}_{g}}}{\omega_{\Sigma}}+\sqrt{-1}\frac{\nabla^{1,0}\phi\wedge\phi^{{\dagger}_{g}}\nabla^{0,1}(\delta\psi)}{\omega_{\Sigma}}\bigg{]},\end{split}$
(22)

and the linearization of $T_{2}$ at $(f,\psi)$ will be

$\begin{split}&DT_{2}(f,\psi,t)[\delta f,\delta\psi]\\\
=&4(|\phi|_{g}^{2}-1)\Delta\delta
f+\bigg{(}2(|\phi|_{g}^{2}-1)+\Big{(}1+2\alpha(1-t)\Big{)}(4r_{2}+2)\bigg{)}\Delta\delta\psi\\\
&-\delta\psi\bigg{[}2|\phi|_{g}^{2}\bigg{(}2\Delta
f+\Delta\psi+\Big{(}1+2\alpha(1-t)\Big{)}(2r_{1}+1)\bigg{)}+\\\
&2(2r_{1}+1)(4r_{2}+2)(2\alpha+1)\varepsilon\bigg{]}.\end{split}$ (23)

We shall show that $DT=[DT_{1},DT_{2}]$ is an isomorphism at a point
$(f_{t},\psi_{t})$, for $t\in I_{2}$. Then by the implicit function theorem
for Banach manifolds, we can conclude that $I_{2}$ is open.

Suppose $(\delta f,\delta\psi)\in Ker(DT(f,\psi,t))$ where $t\in I_{2}$, then
we have $DT_{1}(f,\psi,t)[\delta f,\delta\psi]=0$ and $DT_{2}(f,\psi,t)[\delta
f,\delta\psi]=0$. Now solving $DT_{2}[\delta f,\delta\psi]=0$ for
$\Delta\delta f$ and substituting the value in $DT_{1}[\delta f,\delta\psi]=0$
we get,

$\begin{split}&\Bigg{[}\Big{(}2r_{2}+|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\bigg{\\{}\Big{(}(2r_{2}+2)-|\phi|_{g}^{2}+\\\
&\alpha(1-t)(4r_{2}+2)\Big{)}\Big{(}2\Delta
f+\Delta\psi+(1+2\alpha(1-t))(2r_{1}+1)\Big{)}\\\
&+\sqrt{-1}\frac{\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g}}}{\omega_{\Sigma}}\bigg{\\}}\Big{(}2(|\phi|_{g}^{2}-1)+\left(1+2\alpha(1-t)\right)(4r_{2}+2)\Big{)}\\\
&+4\left(1-|\phi|^{2}_{g}\right)\Big{(}2r_{2}+|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\Big{(}\Delta
f+r_{1}\\\
&+\alpha(1-t)(2r_{1}+1)\Big{)}\Big{(}(2r_{2}+2)-|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\Bigg{]}\Delta\delta\psi\\\
=&\delta\psi\Bigg{[}\Big{(}2r_{2}+|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\bigg{\\{}\Big{(}(2r_{2}+2)-|\phi|_{g}^{2}+\\\
&\alpha(1-t)(4r_{2}+2)\Big{)}\Big{(}2\Delta
f+\Delta\psi+(1+2\alpha(1-t))(2r_{1}+1)\Big{)}\\\
&+\sqrt{-1}\frac{\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g}}}{\omega_{\Sigma}}\bigg{\\}}\bigg{\\{}2|\phi|_{g}^{2}\Big{(}2\Delta
f+\Delta\psi+(1+2\alpha(1-t))(2r_{1}+1)\Big{)}\\\
&+2(2r_{1}+1)(4r_{2}+2)(2\alpha+1)\varepsilon\bigg{\\}}+4\big{(}1-|\phi|^{2}_{g}\big{)}|\phi|_{g}^{2}\Big{(}\Delta
f+r_{1}+\\\
&\alpha(1-t)(2r_{1}+1)\Big{)}\bigg{\\{}2(1-|\phi|_{g}^{2})\Big{(}\Delta
f+\Delta\psi+(r_{1}+1)+\\\
&\alpha(1-t)(2r_{1}+1)\Big{)}+\sqrt{-1}\frac{\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g}}}{\omega_{\Sigma}}\bigg{\\}}+\sqrt{-1}\frac{\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g}}}{\omega_{\Sigma}}\\\
&\Big{(}2r_{2}+|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\Bigg{]}+\Big{(}\Delta
f+r_{1}+\alpha(1-t)(2r_{1}+1)\Big{)}\\\
&\bigg{\\{}\sqrt{-1}\frac{\partial(\delta\psi)\wedge\nabla^{0,1}\phi^{{\dagger}_{g}}}{\omega_{\Sigma}}+\sqrt{-1}\frac{\nabla^{1,0}\phi\wedge\phi^{{\dagger}_{g}}\nabla^{0,1}(\delta\psi)}{\omega_{\Sigma}}\bigg{\\}}.\end{split}$
(24)

Lemma (3.4) shows that for $t\in I_{2}$, we can choose $\varepsilon$ large
enough so that

$\displaystyle\Big{(}\Delta\psi+1+2(2r_{1}+1)(4r_{2}+2)(2\alpha+1)\varepsilon\Big{)}>0.$

Using the maximum principle on the equation (24), we have $\delta\psi=0$. Now
putting $\delta\psi=0$ in $DT_{2}[\delta f,\delta\psi]=0$ gives $\delta f=0$.
Hence for $t\in I_{2}$ we get $Ker(DT(f_{t},\psi_{t},t))=0$. Now we shall
prove that $Ker(DT^{*}(f_{t},\psi_{t},t))$ is also trivial.

As one can imagine, computing the operator $DT^{*}$ will be very complicated.
Since we are only interested in the kernel of $DT^{*}$ and we already know the
kernel of $DT$, it is enough to calculate the index of the operator $DT$. To
do so, let us define the following.

For $t\in I_{2}$ and $s\in[0,1]$, let
$T^{s}\big{(}f,\psi,t\big{)}=\Big{(}T_{1}^{s}\big{(}f,\psi,t\big{)},T_{2}^{s}\big{(}f,\psi,t\big{)}\Big{)}$,
where

$\begin{split}&T^{s}_{1}(f,\psi,t)\\\
=&\Big{(}2r_{2}+s|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\bigg{[}\Big{(}(2r_{2}+2)-s|\phi|_{g}^{2}+\\\
&\alpha(1-t)(4r_{2}+2)\Big{)}\Big{(}2s\Delta
f+s\Delta\psi+\big{(}1+2\alpha(1-t)\big{)}(2r_{1}+1)\Big{)}+\\\
&\sqrt{-1}s\frac{\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g}}}{\omega_{\Sigma}}\bigg{]}\Delta\delta
f+\Delta\delta\psi\Big{(}2r_{2}+s|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\\\
&\Big{(}s\Delta
f+r_{1}+\alpha(1-t)(2r_{1}+1)\Big{)}\Big{(}(2r_{2}+2)-s|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)},\end{split}$
(25)

and

$\begin{split}&T^{s}_{2}(f,\psi,t)\\\ =&4s(|\phi|_{g}^{2}-1)\Delta\delta
f+\bigg{(}2s(|\phi|_{g}^{2}-1)+\Big{(}1+2\alpha(1-t)\Big{)}(4r_{2}+2)\bigg{)}\Delta\delta\psi.\end{split}$
(26)

The following lemma says that we can talk about the index of operator $T^{s}$.

######  Lemma 3.7.

$T^{s}$ are elliptic system for $s\in[0,1]$.

###### Proof.

Let

$\displaystyle A=\begin{bmatrix}A_{11}&A_{12}\\\ A_{21}&A_{22}\end{bmatrix}$

where,

$\begin{split}A_{11}&=\Big{(}2r_{2}+s|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\\\
&\bigg{\\{}\Big{(}(2r_{2}+2)-s|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\\\
&\Big{(}2s\Delta
f+s\Delta\psi+\big{(}1+2\alpha(1-t)\big{)}(2r_{1}+1)\Big{)}\\\
&+s\sqrt{-1}\frac{\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g}}}{\omega_{\Sigma}}\bigg{\\}},\\\
A_{12}&=\Big{(}2r_{2}+s|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\\\
&\Big{(}s\Delta f+r_{1}+\alpha(1-t)(2r_{1}+1)\Big{)},\\\
&\Big{(}(2r_{2}+2)-s|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)},\\\
A_{21}&=4s(|\phi|_{g}^{2}-1),\\\
A_{22}&=2s(|\phi|_{g}^{2}-1)+\Big{(}1+2\alpha(1-t)\Big{)}(4r_{2}+2).\end{split}$

Then

$\displaystyle det(A)$ $\displaystyle=\begin{vmatrix}A_{11}&A_{12}\\\
A_{21}&A_{22}\end{vmatrix}$
$\displaystyle=\Big{(}2r_{2}+s|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\Bigg{\\{}\Big{(}1+2\alpha(1-t)\Big{)}$
$\displaystyle(4r_{2}+2)\Big{[}\Big{(}(2r_{2}+2)-s|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}$
$\displaystyle\Big{(}2s\Delta
f+s\Delta\psi+\big{(}1+2\alpha(1-t)\big{)}(2r_{1}+1)\Big{)}+s\sqrt{-1}\frac{\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g}}}{\omega_{\Sigma}}\Big{]}$
$\displaystyle+2s(|\phi|_{g}^{2}-1)\Big{[}\Big{(}(2r_{2}+2)-s|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}$
$\displaystyle\Big{(}s\Delta\psi+1\big{)}+s\sqrt{-1}\frac{\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g}}}{\omega_{\Sigma}}\Big{]}\Bigg{\\}}$

Case (a). If

$\begin{split}&\Big{[}\Big{(}(2r_{2}+2)-s|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\big{(}s\Delta\psi+1\big{)}\\\
&\qquad+s\sqrt{-1}\frac{\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g}}}{\omega_{\Sigma}}\Big{]}\leq
0.\end{split}$

As $t\in I_{2}$ i.e., $\Delta f+r_{1}+\alpha(1-t)(2r_{1}+1)>0$, therefore
$det(A)>0$.

Case (b). If

$\begin{split}&\Big{[}\Big{(}(2r_{2}+2)-s|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\big{(}s\Delta\psi+1\big{)}\\\
&\qquad\qquad+s\sqrt{-1}\frac{\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g}}}{\omega_{\Sigma}}\Big{]}>0.\end{split}$

Then,

$\begin{split}det(A)=&\Bigg{[}\Big{(}2r_{2}+s|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\Big{(}1+2\alpha(1-t)\Big{)}(4r_{2}+2)\\\
&\Big{(}(2r_{2}+2)-s|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}2\Big{(}s\Delta
f+r_{1}+\\\
&\alpha(1-t)(2r_{1}+1)\Big{)}\Bigg{]}+\Bigg{[}\Big{(}2r_{2}+s|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\\\
&\Big{\\{}\Big{(}(2r_{2}+2)-s|\phi|_{g}^{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\big{(}s\Delta\psi+1\big{)}+\\\
&s\sqrt{-1}\frac{\nabla^{1,0}\phi\wedge\nabla^{0,1}\phi^{{\dagger}_{g}}}{\omega_{\Sigma}}\Big{\\}}\Big{\\{}2s(|\phi|_{g}^{2}-1)+\Big{(}1+2\alpha(1-t)\Big{)}(4r_{2}+2)\Big{\\}}\Bigg{]}>0,\end{split}$

the first term is positive as $t\in I_{2}$, and the second is positive because
of the assumption. Hence $det(A)>0$ for $t\in I_{2}$. ∎

We see that $T^{s}:T^{0}\simeq T^{1}$ defines a homotopy. So the index of
Fredholm operators $T^{s}$ defined by
$Ind(T^{s})=dim(Ker(T^{s}))-dim(Coker(T^{s}))$ will be constant for $0\leq
s\leq 1$, in particular $Ind(T^{0})=Ind(T^{1})$. Now

$\begin{split}&T^{0}(f,\psi,t)\\\
=&\Bigg{(}\bigg{\\{}\Big{(}2r_{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\Big{(}(2r_{2}+2)+\alpha(1-t)(4r_{2}+2)\Big{)}\\\
&\big{(}1+2\alpha(1-t)\big{)}(2r_{1}+1)\Delta\delta
f+\Big{(}2r_{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\\\
&\Big{(}r_{1}+\alpha(1-t)(2r_{1}+1)\Big{)}\Big{(}(2r_{2}+2)+\alpha(1-t)(4r_{2}+2)\Big{)}\Delta\delta\psi\bigg{\\}},\\\
&\big{(}1+2\alpha(1-t)\big{)}(4r_{2}+2)\Delta\delta\psi\Bigg{)},\end{split}$

and

$\begin{split}&{T^{0}}^{*}(f,\psi,t)\\\
=&\Bigg{(}\bigg{\\{}\big{(}1+2\alpha(1-t)\big{)}(4r_{2}+2)\Delta\delta
f-\Big{(}2r_{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\\\
&\Big{(}r_{1}+\alpha(1-t)(2r_{1}+1)\Big{)}\Big{(}(2r_{2}+2)+\alpha(1-t)(4r_{2}+2)\Big{)}\Delta\delta\psi\bigg{\\}},\\\
&\Big{(}2r_{2}+\alpha(1-t)(4r_{2}+2)\Big{)}\Big{(}(2r_{2}+2)+\alpha(1-t)(4r_{2}+2)\Big{)}\\\
&\big{(}1+2\alpha(1-t)\big{)}(2r_{1}+1)\Delta\delta\psi\Bigg{)}.\end{split}$

A simple calculation gives $Ind(T^{0})=0$. Since $DT$ and $T^{1}$ have the
same principal symbol, their index must be the same. Hence, $Ind(DT)=0$ and
consequently $Ker(DT)=Ker(DT^{*})=0$. Therefore Fredholm’s alternative implies
that $DT$ is an isomorphism. This completes the proof of the theorem (2).

## Acknowledgements

I am grateful to my advisor, Vamsi Pritham Pingali, for suggesting this
problem. I also thank him for the invaluable and fruitful discussion about the
same. This work is supported by a scholarship from the Indian Institute of
Science.

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