Also at] Graduate School of Engineering Science, Osaka University, Osaka
560-8531, Japan

# A thermal convection limit of spiral state in wide-gap spherical Couette
flow

Tomoaki Itano<EMAIL_ADDRESS>Fumitoshi Goto  Kazuki Yoshikawa
Department of Pure and Applied Physics, Faculty of Engineering Science, Kansai
University, Osaka, 564-8680, Japan  Masako Sugihara-Seki  [  Department of
Pure and Applied Physics, Faculty of Engineering Science, Kansai University,
Osaka, 564-8680, Japan

###### Abstract

The symmetries of flow structures are often prescribed by their mechanical
instability and geometry. Here, as an example, we present the homotopy of the
rotating 3-fold spiral state that is robust in a spherical Couette flow
towards the hybrid system with a thermal stratification effect. It has not yet
been confirmed that the rotating wave state smoothly connects to the thermal
stratification system. Through continuation, the most dangerous mode at a
purely spherical Couette flow of $m=4$ modes of spherical harmonics is
replaced by $l=4$ and $m=3$ in a purely thermal convective system. For the
state obtained at the limit under only the thermal effect, the residual
quantities of both the torque to the outer sphere and meridional circulation
are discussed in detail.

###### pacs:

47.27.De, 47.20.Ky

††preprint: AIP/123-QED

##  I  Introduction

Fluid motion subject to multiple diffusion processes on a sphere exhibits rich
spatiotemporal properties despite their geometrical restrictions. The
mathematical similarity underlying their independent mechanisms has attracted
significant attention from researchers because of the importance of resolving
the formation of inhomogeneous structures in biological organismsAla52  and
spatial patterns in geophysical and astrophysical bodiesFow04 . Conventionally
studied thermal convective systems, which are configurations that are
applicable in astrophysical and geophysical flows, may also be categorized
into the same class in which two conservative quantities, momentum and heat
that function as suppression and activation factors, competitively diffuse.

Thermal convection in a spherical geometry has been studied with the addition
of other mechanical or chemical effectsCha61 . For instance, the influence of
the radial-dependence proportionality of the gravity field on convection has
been studied experimentally and numerically under microgravity conditions at
the International Space StationFeu11 . The addition of both electric
conductivity and system rotation to the spherical thermal convection system
may generate pairs of cyclonic and anticyclonic vortex columns in the
shellKid97 . They can reproduce the self-generation and reversal of magnetic
fields related to geophysical phenomenaSak99 . According to the Taylor-
Proudman theorem, vortex columns induced by thermal convection under the
rotation of the entire system align with the rotation axis via the Coriolis
effect, which would be realizable even in the absence of an electromagnetic
effectKim11 . From the viewpoint of geophysics, it is interesting that the
generated vortex columns rotate either faster or slower than the system Feu13
; Feu15 . In these examples, the transition is initiated by vortices
satisfying the mirror symmetry on the equatorial plane because the Coriolis
effect suppresses the freedom of the system in the polar direction.

However, the flow that is confined between spherical boundaries that rotate
differentially, namely, spherical Couette flow (SCF)Egb95 ; Wul99 ; Nak02 ;
Abb18a , is of interest because it can reproduce the turbulent state even
without the thermal effect. In particular, the case between a rotating inner
sphere and a stationary outer sphere has been extensively investigated. Past
investigations with several different combinations of spherical boundaries
with various radiiMun75 ; Bel84 ; Egb95  have shown that SCFs with a
relatively wide gap are different from cylindrical Taylor-Couette flows
despite the apparent similarity. Although SCF also leads to vortex columns
induced under the Coriolis effect, the transition typical in shear applied
between spheres that rotate differently is initiated by vortices generated in
a staggered manner about the equatorial plane. This may be attributed to the
presence of meridional circulations due to differential rotation, which is
more influential in the case of a larger gap between spherical boundaries.
Thus, the symmetry that appears in the transitional stage is distinct from
vortex columns induced by the rotation of the entire systemHol06 . In SCF,
rotating wave states with a few azimuthal wavenumbers are robust and sustained
as temporally quasi-periodic states under an external axial magnetic
fieldGar18 ; Gar20 .

Here, we examine the robustness of the rotational wave state in a wide-gap SCF
against unstable thermal stratification. Through the current study, we also
hope to apply the authenticity of the state in SCF (confirmed both numerically
and experimentally) to our uncertain knowledge of spherical Bénard convection
(SBC) under the radial gravitational field, which has only been investigated
numerically. The second purpose was to confirm the potential of residual shear
stress between the boundaries of the SCF under thermal convection. Examining
the transition of laminar flow to rotating wave states perturbed by the
thermal factor reconfirmed a phase diagram, which was obtained in Ref.Ina19 ,
in the extended control parameter space.

The remainder of this paper is organized as follows. We first formulate our
SCF and SBC hybrid system. We describe the numerical method used for time
development followed by describing the homotopy continuation to specify the
rotating wave state in the hybrid system. After commenting on the measures of
symmetry, we explain the path in the parameter space. We then give details of
the numerically obtained states. We then discuss the extent to which non-
axisymmetry and rotation are destroyed around the polar axis at the
continuation limit.

##  II  Numerical Method

We consider a radioactive Boussinesq fluid confined in a spherical shell with
a thickness of $2\Delta\tilde{r}$ and mean radius $\tilde{r}_{0}$. The fluid
was subjected to shear applied by the differential rotation of the boundaries.
We assume that the mass and internal heat sources are distributed
homogeneously inside the entire sphere as well as within the shell. The heat
radiation at the outer surface of the shell may generate a nonequilibrium
state in the shell with a temperature difference of $2\Delta\tilde{T}$ between
its inner and outer surfaces. The length, time, and temperature may be
nondimensionalized by $\Delta\tilde{r}$, diffusion time
$\Delta\tilde{r}^{2}/\tilde{\nu}$, and $\Delta\tilde{T}$, respectively, where
$\tilde{\nu}$ is the kinematic viscosity. The nondimensionalized equations
Ina19  that govern the velocity and temperature fields, ${\bm{u}}$ and
$\varTheta$, are

$\frac{{\rm D}{\bm{u}}}{{\rm
D}t}=-{\bm{\nabla}}p+{\bm{\nabla}}^{2}{\bm{u}}+\mbox{{\rm Gr}}\varTheta
r{\bm{e}}_{r}\ \ ,\frac{{\rm D}\Theta}{{\rm D}t}=\frac{1}{\mbox{{\rm
Pr}}}\Bigl{(}{\bm{\nabla}}^{2}\varTheta+3\frac{1-\eta}{1+\eta}\Bigr{)}\ \ .$

The third term on the right-hand side of the first equation, the gravity
acceleration proportional to the radius, is attributed to the homogeneity of
the mass distribution, while the first term on the right-hand side of the
second equation is the homogeneity of the heat source in the inner sphere as
well as the shell. The no-slip boundary and isothermal conditions are adopted,
that is, ${\bm{u}}=(r_{0}-1)\Omega_{\rm in}{\bm{e}}_{\phi}$ and $\Theta=1$ at
the inner surface, and ${\bm{u}}=0$ and $\Theta=-1$ at the outer surface of
the shell, where $r_{0}=\tilde{r}_{0}/\Delta\tilde{r}$. The system is uniquely
determined by four dimensionless parameters: Prandtl number Pr, Grashof number
Gr that is proportional to the applied radial temperature difference, radius
ratio $\displaystyle\eta=\frac{r_{0}-1}{r_{0}+1}$, and Reynolds number
$\displaystyle\mbox{{\rm Re}}=\frac{(r_{0}-1)^{2}\Omega_{\rm in}}{\nu}$. In
this study, we focused on the cases of $\mbox{{\rm Pr}}=1$ and $\eta=1/2$.

For the divergence-free constraint on ${\bm{u}}$, we invoke the conventional
toroidal and poloidal decompositions for the radial direction:

${\bm{u}}={\bm{u}}_{\rm
S}+{\bm{\nabla}}\times\bigl{(}-{\bm{r}}\varPsi+{\bm{\nabla}}\times({\bm{r}}\varPhi)\bigr{)}\
\ ,$

where ${\bm{u}}_{\rm S}=-{\bm{\nabla}}\times\bigl{(}{\bm{r}}\varPsi_{\rm
S}\bigr{)}$ is Stokes flow, ${\bm{u}}_{\rm S}=u_{\rm
S}(r,\theta){\bm{e}}_{\phi}$, which contains neither radial nor polar
components and satisfies the aforementioned no-slip boundary conditions. Note
that the function $\varPsi_{\phi}$ is determined only by the value of Re;
thus, the system is implicitly dependent on Re via the ${\bm{u}}_{\rm S}$
introduced into the governing equation. We spatially expanded the scalar
fields $\varPhi$, $\varPsi$, and $\varTheta$ in terms of Chebyshev polynomials
and spherical harmonics with the aid of open numerical librariesSch13 ; Fri05
. For instance,
$\varPhi(r,\theta,\phi)=\sum_{l,m,n}\varPhi_{l,m,n}Y_{l}^{m}(\theta,\phi)T_{n}(y)$
using the relative radius $y=[-1,1]$, where $r=r_{0}+y$.

By adapting the second-order Adams-Bashforth method accompanied by the Crank-
Nicolson method for temporal discretization and the Gauss-Lobatto collocation
method for the evaluation of nonlinear terms, we can convert the dimensionless
governing equation to an equivalent inhomogeneous Helmholtz equation with at
first order for $\varPsi$ and $\varTheta$ and second order for $\varPhi$.
These Helmholtz equations are equivalent to a set of linear algebraic
equations for the expansion coefficients $\varPsi_{l,m,n}$, $\varPhi_{l,m,n}$,
and $\varTheta_{l,m,n}$, which are solved using the LAPACK librariesAnd99 .
Unless indicated, we used the truncation levels $(l_{\rm max},m_{\rm
max},n_{\rm max})=(30,30,32)$. According to Ref.Wul99 ; Hol06 ; Nak02 ; Jun00
, for the purely spherical Couette flow with $\eta=1/2$, the basic
axisymmetric state, namely “0-vortex” state, is known to lose stability to a
non-axisymmetric rotating wave perturbation when the Grashof number is
increased above a threshold value denoted by $\mbox{{\rm Re}}_{\rm cr}$. By
starting the numerical integration with a Stokes flow with a small disturbance
that was artificially generated by a series of random numbers, we also
obtained nonequilibrium and rotating wave states around the transitional
Reynolds numbers for $\eta=1/2$. The developed numerical codeIna19  was
validated for $\mbox{{\rm Re}}<600$ and $\mbox{{\rm Gr}}=0$ using quantitative
comparison with previous experimental and numerical results. In this study,
our attention was restricted to the rotating wave state generated over the
transitional Reynolds number for a purely spherical Couette flow with a wide
gap.

We demonstrated a purely spherical Couette flow experiment with $\eta=1/2$ and
reproduced the rotating wave states over transitional Reynolds numbers. In the
expreriment, the rotating wave states were actually illuminated as steady
pattern of aluminium flakes on a laser light sheet projected either on the
azimuthal or meridional planes. The angular velocity of the experimentally
obtained state is in qualitative agreement with that of the numerically
obtained state. It should be emphasized that the rotating wave state with a
constant angular velocity can be realized both numerically and experimentally.
Here, considering that this wave solution rotates with an unknown but constant
angular velocity $\Omega_{\phi}$, we may evaluate $\frac{\partial}{\partial
t}(\cdot)$ as $-\Omega_{\phi}\frac{\partial}{\partial\phi}(\cdot)$ in the set
of linear algebraic equations for the constant expansion coefficients.
Additionally, the freedom of $\Omega_{\phi}$ for the solution can be specified
by fixing the azimuthal phase of the solution. By solving the deduced
quadratic equations using the iterative Newton-Raphson method with the aid of
LAPACK libraries, we can uniquely specify all the coefficients of the rotating
solution and azimuthal angular velocity $\Omega_{\phi}$ simultaneously.

In the following section, we measure the degree of asymmetry for the states
obtained numerically. The degree of non-axisymmetry for the rotation axis and
of the anti-symmetry for the equatorial plane may be measured by the following
$L_{2}$ norm normalized by the corresponding quantity of Stokes flow:

$L_{2}:=\frac{\int|\varPsi_{\rm AM,\rm 3D}|^{2}dv}{\int|\varPsi_{\rm
S}|^{2}dv}\ \ ,$

where $\displaystyle\varPsi_{\rm AM,\rm 3D}:=\sum_{l+m\in{\rm odd},m\neq
0,n}\varPsi_{l,m,n}Y_{l}^{m}(\theta,\phi)T_{n}(y)$ and we termed anti-
mirroring as “AM”. The value of $L_{2}$ is zero for the axisymmetric state
such as in Stokes flow, satisfying $\partial/\partial_{\phi}=0$. Moreover, we
measured the degree of rotation around the polar axis either by the flow rate
across the meridional section $q_{\phi}$ or by the torque exerted on the outer
sphere $T_{z}$. After they are normalized by the corresponding quantity of the
Stokes flow, these values are defined as

$q_{\phi}:=\frac{\int_{{\rm S}_{1}}d{\bm{s}}\cdot{\bm{u}}}{\int_{{\rm
S}_{1}}d{\bm{s}}\cdot{\bm{u}}_{\rm S}}\ \ ,\ \
T_{z}=\frac{{\bm{e}}_{z}\cdot\int_{{\rm S}_{2}}{\bm{r}}\times
d{\bm{s}}\cdot{\bm{\nabla}}{\bm{u}}}{{\bm{e}}_{z}\cdot\int_{{\rm
S}_{2}}{\bm{r}}\times d{\bm{s}}\cdot{\bm{\nabla}}{\bm{u}}_{\rm S}}\ \ ,$

where the surfaces for calculating the integral ${\rm S}_{1}$ and ${\rm
S}_{2}$ are defined as $r=[r_{0}-1,r_{0}+1],\theta=[0,\pi],\phi=0$, and
$r=r_{0}+1,\theta=[0,\pi],\phi=[0,2\pi]$, respectively. Stokes flow should be
a good approximation of purely spherical Couette flow, particularly at low
Reynolds numbers. The polar component of the torque acting on the outer sphere
by the flow is analytically provided in Ref.Lan87 .

##  III  Results

As mentioned before, over the transitional Reynolds number in the system with
$\mbox{{\rm Gr}}=0$ (SCF), the system prefers non-axisymmetric rotating waves
compared to the basic axisymmetric state, which has been performed even in
experimentsEgb95 . We refer to the rotating wave state with the wavenumber $m$
as the $m$-fold (spiral) state. Here, note that the $m$-fold state satisfies
the symmetries
$\displaystyle\varPhi(r,\theta,\phi+2\pi/m)=\varPhi(r,\theta,\phi)$ and
$\varPsi(r,\theta,\phi+2\pi/m)=\varPsi(r,\theta,\phi)$. The preferred state
also satisfies shift-and-reflection symmetry with respect to the equatorial
plane; that is, the fields in both hemispheres shift toward each other by half
the wavelength,
$\displaystyle\varPhi(r,\pi-\theta,\phi+\pi/m)=\varPhi(r,\theta,\phi)$ and
$\displaystyle\varPsi(r,\pi-\theta,\phi+\pi/m)=-\varPsi(r,\theta,\phi)$. At
the limit of $\mbox{{\rm Gr}}=0$, the temperature field passively advected in
the governing equation in the SCF inherits symmetry from $\varPhi$. Thus, it
is natural to assume that
$\displaystyle\varTheta(r,\theta,\phi+2\pi/m)=\varTheta(r,\theta,\phi)$ and
$\displaystyle\varTheta(r,\pi-\theta,\phi+\pi/m)=\varTheta(r,\theta,\phi)$ are
satisfied in continuation of the $m$-fold state to the hybrid system with
$\mbox{{\rm Gr}}\neq 0$. In a previous studyGot21 , we showed that the
$3$-fold state bifurcates from the axisymmetric state at a slightly higher
Reynolds number than the 4-fold state. We also showed that, with an increase
in the Reynolds number, the formation of a basin of attraction of the $3$-fold
state rapidly expands compared to that of the $4$-fold state. In particular,
the latter result should be emphasized, which explains how the latecomer, the
$3$-fold state, tends to be established experimentally as a stable state. This
bifurcation aspect is expected even in the system $\mbox{{\rm Gr}}\approx 0$.

Figure 1:  A square path starting at $(\mbox{{\rm Re}},\mbox{{\rm
Gr}})=(490,0)$ was used for the continuation of the 3 and 4-fold states. The
path consists of two segments: (A) $\mbox{{\rm Re}}=490$, $\mbox{{\rm
Gr}}=0\to 320$, and (B) $\mbox{{\rm Re}}=490\to 0$, $\mbox{{\rm Gr}}=320$,
which has the three points P(490,320), Q(200,320), and R(0,320). The dashed
and dotted curves indicate the neutral curves for the $m$-fold state
$\mbox{{\rm Re}}=\mbox{{\rm Re}}_{\rm cr}^{m}(\mbox{{\rm Gr}})$.

Hereafter, we will extend the 4- and 3-fold states obtained at $\mbox{{\rm
Gr}}=0$ towards the region of the $\mbox{{\rm Gr}}>0$ space. The phase diagram
in the space $(\mbox{{\rm Re}},\mbox{{\rm Gr}})$ has been tentatively explored
through a numerical survey in Ref.Ina19 , although there was no guarantee
whether the m-fold states are sustained in $\mbox{{\rm Gr}}>0$. Here, the
continuation of the 4- and 3-fold states is performed through a square path
starting at $(\mbox{{\rm Re}},\mbox{{\rm Gr}})=(490,0)$, as shown in Fig.1.
The path consists of two segments: (A) $\mbox{{\rm Re}}=490$, $\mbox{{\rm
Gr}}=0\to 320$, and (B) $\mbox{{\rm Re}}=490\to 0$, $\mbox{{\rm Gr}}=320$,
which has the three representative points P(490,320), Q(200,320), and
R(0,320). In the present study, the neutral curves $\mbox{{\rm Re}}=\mbox{{\rm
Re}}_{\rm cr}^{m}(\mbox{{\rm Gr}})$ were calculated independently for the 4-
and 3-fold states, respectively, which are indicated by the dashed and dotted
curves in the figure. The $m$-fold states bifurcate supercritically from the
axisymmetric state on the neutral curves, and their infinitesimal
nonaxisymmetric components exponentially decay for $\mbox{{\rm Re}}<\mbox{{\rm
Re}}_{\rm cr}^{m}(\mbox{{\rm Gr}})$ at a given Gr. A previous studyDum94 ;
Jun00 ; Got21  showed that 4- and 3-fold states bifurcate from the
axisymmetric state at $\mbox{{\rm Re}}_{\rm cr}^{4}(0)=489$ and $\mbox{{\rm
Re}}_{\rm cr}^{3}(0)=491$ at SCF. The present study shows that $\mbox{{\rm
Re}}=\mbox{{\rm Re}}_{\rm cr}^{m}(\mbox{{\rm Gr}})$ for both $m=3$ and $4$
monotonically decreased as Gr increased with an intersection at $(\mbox{{\rm
Re}},\mbox{{\rm Gr}})=(406,167)$. The decrease in $\mbox{{\rm Re}}=\mbox{{\rm
Re}}_{\rm cr}^{m}(\mbox{{\rm Gr}})$ is intuitively consistent with the fact
that either the increase in Re or the increase in Gr triggers the instability
of the system. Although $\mbox{{\rm Re}}=\mbox{{\rm Re}}_{\rm
cr}^{4}(\mbox{{\rm Gr}})$ passing once over segment A around $(261,320)$
reaches $\mbox{{\rm Gr}}>320$, $\mbox{{\rm Re}}=\mbox{{\rm Re}}_{\rm
cr}^{3}(\mbox{{\rm Gr}})$ exhibits a maximum value of approximately
$200<\mbox{{\rm Re}}<250$. It is expected that, through a path that
circumvents the regionIna19 , the continuation of the $m$-fold state educed at
the SCF towards the SBC limit is possible.

The non-axisymmetric component of the 4-fold state monotonically increases at
the starting point of segment A, although it finally attenuates with a
decrease in Re on segment B to degenerate to the axisymmetric state around
$(\mbox{{\rm Re}},\mbox{{\rm Gr}})=(261,320)$ (see also Fig.5). However, the
3-fold state survives up to the end of segment B. Thus, we only achieved the
continuation of the 3-fold state from SCF to SBC, but not that of the 4-fold
state. The linear stability analysis at the purely thermal convective
limitCha61 ; Ita15  showed that the most dangerous mode in the conduction
state consists of $Y_{l=4}^{m}$ and not $Y_{l=3}^{m}$ at $\mbox{{\rm Gr}}_{\rm
cr}$ ($m=0,1,\cdots,l$) in the classical Rayleigh-Bénard system in spherical
geometry with $\eta=1/2$. The replacement of the most dangerous mode from the
4-fold state at the SCF to the 3-fold state at the SBC on the path would seem
to be somewhat puzzling. However, it should be noted that the wavenumber of
the most dangerous wave at the SCF is related to the azimuthal wave number
$m$, but not to the polar wave number $l$ calculated by the linear stability
analysis. The principal mode of the state achieved at point R based on the
continuation of the 3-fold state is $Y_{l=4}^{m=3}$, as explained below.

Figure 2:  Polar and equatorial views of the 3-fold state at P (a,b), Q (c,d),
and R (e,f) on segment B. The isosurface was $\varPhi_{\rm 3D}=C$, where $C$
was a tenth of the median of $\varPhi_{\rm 3D}$.

From the starting point of segment A to the middle of segment B, the 3-fold
state sustains three spiral vortical arms alternately extending from the
equator to the poles in each hemisphere, which are the same as those
visualized in purely spherical Couette experiments at transitional Reynolds
numbersWul99 . Fig.2 shows the isosurface of $\varPhi_{\rm 3D}=C$ at point
P,Q, and R on segment B, where $\varPhi_{\rm 3D}$ is the non-axisymmetric
component of $\varPhi$. Note that an isosurface qualitatively corresponds to a
region of lift-up flow from the inner to the outer spheres for the non-
axisymmetric component. Close to point R from Q, the isosurface of the
disturbance seems to gradually shift from a shear-related flow to a state
originating at the heat convective flow. As Re vanishes, the amplitude of the
disturbance in the polar zone diminishes and the 3-fold state continuously
changes from a spiral pattern to a regularly convective pattern as six
slightly skewed triangular pillars are positioned on the faces of a regular
hexahedron, as can be seen in Fig.2(f), which could be related to the mode
$Y_{l=4}^{m=3}$.

At SBC ($\mbox{{\rm Re}}=0$), the unstable modes that degenerate owing to
spherical homogeneity emerge simultaneously from the static thermally
conductive state over the critical Grashof number $\mbox{{\rm Gr}}_{\rm cr}$.
Owing to the nonlinear interactions among these modes, highly symmetric steady
states that are invariant under a set of transformations of point groups, such
as axisymmetric or polyhedral patterns, may bifurcate directly from the static
stateCha61 ; Bus75 ; Zeb83 . For the top view (Fig.2(e)), it must be
considered that the 3-fold state achieved at the point R additionally
satisfies the mirror symmetry with respect to the $\phi=0$ plane,
$(\varPhi,\varPsi,\varTheta)(r,\theta,\phi)=(\varPhi,-\varPsi,\varTheta)(r,\theta,-\phi)$.
However, the equatorial view (Fig.2(f)) suggests that the mirror symmetry is
not practically satisfied by the 3-fold state obtained at R, where the Grashof
number is significantly larger than $\mbox{{\rm Gr}}_{\rm cr}$. The weak
breaking of the mirror symmetry is associated with the nonzero angular
momentum around the polar axis, as discussed later. This was also clear in the
equatorial section of the flow structure of the 3-fold state at R, as shown in
Fig.3.

Figure 3:  The 3-fold state that is achieved at $(\mbox{{\rm Re}},\mbox{{\rm
Gr}})=(0,320)$ by the continuation through the path. The disturbance component
of $\varTheta$ and the flow field $(u_{\phi},u_{r})$ on the equatorial plane
are described in the contour and vector fields, respectively.  Figure 4:  A
saturated state was obtained by imposing mirror symmetry with respect to the
$\phi=0$ plane on the 3-fold state obtained at R, as in Fig.3.

The flow structure of the 3-fold state is a set of sinusoidal waves travelling
in the azimuthal direction at the equatorial plane, which connects the spiral
disturbance extending into each hemisphere from the poles. For comparison, we
examined the following computation. Imposing the mirror symmetry with respect
to the $\phi=0$ plane, time integration using the 3-fold state obtained at R
as the initial condition saturates to another state similar to the 3-fold
state at R. The obtained state exactly satisfies mirror symmetry with respect
to the $\phi=0$ plane, which was not satisfied by the 3-fold state. Figs.3 and
4 show the disturbance component of $\varTheta$ and the flow field
$(u_{\phi},u_{r})$ on the equatorial plane, which are described in contour and
vector fields, respectively. While Fig.3 is obtained from the $3$-fold state
at R, Fig.4 is obtained from the additionally computed mirror-symmetric state
at R. A comparison of the figures indicates that the temperature and vector
fields in the 3-fold state at R are distorted in the azimuthal direction and
violate the mirror symmetry with respect to the $\phi=0$ plane. The difference
in magnitude in the temperature and vector fields was quantitatively
nonnegligible; thus, the 3-fold state obtained at R is distinct from the
highly symmetric steady and polyhedral states invariant under a set of
transformations of point groups. This will be discussed in further detail in
the next section.

Figure 5:  The continuous variation of the degrees of asymmetry to the 4-fold
state with a continuous change of the parameter in segment A. The solid curve
is the meridional circulation $q_{\phi}$, the dotted curve is half of the
torque on the outer sphere $T_{z}/2$, the dashed curve is the normalized
angular velocity $\omega_{\phi}$, and the dash-dotted curve represents the
degree of nonaxisymmetric $L_{2}$.  Figure 6:  Continuous variation of the
degrees of asymmetry to the 4-fold state with continuous parameter change in
segment B, as in Fig.5.  Figure 7:  Continuous variation of the degrees of
asymmetry to the 3-fold state with continuous parameter change in segment A,
as win Fig.5.  Figure 8:  Continuous variation of the degrees of asymmetry to
the 3-fold state with continuous parameter change in segment B, as in Fig.5.

##  IV  Discussion

Figs.5 and 6 show the continuous variation of the degrees of asymmetry with
respect to the 4-fold state with continuous parameter change for segments A
($\mbox{{\rm Gr}}:0\to 320$) and B ($\mbox{{\rm Re}}:490\to 0$). Here, as an
index of the degree of rotation with respect to the polar direction, we
plotted the normalized angular phase velocity
$\omega_{\phi}=\Omega_{\phi}/\Omega_{\rm in}$, $q_{\phi}$, and $T_{z}$. Note
that all the indices in both figures change continuously. It does not appear
that the branches of 4-fold states have discontinuous nodes on the path. As
shown in the figures, with an increase in Gr from $0$ to $320$ at $\mbox{{\rm
Re}}=490$ in segment A, the index for the degree of nonaxisymmetry, $L_{2}$,
increases monotonically, whereas $L_{2}$ decreases monotonically until the
degeneration to the axisymmetric state, with a decrease in Gr from $490$ to
$261$ in segment B. Because the system is rotated as far as $\mbox{{\rm
Re}}>0$, it is natural that the other indices describing asymmetry are finite
throughout the path.

Figs.7 and 8 show the continuous variation of the degrees of asymmetry for the
3-fold state with continuous parameter changes in segments A and B. The
branches of 3-fold states also have no discontinuous node on the path. As
shown in Fig.7, with an increase of Gr from $0$ to $320$ at $\mbox{{\rm
Re}}=490$ on segment A, the index for the degree of nonaxisymmetry, $L_{2}$,
increases monotonically, while the degree of rotation with respect to the
polar direction remains almost constant. This suggests that the angular
velocity of the state in the hybrid system with both $\mbox{{\rm Re}}\neq 0$
and $\mbox{{\rm Gr}}\neq 0$ can be estimated only using Re.

In contrast, all the indices of the 3-fold state change continuously but
largely in Fig.8, where Re decreases from $490$ to $0$ in segment B. The index
for the degree of nonaxisymmetry, $L_{2}$, monotonically increases after being
annihilated around $\mbox{{\rm Re}}=240$ with Re decreasing in segment B. The
transient decrease in $L_{2}$ at $\mbox{{\rm Re}}>240$ suggests that segment B
passes through the neutral curve, $\mbox{{\rm Re}}=\mbox{{\rm Re}}_{\rm
cr}(\mbox{{\rm Gr}})$. Note that the denominator of $L_{2}$ is proportional to
the strength of the Stokes flow; that is, Re. Thus, the divergence of $L_{2}$
to infinity at $\mbox{{\rm Re}}\to 0$ in segment B implies that the numerator
of $L_{2}$ converges to a nonzero value in the limit of $\mbox{{\rm Gr}}\to
0$. This is not surprising because a variety of non-axisymmetric equilibrium
states Cha61 ; Bus75 ; Zeb83  can be realized in the purely thermal convective
state, $\mbox{{\rm Re}}=0$.

In general, fluid mixing in the shell induced by the 3-fold state leads to
some momentum exchange between the inner and outer spherical boundaries, so
that the torque exerted on the outer sphere is larger than that of the Stokes
flow; $T_{z}>1$ should be basically satisfied as long as the truncation error
is negligible. The normalized torque $T_{z}$, which is maintained at
approximately 3 on segment A converges to a finite value at $\mbox{{\rm
Re}}\to 0$ on segment B. This convergence suggests that the torque exerted on
the outer sphere vanished at $\mbox{{\rm Re}}\to 0$. Naturally, the torque
exerted on the boundary from the fluid vanishes when the rotation of the
system stops. The symmetry is a coincidence between the solution and equation.

In contrast to the torque, the values of the normalized flow rate and angular
velocity, $q_{\phi}$ and $\omega_{\phi}$, diverge at the limit of $\mbox{{\rm
Re}}\to 0$ in Fig.8. This is because the numerators of these values converged
to finite values at $\mbox{{\rm Re}}\to 0$. Interestingly, the degree of
rotation in the polar direction remains at nonzero values at the limit. In
other words, the orientation of the state did not change when the system
became isotropic. In general, we should clearly distinguish between the
symmetry of the governing equation and the symmetry of the states realized
under the equation. In particular, with respect to thermal convection, there
are many examples in which a realized state breaks the symmetry inherited from
the system. For instance, the oscillative flow reversals intermittently change
the circulation direction in conventional Rayleigh-Bénard system breaking the
temporal homogeneityKid97 . A spiral state realized in either a spherical
reaction-diffusion systemSig11  or Rayleigh-Bénard systemIta15  breaks
orientation homogeneity.

Associated with symmetry breaking, the natural convection in a vertical slot
between parallel plates with a temperature differenceNag83  is suggestive of
the present result. For fluids with a low Prandtl number, such as liquid
metal, the two-dimensional transverse vortex state bifurcated from the basic
steady state principally becomes unstable against subharmonic perturbation at
the second critical Grashof number. If the system is restricted to a harmonic
domain where the subharmonic perturbation cannot manifest itself, the
transverse vortex state loses its stability to the three-dimensional traveling
wave solution at the second critical Grashof number. Surprisingly, the
bifurcated three-dimensional solution is the wave solution traveling not in
the vertical direction but the spanwise direction. The solution has a finite
spanwise momentum but exerts no friction on the platesNagIta03 .
Unfortunately, this phenomenon cannot be confirmed because it is realizable in
a virtual slot with periodic boundaries in both the vertical and spanwise
directions. However, it would have been noticed if it were confirmed in the
spherical shell periodically in both the polar and azimuthal directions.

The thick atmosphere of Venus is known to rotate at speeds up to 60 times that
of slow planetary spinHor20 . One would think that superrotation cannot be
sustained in nature unless the atmospheric angular momentum around its spin
axis is supplied by axially asymmetric waves or turbulence. Otherwise, the
diffusion between the atmosphere and the surface and the mixing due to
circulation in the meridional plane induced by the pole-equator temperature
difference should attenuate the angular momentum. The present result is fluid-
dynamical evidence, implying that such a super- or sub-rotation may be
sustained without some exchange of angular momentum via torque between the
fluid and boundary.

In the spherical thermal convection system, the multiple unstable modes
originally degenerated under homogeneity are revealed simultaneously at a
critical Grashof number, estimating the ratio of the buoyancy to viscous
force. Various types of highly symmetric steady states are invariant under a
set of transformations of point groups, such as axisymmetric or polyhedral
patterns, and bifurcate from the static state via nonlinear interactions
between these modes; thus, the isotropy inherited in the system is first
brokenCha61 ; Bus75 ; Zeb83 . An arbitrary incompressible field may be
decomposed into toroidal and poloidal components in the radial direction. The
velocity fields realizable over the threshold consist only of the poloidal
field, because of the absence of an energy source term that sustains the
toroidal componentIta15 . This context is analogous to the question of how
super-rotation is sustained on Venus and to the deduction that the velocity
field bifurcated sequentially with a quasistatic increase in the Grashof
number never leads to any net angular momentum. The present study provided a
counterexample against such an intuitive deduction, and showed that spherical
thermal convection is a realizable system that can sustain a steady state with
no momentum exchange between the fluid and boundaries, as presented in
Ref.NagIta03 .

##  V  Summary

Here, we revisit the conventional SCF, where a differential rotation of the
spherical boundaries generates a nonequilibrium state of an incompressible
Newtonian fluid in the spherical shell. From previous experimental and
numerical studies on SCFs, it is known that the first transition in a
relatively wide-gap SCF is initiated by the rotating wave states accompanied
by a change in the net angular momentum around the rotation axis. By
introducing the Boussinesq effect and temperature difference between the
boundaries to the equations of SCF, the existence of the rotating wave state
can be extended into the parameter space spanned by the Grashof number. The
main interest of the present study was the homotopy continuation of the
rotating wave state realized experimentally towards the thermal convection
limit in the parameter space. Recalling a clear distinction between the
symmetry of a governing equation and the symmetry of the states realized under
the equation, we should recognize that there may be no basis to deny the
mechanism that sustains a finite net angular momentum. For example, it was
verified in experimentsBen90  that the thermal convection system of binary
fluid mixtures in an annular container with a large aspect ratio forms a
localized traveling wave that breaks reflection symmetry. In the present
study, we obtained a 3-fold spiral state at the vanishing limit of the
system’s rotation. The state obtained at the limit was asymmetric to the
azimuthal equation and we provided a numerically exact solution of a spherical
Rayleigh-Bénard system. Further research is needed to analyze the bifurcation
of the state at $\mbox{{\rm Gr}}=0$.

###### Acknowledgements.

The authors would like to thank Mr. T. Inagaki for his pilot survey and Dr. H.
Yamashita for his valuable comments on the draft. We would also like to thank
Editage (www.editage.com) for English language editing. This work was
supported in part by the Grant-in-Aid for Scientific Research(C), JSPS KAKENHI
Grant No.20K04294. This work also benefited from interaction within RISE-2018
No.824022 ATM2BT of the European Union Horizon 2020-MSCA program, which
includes Kansai University.

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Back to the proof of Theorem 9.1. Now the proof of Theorem 6.1 can proceed
with no difficulty. Following the notations therein, define

$u(z)=\|\nabla^{A}P\|^{2}_{L^{2}(\Sigma)}+\|F\|^{2}_{L^{2}(\Sigma)}\text{ and
}w(z)=\|\nabla^{A}P\|^{2}_{\mathcal{H}_{1}}+\|F\|^{2}_{L^{2}_{1}(\Sigma)}.$

For any $\eta>0$, by the compactness theorem [KM07, Theorem 5.2.1], there
exists a constant $\epsilon(\eta)>0$ such that for any configuration
$\gamma=(\bar{A},P)$ with

$\mathcal{E}_{an}(\gamma,\Omega_{0})=\int_{\Omega_{0}}u(z)dz<\epsilon(\eta),$

we have the point-wise estimate

$0\leq u(z)\leq w(z)\leq\eta,\forall z\in\Omega_{0}^{\prime}$

for a smaller domain $\Omega_{0}^{\prime}\subset\Omega_{0}$. By Proposition
7.4, $O_{*}=\mu^{-1}(\delta)\cap\operatorname{Crit}(L)$ consists of
$\binom{2g-2}{d}$ free $\mathcal{G}(\Sigma)$-orbits. Let them be
$O_{1},\cdots,O_{k}$ and $\kappa_{j}\in O_{j}$ be a reference configuration.
By taking $\eta\ll 1$, we conclude that there exists some $1\leq j\leq k$ such
that for any $z\in\Omega_{0}$, $P(z)\in\Omega(4,j)$ (after a gauge
transformation). Here $\Omega(4,j)$ is the neighborhood of $\kappa_{j}$
obtained in Lemma 9.2 with $\epsilon_{1}=4$.

Replace $\Omega_{0}$ by $\Omega_{n,R}$ for any $n\in\mathbb{Z}$ and $R>1$. By
shrinking the size of $\Omega(4,j)$, we ensure that
$\Omega(4,j)\cap\Omega(4,j^{\prime})=\emptyset$ for any $j\neq j^{\prime}$.
Since $\mathbb{H}^{2}_{+}$ is connected, the subscript $j$ is the same for all
$(n,R)$. As a result, there exists some $1\leq j\leq k$ such that for any
$z\in\mathbb{H}^{2}_{+}$, $P(z)\in\Omega(4,j)$ (after a gauge transformation).
The Bochner-type formula in Lemma 6.4 then implies that

$0\geq\frac{1}{2}\Delta_{\mathbb{H}^{2}_{+}}u+\zeta^{2}w-Cw^{3/2}\geq\frac{1}{2}\Delta_{\mathbb{H}^{2}_{+}}u+\frac{\zeta^{2}}{2}w\geq\frac{1}{2}(\Delta_{\mathbb{H}^{2}_{+}}+\zeta^{2})u,$

for some $\zeta>0$. Now we use Lemma 6.3 to conclude. ∎

###  10\. Finite Energy Solutions on $\mathbb{R}_{s}\times\Sigma$

Fix an $H$-surface $\operatorname{\bf\Sigma}=(\Sigma,g_{\Sigma},\lambda,\nu)$.
In this section, we study the 3-dimensional Seiberg-Witten equations:

(10.1)
$\left\\{\begin{array}[]{r}\frac{1}{2}\rho_{3}(F_{B^{t}})-(\Psi\Psi^{*})_{0}-\rho_{3}(\omega)=0,\\\
D_{B}\Psi=0.\end{array}\right.$

on the cylinder $\mathbb{R}_{s}\times\Sigma$ with
$\omega=\nu+ds\wedge\lambda$. In terms of Landau-Ginzburg models, the
equations (10.1) are equivalent to the downward gradient flow equation of
$L=\operatorname{Re}W_{\lambda}$:

(10.2) $\frac{d}{ds}p(s)+\nabla L\circ p=0.$

where $p(s)=(\check{b}(s),\check{\Psi}(s)):\mathbb{R}\to
M(\Sigma,\mathfrak{s})$ is a smooth path in the Kähler manifold
$M(\Sigma,\mathfrak{s})$. Its relation with (10.1) can be seen by setting

$B=\frac{d}{ds}+\check{B}_{0}+\check{b}(s)\text{ and
}\Psi=\check{\Psi}(s)\text{ on }\\{s\\}\times\Sigma.$

We require the path $p$ to have finite analytic energy:

(10.3)
$\mathcal{E}_{an}(p)\colonequals\int_{\mathbb{R}_{s}}|\frac{d}{ds}p|^{2}+|\nabla
L|^{2}<\infty.$

Under (10.3), it is easily seen that the path $p$ has a finite length and

$q_{\pm}=\lim_{s\to\pm\infty}p(s)$

lies in $\operatorname{Crit}(L)$. By the Cauchy-Riemann equation (2.1), (10.2)
is also the Hamiltonian flow equation for $H=\operatorname{Im}W_{\lambda}$.
Hence, a flowline connecting $q_{-}$ and $q_{+}$ exists only if

(10.4) $L(q_{-})\geq L(q_{+})\text{ and }H(q_{-})=H(q_{+}).$

This motivates the following definition:

######  Definition 10.1.

An $H$-surface $\operatorname{\bf\Sigma}=(\Sigma,g_{\Sigma},\lambda,\nu)$ is
called good if any critical values $a_{1}\neq a_{2}$ of $W_{\lambda}$
satisfies $\operatorname{Im}a_{1}\neq\operatorname{Im}a_{2}$. ∎

One can always make $\operatorname{\bf\Sigma}$ good by replacing $\lambda$ by
$e^{i\theta}\cdot\lambda$ for some $e^{i\theta}\in S^{1}$, since the set of
critical values of $W_{\lambda}$ is at most countable.

######  Proposition 10.2.

If an $H$-surface $\operatorname{\bf\Sigma}$ is good, then any finite energy
solutions to (10.1) are necessarily $\mathbb{R}_{s}$-invariant.

When $g(\Sigma)=1$, we can understand good $H$-surfaces more concretely:

######  Proposition 10.3.

If $g(\Sigma)=1$ and $\lambda\in\Omega^{1}_{h}(\Sigma)\cong
H^{1}(\Sigma,i\mathbb{R})$ is not a multiple of any integral classes, then
$\operatorname{\bf\Sigma}$ is good. In particular, any finite energy solution
of $(\ref{3DDSWEQ})$ has to be $\mathbb{R}_{s}$-invariant, i.e $p(s)\equiv
q_{-}=q_{+}$.

###### Proof.

By Proposition 7.4, $\mu^{-1}(\vec{\delta})\cap\operatorname{Crit}(L)$
consists of a single $\mathcal{G}(\Sigma)$-orbit, so $q_{+}=u\cdot q_{-}$ for
some $u:\Sigma\to S^{1}$. Hence,

$W_{\lambda}(q_{-})-W_{\lambda}(q_{+})=-\int_{\Sigma}\langle
u^{-1}du,\lambda\rangle_{h_{M}}.$

In particular, $H(q_{-})-H(q_{+})=4\pi^{2}([\frac{u^{-1}du}{2\pi
i}]\cup[\frac{\lambda}{2\pi i}])[\Sigma]$. If $\lambda$ is not a multiple of
an integral class, this pairing is nonzero unless $[\frac{u^{-1}du}{2\pi
i}]=0\in H^{1}(\Sigma,\mathbb{Z})$. This implies that

$\mathcal{E}_{an}(p)=2(L(q_{-})-L(q_{+}))=0,$

so the path $p$ has to be $\mathbb{R}_{s}$-translation invariant. ∎

######  Remark 10.4.

A solution of (10.1) can be viewed as an $S^{1}$-invariant solution of the
4-dimensional equations (7.1) on $S^{1}\times\mathbb{R}_{s}\times\Sigma$. When
$g_{\Sigma}$ is flat, Proposition 10.3 follows from a theorem of Taubes
[Tau01, Proposition 4.4]. ∎

Taubes’ theorems provide another simple condition that precludes non-trivial
solutions.

######  Proposition 10.5 ([Tau01, Proposition 4.7]).

Let $(\Sigma,g_{\Sigma})$ be a flat 2-torus and consider the $H$-surface
$\operatorname{\bf\Sigma}=(\Sigma,g_{\Sigma},\lambda,\nu)$ with $\nu$
harmonic. If $\langle\nu,[\Sigma]\rangle\neq 0$, then any finite energy
solution of $(\ref{3DDSWEQ})$ has to be $\mathbb{R}_{s}$-invariant, i.e.
$p(s)\equiv q_{-}=q_{+}$.

###### Proof.

The proof is borrowed from [Tau01, P. 486-487]. Since $g_{\Sigma}$ is flat,
the closed 2-form $\omega=ds\wedge\lambda+\nu$ in (10.1) is parallel on
$\mathbb{R}_{s}\times\Sigma$. The spin bundle $S^{+}$ splits as

(10.5) $L^{+}_{\omega}\oplus L^{-}_{\omega}$

with $\rho_{3}(\omega)$ acting on by a diagonal matrix

$m\begin{pmatrix}-1&0\\\ 0&1\end{pmatrix}$

where $m=\sqrt{|\nu|^{2}+|\lambda|^{2}}$ is a positive number. The splitting
(10.5) is parallel. Let $p(t)=(b(t),\Psi(t))$ be a solution of (10.2) on
$\mathbb{R}_{s}\times\Sigma$. Write $\Psi(t)=\sqrt{2m}(\alpha(t),\beta(t))$
with respect to the decomposition (10.5). By Witten’s vanishing spinor
argument [Wit94, Section 4], $\beta\equiv 0$. The first equation of (10.1)
becomes

$\frac{1}{2}F_{B^{t}}=(1-|\alpha|^{2})\omega.$

The curvature form $F_{B^{t}}$ is closed, so
$d(1-|\alpha|^{2})\wedge\omega=0$, which is also $\langle
d|\alpha|^{2},*_{3}\omega\rangle=0$. The dual tangent vector of $i*_{3}\omega$
generates a flow on $\mathbb{R}_{s}\times\Sigma$ along which $|\alpha|^{2}$
stays constant. Since $\nu\neq 0$, this flow translates the spatial coordinate
$s$ as time varies. Since $|\alpha|\to 1$ as $s\to\pm\infty$, $|\alpha|\equiv
1$. This completes the proof. ∎

When $(\Sigma,g_{\Sigma})$ is flat, $\nu=0$ and $\lambda$ is a multiple of an
integral class, there is a non-trivial moduli space of flowlines for any pair
$(q_{-},q_{+})$ subject to (10.4). They are pulled back from vortices on the
cylinder $\mathbb{R}_{s}\times S^{1}$. These moduli spaces are not regular;
their expected dimensions are always zero by the index computation. For more
details, see [Tau01, Section 4(d)(e)]. Here is an immediate corollary of
Proposition 10.5.

######  Corollary 10.6.

Let $(\Sigma,g_{\Sigma})$ be a flat 2-torus and consider the $H$-surface
$\operatorname{\bf\Sigma}=(\Sigma,g_{\Sigma},\lambda,\nu)$ with $\nu$
harmonic. If $\langle\nu,[\Sigma]\rangle\neq 0$, then for any $e^{i\theta}\in
S^{1}$, any down-ward gradient flowline of the functional
$\operatorname{Re}(e^{i\theta}W_{\lambda})$:

$\partial_{s}p(s)+\nabla\operatorname{Re}(e^{i\theta}W_{\lambda})=0,$

has to be a constant path.

###### Proof.

This corollary follows from Proposition 10.5 by noting that

$e^{i\theta}W_{\lambda}(\check{b},\check{\Psi}_{+},\check{\Psi}_{-})=W_{0}(\check{b},\check{\Psi}_{+},e^{-i\theta}\check{\Psi}_{-})+\langle\check{b},e^{-i\theta}\lambda\rangle_{h_{M}}.\qed$

###  Appendix A The Maximum Principle

In this appendix, we prove a version of maximum principle, from which one can
easily deduce exponential decay. The author is greatly indebted to Ao Sun for
teaching him this elementary and enlightening proof of Lemma A.1. The
Laplacian operator is always assumed to have a non-negative spectrum. In
particular, over the complex plane,

$\Delta_{\mathbb{C}}\colonequals-\partial_{t}^{2}-\partial_{s}^{2}.$

This sign convention is adopted throughout this paper.

######  Proposition A.1.

Take $\zeta>0$. Suppose
$u:\mathbb{H}^{2}_{+}=\mathbb{R}_{t}\times[0,\infty)_{s}\to\mathbb{R}$ is a
bounded $\mathcal{C}^{2}$-function on the upper half plane such that

  1. (1)

$(\Delta_{\mathbb{C}}+\zeta^{2})u\leq 0$, and

  2. (2)

$u(t,0)\leq 0$ for any $t\in\mathbb{R}_{t}$.

Then $u(t,s)\leq 0$ for any $(t,s)\in\mathbb{H}^{2}_{+}$.

###### Proof.

Choose a smooth cut-off function $\psi:[0,\infty)\to[0,\infty)$ such that

  * •

$\psi\equiv 1$ on $[0,1]$,

  * •

$\psi\equiv 0$ on $[2,\infty)$ and

  * •

$0\leq\psi\leq 1$.

Let $\phi_{R}:\mathbb{C}\to\mathbb{R}$ such that $\phi_{R}(z)=\psi(|z|/R)$.
Then

  * •

$\phi_{R}\equiv 1$, $\nabla\phi_{R}=0$ and $\Delta_{\mathbb{C}}\phi_{R}\equiv
0$ when $|z|<R$;

  * •

$\phi_{R}\equiv 0$, $\nabla\phi_{R}=0$ and $\Delta_{\mathbb{C}}\phi_{R}\equiv
0$ when $|z|>2R$;

  * •

for some $L>0$, $|\nabla\phi_{R}|<\frac{L}{R}$ and
$|\Delta_{\mathbb{C}}\phi_{R}|<\frac{L}{R^{2}}$.

Only the last property requires some explanation. In general, we have

$\displaystyle\nabla\phi_{R}$
$\displaystyle=\frac{1}{R}\phi^{\prime}(\frac{|z|}{R})\partial_{r},$
$\displaystyle\Delta_{\mathbb{C}}\phi_{R}$
$\displaystyle=-(\partial_{r}^{2}\phi_{R}+\frac{1}{|z|}\partial_{r}\phi_{R})=-\frac{1}{R^{2}}\psi^{\prime\prime}(\frac{|z|}{R})-\frac{1}{|z|R}\psi^{\prime}(\frac{|z|}{R}).$

Suppose $u(z_{0})>0$ at some $z_{0}\in\mathbb{H}^{2}_{+}$. Consider
$u_{R}(z)\colonequals u(z)\phi_{R}(z-z_{0})$. Then $u_{R}(z)\equiv 0$ when
$|z-z_{0}|>2R$ and

$u_{R}(t,0)\leq 0.$

Hence, $\max u_{R}$ is attained at some $z_{1}$ in the interior of
$\mathbb{H}^{2}_{+}$. Let $N=\|u\|_{\infty}$, so

(A.1) $0<u(z_{0})=u_{R}(z_{0})\leq u_{R}(z_{1})\leq N\phi_{R}(z_{1}-z_{0}).$

At $z_{1}\in\mathbb{H}^{2}_{+}$, we have

$0=(\nabla u_{R})(z_{1})=(\nabla
u\cdot\phi_{R}(z_{1}-z_{0})+u\nabla\phi_{R}(z_{1}-z_{0}),$

so $\nabla
u(z_{1})=-u(z_{1})\nabla\phi_{R}(z_{1}-z_{0})/\phi_{R}(z_{1}-z_{0})$. The
relation $\Delta_{\mathbb{C}}u\leq-\zeta^{2}u$ then gives:

$\displaystyle 0\leq\Delta_{\mathbb{C}}u_{R}(z_{1})$
$\displaystyle=\phi_{R}(z_{1}-z_{0})(\Delta_{\mathbb{C}}u)+u(\Delta_{\mathbb{C}}\phi_{R}(z_{1}-z_{0}))-2\nabla\phi_{R}(z_{1}-z_{0})\cdot\nabla
u,$ $\displaystyle\leq
u(-\zeta^{2}\phi_{R}+\Delta_{\mathbb{C}}\phi_{R}+\frac{2|\nabla\phi_{R}|^{2}}{\phi_{R}})(z_{1}-z_{0}).$

However, this inequality is violated when $R\gg 0$ which yields a
contradiction. By (A.1), when $R\gg 1$,

$|\Delta_{\mathbb{C}}\phi_{R}+\frac{2|\nabla\phi_{R}|^{2}}{\phi_{R}}|(z_{1}-z_{0})\leq\frac{L}{R^{2}}+\frac{2L^{2}}{R^{2}}\cdot\frac{N}{u(z_{0})}<\frac{\zeta^{2}u(z_{0})}{N}\leq\zeta^{2}\phi_{R}(z_{1}-z_{0}).$

Moreover, $u(z_{1})>0$. This completes the proof. ∎

######  Corollary A.2.

Take $\zeta>0$. Suppose
$u:\mathbb{H}^{2}_{+}=\mathbb{R}_{t}\times[0,\infty)_{s}\to\mathbb{R}$ is a
bounded $\mathcal{C}^{2}$-function on the upper half plane such that

  1. (1)

$(\Delta_{\mathbb{C}}+\zeta^{2})u\leq 0$, and

  2. (2)

$u(t,0)\leq K$ for some $K>0$ and any $t\in\mathbb{R}_{t}$.

Then $u(t,s)\leq Ke^{-\zeta s}$ for any $(t,s)\in\mathbb{H}^{2}_{+}$.

###### Proof.

Let $v(t,s)=Ke^{-\zeta s}$. Then $(\Delta_{\mathbb{C}}+\zeta^{2})v=0$ and
$v(t,0)=K$ for any $t\in\mathbb{R}_{t}$. Apply Proposition A.1 to $u-v$ to
conclude. ∎

There are analogous statements for a strip of finite length. Their proofs are
similar and omitted here.

######  Proposition A.3.

Take $\zeta>0$. Suppose $u:\mathbb{R}_{t}\times[0,2R]_{s}\to\mathbb{R}$ is a
bounded $\mathcal{C}^{2}$-function such that

  1. (1)

$(\Delta_{\mathbb{C}}+\zeta^{2})u\leq 0$, and

  2. (2)

$u(t,s)\leq 0$ for any $t\in\mathbb{R}_{t}$ and $s\in\\{0,2R\\}$.

Then $u(t,s)\leq 0$ for any $(t,s)\in\mathbb{R}_{t}\times[0,2R]_{s}$.

######  Corollary A.4.

Take $\zeta>0$. Suppose $u:\mathbb{R}_{t}\times[0,2R]_{s}\to\mathbb{R}$ is a
bounded $\mathcal{C}^{2}$-function such that

  1. (1)

$(\Delta_{\mathbb{C}}+\zeta^{2})u\leq 0$, and

  2. (2)

$u(t,s)\leq K$ for any $t\in\mathbb{R}_{t}$ and $s\in\\{0,2R\\}$.

Then $u(t,s)\leq K\cdot\frac{\cosh(\zeta(s-R))}{\cosh(\zeta R)}$ for any
$(t,s)\in\mathbb{R}_{t}\times[0,2R]_{s}$.

###  Appendix B A Bochner-Type Formula

The purpose of this appendix is to summarize some differential geometric
formulae for a gauged Landau-Ginzburg model $(M,W,G,\rho)$. In particular, we
will prove a Bochner-type formula for a solution $(A,P)$ to the gauged Witten
equations (4.1) on $\mathbb{H}^{2}_{+}$. For our primary application, we will
take $M$ to be a complex linear space, in which case many formulae become much
simpler.

####  B.1. Some Useful Formulae

Recall that $(M,\omega,J,g)$ is a Kähler manifold and $G$ is a compact abelian
Lie group acting on $M$ holomorphically and isometrically. $(G,\rho)$ admits a
moment map $\mu:M\to\mathfrak{g}$ which is $G$-invariant. $W=L+iH$ is a
$G_{\mathbb{C}}$-invariant holomorphic function on $M$, called the
superpotential.

For any $\xi\in\mathfrak{g}$, let $\tilde{\xi}$ be the vector field on $M$
induced from the action $(G,\rho)$. The convention of the moment map used in
our paper is that

$\iota(\tilde{\xi})\omega=-d\langle\mu,\xi\rangle_{\mathfrak{g}},$

or equivalently,

(B.1) $\tilde{\xi}=J\langle\nabla\mu,\xi\rangle_{\mathfrak{g}},$

since $\omega(\cdot,\cdot)=g(J\cdot,\cdot)$. Here
$\nabla\mu\in\Gamma(M,TM\otimes\mathfrak{g})$ is viewed as a
$\mathfrak{g}$-valued vector field on $M$.

######  Lemma B.1.

For a gauged Landau-Ginzburg model $(M,W,G,\rho)$ defined as in Definition
3.1, we have the following identities:

  1. (1)

$\nabla L+J\nabla H=0.$

  2. (2)

$\operatorname{Hess}L+J\circ\operatorname{Hess}H=0$.

  3. (3)

$J\circ\operatorname{Hess}H+\operatorname{Hess}H\circ J=0$, i.e. the Hessian
$\operatorname{Hess}H$ of $H$ anti-commutes with $J$.

  4. (4)

$J\circ\operatorname{Hess}\mu=\operatorname{Hess}\mu\circ J$, i.e. the Hessian
$\operatorname{Hess}\mu$ commutes with $J$.

  5. (5)

$\langle\nabla\mu,\nabla H\rangle=\langle J\nabla\mu,\nabla H\rangle=0$.

  6. (6)

$\langle\nabla\mu,\tilde{\xi}\rangle=0$ for any $\xi\in\mathfrak{g}$.

###### Proof.

The first identity (1) is the Cauchy-Riemann equation.

Since $M$ is Kähler, the almost complex structure $J$ is parallel, i.e.
$\nabla J=0$, so (2) follows from (1) by taking the covariant derivative.

Both $\operatorname{Hess}L$ and $\operatorname{Hess}H$ are symmetric operators
with respect to the metric $g$, so by (2), we have

$J\circ\operatorname{Hess}H=(J\circ\operatorname{Hess}H)^{T}=(\operatorname{Hess}H)^{T}\circ
J^{T}=-\operatorname{Hess}H\circ J.$

Since the metric $g$ is $G$-invariant, for any $\xi\in\mathfrak{g}$, the Lie
derivative of $g$ is zero:

$\mathcal{L}_{\tilde{\xi}}g=0.$

This implies that for any vector fields $V,U$,

(B.2)
$\langle\nabla_{V}\tilde{\xi},U\rangle+\langle\nabla_{U}\tilde{\xi},V\rangle=0.$

Using the defining equation (B.1), we conclude that $\langle
J\operatorname{Hess}\mu,\xi\rangle_{\mathfrak{g}}$ is an anti-symmetric
operator, so $\operatorname{Hess}\mu$ commutes with $J$. This proves (4)

Finally, since $H$ is $G$-invariant, $\langle\tilde{\xi},\nabla H\rangle=0$.
By (B.1), $\langle J\nabla\mu,\nabla H\rangle=0$. The other identity in (5)
follows from the $G$-invariance of $L$ and the first identity (1).

(6) follows from the fact that $\mu:M\to\mathfrak{g}$ is $G$-invariant. ∎

A smooth connection $A=d+a$ in the trivial principal bundle
$\mathbb{H}^{2}_{+}\times G\to\mathbb{H}^{2}_{+}$ allows us to take covariant
derivatives of a map $P:\mathbb{H}^{2}_{+}\to M$. It is also important to know
covariant derivatives of a vector field $v$ along $P$, i.e. a smooth map
$v:\mathbb{H}^{2}_{+}\to TM$ such that $v(x)\in T_{P(x)}M$.

Recall that for any tangent vector $(x,V)\in T\mathbb{H}^{2}_{+}$,
$\nabla^{A}_{V}P$ is defined by the property:

(B.3) $\nabla^{A}_{V}P=\frac{d}{dt}\rho(e^{t\cdot
a(V)})P(\gamma(t))\bigg{|}_{t=0}$

where $\gamma:[0,1]\to\mathbb{H}^{2}_{+}$ is a path with $\gamma(0)=x$ and
$\gamma^{\prime}(0)=V$. The action of $G$ extends to the tangent bundle $TM$
of $M$:

$\rho_{TM}(g)(p,v)\mapsto(\rho(g)p,\rho(g)_{*}v).$

If $v$ is a vector field along $P\circ\gamma(t)$, it is reasonable to define
its derivative $\nabla^{A}_{V}v$ as:

(B.4) $\nabla^{A}_{V}v\colonequals\nabla_{\nabla^{A}_{V}P}\rho(e^{t\cdot
a(V)})_{*}v.$

######  Remark B.2.

When ${\nabla^{A}_{V}P}=0$, the formula (B.4) is understood as follows: in
terms of local coordinates $\\{y^{i}\\}_{1\leq i\leq n}$,

(B.5)
$\nabla_{P_{*}V}v\colonequals\partial_{t}v^{i}(t)\partial_{i}+\frac{\partial
y^{j}(P\circ\gamma(t))}{\partial t}v^{i}(t)\Gamma_{ij}^{k}\partial_{k}.$

where $\Gamma_{ij}^{k}$ are Christoffel symbols and
$v=v^{i}\partial_{i}:[0,1]_{t}\to TM$ is the smooth path lying over
$P\circ\gamma$. (B.5) makes sense even if $P\circ\gamma(t)\equiv P(x)$ is a
constant path. ∎

It is enlightening to find a concrete formula of $\nabla^{A}_{V}v$ without
using the group action. By the defining property of the moment map (B.1), we
have

(B.6) $\nabla^{A}_{V}P=V\cdot
P+\tilde{a}(V)=P_{*}(V)+J\langle\nabla\mu,a(V)\rangle.$

######  Lemma B.3.

The covariant derivative of a vector field $v$ equals:

$\nabla^{A}_{V}v=\nabla_{P_{*}V}v+J\langle\operatorname{Hess}\mu(v),a(V)\rangle.$

where $\nabla_{P_{*}V}v$ denotes the covariant derivative with respect to the
Levi-Civita connection.

######  Remark B.4.

The correction term $J\langle\operatorname{Hess}\mu(v),a(V)\rangle$ reflects
the dependence on the connection 1-form $a$. It is linear in $a$, $v$ and $V$
as expected. ∎

###### Proof.

The formal proof is to linearize the equation (B.6) along the tangent vector
$v(x)\in T_{P(x)}M$. Let us make this intuition precise. Consider a variation
of $P\circ\gamma$ along the vector field $v$:

$Q(r,t,s)=\rho(e^{r\cdot a(V)})\exp_{P\circ\gamma(t)}(sv(t)).$

When $r\equiv 0$, $Q(0,t,s)$ is a variation of the path $P\circ\gamma(t)$.
Indeed, $Q(0,t,0)\equiv P\circ\gamma(t)$. When $s\equiv 0$, the covariant
derivative of $P$ is defined as (comparing (B.3)):

$\displaystyle\frac{d}{dt}Q(t,t,0)\bigg{|}_{t=0}=\nabla^{A}_{V}P$
$\displaystyle=\frac{d}{dt}Q(0,t,0)+\frac{d}{dt}Q(t,0,0)\bigg{|}_{t=0}$
$\displaystyle=P_{*}(V)+J\langle\nabla\mu,a(V)\rangle.$

Let $U_{1}=\frac{d}{dt}Q(t,t,s)$ and $U_{2}=\partial_{s}Q(t,t,s)$. Then
$U_{1}=U_{3}+U_{4}$ with

$U_{3}=(\partial_{2}Q)(t,t,s),U_{4}=(\partial_{1}Q)(t,t,s).$

$U_{4}=J\langle\nabla\mu,a(V)\rangle$. When $t=s=0$, $U_{3}=P_{*}V$. Moreover,
$[U_{2},U_{4}]=0$. By (B.4), we have

$\displaystyle\nabla^{A}_{V}v=\nabla_{U_{1}}U_{2}\bigg{|}_{t=s=0}$
$\displaystyle=\nabla_{U_{3}}U_{2}+\nabla_{U_{4}}U_{2}\bigg{|}_{t=s=0}=\nabla_{P_{*}V}v+\nabla_{U_{2}}U_{4}\bigg{|}_{t=s=0}$
$\displaystyle=\nabla_{P_{*}V}v+J\langle\operatorname{Hess}\mu(v),a(V)\rangle.\qed$

The next lemma concerns the curvature tensor of $\nabla^{A}$. Since we are
merely interested in the manifold $\mathbb{H}^{2}_{+}$, it suffices to work
with vector fields $\partial_{t}$ and $\partial_{s}$.

######  Lemma B.5.

Write $T=\nabla^{A}_{\partial_{t}}P$ and $S=\nabla^{A}_{\partial_{s}}P$ for
short. The following properties hold for any configuration $(A,P)$ and any
vector field $v$ along $P$:

  1. (1)

The connection $\nabla^{A}$ is equivariant under the gauge transformation
$u(A,P)=(A-u^{-1}du,u\cdot P)$, i.e.

$u_{*}(\nabla^{A}_{V}P)=\nabla^{u(A)}_{V}u(P),\
u_{*}(\nabla^{A}_{V}v)=\nabla^{u(A)}_{V}u_{*}v.$

where $u_{*}v$ is the vector field along $u(P)$.

  2. (2)

If $v$ is the pull-back of a $G$-invariant vector field on $M$, then
$\nabla^{A}_{\partial_{s}}v=\nabla_{S}v$.

  3. (3)

$(\nabla^{A}_{\partial_{t}}\nabla^{A}_{\partial_{s}}-\nabla^{A}_{\partial_{s}}\nabla^{A}_{\partial_{t}})P=J\langle\nabla\mu,F_{A}(\partial_{t},\partial_{s})\rangle=-\tilde{F}$
where $F=-*_{2}F_{A}$.

  4. (4)

For any vector fields $v,w$ on $\operatorname{Im}P\subset M$,

$\partial_{s}\langle
v,w\rangle=\langle\nabla^{A}_{\partial_{s}}v,w\rangle+\langle
v,\nabla^{A}_{\partial_{s}}w\rangle,$

i.e. the connection $\nabla^{A}$ is unitary.

  5. (5)

The curvature tensor of $\nabla^{A}$ is given by

$(\nabla^{A}_{\partial_{t}}\nabla^{A}_{\partial_{s}}-\nabla^{A}_{\partial_{s}}\nabla^{A}_{\partial_{t}})v=R_{M}(T,S)v+J\langle\operatorname{Hess}\mu(v),F_{A}(\partial_{t},\partial_{s})\rangle.$

where $R_{M}(\cdot,\cdot)\cdot$ denotes the Riemannian curvature tensor on
$M$.

###### Proof.

The property (1) follows from the defining property (B.3) and (B.4) of
$\nabla_{A}$.

If $v$ is induced from a $G$-variant vector field on $M$, then for any $g\in
G$, $\rho(g)_{*}v=v$. By (B.4),

$\nabla^{A}_{V}v\colonequals\nabla_{\nabla^{A}_{V}P}\rho(e^{t\cdot
a(V)})_{*}v=\nabla_{\nabla^{A}_{V}P}v.$

This proves (2). For (3), if $F_{A}\equiv 0$ near a point
$x\in\mathbb{H}^{2}_{+}$, then we apply a gauge transformation $u$ so that the
connection 1-form $a\equiv 0$ near $x$. Thus,

$u_{*}(\nabla^{A}_{\partial_{t}}\nabla^{A}_{\partial_{s}}-\nabla^{A}_{\partial_{s}}\nabla^{A}_{\partial_{t}})P=\nabla_{\partial_{t}}\partial_{s}u(P)-\nabla_{\partial_{s}}\partial_{t}u(P)=0.$

This shows the commutator is at least proportional to $*F_{A}$. To work out
the general case, we apply (B.6) and Lemma B.3:

$\displaystyle\nabla^{A}_{\partial_{t}}\nabla^{A}_{\partial_{s}}P$
$\displaystyle=\nabla_{\partial_{t}P}S+J\langle\operatorname{Hess}\mu(S),a(\partial_{t})\rangle$
$\displaystyle=\nabla_{\partial_{t}P}\partial_{s}P+\nabla_{\partial_{t}P}J\langle\nabla\mu,a(\partial_{s})\rangle+J\langle\operatorname{Hess}\mu(S),a(\partial_{t})\rangle$
$\displaystyle=\nabla_{\partial_{t}P}\partial_{s}P+J\langle\nabla\mu,\partial_{t}a(\partial_{s})\rangle-\langle\operatorname{Hess}\mu(\langle\nabla\mu,a(\partial_{s})\rangle),a(\partial_{t})\rangle$
$\displaystyle\qquad+J\langle\operatorname{Hess}\mu(\partial_{t}P),a(\partial_{s})\rangle+J\langle\operatorname{Hess}\mu(\partial_{s}P),a(\partial_{t})\rangle.$

At this point, we need the following fact. For any $\xi,\eta\in\mathfrak{g}$,

$\langle\operatorname{Hess}\mu(\langle\nabla\mu,\xi\rangle),\eta\rangle=\langle\operatorname{Hess}\mu(\langle\nabla\mu,\eta\rangle),\xi\rangle.$

This follows from the fact that
$\nabla_{\tilde{\xi}}\tilde{\eta}-\nabla_{\tilde{\xi}}\tilde{\eta}=[\tilde{\xi},\tilde{\eta}]=\widetilde{[\xi,\eta]}=0$.
This proves $(3)$.

As for (4), we apply the gauge invariance of $\nabla^{A}$. Alternatively, one
uses Lemma B.3 and the fact that

(B.7) $\langle J\operatorname{Hess}\mu(v),w\rangle+\langle
J\operatorname{Hess}\mu(w),v\rangle=0.$

which is obtained by differentiating (B.2).

The expression of the curvature tensor $(\ref{5})$ requires some work. Again,
if $F_{A}\equiv 0$, we use the gauge invariance of $\nabla_{A}$, and $(5)$
follows from the definition of $R_{M}$. The actually proof is not very tidy.
We follow the strategy of (3):

$\displaystyle\nabla^{A}_{\partial_{t}}\nabla^{A}_{\partial_{s}}v$
$\displaystyle=\nabla_{\partial_{t}P}\nabla^{A}_{\partial_{s}}v+J\langle\operatorname{Hess}\mu(\nabla^{A}_{\partial_{s}}v),a(\partial_{t})\rangle$
$\displaystyle=\nabla_{\partial_{t}P}\nabla_{\partial_{s}P}v+J\nabla_{\partial_{t}P}\langle\operatorname{Hess}\mu(v),a(\partial_{s})\rangle+J\langle\operatorname{Hess}\mu(\nabla^{A}_{\partial_{s}}v),a(\partial_{t})\rangle$
$\displaystyle=\nabla_{\partial_{t}P}\nabla_{\partial_{s}P}v+J\langle\operatorname{Hess}\mu(v),\partial_{t}a(\partial_{s})\rangle+J\langle(\nabla_{\partial_{t}P}\operatorname{Hess}\mu)(v),a(\partial_{s})\rangle$
$\displaystyle\qquad+J\langle\operatorname{Hess}\mu(\nabla_{\partial_{t}P}v),a(\partial_{s})\rangle+J\langle\operatorname{Hess}\mu(\nabla_{\partial_{s}P}v),a(\partial_{t})\rangle$
$\displaystyle\qquad+J\langle\operatorname{Hess}\mu(\langle
J\operatorname{Hess}\mu(v),a(\partial_{s})\rangle),a(\partial_{t})\rangle$

There are six terms in the expression. The fourth and fifth ones will also
occur in that of $\nabla^{A}_{\partial_{s}}\nabla^{A}_{\partial_{t}}v$, so
canceled out. The second term contributes to

$J\langle\operatorname{Hess}\mu(v),F_{A}(\partial_{t},\partial_{s})\rangle.$

The first one contributes to $R_{M}(\partial_{t}P,\partial_{s}P)$. To better
organize the proof, we point out two lemmas from which the identity (5) will
follow. Note that

$R_{M}(T,S)-R_{M}(\partial_{t}P,\partial_{s}P)=R_{M}(\partial_{t}P,\tilde{a}(\partial_{s}))+R_{M}(\tilde{a}(\partial_{t}),\partial_{s}P)+R_{M}(\tilde{a}(\partial_{t}),\tilde{a}(\partial_{s})).$

We have to identify these terms in the expression of
$\nabla^{A}_{\partial_{t}}\nabla^{A}_{\partial_{s}}v-\nabla^{A}_{\partial_{s}}\nabla^{A}_{\partial_{t}}v$.∎

######  Lemma B.6.

For any $\xi\in\mathfrak{g}$ and vector fields $u,w$ on $M$, we have

$R_{M}(w,\tilde{\xi})u=\langle\nabla_{w}(J\operatorname{Hess}\mu)(u),\xi\rangle.$

###### Proof of Lemma.

Differentiating (B.7) yields that

$\langle\nabla_{u}(J\operatorname{Hess}\mu)(v),w\rangle+\langle\nabla_{u}(J\operatorname{Hess}\mu)(w),v\rangle=0.$

The key observation is that for any vectors $u,v,w$, we have

$\langle
R_{M}(u,v)w,J\nabla\mu\rangle=-\langle\nabla_{u}(J\operatorname{Hess}\mu)(v),w\rangle-\langle\nabla_{v}(J\operatorname{Hess}\mu)(w),u\rangle.$

Indeed, we use the symmetry of curvature tensor to compute:

$\displaystyle\langle R_{M}(u,v)w,J\nabla\mu\rangle$ $\displaystyle=-\langle
R_{M}(u,v)J\nabla\mu,w\rangle$
$\displaystyle=-\langle\nabla_{u}(J\operatorname{Hess}\mu)(v),w\rangle+\langle\nabla_{v}(J\operatorname{Hess}\mu)(u),w\rangle$
$\displaystyle=-\langle\nabla_{u}(J\operatorname{Hess}\mu)(v),w\rangle-\langle\nabla_{v}(J\operatorname{Hess}\mu)(w),u\rangle.$

This expression is unchanged if we permute $(u,v,w)$. Using the symmetry

$R_{M}(u,v)w+R_{M}(v,w)u+R_{M}(w,u)v=0$

from Riemannian geometry, we conclude that

$\langle\nabla_{u}(J\operatorname{Hess}\mu)(v),w\rangle+\langle\nabla_{v}(J\operatorname{Hess}\mu)(w),u\rangle+\langle\nabla_{w}(J\operatorname{Hess}\mu)(u),v\rangle=0.$

In particular, this implies $\langle
R_{M}(u,v)w,J\nabla\mu\rangle=\langle\nabla_{w}(J\operatorname{Hess}\mu)(u),v\rangle.$
Finally, note that $\tilde{\xi}=\langle\nabla\mu,\xi\rangle$ and

$\langle R_{M}(w,\tilde{\xi})u,v\rangle=\langle
R_{M}(u,v)w,\tilde{\xi}\rangle=\langle\nabla_{w}(J\operatorname{Hess}\mu)(u),v\otimes\xi\rangle.\qed$

######  Lemma B.7.

For any $\xi,\eta\in\mathfrak{g}$ and any vector field $v$ on $M$,

$R_{M}(\tilde{\xi},\tilde{\eta})v=J\langle\operatorname{Hess}\mu(\langle
J\operatorname{Hess}\mu(v),\eta\rangle_{\mathfrak{g}}),\xi\rangle_{\mathfrak{g}}-J\langle\operatorname{Hess}\mu(\langle
J\operatorname{Hess}\mu(v),\xi\rangle_{\mathfrak{g}}),\eta\rangle_{\mathfrak{g}}.$

###### Proof of Lemma.

This identity is equivalent to that

(B.8) $\langle
R_{M}(\tilde{\xi},\tilde{\eta})v,w\rangle=-\langle\nabla_{v}\tilde{\eta},\nabla_{w}\tilde{\xi}\rangle+\langle\nabla_{v}\tilde{\xi},\nabla_{w}\tilde{\eta}\rangle.$

Recall that $\mathcal{L}_{\tilde{\xi}}g=0$ implies
$\langle\nabla_{v}\tilde{\xi},w\rangle=-\langle\nabla_{w}\tilde{\xi},v\rangle$.
Hence, the right hand side of (B.8) equals

$\displaystyle I$
$\displaystyle\colonequals-\langle\nabla_{v}\tilde{\eta},\nabla_{w}\tilde{\xi}\rangle+\langle\nabla_{v}\tilde{\xi},\nabla_{w}\tilde{\eta}\rangle$
$\displaystyle=-v\cdot\langle\tilde{\eta},\nabla_{w}\tilde{\xi}\rangle+\langle\tilde{\eta},\nabla_{v}\nabla_{w}\tilde{\xi}\rangle+w\cdot\langle\tilde{\eta},\nabla_{v}\tilde{\xi}\rangle-\langle\tilde{\eta},\nabla_{w}\nabla_{v}\tilde{\xi}\rangle$

Since $\nabla_{\tilde{\xi}}\tilde{\eta}=\nabla_{\tilde{\eta}}\tilde{\xi}$, we
have

$\langle\nabla_{v}\tilde{\xi},\tilde{\eta}\rangle=-\langle\nabla_{\tilde{\eta}}\tilde{\xi},v\rangle=-\langle\nabla_{\tilde{\xi}}\tilde{\eta},v\rangle=\langle\nabla_{v}\tilde{\eta},\tilde{\xi}\rangle.$

Therefore,
$v\cdot\langle\tilde{\eta},\tilde{\xi}\rangle=2\langle\nabla_{v}\tilde{\xi},\tilde{\eta}\rangle$.
Moreover,

$\displaystyle-v\cdot\langle\tilde{\eta},\nabla_{w}\tilde{\xi}\rangle+w\cdot\langle\tilde{\eta},\nabla_{v}\tilde{\xi}\rangle$
$\displaystyle=-\frac{1}{2}v\cdot
w\langle\tilde{\eta},\tilde{\xi}\rangle+\frac{1}{2}w\cdot
v\langle\tilde{\eta},\tilde{\xi}\rangle.$
$\displaystyle=-\frac{1}{2}[v,w]\langle\tilde{\eta},\tilde{\xi}\rangle=-\langle\tilde{\eta},\nabla_{[v,w]}\tilde{\xi}\rangle.$

Finally, we conclude that $I=\langle
R_{M}(v,w)\tilde{\xi},\tilde{\eta}\rangle=\langle
R_{M}(\tilde{\xi},\tilde{\eta})v,w\rangle$. ∎

####  B.2. A Bochner-Type Formula

It will be convenient to introduce an operator $D_{p}$ for any $p\in M$. For
any tangent vector $v\in T_{p}M$, its image $D_{p}(v)$ is defined as:

$v\in T_{p}M\mapsto
D_{p}(v)\colonequals(\operatorname{Hess}_{p}H(v),\langle\nabla\mu,v\rangle,\langle\nabla\mu,Jv\rangle)\in
T_{p}M\oplus\mathfrak{g}\oplus\mathfrak{g}.$

The family of operators $D_{p}$ forms a bundle map $TM\to
TM\oplus\mathfrak{g}\oplus\mathfrak{g}$.

The Bochner’s formula [Pet16, P. 334] was originally stated for a harmonic
function $u:M\to\mathbb{R}$ on a Riemannian manifold $M$. It computes the
Laplacian of $|\nabla u|^{2}$:

$0=\frac{1}{2}\Delta_{M}|\nabla
u|^{2}+|\operatorname{Hess}u|^{2}+\operatorname{Ric}(\nabla u,\nabla u).$

We provide a formula in the same spirit for a solution $(A,P)$ to the gauged
Witten equation (4.1) on $\mathbb{H}^{2}_{+}$, with $D_{p}$ playing the role
of $\operatorname{Ric}(\cdot,\cdot)$. The Laplacian operator $\Delta_{M}$ or
$\Delta_{\mathbb{H}^{2}_{+}}$ is always assumed to have a non-negative
spectrum. In particular,

$\Delta_{\mathbb{H}^{2}_{+}}=-(\partial_{t}^{2}+\partial_{s}^{2}).$

######  Theorem B.8.

Write $T=\nabla^{A}_{\partial_{t}}P$, $S=\nabla^{A}_{\partial_{s}}P$ and
$F=-*_{2}F_{A}$ for short. For a solution $(A,P)$ to the gauged Witten
equation $(\ref{generalized-vortex-equation})$ on $\mathbb{H}^{2}_{+}$, we
have identities:

  1. (1)

The Laplacian
$\frac{1}{2}(-\Delta_{\mathbb{H}^{2}_{+}})|T|^{2}=\frac{1}{2}(\partial_{s}^{2}+\partial_{t}^{2})|T|^{2}$
of $|T|^{2}$ is equal to

$\displaystyle|\nabla^{A}_{\partial_{s}}T|^{2}+|\nabla^{A}_{\partial_{t}}T|^{2}+|D_{P}(T)|^{2}+\langle
R_{M}(S,T)S,T\rangle$
$\displaystyle\qquad+\langle(\nabla_{T}\operatorname{Hess}H)(\nabla
H),T\rangle+\langle\operatorname{Hess}\mu(2JS-T),T\otimes F\rangle.$

  2. (2)

Similarly,
$\frac{1}{2}(-\Delta_{\mathbb{H}^{2}_{+}})|S|^{2}=\frac{1}{2}(\partial_{s}^{2}+\partial_{t}^{2})|S|^{2}$
is equal to

$\displaystyle|\nabla^{A}_{\partial_{t}}S|^{2}+|\nabla^{A}_{\partial_{s}}S|^{2}+|D_{P}(S)|^{2}+\langle
R_{M}(T,S)T,S\rangle$
$\displaystyle\qquad+\langle(\nabla_{S}\operatorname{Hess}H)(\nabla
H),S\rangle+\langle\operatorname{Hess}\mu(-2JT-S),S\otimes F\rangle.$

  3. (3)

The Laplacian
$\frac{1}{2}(-\Delta_{\mathbb{H}^{2}_{+}})|F|_{\mathfrak{g}}^{2}$ of
$|F|_{\mathfrak{g}}^{2}$ is equal to

$\displaystyle|\partial_{s}F|^{2}_{\mathfrak{g}}+|\partial_{t}F|^{2}_{\mathfrak{g}}+|\langle\nabla\mu,F\rangle_{\mathfrak{g}}|^{2}+2\langle\operatorname{Hess}\mu(JS),T\otimes
F\rangle$

Define $\nabla^{A}P=dt\otimes T+ds\otimes S$ and $\operatorname{Hess}_{A}P$ as
$\omega_{i}\otimes\omega_{j}\otimes\nabla^{A}_{e_{i}}\nabla^{A}_{e_{j}}P$ with
$(e_{1},e_{2})=(\partial_{t},\partial_{s})$ and
$(\omega_{1},\omega_{2})=(dt,ds)$.

######  Corollary B.9.

We have the following identity for
$\Delta_{\mathbb{H}^{2}_{+}}(|\nabla^{A}P|^{2}+|F|^{2}_{\mathfrak{g}})$:

$\displaystyle 0$
$\displaystyle=\frac{1}{2}\Delta_{\mathbb{H}^{2}_{+}}(|\nabla^{A}P|^{2}+|F|^{2}_{\mathfrak{g}})+|\operatorname{Hess}_{A}P|^{2}+|\nabla
F|^{2}_{\mathfrak{g}}+|D_{P}(\nabla^{A}P)|^{2}+|\langle\nabla\mu,F\rangle_{\mathfrak{g}}|^{2}$
$\displaystyle\qquad+2\langle
R(S,T)S,T\rangle+\langle(\nabla_{T}\operatorname{Hess}H)(\nabla
H),T\rangle+\langle(\nabla_{S}\operatorname{Hess}H)(\nabla H),S\rangle$
$\displaystyle\qquad+6\langle\operatorname{Hess}\mu(JS),T\otimes
F\rangle-\langle\operatorname{Hess}\mu(T),T\otimes
F\rangle-\langle\operatorname{Hess}\mu(S),S\otimes F\rangle.$

One thinks of $-(\nabla^{A}_{\partial_{t}}T+\nabla^{A}_{\partial_{s}}S)$ as
the connection Laplacian of $P$. Let us define

$-\Delta_{A}P\colonequals\nabla^{A}_{\partial_{t}}\nabla^{A}_{\partial_{t}}P+\nabla^{A}_{\partial_{s}}\nabla^{A}_{\partial_{s}}P=\nabla^{A}_{\partial_{t}}T+\nabla^{A}_{\partial_{s}}S.$

To prove Theorem B.8, we start with a useful formula of $\Delta_{A}P$.

######  Lemma B.10.

$-\Delta_{A}P=J\tilde{F}+\operatorname{Hess}H(\nabla H)$.

###### Proof.

Apply the operator $-\nabla^{A}_{\partial_{t}}+J\nabla^{A}_{\partial_{s}}$ to
the equation (4.7b). Using the fact that $\operatorname{Hess}H$ anti-commutes
with $J$ (Lemma B.1 (3)) and Lemma B.5 (2)(3), we have

$\displaystyle 0$
$\displaystyle=(-\nabla^{A}_{\partial_{t}}+J\nabla^{A}_{\partial_{s}})(T+JS+\nabla
H)$
$\displaystyle=-(\nabla^{A}_{\partial_{t}}T+\nabla^{A}_{\partial_{s}}S)-J[\nabla^{A}_{\partial_{t}}S-\nabla^{A}_{\partial_{s}}T]-\operatorname{Hess}H(T+JS)$
$\displaystyle=\Delta_{A}P+J(\tilde{F})+\operatorname{Hess}H(\nabla H).\qed$

######  Remark B.11.

It is enlightening to work out Lemma B.10 and Theorem B.8 concretely in some
special cases. In Example 3.2, we have $M=\mathbb{C},W\equiv 0$ and
$\mu=\frac{i}{2}|z|^{2}$. The metric on $\mathbb{C}$ is flat, and we identify
$T_{z}M$ with $\mathbb{C}$ for each $z\in M$. Hence,

$\nabla\mu(z)=z\otimes i\text{ and
}\operatorname{Hess}\mu=1_{\mathbb{C}}\otimes i.$

Take $\delta=\frac{i}{2}\in i\mathbb{R}=\mathfrak{g}$. In this case, the
equation (4.1) recovers the vortex equation on
$\mathbb{H}^{2}_{+}\subset\mathbb{C}$:

$\left\\{\begin{array}[]{rl}\bar{\partial}_{A}\Phi&=0\\\
-*F_{A}+\mu&=\frac{i}{2}\end{array}\right.$

where $A$ is a unitary connection and $\Phi:\mathbb{H}^{2}_{+}\to\mathbb{C}$
is a smooth function. In this case, Lemma B.10 says

$\Delta_{A}\Phi=\langle F,i\rangle_{\mathfrak{g}}\Phi$

which follows from the Weitzenböck formula. Since $\nabla H\equiv 0$, $T=-JS$.
The first two identities in Theorem B.8 yield:

$0=\frac{1}{2}\Delta_{\mathbb{H}^{2}_{+}}|\nabla_{A}\Phi|^{2}+|\operatorname{Hess}_{A}\Phi|^{2}+|D(\nabla_{A}\Phi)|^{2}-3\langle
F,i\rangle_{\mathfrak{g}}|\nabla_{A}\Phi|^{2}.$

In this case, $D(v)=(\langle\nabla\mu,v\rangle,\langle\nabla\mu,Jv\rangle)$
and $|D(v)|^{2}_{\mathfrak{g}}=|\Phi|^{2}|v|^{2}$. In the meanwhile, we have

$0=\frac{1}{2}\Delta_{\mathbb{H}^{2}_{+}}|F|^{2}+|\nabla
F|^{2}_{\mathfrak{g}}+|\Phi|^{2}|F|^{2}_{\mathfrak{g}}-|\nabla_{A}\Phi|^{2}\langle
F,i\rangle_{\mathfrak{g}}.$

These formulae were first proved in [JT80, Proposition 6.1]. ∎

###### Proof of Theorem B.8.

Let us start with $\partial_{s}^{2}|T|^{2}$. By Lemma B.5 (4),

$\displaystyle\frac{1}{2}\partial_{s}^{2}|T|^{2}$
$\displaystyle=\partial_{s}\langle\nabla^{A}_{\partial_{s}}T,T\rangle=|\nabla^{A}_{\partial_{s}}T|^{2}+\langle
T,\nabla^{A}_{\partial_{s}}\nabla^{A}_{\partial_{s}}T\rangle.$

By Lemma B.5 (3)(5), we have

$\displaystyle\nabla^{A}_{\partial_{s}}\nabla^{A}_{\partial_{s}}T$
$\displaystyle=\nabla^{A}_{\partial_{s}}(\nabla^{A}_{\partial_{t}}S+\tilde{F})$
$\displaystyle=\nabla^{A}_{\partial_{t}}\nabla^{A}_{\partial_{s}}S+R_{M}(S,T)S+\langle
J\operatorname{Hess}\mu(S),F\rangle+\nabla^{A}_{\partial_{s}}\tilde{F},$

which implies that

$\displaystyle\frac{1}{2}(\partial_{t}^{2}+\partial_{s}^{2})|T|^{2}$
$\displaystyle=|\nabla^{A}_{\partial_{s}}T|^{2}+|\nabla^{A}_{\partial_{t}}T|^{2}+\langle\nabla^{A}_{\partial_{t}}(-\Delta_{A}P),T\rangle+\langle
R_{M}(S,T)S,T\rangle$ $\displaystyle\qquad+\langle
J\operatorname{Hess}\mu(S),F\otimes
T\rangle+\langle\nabla^{A}_{\partial_{s}}\tilde{F},T\rangle.$

By using the equation (4.7a), we attempt to extract some positivity from the
last term:

$\displaystyle\langle\nabla^{A}_{\partial_{s}}\tilde{F},T\rangle$
$\displaystyle=\langle\nabla^{A}_{\partial_{s}}\langle
J\nabla\mu,F\rangle,T\rangle.$ $\displaystyle=\langle
J\operatorname{Hess}\mu(S),F\otimes T\rangle+\langle
J\nabla\mu,\partial_{s}F\rangle,T\rangle.$ $\displaystyle=\langle
J\operatorname{Hess}\mu(S),F\otimes
T\rangle+\langle\langle\nabla\mu,JT\rangle,\langle\nabla\mu,S\rangle\rangle_{\mathfrak{g}}$

By Lemma B.1 (5), $\langle\nabla\mu,J\nabla H\rangle=0$. Using (4.7b), we have

$\langle\nabla\mu,JT\rangle=\langle\nabla\mu,S-J\nabla
H\rangle=\langle\nabla\mu,S\rangle.$

Hence, $\langle\nabla^{A}_{\partial_{s}}\tilde{F},T\rangle=\langle
J\operatorname{Hess}\mu(S),F\otimes
T\rangle+|\langle\nabla\mu,JT\rangle|_{\mathfrak{g}}$.

Now we deal with the term involving $\Delta_{A}P$, using Lemma B.10. We
exploit the fact that $\operatorname{Hess}H(\nabla H)$ is a $G$-invariant
vector field on $M$ and Lemma B.5 (2):

$\displaystyle\nabla^{A}_{\partial_{t}}(-\Delta_{A}P)$
$\displaystyle=\nabla^{A}_{\partial_{t}}(J\tilde{F}+\operatorname{Hess}H(\nabla
H))$
$\displaystyle=-\nabla^{A}_{\partial_{t}}\langle\nabla\mu,F\rangle+(\nabla_{T}\operatorname{Hess}H)(\nabla
H)+\operatorname{Hess}H^{2}(T).$
$\displaystyle=-\langle\operatorname{Hess}\mu(T),F\rangle-\langle\nabla\mu,\partial_{t}F\rangle+(\nabla_{T}\operatorname{Hess}H)(\nabla
H)+\operatorname{Hess}H^{2}(T).$

Note that
$-\langle\langle\nabla\mu,\partial_{t}F\rangle,T\rangle=\langle\langle\nabla\mu,\langle\nabla\mu,T\rangle\rangle,T\rangle=|\langle\nabla\mu,T\rangle|_{\mathfrak{g}}^{2}$.

Combining all these together, we obtain

$\displaystyle\frac{1}{2}(\partial_{t}^{2}+\partial_{s}^{2})|T|^{2}$
$\displaystyle=|\nabla^{A}_{\partial_{s}}T|^{2}+|\nabla^{A}_{\partial_{t}}T|^{2}+\langle
R_{M}(S,T)S,T\rangle$
$\displaystyle\qquad+|\langle\nabla\mu,T\rangle|_{\mathfrak{g}}^{2}+|\langle\nabla\mu,JT\rangle|_{\mathfrak{g}}^{2}+|\operatorname{Hess}H(T)|^{2}$
$\displaystyle\qquad+\langle(\nabla_{T}\operatorname{Hess}H)(\nabla
H),T\rangle+\langle\operatorname{Hess}\mu(2JS-T),F\otimes T\rangle.$

The formula of $\frac{1}{2}(\partial_{t}^{2}+\partial_{s}^{2})|S|^{2}$ is
proved in a similar may.

Finally, let us deal with the Laplacian of $|F|^{2}_{\mathfrak{g}}$. By
(4.7a), we have

$\displaystyle\frac{1}{2}\partial_{s}^{2}|F|^{2}_{\mathfrak{g}}$
$\displaystyle=|\partial_{s}F|^{2}_{\mathfrak{g}}+\langle
F,\partial_{s}^{2}(-\mu)\rangle_{\mathfrak{g}}.$
$\displaystyle=|\partial_{s}F|^{2}_{\mathfrak{g}}-\langle\operatorname{Hess}\mu(S),S\otimes
F\rangle-\langle\nabla\mu,\nabla^{A}_{\partial_{s}}S\otimes F\rangle.$

By Lemma B.10, we have

$\displaystyle\langle-\nabla\mu,(\nabla^{A}_{\partial_{t}}T+\nabla^{A}_{\partial_{s}}S)\otimes
F\rangle$ $\displaystyle=\langle-\nabla\mu,J\tilde{F}\otimes
F\rangle+\langle-\nabla\mu,\operatorname{Hess}H(\nabla H)\otimes F\rangle$
$\displaystyle=|\langle\mu,F\rangle_{\mathfrak{g}}|^{2}+\langle\operatorname{Hess}\mu(\nabla
H),\nabla H\otimes F\rangle.$

At the last step, we used the identity:

(B.9) $\langle\operatorname{Hess}\mu(X),\nabla
H\rangle+\langle\nabla\mu,\operatorname{Hess}H(X)\rangle=0$

with $X=\nabla H$. To see (B.9), notice that by Lemma B.1 (5),
$\langle\nabla\mu,\nabla H\rangle\equiv 0$. Expanding the expression
$X\cdot\langle\nabla\mu,\nabla H\rangle\equiv 0$ yields (B.9). This completes
the proof of Theorem B.8. ∎

###  Appendix C Surfaces with Cylindrical Ends

In this appendix, we generalize Proposition 7.3 for oriented surfaces with
cylindrical ends. To start, let us fix a conformal structure of the genus-$g$
surface $\Sigma_{g}$ and consider the punctured surface

$\Sigma_{g,n}\colonequals\Sigma_{g}\setminus\\{p_{1},\cdots,p_{n}\\},$

where $\\{p_{j}\\}_{1\leq j\leq n}\subset\Sigma_{g}$ is also viewed as a
positive divisor $\mathcal{D}\colonequals\sum p_{j}$. Later, we will specify
the metric of $\Sigma_{g,n}$ compatible with this complex structure. We
require that

$\chi(\Sigma_{g,n})=2-2g-n\leq 0.$

Thus one may take $g=0$ if $n\geq 2$. The harmonic 1-form $\lambda$ is allowed
to have poles at each puncture $p_{j},1\leq j\leq n$, but further conditions
are required:

######  Assumption C.1.

Let $\lambda\in\Omega^{1}(\Sigma_{g,n},i\mathbb{R})$ and assume that its
$(1,0)$-part $\lambda^{1,0}\in H^{0}(\Sigma_{g},\mathcal{K}(\mathcal{D}))$
extends to a holomorphic section of the canonical bundle
$\mathcal{K}\colonequals\Lambda^{1,0}\Sigma_{g}\to\Sigma_{g}$ twisted by
$\mathcal{D}$. We require that

  * •

the residue of $\lambda^{1,0}$ at each puncture $p_{j}\in\mathcal{D}$ is non-
zero, and

  * •

$\lambda^{1,0}$ has $2g-2+n$ simple zeros in $\Sigma_{g,n}$.

For such a 1-form $\lambda$ to exist, we necessarily have $n\geq 2$. ∎

In particular, for some holomorphic coordinate $z_{j}$ near
$p_{j}\in\Sigma_{g}$ and $a_{j}\neq 0\in\mathbb{C}$,

$\lambda^{1,0}=-\frac{a_{j}dz_{j}}{z_{j}}.$

By rescaling the coordinate function $z_{j}$ if necessary, we assume that the
balls $B(p_{j},1),1\leq j\leq n$ are disjoint in $\Sigma_{g}$. In terms of the
polar coordinate $z_{j}=\exp(-(s_{j}+i\theta_{j}))$ with $s_{i}\in[0,\infty)$
and $\theta_{j}\in\mathbb{R}/2\pi\mathbb{Z}$, we have

$\lambda^{1,0}=a_{j}(ds_{j}+id\theta_{j}).$

Pick a Riemannian metric of $\Sigma_{g,n}$ which restricts to the product
metric on each end

$U_{j}\colonequals[0,\infty)_{s_{j}}\times(\mathbb{R}/2\pi\mathbb{Z})_{\theta_{j}}\subset\Sigma_{g,n},1\leq
j\leq n,$

and is compatible to the given complex structure. Then
$\lambda=2i(\operatorname{Im}a_{j}ds_{j}+\operatorname{Re}a_{j}d\theta_{j})$
is covariantly constant on $\coprod U_{j}$.

Because the surface $\Sigma_{g,n}$ is not compact, we have to work instead
with a relative $spin^{c}\ $structure
$\widehat{\mathfrak{s}}=(\mathfrak{s},\varphi)$, where
$\mathfrak{s}=(S^{+}=L^{+}\oplus L^{-},\rho_{2})$ is a $spin^{c}\ $structure
of $\Sigma_{g,n}$ and

$\varphi:L^{+}|_{\coprod U_{j}}\cong\mathbb{C}$

is a trivialization of $L^{+}$ over the ends $\coprod U_{j}$. We set
$L^{-}=L^{+}\otimes\Lambda^{0,1}\Sigma_{g,n}$. In this case, we can define the
relative Chern classes and the relative degrees of
$\widehat{\mathfrak{s}},L^{+}$ and $\Lambda^{0,1}\Sigma_{g,n}$ respectively.
In particular, $\deg\Lambda^{0,1}\Sigma_{g,n}=-(2g-2+n)\leq 0$ and

$c_{1}(\widehat{\mathfrak{s}})[\Sigma_{g,n},\partial\Sigma_{g,n}]=2\deg
L^{+}+\deg\Lambda^{0,1}\Sigma_{g,n}.$

We are interested in the case when $d\colonequals\deg L^{+}\in[0,2g-2+n]$.

When $g=0$ and $n=2$, one may simply take
$\Sigma_{0,2}=\mathbb{R}_{s}\times(\mathbb{R}/2\pi\mathbb{Z})_{\theta}$ to be
the product manifold. For the standard relative $spin^{c}\ $structure
$\widehat{\mathfrak{s}}_{std}=(S^{+}_{std},\rho_{2})$, we have

$S^{+}_{std}=\mathbb{C}\oplus\Lambda^{0,1}\Sigma_{0,2}.$

We take $(d,\nabla^{LC})$ as the reference $spin^{c}\ $connection. Then for
any $\delta\in i\mathbb{R}$ and
$\lambda=2i((\operatorname{Im}a)ds+(\operatorname{Re}a)d\theta)$, there is a
unique covariantly constant spinor
$(\check{\Psi}_{+},\check{\Psi}_{-})\in\Gamma(\Sigma_{0,2},S^{+}_{std})$ (up
to the action of $S^{1}$) that solves the equations

(C.1)
$\left\\{\begin{array}[]{rl}\check{\Psi}_{+}^{*}\otimes\check{\Psi}_{-}&=\sqrt{2}\lambda^{0,1},\\\
\frac{i}{2}(|\check{\Psi}_{+}|^{2}-|\check{\Psi}_{-}|^{2})&\equiv\delta.\end{array}\right.$

Back to the case of $\Sigma_{g,n}$, the gauged Landau-Ginzburg model

$(M(\Sigma_{g,n},\widehat{\mathfrak{s}};\vec{\delta}_{*}),W_{\lambda},\mathcal{G}(\Sigma_{g,n}))$

now relies on an auxiliary function $\vec{\delta}_{*}$ and is constructed as
follows:

  * •

$\vec{\delta}_{*}:\Sigma_{g,n}\to i\mathbb{R}$ is a smooth function such that
$\vec{\delta}_{*}\equiv\delta_{j}$ for some constant $\delta_{j}\in
i\mathbb{R}$ on each end $U_{j}\subset\Sigma_{g,n}$;

  * •

the reference $spin^{c}\ $connection $\check{B}_{*}$ is identified with
$(d,\nabla^{LC})$ under the trivialization $\varphi$ on each end $U_{j}$;

  * •

the reference spinor $\check{\Psi}_{*}$ is identified with the solution
$(\check{\Psi}_{+,j},\check{\Psi}_{-,j})$ of (C.1) associated to the data
$(\delta_{j},a_{j})$ on each end $U_{j}$ under the trivialization $\varphi$.

  * •

The Kähler manifold $M(\Sigma,\widehat{\mathfrak{s}};\vec{\delta}_{*})$ is the
configuration space on $\Sigma$:

$(\check{B}_{*},\check{\Psi}_{*})+L^{2}_{k}(\Sigma_{g,n},i\Lambda^{1}\Sigma_{g,n}\oplus
S^{+}),$

for some $k\geq 1$, where
$\kappa_{*}\colonequals(\check{B}_{*},\check{\Psi}_{*})$ is the reference
configuration defined above.

  * •

The complex gauge group acting on $M$ is
$\mathcal{G}_{\mathbb{C}}(\Sigma_{g,n})=\\{u:\Sigma_{g,n}\to\mathbb{C}^{*}:u-1\in
L^{2}_{k+1}(\Sigma_{g,n})\\}$ whose Lie algebra is
$L^{2}_{k}(\Sigma_{g,n},\mathbb{C})$.

  * •

The moment map $\mu:M(\Sigma_{g,n},\widehat{\mathfrak{s}};\vec{\delta}_{*})\to
L^{2}_{k-1}(\Sigma_{g,n},i\mathbb{R})$ is defined by the formula

$\displaystyle\mu(\check{B},\check{\Psi})$
$\displaystyle=-\frac{1}{2}*_{\Sigma}F_{\check{B}^{t}}+\frac{i}{2}(|\check{\Psi}_{+}|^{2}-|\check{\Psi}_{-}|^{2})-\vec{\delta}_{*}.$

  * •

the superpotential $W_{\lambda}$ is the Dirac functional perturbed by the
harmonic 1-form $\lambda$, cf. Subsection 7.2.

This setup will allow us to generalize Propositions 7.4.

######  Proposition C.2.

Suppose that the harmonic 1-form $\lambda\neq
0\in\Omega^{1}_{h}(\Sigma_{g,n},i\mathbb{R})$ is chosen as above and
Assumption C.1 holds. Then for the gauged Landau-Ginzburg model

$(M(\Sigma_{g,n},\widehat{\mathfrak{s}};\vec{\delta}_{*}),W_{\lambda},\mathcal{G}(\Sigma_{g,n}))$

defined above, the critical locus $\operatorname{Crit}(L)$ consists of
$\binom{2g-2+n}{d}$ free $\mathcal{G}_{\mathbb{C}}$-orbits with $d=\deg
L^{+}$. For any function $\vec{\delta}\in
L^{2}_{k-1}(\Sigma_{g,n},i\mathbb{R})$,
$\mu^{-1}(\vec{\delta})\cap\operatorname{Crit}(L)$ contains a unique
$\mathcal{G}(\Sigma)$-orbit within each $\mathcal{G}_{\mathbb{C}}$-orbit of
$\operatorname{Crit}(L)$. Moreover, $W_{\lambda}$ is a Morse-Bott function. In
this sense, we say that $W_{\lambda}$ is stable and any
$\vec{\delta}\in\operatorname{Lie}(\mathcal{G})$ is $W_{\lambda}$-stable.

###### Proof.

We point out the place where the proof of Proposition 7.4 is modified. To
construct a configuration in
$\mu^{-1}(\vec{\delta})\cap\operatorname{Crit}(L)$, we have applied the a
priori estimate from Lemma 7.6 with

$w_{+}=|\check{\Psi}_{+}|\text{ and }w_{-}=|\check{\Psi}_{-}|.$

Here $\kappa=(\check{B},\check{\Psi}_{+},\check{\Psi}_{-})$ is some
representative in a free $\mathcal{G}_{\mathbb{C}}$-orbit of
$\operatorname{Crit}(L)$. However, Lemma 7.6 is stated only for closed
surfaces. We must adapt its proof to the non-compact surface $\Sigma_{g,n}$.
To start, we choose $\kappa$ such that

$\kappa|_{U_{j}}=(\check{B}_{*},\check{\Psi}_{*}),$

i.e., it agrees with the reference configuration on each end
$U_{j}\subset\Sigma_{g,n}$. As a result, for some $c>0$, we have
$w_{+},w_{-}>c$ on the union $\coprod U_{j}$. This allows us to derive the
estimate:

$\|d\alpha_{\pm}\|_{2}^{2}+\int_{\Sigma_{g,n}}\alpha_{\pm}^{2}w_{\pm}^{2}\geq
c_{\pm}\|\alpha_{\pm}\|_{L^{2}_{1}}^{2},$

as required in the proof of Lemma 7.6. Finally, Trudinger’s inequality also
holds for non-compact spaces; cf. [Tay11, Proposition 4.2] or [Wan18,
Proposition A.3]. Now the proof of Proposition 7.4 can proceed with no
difficulty. ∎

######  Remark C.3.

Proposition C.2 will allow us to compute the monopole Floer homology of the
product manifold $\Sigma_{g,n}\times S^{1}$ in the second paper [Wan20a].
Indeed, the group

$\operatorname{\mathit{HM}}_{*}(\Sigma_{g,n}\times
S^{1},\omega;\widehat{\mathfrak{s}})$

will have rank $\binom{2g-2+n}{d}$ for a suitable closed 2-form $\omega$ on
$\Sigma_{g,n}\times S^{1}$ and for the relative $spin^{c}\ $structure
$\widehat{\mathfrak{s}}$ with

$c_{1}(\widehat{\mathfrak{s}})=(2d-2g+2-n)\cdot k\in H^{2}(\Sigma_{g,n}\times
S^{1},\partial\Sigma_{g,n}\times S^{1};\mathbb{Z}),$

where $k$ is the Poincaré dual of $\\{pt\\}\times S^{1}$. ∎

######  Remark C.4.

The idea of [Wan20a] is to complete a 3-manifold with torus boundary into a
manifold with cylindrical ends. Likewise, one may complete a balanced sutured
manifold, which is a 3-manifold with corners, into a manifold with planar
ends. Then Proposition C.2 would be the replacement of Proposition 7.4, if one
attempts to construct the sutured Floer homology analytically. ∎

Finally, let us explain why Assumption C.1 is a generic condition. Let
$\mathcal{K}\colonequals\Lambda^{1,0}\Sigma_{g}$ be the canonical bundle of
$\Sigma_{g}$. Then we have a long exact sequence:

$\begin{array}[]{rcccl}0\to
H^{0}(\Sigma_{g},\mathcal{K})\hookrightarrow&H^{0}(\Sigma_{g},\mathcal{K}(\mathcal{D}))&\xrightarrow{\operatorname{Res}}&\mathbb{C}^{n}&\xrightarrow{\sigma}\mathbb{C}\to
0,\\\ &\eta&\mapsto&(\operatorname{Res}_{p_{j}}(\eta))_{1\leq j\leq
n}&\end{array}$

where $\operatorname{Res}_{p_{j}}(\eta)$ is the residue of $\eta$ at
$p_{j}\in\mathcal{D}$. The penultimate map $\sigma$ takes the sum of all
residues. This sequence is exact, because
$\dim_{\mathbb{C}}H^{0}(\Sigma_{g},\mathcal{K}(\mathcal{D}))=g+n-1$ by the
Riemann-Roch theorem.

######  Lemma C.5.

If $g\geq 1$, then for any vector $v=(a_{1},\cdots,a_{n})\in\mathbb{C}^{n}$
with $a_{j}\neq 0$, $1\leq j\leq n$ and $\sigma(v)=0$, there is a dense open
subset of $\operatorname{Res}^{-1}(v)$ such that any $\eta$ in this subset has
$2g-2+n$ simple zeros.

###### Proof of Lemma.

Let $V$ be the subset of $\operatorname{Res}^{-1}(v)$ consisting of sections
with simples zeros. $V$ is clearly open. To show that $V$ is dense, consider a
form $\eta\in\operatorname{Res}^{-1}(v)$ that has zeros with multiplicities
precisely at $q_{1},\cdots,q_{m}\subset\Sigma_{g}\setminus\mathcal{D}$. It
suffices to find some $\nu\in H^{0}(\Sigma_{g},\mathcal{K})$ such that
$\nu(q_{j})\neq 0$ for any $1\leq j\leq m$; then the sum $\eta+\epsilon\nu$
lies in $V$ for any sufficiently small $\epsilon$. Since $g\geq 1$, the
canonical bundle $\mathcal{K}$ is base-point free. Indeed, by the Riemann-Roch
theorem and Serre duality, for any $q\in\Sigma_{g}$,

$\dim_{\mathbb{C}}H^{0}(\Sigma_{g},\mathcal{K}(-q))=g-1<\dim
H^{0}(\Sigma_{g},\mathcal{K}).$

Now we set $\nu=\sum\epsilon_{j}\nu_{j}$ for some $\epsilon_{j}\in\mathbb{C}$
and $\nu_{j}\in H^{0}(\Sigma_{g},\mathcal{K})$ with $\nu_{j}(q_{j})\neq 0$,
$1\leq j\leq n$. ∎

The case when $g=0$ is more rigid, since $\operatorname{Res}$ is an injection.
For any $v\in\mathbb{C}^{n}$, one may vary the positive divisor $\mathcal{D}$
instead to achieve Assumption C.1:

######  Lemma C.6.

If $g=0$, then for any vector $v=(a_{1},\cdots,a_{n})\in\mathbb{C}^{n}$ with
$a_{j}\neq 0$, $1\leq j\leq n$ and $\sigma(v)=0$, the section
$\eta=\operatorname{Res}^{-1}(v)$ has $n-2$ simple zeros in
$\Sigma_{0,n}=\mathbb{CP}^{1}\setminus\mathcal{D}$ for a generic choice of
$\mathcal{D}$.

###### Proof of Lemma.

By an automorphism of $\mathbb{CP}^{1}$, we set $p_{n}=\infty$. Then the
holomorphic 1-form $\eta=\operatorname{Res}^{-1}(v)$ can be constructed
explicitly as

$\eta(z)=dz\cdot\sum_{j=1}^{n-1}\frac{a_{j}}{z-p_{j}}\text{ for any
}z\in\mathbb{C}.$

As we vary $p_{1}$ in a small neighborhood, the difference

$\frac{1}{z-p_{1}}-\frac{1}{z-p_{1}^{\prime}}=\frac{p_{1}-p_{1}^{\prime}}{(z-p_{1})(z-p_{1}^{\prime})}$

is non-vanishing on $\mathbb{C}$. Now one can argue as in the proof of Lemma
C.5. ∎

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