# Three-dimensional soft streaming

Songyuan Cui1  Yashraj Bhosale1  Mattia Gazzola123<EMAIL_ADDRESS>1Mechanical Sciences and Engineering, University of Illinois at Urbana-
Champaign, Urbana, IL 61801, USA 2National Center for Supercomputing
Applications, University of Illinois at Urbana-Champaign, Urbana, IL 61801,
USA 3Carl R. Woese Institute for Genomic Biology, University of Illinois at
Urbana-Champaign, Urbana, IL 61801, USA

###### Abstract

Viscous streaming is an efficient rectification mechanism to exploit flow
inertia at small scales for fluid and particle manipulation. It typically
entails a fluid vibrating around an immersed solid feature that, by
concentrating stresses, modulates the emergence of steady flows of useful
topology. Motivated by its relevance in biological and artificial settings
characterized by soft materials, recent studies have theoretically elucidated,
in two dimensions, the impact of body elasticity on streaming flows. Here, we
generalize those findings to three dimensions, via the minimal case of an
immersed soft sphere. We first improve existing solutions for the rigid sphere
limit, by considering previously unaccounted terms. We then enable body
compliance, exposing a three-dimensional, elastic streaming process available
even in Stokes flows. Such effect, consistent with two-dimensional analyses
but analytically distinct, is validated against direct numerical simulations
and shown to translate to bodies of complex geometry and topology, paving the
way for advanced forms of flow control.

###### keywords:

viscous streaming, microfluidics, elasticity, flow–structure interaction

##  1 Introduction

This study investigates the effects of body elasticity on three-dimensional
viscous streaming. Viscous streaming, an inertial phenomenon, refers to the
steady, rectified flows that emerge when a fluid oscillates around a localized
microfeature. Given its ability to remodel surrounding flows over short time
and length scales, streaming has found application in multiple aspects of
microfluidics, from particle manipulation Lutz et al. (2003, 2005); Marmottant
& Hilgenfeldt (2004); Lutz et al. (2006); Wang et al. (2011); Chong et al.
(2013); Chen & Lee (2014); Klotsa et al. (2015); Thameem et al. (2017);
Pommella et al. (2021) and chemical mixing Liu et al. (2002); Lutz et al.
(2003, 2005); Ahmed et al. (2009) to vesicle transport and lysis Marmottant &
Hilgenfeldt (2003, 2004). Recently, the use of multi-curvature streaming
bodies has expanded the ability to manipulate flows, leading to compact,
robust, and tunable devices for filtering and separating both synthetic and
biological particles Parthasarathy et al. (2019); Bhosale et al. (2020); Chan
et al. (2022); Bhosale et al. (2022b). More recently yet, motivated by medical
and biological applications, the effect of body compliance has been considered
Pande et al. (2023); Anand & Christov (2020), with one of the studies yielding
a first two-dimensional streaming theory for soft cylinders Bhosale et al.
(2022a). Its major outcome is encapsulated in the relation

$\langle\psi_{1}\rangle=\sin 2\theta~{}\left[\Theta(r)+\Lambda(r)\right]$ (1)

where $\langle\psi_{1}\rangle$ is the time-averaged Stokes streamfunction, $r$
and $\theta$ are the radial and polar coordinates in the cylindrical system.
This relation reveals an additional streaming process $\Lambda(r)$, purely
induced by body elasticity, that is available even in Stokes flows where
rigid-body streaming $\Theta(r)$ cannot exist Holtsmark et al. (1954).
Elasticity modulation has then been shown to achieve streaming configurations
similar to rigid bodies, but at significantly lower frequencies. This
frequency reduction has relevant implications, as it renders viscous streaming
accessible within the limits of biological actuation.

In this work, we extend this understanding to three dimensions by examining
the minimal case of an oscillating, soft sphere. We first present an improved
theoretical solution for the rigid sphere case by augmenting Lane (1955)’s
derivation with a previously unaccounted term related to vortex stretching.
Our formulation is shown to significantly enhance quantitative agreement with
direct numerical simulations and experiments. Next, within the same
theoretical framework, we consider body elasticity and seek a modified
streaming solution dependent on material compliance. We recover an independent
elastic modification term, similar in nature to the above two-dimensional
result but analytically distinct. We then demonstrate the accuracy of our
theory against direct numerical simulations, and show how observed elasticity
effects may translate to bodies of complex geometry and topology, further
expanding the potential utility of soft streaming.

Figure 1:  Problem setup. (a) 3D Elastic solid sphere $\Omega_{e}$ of radius
$a$ with a rigid inclusion (pinned zone $\Gamma$ of radius $b$), immersed in
viscous fluid $\Omega_{f}$. In this study, we deploy a spherical coordinate
system where $(r,\theta,\phi)$ are the radial, polar, and azimuthal
coordinates. The sphere is exposed to an oscillatory flow with far-field
velocity $V(t)=\epsilon a\omega\cos(\omega t)$ in the $x$ direction, along the
axis of symmetry. (b) 2D axisymmetric cross-section of the elastic sphere.

##  2 Problem setup and governing equations

We derive the streaming solution by considering the setup shown in Fig. 1,
where a 3D visco-elastic solid sphere $\Omega_{e}$ of radius $a$ is immersed
in a viscous fluid $\Omega_{f}$. The fluid oscillates with velocity
$V(t)=\epsilon a\omega\cos\omega t$, where $\epsilon$, $\omega$ and $t$
represent the non-dimensional amplitude, angular frequency, and time.
Following our previous setup for a soft 2D cylinder Bhosale et al. (2022a), we
kinematically enforce zero strain and velocity near the sphere’s center by
‘pinning’ the sphere with a rigid inclusion $\Gamma$ of radius $b<a$, the
boundary of which is denoted by $\partial\Gamma$. We further denote by
$\partial\Omega$ the boundary between the elastic solid and the viscous fluid.

Both fluid and solid are assumed to be isotropic and incompressible, where the
fluid is Newtonian with kinematic viscosity $\nu_{f}$ and density $\rho_{f}$,
and the solid follows the Kelvin-Voigt viscoelastic model. Characteristic of
soft biological materials Bower (2009), elastic stresses within the solid are
modeled via neo-Hookean hyperelasticity with shear modulus $G$, kinematic
viscosity $\nu_{e}$, and density $\rho_{e}$. However, we will later show that
the choice of hyperelastic or viscoelastic model does not affect the theory.

The dynamics in the fluid and solid phases are governed by the incompressible
Cauchy momentum equations, non-dimensionalized using the characteristic scales
of velocity $V=\epsilon a\omega$, length $L=a$, time $T=1/\omega$, and
hydrostatic pressure $P=\mu_{f}V/L=\mu_{f}\epsilon\omega$

Incomp.
$\displaystyle\begin{cases}\mbox{\boldmath$\nabla$}\cdot\mbox{\boldmath$v$}=0,~{}\mbox{\boldmath$x$}\in\Omega_{f}\cup\Omega_{e}\end{cases}$
(2) Fluid
$\displaystyle\begin{cases}\frac{\partial{\mbox{\boldmath$v$}}}{\partial{t}}+\epsilon({\mbox{\boldmath$v$}}\cdot{\bm{\nabla}}){\mbox{\boldmath$v$}}=\frac{1}{M^{2}}\left(-{\nabla}{p}+{\nabla^{2}}{\mbox{\boldmath$v$}}\right),~{}{\mbox{\boldmath$x$}}\in\Omega_{f}\end{cases}$
Solid
$\displaystyle\begin{cases}\alpha\mathrm{Cau}\left(\frac{\partial{\mbox{\boldmath$v$}}}{\partial{t}}+\epsilon({\mbox{\boldmath$v$}}\cdot{\bm{\nabla}}){\mbox{\boldmath$v$}}\right)=\frac{\mathrm{Cau}}{M^{2}}\left(-{\nabla}{p}+\beta{\nabla^{2}}{\mbox{\boldmath$v$}}\right)+{\bm{\nabla}}\cdot({\bm{F}}{\bm{F}}^{T})^{\prime},~{}{\mbox{\boldmath$x$}}\in\Omega_{e},\end{cases}$

where $v$ and $p$ are the velocity and pressure fields, and $\bm{F}$ is the
deformation gradient tensor, defined as
$\bm{F}=\bm{I}+\bm{\nabla}\mbox{\boldmath$u$}$, where $\bm{I}$ is the
identity, $\mbox{\boldmath$u$}=\mbox{\boldmath$x$}-\mbox{\boldmath$X$}$ is the
material displacement field, and $x$, $X$ are the position of a material point
after deformation and at rest, respectively. The prime symbol ′ on a tensor
denotes its deviatoric. The key non-dimensional parameters within this system
are the scaled oscillation amplitude $\epsilon$, Womersley number
$M=a\sqrt{\rho_{f}\omega/\mu_{f}}$, Cauchy number
$\mathrm{Cau}=\epsilon\rho_{f}a^{2}\omega^{2}/G$, density ratio
$\alpha=\rho_{e}/\rho_{f}$, and viscosity ratio $\beta=\mu_{e}/\mu_{f}$.
Physically, the Womersley number ($M$) represents the ratio of inertial to
viscous forces, and the Cauchy number ($\mathrm{Cau}$) represents the ratio of
inertial to elastic forces. Therefore, a higher $M$ corresponds to stronger
dominance of inertia in the fluid environment, and higher $\mathrm{Cau}$
values correspond to increasingly soft bodies. We then impose a set of
boundary conditions upon the governing equations, consistent with Lane (1955)

Pinned zone
$\displaystyle\begin{cases}\mbox{\boldmath$u$}=0,\mbox{\boldmath$v$}=0,~{}\mbox{\boldmath$x$}\in\Gamma\end{cases}$
(2.2) Interface velocity
$\displaystyle\begin{cases}\mbox{\boldmath$v$}_{e}=\mbox{\boldmath$v$}_{f},~{}\mbox{\boldmath$x$}\in\partial\Omega\end{cases}$
(2.3) Interface stresses
$\displaystyle\begin{cases}\bm{\sigma}_{f}=-p\bm{I}+(\bm{\nabla}\mbox{\boldmath$v$}+\bm{\nabla}\mbox{\boldmath$v$}^{T}),~{}\mbox{\boldmath$x$}\in\Omega_{f}\\\
\bm{\sigma}_{e}=-p\bm{I}+\beta(\bm{\nabla}\mbox{\boldmath$v$}+\bm{\nabla}\mbox{\boldmath$v$}^{T})+\frac{M^{2}}{\mathrm{Cau}}(\bm{F}\bm{F}^{T})^{\prime},~{}\mbox{\boldmath$x$}\in\Omega_{e},\\\
\mbox{\boldmath$n$}\cdot\bm{\sigma}_{e}\cdot\mbox{\boldmath$n$}=\mbox{\boldmath$n$}\cdot\bm{\sigma}_{f}\cdot\mbox{\boldmath$n$},~{}\mbox{\boldmath$x$}\in\partial\Omega\\\
\mbox{\boldmath$n$}\cdot\bm{\sigma}_{e}\cdot\mbox{\boldmath$t$}=\mbox{\boldmath$n$}\cdot\bm{\sigma}_{f}\cdot\mbox{\boldmath$t$},~{}\mbox{\boldmath$x$}\in\partial\Omega\\\
\end{cases}$ (2.4) Far-field
$\displaystyle\begin{cases}\mbox{\boldmath$v$}(|\mbox{\boldmath$x$}|\to\infty)=\cos
t~{}\hat{i},~{}\mbox{\boldmath$x$}\in\Omega_{f},\end{cases}$ (2.5)

where Eq. 2.2 is the pinned-zone rigidity constraint, Eq. 2.3 is the no-slip
boundary condition between and solid and fluid phases, Eq. 2.4 dictates stress
continuity, and Eq. 2.5 is the far-field flow velocity. We use subscripts $e$
and $f$ to denote elastic and fluid phases, respectively, wherever ambiguity
may arise. Next, we identify ranges of relevant parameters and solve Eq. 2 via
perturbation theory.

##  3 Perturbation series solution

In viscous streaming applications, typically we have small non-dimensional
oscillation amplitudes $\epsilon\ll{1}$ Wang (1965); Bertelsen et al. (1973);
Lutz et al. (2005), density ratio $\alpha$ and viscosity ratio $\beta$ of
$\mathcal{O}\left(1\right)$, and Womersley number
$M\geq\mathcal{O}\left(1\right)$ Marmottant & Hilgenfeldt (2004); Lutz et al.
(2006). For the Cauchy number $\mathrm{Cau}$, we apply the same treatment as
Bhosale et al. (2022a), where we use $\mathrm{Cau}=0$ for a rigid body and
$\mathrm{Cau}=\kappa\epsilon$ with $\kappa=\mathcal{O}\left(1\right)$ for
elastic bodies. The latter assumption implies that $\mathrm{Cau}\ll 1$, which
physically means that the body is weakly elastic. We make this assumption for
two reasons. First, we choose $\mathrm{Cau}$ to be small to simplify the
treatment of hyperelastic materials, whose non-linearities become
mathematically challenging for $\mathrm{Cau}\geq\mathcal{O}\left(1\right)$.
Second, matching $\mathrm{Cau}$ with $\epsilon$ simplifies the asymptotic
expansion, while preserving the practical generality of the results (for
details, see supplementary material §2 of Bhosale et al. (2022a)).

We then seek a perturbation series solution of Eqs. (2) by asymptotically
expanding all relevant fields in powers of $\epsilon$. Our derivation closely
follows the approach taken by Bhosale et al. (2022a) for 2D elastic cylinders,
while augmenting it to encompass 3D settings. Then, the zeroth order solution
reduces to a rigid sphere in a purely oscillatory flow governed by the
unsteady Stokes equation Lane (1955). The first order solution is subsequently
derived in two stages. First, we obtain the deformation of the elastic body
resulting from the flow field at zeroth order. Second, we incorporate the
elastic feedback into the streaming solution by using the obtained body
deformations as boundary conditions for the flow at
$\mathcal{O}\left(\epsilon\right)$. The steps are outlined below, with details
listed in the supplementary material.

We start by perturbing to $\mathcal{O}\left(\epsilon\right)$ all physical
quantities $q$, which include $v$, $u$, $p$, $\Omega$, $n$, $t$, as

$q\sim q_{0}+\epsilon q_{1}+\mathcal{O}\left(\epsilon^{2}\right)$ (3)

and substitute them into the governing equations Eqs. (2) and boundary
conditions Eqs. (2.2)–(2.5). Steps are explicitly reported in the
supplementary material (Eq.1.17–1.26), where subscripts (0, 1, …) refer to the
solution order. Next, we adopt the geometrically convenient spherical
coordinate system $(r,\theta,\phi)$, with radial coordinate $r$, polar angle
$\theta$, azimuthal angle $\phi$, and origin at the center of the sphere. The
horizontal axis direction $i$ corresponds to $\theta=0$.

###  3.1 Zeroth order $\mathcal{O}\left(1\right)$ solution

The governing equations and boundary conditions in the solid phase at zeroth
order $\mathcal{O}\left(1\right)$ simplify to

$\bm{\nabla}\cdot((\bm{I}+\bm{\nabla}\bm{u}_{0})(\bm{I}+\bm{\nabla}\bm{u}_{0})^{T})^{\prime}=0,~{}~{}r\leq
1;~{}~{}~{}\mbox{\boldmath$u$}_{0}|_{r=\zeta}=0$ (4)

where $\zeta=b/a$ is the non-dimensional radius of the pinned zone. Since at
this order $\mathrm{Cau}=\kappa\epsilon=0$, the solution of Eq. 4 physically
corresponds to a fixed, rigid sphere with

$\partial\Omega_{0}=r=1;~{}~{}\mbox{\boldmath$u$}_{0}=0,~{}~{}\mbox{\boldmath$v$}_{0}=\frac{\partial\mbox{\boldmath$u$}_{0}}{dt}=0,~{}~{}r\leq
1.$ (5)

Thus the fluid phase governing equations and boundary conditions reduce to

$\displaystyle
M^{2}\frac{\partial\bm{\nabla}^{2}\mbox{\boldmath$\varphi$}_{0}}{\partial t}$
$\displaystyle=\bm{\nabla}^{4}\mbox{\boldmath$\varphi$}_{0}~{}~{}~{}~{}r\geq
1$ (6) $\displaystyle
v_{0,r}|_{r=1}=\frac{1}{r\sin\theta}\frac{\partial(\varphi_{0}\sin\theta)}{\partial\theta}\biggr{|}_{r=1}$
$\displaystyle=0;~{}v_{0,\theta}|_{r=1}=-\frac{1}{r}\frac{\partial(r\varphi_{0})}{\partial
r}\biggr{|}_{r=1}$ $\displaystyle=0$ $\displaystyle
v_{0,r}|_{r\to\infty}=\cos\theta~{}\cos t$
$\displaystyle;~{}v_{0,\theta}|_{r\to\infty}=-\sin\theta~{}\cos t,$

where $\mbox{\boldmath$\varphi$}=\varphi\mbox{\boldmath$\hat{\phi}$}$ is the
vector potential defined as
$\mbox{\boldmath$v$}=\bm{\nabla}\times\mbox{\boldmath$\varphi$}$, with
$\hat{\phi}$ being the unit vector in the azimuthal direction. We note that
the bold font $\bm{\nabla}^{2}$ refers to the vector Laplacian operator, which
is distinct from the scalar Laplacian operator in spherical coordinates. This
system (Eqs. (5), (6)) defines a rigid sphere immersed in an oscillating
unsteady Stokes flow, which has an exact analytical solution Lane (1955)

$\varphi_{0}=-\frac{\sin\theta}{4}\left(3\frac{h_{1}(mr)}{mh_{0}(m)}-r-\frac{h_{2}(m)}{r^{2}h_{0}(m)}\right)e^{-it}+c.c.,~{}r\geq
1$ (7)

where $i=\sqrt{-1}$, and $m=\sqrt{i}M$. Here, $h_{n}$ and $c.c.$ refer to the
$n^{\textrm{th}}$ order spherical Hankel function of the first kind and
complex conjugate, respectively. As observed in Lane (1955), the zeroth order
vector potential field $\mbox{\boldmath$\varphi$}_{0}$ in the fluid phase is
purely oscillatory in time, and thus no steady streaming appears at this
order. Moreover, the flow at $\mathcal{O}\left(1\right)$ is unaffected by
elasticity.

###  3.2 First order $\mathcal{O}\left(\epsilon\right)$ solution

We then proceed to the next order approximation
$\mathcal{O}\left(\epsilon\right)$, where we expect time-independent steady
streaming to emerge Lane (1955). At $\mathcal{O}\left(\epsilon\right)$, the
solid governing equations reduce (supplementary material, Eqs. 1.38-1.42,
1.49) to the homogeneous biharmonic equation

$\bm{\nabla}^{4}\mbox{\boldmath$\varphi$}_{e,1}=0,~{}~{}~{}~{}\mbox{\boldmath$x$}\in\Omega_{e}$
(8)

where we have defined the strain function
$\mbox{\boldmath$\varphi$}_{e}=\varphi_{e}\hat{\mbox{\boldmath$\phi$}}$
similar to the fluid phase, so that the displacement field is
$\mbox{\boldmath$u$}=\mbox{\boldmath$\nabla$}\times\mbox{\boldmath$\varphi$}_{e}$.
Equation 8 demonstrates how the specific choice of solid elasticity model used
at $\mathcal{O}\left(\epsilon\right)$ becomes irrelevant, since all nonlinear
stress-strain responses drop out as a result of linearization (supplementary
material, Eqs. 1.38–1.42). Equation 8 is complemented by the Dirichlet
boundary conditions at the pinned zone interface

$u_{1,r}=\left.\frac{1}{r\sin\theta}\frac{\partial(\varphi_{e,1}\sin\theta)}{\partial\theta}\right|_{r=\zeta}=0;~{}~{}~{}~{}u_{1,\theta}=\left.-\frac{1}{r}\frac{\partial(r\varphi_{e,1})}{\partial
r}\right|_{r=\zeta}=0.$ (9)

Additionally, in accordance with Eq. 2.4, the $\mathcal{O}\left(1\right)$ flow
solution exerts interfacial stresses on the solid, which at
$\mathcal{O}\left(\epsilon\right)$ is no longer rigid but instead deformable.
This results in the following boundary conditions for the radial and
tangential stresses at the interface $\partial\Omega_{0}$

$\displaystyle\frac{M^{2}}{\kappa}\left.\frac{\partial u_{1,r}}{\partial
r}\right|_{r=1}$ $\displaystyle=\left.\frac{\partial v_{0,r}}{\partial
r}\right|_{r=1}$ (10)
$\displaystyle\frac{M^{2}}{\kappa}\left.\left(\frac{1}{r}\frac{\partial
u_{1,r}}{\partial\theta}+\frac{\partial u_{1,\theta}}{\partial
r}-\frac{u_{1,\theta}}{r}\right)\right|_{r=1}$
$\displaystyle=\left.\left(\frac{1}{r}\frac{\partial
v_{0,r}}{\partial\theta}+\frac{\partial v_{0,\theta}}{\partial
r}-\frac{v_{0,\theta}}{r}\right)\right|_{r=1},$

where the LHS corresponds to the elastic stresses in the solid phase
($\frac{M^{2}}{\mathrm{Cau}}(\bm{F}\bm{F}^{T})^{\prime}$, Eq. 2.4) and the RHS
to the viscous stresses in the fluid phase
($\bm{\nabla}\mbox{\boldmath$v$}+\bm{\nabla}\mbox{\boldmath$v$}^{T}$, Eq.
2.4), both evaluated at the zeroth order interface $r=1$. The pressure term
($-p\bm{I}$, Eq. 2.4) cancels out due to pressure continuity at the interface,
hence its absence in Eq. 10 (supplementary material, Eq. 1.31). We point out
that the use of $r=1$ in Eq. 10 is consistent despite the fact that the solid
interface deforms at this order. Indeed, as demonstrated in Bhosale et al.
(2022a) and supplementary material Eqs. 1.44–1.46, errors associated with the
$r=1$ approximation all appear at the higher order
$\mathcal{O}\left(\epsilon^{2}\right)$ and thus do not affect our solution. In
addition, we note that the viscous stress term
($\beta(\bm{\nabla}\mbox{\boldmath$v$}+\bm{\nabla}\mbox{\boldmath$v$}^{T})$,
Eq. 2.4) is also of higher order $\mathcal{O}\left(\epsilon^{2}\right)$ and
thus absent in Eq. 10, implying that the specific choice of viscosity model is
irrelevant at $\mathcal{O}\left(\epsilon\right)$. Next, we use the
$\mathcal{O}\left(1\right)$ flow velocity at the interface, Eqs. 6 and 7, to
directly evaluate the RHS of Eq. 10

$\displaystyle\left.\frac{\partial v_{0,r}}{\partial r}\right|_{r=1}$
$\displaystyle=0$ (11) $\displaystyle\left.\left(\frac{1}{r}\frac{\partial
v_{0,r}}{\partial\theta}+\frac{\partial v_{0,\theta}}{\partial
r}-\frac{v_{0,\theta}}{r}\right)\right|_{r=1}$
$\displaystyle=\sin{\theta}~{}F(m)~{}e^{-it}+c.c.$

with

$F(m)=-\frac{3mh_{1}(m)}{4h_{0}(m)}.$ (12)

With the boundary conditions Eqs. (9) – (12) resolved, the homogeneous
biharmonic Eq. 8 can be solved exactly to obtain the
$\mathcal{O}\left(\epsilon\right)$ solid strain function

$\varphi_{e,1}=\frac{\kappa}{M^{2}}\sin{\theta}\left(c_{1}r+\frac{c_{2}}{r^{2}}+c_{3}r^{3}+c_{4}\right)F(m)~{}e^{-it}+c.c.,$
(13)

where the exact expressions for $c_{1},c_{2},c_{3},c_{4}$ (functions of
$\zeta$) are reported in the supplementary material (Eq. 1.56). The
$\mathcal{O}\left(\epsilon\right)$ solid displacement field
$\mbox{\boldmath$u$}_{1}$, both in the bulk $\Omega_{e}$ and at the boundary
$\partial\Omega$, can then be directly obtained from Eq. 13. This, in turn,
kinematically affects the flow at $\mathcal{O}\left(\epsilon\right)$ via the
interfacial boundary conditions, as we will see.

The flow governing equation at $\mathcal{O}\left(\epsilon\right)$, in vector
potential form, reads

$M^{2}\frac{\partial\bm{\nabla}^{2}\mbox{\boldmath$\varphi$}_{1}}{\partial
t}+M^{2}\left(\left(\mbox{\boldmath$v$}_{0}\cdot\bm{\nabla}\right)\bm{\nabla}^{2}\mbox{\boldmath$\varphi$}_{0}\right)-M^{2}\left((\bm{\nabla}^{2}\mbox{\boldmath$\varphi$}_{0}\cdot\bm{\nabla})\mbox{\boldmath$v$}_{0}\right)=\bm{\nabla}^{4}\mbox{\boldmath$\varphi$}_{1},~{}~{}~{}~{}r\geq
1.$ (14)

We note that the term
$M^{2}(\bm{\nabla}^{2}\mbox{\boldmath$\varphi$}_{0}\cdot\bm{\nabla})\mbox{\boldmath$v$}_{0}$
in Eq. 14, which corresponds to vortex stretching, is absent in the rigid
sphere streaming derivation of Lane (1955). By considering this unaccounted
term, our work improves upon the existing theory, as demonstrated in Section
4.

Next, since we are interested in steady streaming, we consider the time-
averaged form of Eq. 14

$\displaystyle\bm{\nabla}^{4}\langle\mbox{\boldmath$\varphi$}_{1}\rangle$
$\displaystyle=M^{2}\underbrace{\langle\left(\mbox{\boldmath$v$}_{0}\cdot\bm{\nabla}\right)\bm{\nabla}^{2}\mbox{\boldmath$\varphi$}_{0}-(\bm{\nabla}^{2}\mbox{\boldmath$\varphi$}_{0}\cdot\bm{\nabla})\mbox{\boldmath$v$}_{0}\rangle}_{\textrm{RHS}},~{}~{}~{}~{}r\geq
1,$ (15)

where we substitute Eq. 7 into the RHS to yield

$\displaystyle\bm{\nabla}^{4}\langle\mbox{\boldmath$\varphi$}_{1}\rangle$
$\displaystyle=\sin
2\theta~{}\rho(r)\mbox{\boldmath$\hat{{\phi}}$},~{}~{}~{}~{}r\geq 1,$ (16)
$\displaystyle\rho(r)$
$\displaystyle=\frac{1}{16r^{4}}~{}\left(r^{3}J^{(3)}+r^{2}J^{(2)}-6rJ^{(1)}+6J\right)J^{*}+c.c.$
$\displaystyle J(r)$
$\displaystyle=3\frac{h_{1}(mr)}{mh_{0}(m)}-r-\frac{h_{2}(m)}{r^{2}h_{0}(m)}~{}.$

Here, $J$ is the radially dependent term of Eq. 7, with $J^{(n)}$ and $J^{*}$
being its $n^{\text{th}}$ derivative and complex conjugate, respectively.
Solving this inhomogeneous biharmonic equation requires four independent
boundary conditions. The first two are the radial and tangential, time-
averaged, far-field velocity

$\displaystyle\left.\frac{1}{r\sin\theta}\frac{\partial(\langle\varphi_{1}\rangle\sin\theta)}{\partial\theta}\right|_{r\to\infty}=\left.\frac{1}{r}\frac{\partial(r\langle\varphi_{1}\rangle)}{\partial
r}\right|_{r\to\infty}=0.$ (17)

Next, we recall the no-slip boundary condition of Eq. 2.3 that needs to be
enforced at the $\mathcal{O}\left(\epsilon\right)$ accurate solid–fluid
interface

$\left.\mbox{\boldmath$v$}_{e}\right|_{\partial\Omega}=\left.\mbox{\boldmath$v$}_{e}\right|_{r=1+\epsilon
u_{1,r}}+\mathcal{O}\left(\epsilon^{2}\right)=\left.\mbox{\boldmath$v$}_{f}\right|_{\partial\Omega}=\left.\mbox{\boldmath$v$}_{f}\right|_{r=1+\epsilon
u_{1,r}}+\mathcal{O}\left(\epsilon^{2}\right).$ (18)

We note that the sphere interface at $\mathcal{O}\left(\epsilon\right)$
deforms as $r^{\prime}=1+\epsilon u_{1,r}$, where $u_{1,r}$ is the radial
component of the $\mathcal{O}\left(\epsilon\right)$ accurate displacement
field obtained by taking the curl of the strain function
$\mbox{\boldmath$u$}_{1}=\bm{\nabla}\times\mbox{\boldmath$\varphi$}_{e,1}$.
Similar to Bhosale et al. (2022a), we enforce the no-slip boundary condition
in Eq. 18 by deploying the technique presented in Longuet-Higgins (1998),
where $\left.\mbox{\boldmath$v$}_{f}\right|_{r=r^{\prime}}$ is Taylor expanded
about $r=1$ (supplementary material, Eqs. 1.64–1.66)

$\displaystyle\left.\mbox{\boldmath$v$}_{f}\right|_{r=1+\epsilon u_{1,r}}$
$\displaystyle=\left.\left(\mbox{\boldmath$v$}_{f,1}+\epsilon\frac{\partial\mbox{\boldmath$v$}_{f,0}}{\partial
r}u_{1,r}\right)\right|_{r=1}+\mathcal{O}\left(\epsilon^{2}\right).$ (19)

The boundary solid velocity
$\left.\mbox{\boldmath$v$}_{e}\right|_{r=1+\epsilon u_{1,r}}$ (LHS of Eq. 18)
can be instead computed to $\mathcal{O}\left(\epsilon\right)$ accuracy as
$\left.\partial u_{1,r}/\partial t\right|_{r=1}$ (supplementary material, Eq.
1.63). We note that both $u_{1,r}$ and
$\partial\mbox{\boldmath$v$}_{f,0}/\partial r$ are known from Eqs. 13 and 7.
Thus, the $\mathcal{O}\left(\epsilon\right)$ flow velocity
$\mbox{\boldmath$v$}_{f,1}$ at $r=1$, denoted by $\mbox{\boldmath$v$}_{1}$
henceforth, can be obtained by substituting Eq. 19 into Eq. 18 (supplementary
material, Eqs. 1.62–1.67). Time averaging then yields the remaining two
boundary conditions for Eq. 16

$\displaystyle\left.\langle v_{1,r}\rangle\right|_{r=1}$
$\displaystyle=\left.\frac{1}{r\sin\theta}\frac{\partial(\langle\varphi_{1}\rangle\sin\theta)}{\partial\theta}\right|_{r=1}=0~{}$
(20) $\displaystyle\left.\langle v_{1,\theta}\rangle\right|_{r=1}$
$\displaystyle=-\left.\frac{1}{r}\frac{\partial(r\langle\varphi_{1}\rangle)}{\partial
r}\right|_{r=1}=-\frac{\kappa}{M^{2}}\sin 2\theta~{}G_{1}(\zeta)F(m)F^{*}(m)$

with

$G_{1}(\zeta)=\frac{(\zeta-1)^{2}(4\zeta^{2}+7\zeta+4)}{3\zeta(2\zeta^{3}+4\zeta^{2}+6\zeta+3)}.$
(21)

Equation 20 physically implies a rectified tangential slip velocity ($\langle
v_{1,\theta}\rangle|_{r=1}\neq 0$) in the fluid phase at the zeroth-order
fixed interface $r=1$. This slip velocity captures the effect of body elastic
deformation ($\langle v_{1,\theta}\rangle|_{r=1}=0$ for rigid bodies) by
equivalently modifying the fluid Reynolds stresses ($\sin 2\theta\rho(r)$ in
Eq. 16), thus impacting the resulting streaming flow. We further remark that,
in contrast to rigid body streaming, such modification is accessible even in
the Stokes limit as it is independent of the Navier-Stokes nonlinear inertial
advection, a conclusion similarly drawn in our previous work on 2D soft
cylinder streaming Bhosale et al. (2022a). This phenomenon shares
characteristics with the artificial mixed-mode streaming of pulsating bubbles
Longuet-Higgins (1998); Spelman & Lauga (2017), whereas the streaming process
derived here arises spontaneously from the coupling between viscous fluid and
elastic solid.

Figure 2:  Elastic sphere and streaming flow response. (a-c) 3D time-averaged
Lagrangian (i.e. Stokes-drift corrected, supplementary material §3) Stokes
streamfunction depicting the streaming response at $M=6$ with increasing
softness $\mathrm{Cau}$. (a) Rigid limit $\mathrm{Cau}=0$, (b)
$\mathrm{Cau}=0.025$, and (c) $\mathrm{Cau}=0.05$. Note that blue/orange
represent clockwise/counterclockwise rotating regions. The non-dimensional
radius of the pinned zone is set at $\zeta=0.4$ throughout the study, to
maintain the tangential slip velocity magnitude (Eq. 20) at
$\mathcal{O}\left(1\right)$, consistent with the asymptotic analysis. Lowering
the pinned zone radius rapidly increases the prefactor $G_{1}(\zeta)$ (Eq.
21), which results in a slip velocity much greater than
$\mathcal{O}\left(1\right)$, thereby weakening the asymptotic assumption. The
opposite holds true for large $\zeta$, rendering $\zeta=0.4$ a robust
compromise. The effect of pinned zone radius on streaming flow is detailed in
Section §4 of the supplementary material. (d) Normalized DC layer thickness
($\delta_{DC}/a$) vs. inverse Womersley number ($1/M$) from theory and
simulations, for varying body elasticity $\mathrm{Cau}$. Incomplete theory
(dashed line) Lane (1955) and experimental results (grey squares) Kotas et al.
(2007) in the rigidity limit are plotted for reference. We characterize
streaming via the thickness of the DC layer, which refers to the innermost
recirculation zone, for its utility in trapping, filtration and chemical
mixing, and because of its robust nature. (e-g) Radial decay of velocity
magnitude along $\theta=90^{\circ}$ from theory and simulations at $M=6$, with
increasing softness $\mathrm{Cau}$. (e) Rigid limit $\mathrm{Cau}=0$, (f)
$\mathrm{Cau}=0.0125$, and (g) $\mathrm{Cau}=0.025$. Simulations are performed
using a vortex-method based formulation Gazzola et al. (2011); Bhosale et al.
(2021, 2023) that solves the vorticity form of the Navier–Stokes equations in
an axisymmetric cylindrical coordinate system. Within this framework, fluid
and solid are modeled to be density matched ($\rho_{f}=\rho_{e}=1$). The rigid
sphere and pinned zone are modeled as Brinkman solids, and the elastic phase
as a viscoelastic Kelvin-Voigt solid with shear modulus $G$. Soft body
deformations are tracked using an inverse-map technique Bhosale et al. (2021);
Kamrin & Nave (2009); Kamrin et al. (2012). The far-field velocity is
$V(t)=V_{0}\cos\omega t$ with characteristic velocity $V_{0}=\epsilon
a\omega$, where $\epsilon=0.1$, $a=0.1$, and $\omega=32\pi$. The fluid dynamic
viscosity $\mu_{f}$ and elastic body’s shear modulus $G$ are determined based
on the Womersley number $M=a\sqrt{\rho_{f}\omega/\mu_{f}}$ and Cauchy number
$\mathrm{Cau}=\epsilon\rho_{f}a^{2}\omega^{2}/G$. Additional simulation
parameters include: domain $[0,1]\times[0,0.5]$, uniform grid spacing
$h=1/1024$, Brinkman penalization factor $\lambda=1e6$, mollification length
$\epsilon_{moll}=2h$, Courant–Friedrichs–Lewy number CFL = $0.1$. For details
on these parameters, refer to Gazzola et al. (2011); Bhosale et al. (2021);
Parthasarathy et al. (2022); Bhosale et al. (2023).

Given the steady flow of Eq. (16) and boundary conditions of Eqs. 17 and 20,
the streaming solution can finally be written as

$\langle\varphi_{1}\rangle=\sin 2\theta~{}\left[\Theta(r)+\Lambda(r)\right],$
(22)

where $\Theta(r)$ is the rectified rigid-body solution

$\displaystyle\Theta(r)$
$\displaystyle=-\frac{r^{4}}{70}\int_{r}^{\infty}\frac{\rho(\tau)}{\tau}~{}\mathrm{d}\tau+\frac{r^{2}}{30}\int_{r}^{\infty}\tau\rho(\tau)~{}\mathrm{d}\tau$
(23)
$\displaystyle+\frac{1}{r}\left(\frac{1}{30}\int_{1}^{r}\tau^{4}\rho(\tau)~{}\mathrm{d}\tau+\frac{1}{20}\int_{1}^{\infty}\frac{\rho(\tau)}{\tau}~{}\mathrm{d}\tau-\frac{1}{12}\int_{1}^{\infty}\tau\rho(\tau)~{}\mathrm{d}\tau\right)$
$\displaystyle+\frac{1}{r^{3}}\left(-\frac{1}{70}\int_{1}^{r}\tau^{6}\rho(\tau)~{}\mathrm{d}\tau-\frac{1}{28}\int_{1}^{\infty}\frac{\rho(\tau)}{\tau}~{}\mathrm{d}\tau\right.\left.+\frac{1}{20}\int_{1}^{\infty}\tau\rho(\tau)~{}\mathrm{d}\tau\right)$

and $\Lambda(r)$ is the new elastic modification

$\Lambda(r)=0.5\frac{\kappa}{M^{2}}~{}G_{1}(\zeta)F(m)F^{*}(m)\left(\frac{1}{r}-\frac{1}{r^{3}}\right)$
(24)

with $G_{1}(\zeta)$ and $F(m)$ given in Eq. 21 and Eq. 12, respectively. We
note that while Eq. 23 is of the same form as Lane (1955)’s solution, the
explicit expression of $\rho(r)$ is different (Eq. 16) because of the vortex
stretching term of Eq. 14. This concludes our theoretical analysis.

##  4 Numerical validation and extension to complex bodies

Next, we compare our theory against known experimental and analytical results
in the rigidity limit Kotas et al. (2007); Lane (1955), as well as direct
numerical simulations performed using an axisymmetric vortex-method based
formulation Gazzola et al. (2012); Bhosale et al. (2021, 2023) (see also the
caption of Fig. 2). The Stokes streamfunction pattern for a rigid sphere
($\mathrm{Cau}=0$) oscillating at $M\approx 6$ is shown in Fig. 2a . We
highlight the two-fold symmetry on top of the axisymmetry, and the presence of
a well-defined direct circulation (DC) layer of thickness $\delta_{DC}$. This
characteristic flow configuration, as well as the divergence of the DC layer
thickness ($\delta_{DC}\rightarrow\infty$) with increasing $1/M$, is
consistent with Lane (1955). This qualitative behavior is recovered by our
theory at $\mathrm{Cau}=0$ (grey line in Fig. 2d), and by simulations (black
dots in Fig. 2d). However, we note the significant quantitative difference
between the results from Lane (1955) (black dashed line in Fig. 2d) and our
simulations/theory, which are instead found to be in close agreement with
experimental results (grey dots) by Kotas et al. (2007) (grey squares). The
additional accuracy of our theory directly stems from including the vortex
stretching term of Eq. 15, as previously discussed. Next, as we enable solid
compliance ($\mathrm{Cau}>0$), we observe that the two-fold symmetry is
preserved ($\sin 2\theta$ in Eq. 22) while $\delta_{DC}$ contracts due to the
elastic modification term $\Lambda\neq 0$, in agreement with numerical
simulations across a range of $\mathrm{Cau}$ (Fig. 2(b-d)). These observations
are consistent with our previous work on streaming for a 2D soft cylinder
Bhosale et al. (2022a), and thus a similar, intuitive explanation exists. The
flow receives feedback deformation velocities on account of the deformable
sphere surface, which acts as an additional source of inertia. Since the
Womersley number (M) is the ratio between inertial and viscous forces, this is
equivalent to rigid body streaming with a larger $M$, hence the decrease of DC
layer thickness with increasing elasticity. This implies that an elastic body
can access the streaming flow configurations of its rigid counterpart with
significantly lower oscillation frequencies. Such frequency reduction is shown
in Fig. 2d where, for example at $\mathrm{Cau}=0.05$, the same DC layer
thickness is achieved at $\sim 2\times$ lower frequency. Similar to rigid
objects, the $\delta_{DC}$ of a soft sphere still diverges with decreasing
$M$, albeit at lower values, since the elastic modification $\Lambda(r)$ does
not alter the asymptotic behavior of the rigid contribution $\Theta(r)$ (see
supplementary material, Section 6 for details). We further note that as
$\mathrm{Cau}$ increases and $M$ decreases, the deviation between simulations
and theory grows. This follows from the fact that as $\kappa/M^{2}$ in Eq. 20
increases, the tangential slip velocity assumed to be of
$\mathcal{O}\left(\epsilon\right)$ can exceed $\mathcal{O}\left(1\right)$,
eventually leading to the breakdown of the asymptotic analysis. We conclude
our validation by showing close agreement between theoretical and simulated
radially-varying, time-averaged velocities $\langle v\rangle$ at
$\theta=90^{\circ}$ (Fig. 2(e–g)). For a detailed analysis concerning the
effect of inertia ($M$) and elasticity (Cau) on velocity magnitudes (flow
strength), refer to Section 4 of the supplementary material.

Figure 3:  Extension to complex bodies. Here we consider compliance-induced
streaming in a soft torus, a complex shape entailing multiple curvatures and
distinct topology relative to the sphere. Numerically simulated time-averaged
Eulerian flow topologies for a torus of core radius $r$ and cross-sectional
radius $a=r/3$, at $M\approx 4$, with varying body elasticity $\mathrm{Cau}$.
(a) Rigid limit $\mathrm{Cau}=0$, (b) $\mathrm{Cau}=0.025$, and (c)
$\mathrm{Cau}=0.05$. The viscous fluid oscillates with velocity $V(t)=\epsilon
a\omega\cos\omega t$. The torus is ‘pinned’ at the center of its circular
cross-section by a rigid toroidal inclusion of radius $0.4a$. All other
physical and simulation parameters are consistent with sphere streaming (see
captions of Fig. 2 for details).

Finally, we demonstrate how gained theoretical insights translate to 3D
geometries characterized by multiple curvatures and distinct topology,
illustrated here by means of a torus, a shape of interest due to its
microfluidic properties Chan et al. (2022) and recent bioengineered
demonstrations Dou et al. (2022). Figure 3a presents the streaming flow
generated for a rigid torus immersed in an oscillatory flow field at $M\approx
4$. As can be seen, the highlighted recirculating flow features (inner/outer
flow rings that may be used for particle manipulation) are weak for practical
applications in the rigid limit. This can be remedied by increasing elasticity
($\mathrm{Cau}$), for which we indeed observe enhanced flow strengths (Fig.
3(b,c)). Finally, we highlight that to obtain a flow topology similar to Fig.
3c, but with a rigid torus, oscillation frequency $\sim 4\times$ higher are
necessary (supplementary, section §7), in conformity with the intuition gained
via the soft sphere streaming analysis.

##  5 Conclusion

In summary, this study improves existing three-dimensional rigid sphere
streaming theory, expands it to the case of elastic materials, and further
corroborates it by means of direct numerical simulations. Our work reveals, in
keeping with our previous work on 2D soft cylinders, an additional streaming
mode accessible through material compliance and available even in Stokes flow.
It further demonstrates how body elasticity strengthens streaming or enables
it at significantly lower frequencies relative to rigid bodies. Finally, we
show how theoretical insights extend to geometries other than the sphere,
highlighting the practical generality of our theory. Overall, these findings
advance our fundamental understanding of streaming flows, with potential
implications in both biological and engineering domains.

Funding. This work was supported by the NSF CAREER Grant No. CBET-1846752
(MG).

Declaration of Interests. The authors report no conflict of interests.

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# Lower bound for the Poisson bracket invariant
on surfaces

Kun Shi and Guangcun Lu Corresponding author Partially supported by the NNSF
11271044 of China. 2010 Mathematics Subject Classification. 53Dxx, 37J05,
57R17 (secondary).

(May 20, 2019)

###### Abstract

Recently, L. Buhovsky, A. Logunov and S. Tanny proved the (strong) Poisson
bracket conjecture by Leonid Polterovich in dimension $2$. In this note,
instead of open cover constituted of displaceable sets in their work,
considering open cover constituted of topological discs we give a sufficient
and necessary condition that Poisson bracket invariants of these covers are
positive.

###### Contents

  1. 1 Introduction and results
  2. 2 Proof of Theorem 1.6
  3. 3 Proof of Theorem 1.7

##  1 Introduction and results

Throughout this note we always assume that $(M,\omega)$ is a closed connected
symplectic manifold and that $\|\cdot\|:C^{\infty}(M)\rightarrow[0,\infty)$
denotes the $C^{0}$-norm or $L^{\infty}$-norm of smooth functions on $M$.
Recall that a subset $X$ of $M$ is called displaceable if there exists a
Hamiltonian diffeomorphism $\phi\in{\rm Ham}(M,\omega)$ such that $\phi(X)$ is
disjoint with $X$ ([8]). Entov and Polterovich proved in [4] a surprising link
between nondisplaceability and Poisson commutativity, the non-displaceable
fiber theorem, which claimed that some fiber $(\vec{f})^{-1}(p)\neq\emptyset$
and is nondisplaceable for a smooth map
$\vec{f}=(f_{1},\cdots,f_{N}):M\to\mathbb{R}^{N}$ with pairwise Poisson
commuting coordinate functions $f_{i}$, $i=1,\cdots,N\in\mathbb{N}$. This
theorem is equivalent to the following rigidity of partitions of unity [5]:
any finite open cover of $M$ by open displaceable sets does not admit a
Poisson commuting partition of unity.

We say an open cover $\mathcal{U}$ of $M$ to be connected (resp. displaceable,
resp. connected and displaceable if each element of $\mathcal{U}$ is connected
(resp. displaceable, resp. connected and displaceable) in $M$.

In order to measure the Poisson noncommutativity of a smooth partition of
unity $\mathcal{F}:=\\{f_{i}\\}_{i=1}^{N}$ subordinated to a connected and
displaceable finite open cover $\mathcal{U}:=\\{U_{i}\\}_{i=1}^{N}$ of $M$,
Polterovich [13] defined the magnitude of its Poisson noncommutativity by

$\nu_{c}(\mathcal{F}):=\max_{a,b\in[-1,1]^{N}}\left\|\left\\{\sum_{i=1}^{N}a_{i}f_{i},\sum_{j=1}^{N}b_{j}f_{j}\right\\}\right\|,$

and the Poisson bracket invariant of $\mathcal{U}$ by

$pb(\mathcal{U}):=\inf_{\mathcal{F}}\nu_{c}(\mathcal{F}),$

where the infimum is taken over all partitions of unity $\mathcal{F}$
subordinate to $\mathcal{U}$. Polterovich gave several lower bounds for
$pb(\mathcal{U})$ and proposed:

######  Question 1.1 ([14, Question 8.1]).

Is it true that $pb(\mathcal{U})\geq C/e(\mathcal{U})$, where the constant $C$
depends only on the symplectic manifold $(M,\omega)$, and where
$e(\mathcal{U}):=\max_{i\in I}e(U_{i})$ and $e(U_{i})$ is the displacement
energy of $U_{i}$?

This was called the (strong) Poisson bracket conjecture by J. Payette [12],
and studied in references [1, 2, 3, 6, 9, 11, 12, 16].

Recently, Buhovsky, Logunov and Tanny [2] affirmatively answered this question
in dimension $2$. Let $\mathcal{U}$ be a finite open cover on smooth manifold
$M$. A set $U$ in $\mathcal{U}$ is called essential if
$\mathcal{U}\setminus\\{U\\}$ is not a cover [2, Def.1.6]. Denote by
$\mathcal{J}(\mathcal{U})\subset\mathcal{U}$ the collection of essential sets
of $\mathcal{U}$. Let $|\mathcal{U}|$ denote the number of open subsets in
$\mathcal{U}$. A subcover $\mathcal{U}_{0}$ of $\mathcal{U}$ is said to be
smallest if $|\mathcal{U}_{0}|$ is equal to

$\kappa(\mathcal{U}):=\min_{\mathcal{U}^{\prime}\subset\mathcal{U}}\\{|\mathcal{U}^{\prime}|\,:\,\mathcal{U}^{\prime}\;\text{is
also a cover of }\;M\\}.$

Clearly, the smallest subcover of $\mathcal{U}$ is not necessarily unique, and
each of them is a minimal cover of $M$. Moreover, a subcover of $\mathcal{U}$
which is a minimal cover of $M$ is not always a smallest subcover of
$\mathcal{U}$ because the cardinality of a minimal cover of $M$ may be greater
than $\kappa(\mathcal{U})$. Note that each element of $\mathcal{U}_{0}$ is the
essential set of $\mathcal{U}_{0}$ but not always the essential set of
$\mathcal{U}$. Buhovsky, Logunov and Tanny [2] proved the following results.

######  Theorem 1.2 ([2, Theorem 1.7 and Remark 1.8]).

There exists a constant $c>0$ such that for any closed and connected
symplectic surface $(M,\omega)$ and any finite open cover
$\mathcal{U}=\\{U_{i}\\}_{i=1}^{N}$ of $M$ made by topological discs of area
less than ${\rm Area}(M)/2$,

$pb(\mathcal{U})\geq\frac{|\mathcal{J}(\mathcal{U})|c}{{\rm
Area}(M)}\quad\hbox{and}\quad
pb(\mathcal{U})\geq\frac{c}{\min_{U\in\mathcal{J}(\mathcal{U})}{\rm
Area}(U)},$

where we set the minimum of an empty set to be infinity.

######  Theorem 1.3 ([2, Theorem 1.5’]).

Let $(M,\omega)$ be a closed and connected symplectic surface. Let
$\mathcal{U}=\\{U_{i}\\}_{i\in I}$, $\mathcal{V}=\\{V_{j}\\}_{j\in J}$ be
finite open covers of $M$, and let $\\{f_{i}\\}_{i\in I}$, $\\{g_{j}\\}_{j\in
J}$ be partitions of unity subordinate to $\mathcal{U}$, $\mathcal{V}$
correspondingly. Then

$\int_{M}\sum_{i\in I}\sum_{j\in J}|\\{f_{i},g_{j}\\}|\ \omega\geq\frac{{\rm
Area}(M)}{2\cdot\max(e(\mathcal{U}),e(\mathcal{V}))}$ (1.1)

Note that the definition of displaceability at the bottom of [2, page 1] is
different from ours. They called a subset $X$ of $M$ displaceable if the
closure $\bar{X}$ of $X$ is displaceable in our definition above. Thus in our
language [2, Corollary 1.9] may be reformulated as

######  Corollary 1.4 ([2, Corollary 1.9]).

Let $(M,\omega)$ be a closed and connected symplectic surface. Let
$\mathcal{U}=\\{U_{i}\\}_{i\in I}$ be a finite connected open cover of $M$
such that the closure of each $U_{i}$ is displaceable, then for an absolute
constant $c>0$ we have

$pb(\mathcal{U})\geq\frac{c}{e(\mathcal{U})}.$ (1.2)

In Theorem 1.2, since the second inequality holds trivially if
$\mathcal{J}(\mathcal{U})=\emptyset$, it is necessary to assume that the cover
$\mathcal{U}$ contains at least an essential set so that
$\min_{U\in\mathcal{J}(\mathcal{U})}{\rm Area}(U)$ is a positive number. Note
also that the condition “$\max_{1\leq i\leq N}{\rm Area}(U_{i})<{\rm
Area}(M)/2$” in Theorem 1.2 implies $\kappa(\mathcal{U})\geq 3$ (because $M$
is closed and each $U_{i}$ is a topological disc of area less than ${\rm
Area}(M)/2$). It is obvious that the converse is not true in general.

######  Example 1.5.

Let
$M=S^{2}=\\{(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}\,|\,x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1\\}$,
$\omega$ an area form on $S^{2}$, and $\mathcal{U}=\\{U_{1},U_{2},U_{3}\\}$,
where

$\left.\begin{array}[]{ll}&U_{1}=\\{(x_{1},x_{2},x_{3})\in
S^{2}\,|\,x_{3}<1/2\\},\\\ &U_{2}=\\{(x_{1},x_{2},x_{3})\in
S^{2}\,|\,x_{3}>0,\;x_{1}>-1/4\\},\\\ &U_{3}=\\{(x_{1},x_{2},x_{3})\in
S^{2}\,|\,x_{3}>0,\;x_{1}<1/4\\}.\end{array}\right\\}$ (1.3)

Clearly, $\kappa(\mathcal{U})=3$, but ${\rm Area}(U_{1})>{\rm Area}(M)/2$.

The following result claims that the condition “$\max_{1\leq i\leq N}{\rm
Area}(U_{i})<{\rm Area}(M)/2$” in Theorem 1.2 may be replaced by
“$\kappa(\mathcal{U})\geq 3$” and “$\mathcal{J}(\mathcal{U})\neq\emptyset$”.

######  Theorem 1.6.

There exists a constant $c>0$ such that for any closed and connected
symplectic surface $(M,\omega)$ and any finite open cover
$\mathcal{U}=\\{U_{i}\\}_{i=1}^{N}$ made by topological discs and containing
an essential set and with $\kappa(\mathcal{U})\geq 3$,

$pb(\mathcal{U})\geq\frac{|\mathcal{J}(\mathcal{U})|c}{{\rm
Area}(M)}\quad\hbox{and}\quad
pb(\mathcal{U})\geq\frac{c}{\min_{U\in\mathcal{J}(\mathcal{U})}{\rm
Area}(U)}.$

If the condition “$\mathcal{J}(\mathcal{U})\neq\emptyset$” is cast away, as an
improvement of a special case of [2, Theorem 1.5 or 1.5’] we have the
following analogue of Theorem 1.3 ([2, Theorem 1.5’]).

######  Theorem 1.7.

Let $(M,\omega)$ be a closed connected symplectic surface and let
$\mathcal{U}=\\{U_{i}\\}_{i=1}^{N}$ be a finite open cover consisted of
topological discs. If $\kappa(\mathcal{U})\geq 3$, then for any partition of
unity subordinate to $\mathcal{U}$, $\mathcal{F}:=\\{f_{i}\\}_{i=1}^{N}$,
there holds

$\int_{M}\sum_{i,j=1}^{N}|\\{f_{i},f_{j}\\}|\omega\geq 1$

and hence

$pb(\mathcal{U})\geq\frac{c}{{\rm Area}(M,\omega)}$ (1.4)

for some absolute constant $c>0$.

Clearly, if an open cover $\mathcal{U}=\\{U_{i}\\}_{i=1}^{N}$ of a closed
connected symplectic surface $(M,\omega)$ consists of topological discs, then
$\kappa(\mathcal{U})\geq 2$. We claim $pb(\mathcal{U})=0$ if
$\kappa(\mathcal{U})<3$. In fact, in this case any smallest subcover
$\mathcal{U}_{0}$ of $\mathcal{U}$ only contains two sets, saying
$\mathcal{U}_{0}=\\{U_{1},U_{2}\\}$, and any smooth partition of unity
$\mathcal{F}_{0}=\\{f_{1},f_{2}\\}$ subordinated to $\mathcal{U}_{0}$
satisfies $\\{f_{1},f_{2}\\}=\\{f_{1},1-f_{1}\\}=0$ and so
$0=pb(\mathcal{U}_{0})\geq pb(\mathcal{U})\geq 0$. From these and Theorem 1.7
we arrive at:

######  Corollary 1.8.

Let $(M,\omega)$ be a closed connected symplectic surface and
$\mathcal{U}=\\{U_{i}\\}_{i=1}^{N}$ be a open cover of $M$ consisted of
topological discs. Then $pb(\mathcal{U})>0$ if and only if
$\kappa(\mathcal{U})\geq 3$.

Under the assumptions of Corollary 1.4([2, Corollary 1.9]) the inequality in
(1.4) may follow from (1.2) since $e(\mathcal{U})\leq{\rm Area}(M,\omega)/2$.
Obverse that the conditions of Theorem 1.7 and those of Corollary 1.4 cannot
be contained each other. Actually, Theorem 1.7 can also lead to

######  Corollary 1.9.

Let $(M,\omega)$ be a closed and connected symplectic surface and let
$\mathcal{U}=\\{U_{i}\\}^{N}_{i=1}$ be a finite connected open cover of $M$
such that the closure of each $U_{i}$ is displaceable. Then for an absolute
constant $c>0$ there holds

$pb(\mathcal{U})\geq\frac{c}{{\rm Area}(M,\omega)}.$ (1.5)

###### Proof.

Since $U_{i}$ is connected, so is the closure $\bar{U}_{i}$ of it. Moreover
$\bar{U}_{i}$ is displaceable, and compact due to the fact that $M$ is a
closed surface. By [2, Remark 1.2], we have embedded open topological discs
$V_{i}\subset M$ with smooth boundaries such that $U_{i}\subset V_{i}$ and
${\rm Area}(V_{i})\leq{\rm Area}(M)/2$, $i=1,\cdots,N$. (In fact, by Exercise
3 in [7, Chap. 3, §1] $\bar{U}_{i}$ has a closed neighborhood $W_{i}$ is a
smooth submanifold of $M$. Moreover, $W_{i}$ can be required to be arbitrarily
close $\bar{U}_{i}$, and so displaceable connected and has area less than
${\rm Area}(M)/2$. By [12, Theorem 4.1(2)-(i)] we get a smoothly embedded
closed disc $\bar{D}_{i}$ of area ${\rm Area}(\bar{D}_{i})\leq{\rm
Area}(M)/2$. Then $V_{i}:={\rm Int}(\bar{D}_{i})$ satisfies our requirements.)
Clearly, $\mathcal{V}=\\{V_{i}\\}^{N}_{i=1}$ is an open cover of $M$, and
$\kappa(\mathcal{V})\geq 3$. Hence (1.5) follows from Theorem 1.7 since
$pb(\mathcal{U})\geq pb(\mathcal{V})$ by definition. ∎

Finally, it should be pointed out that Example 1.5 satisfies the conditions of
Theorem 1.7, but not those of Corollary 1.4([2, Corollary 1.9]).

We shall complete proofs of Theorems 1.6, 1.7 in Sections 2, 3, respectively.

Acknowledgements. Both authors are deeply grateful to Lev Buhovsky for
wonderful reports and explanations on their work.

##  2 Proof of Theorem 1.6

This can be completed with the similar way to [2, Theorem 1.7].

######  Lemma 2.1 ([2, Lemma 1.3]).

Let $(M^{2n},\omega)$ be a closed symplectic manifold of dimension $2n$. Then,
there exists a constant $c(n)>0$ depending only on the dimension $2n$ of $M$,
such that for every finite collection of smooth functions
$\\{f_{i}\\}_{i=1}^{N}$ on $M$,

$\max_{a,b\in[-1,1]^{N}}\left\|\left\\{\sum_{i=1}^{N}a_{i}f_{i},\sum_{j=1}^{N}b_{j}f_{j}\right\\}\right\|\geq
c(n)\cdot\max_{M}\sum_{i,j=1}^{N}|\\{f_{i},f_{j}\\}|.$

######  Lemma 2.2 ([2, Lemma 2.1]).

For given two smooth functions $f,g:M\rightarrow\mathbb{R}$ let
$\Phi:=(f,g):M\rightarrow\mathbb{R}^{2}$. Consider the function
$K:\mathbb{R}^{2}\rightarrow\mathbb{R}\cup\\{\infty\\}$ defined by

$K(s,t):=\\#(f^{-1}(s)\cap g^{-1}(t))=\\#\Phi^{-1}(s,t).$

Then for any Lebesgue measurable set $\Omega\subset\mathbb{R}^{2}$,

$\int_{\Phi^{-1}(\Omega)}|\\{f,g\\}|\omega=\int_{\Omega}K(s,t)dsdt.$

Let $\mathcal{F}=\\{f_{i}\\}_{i=1}^{N}$ be a partition of unity subordinated
to a given open cover $\mathcal{U}=\\{U_{i}\\}^{N}_{i=1}$ of $M$. For $t\geq
0$ we define

$U_{i}(t):=\\{x\in M:f_{i}>t\\},\quad i=1,\cdots,N.$ (2.1)

Theorem 1.6 may be derived from the last two lemmas and the following:

######  Lemma 2.3.

Under the assumptions of Theorem 1.6, for any partition of unity
$\mathcal{F}=\\{f_{i}\\}_{i=1}^{N}$ subordinated to $\mathcal{U}$,

$\int_{M}\sum_{i,j=1}^{N}|\\{f_{i},f_{j}\\}|\omega\geq|\mathcal{J}(\mathcal{U})|,$
$\max_{M}\sum_{i,j=1}^{N}|\\{f_{i},f_{j}\\}|\geq\frac{1}{\min_{U_{i}\in\mathcal{J}(\mathcal{U})}{\rm
Area}(U_{i})},$

where the minimum of an empty set is understand to be infinity.

###### Proof.

This will be proved as in [2, Theorem 1.7]. Since the open cover
$\mathcal{U}=\\{U_{i}\\}_{i=1}^{N}$ contains at least an essential set, we can
fix an essential set $U_{i}\in\mathcal{U}$. Then there exists a point
$z_{i}\in U_{i}$ such that $z_{i}\notin U_{j}$ for all
$j\in\\{1,\cdots,N\\}\setminus\\{i\\}$. It follows that $f_{i}(z_{i})=1$ and
hence $z_{i}\in U_{i}(s)$ for all $s\in(0,1)$. For every regular value
$s\in(0,1)$ of $f_{i}$, let $V_{i}(s)$ denote the connected component of
$U_{i}(s)$ that contain $z_{i}$. Denote by $\tilde{V}_{i}(s)$ the open
topological disc of minimal area which contains $V_{i}(s)$ and is contained in
$U_{i}$, and by $\gamma^{s}:=\partial\tilde{V}_{i}(s)$. Then $\gamma^{s}$ is
connected and is contained in $\\{f_{i}=s\\}\cap\partial V_{i}(s)$. Fix
$y^{s}\in\gamma^{s}$ and define $t^{s}_{j}:=f_{j}(y^{s})\in\mathbb{R}$ for
each $j\neq i$.

For a fixed $j\neq i$, and let $t\in(0,t_{j}^{s})$ be a regular value of
$f_{j}$. Then $y^{s}\in U_{j}(t)$. Denote by $D_{j}(t)$ the closure of
connected component of $U_{j}(t)$ that contains $y^{s}$, and by
$\tilde{D}_{j}(t)$ the closed topological disc of minimal area which contains
$D_{j}(t)$ and is contained in $U_{j}$. Then
$\partial\tilde{D}_{j}(t)\subset\partial D_{j}(t)\cap\\{f_{j}=t\\}$.

We claim that $\gamma^{s}$ has at least two points of intersection with
$\partial\tilde{D}_{j}(t)$. In fact, since the interior of $\tilde{D}_{j}(t)$
intersects $\gamma^{s}$, we only need to prove that $\gamma^{s}$ is not
contained in $\tilde{D}_{j}(t)$. Equivalently, it suffice to show that both
$\tilde{V}_{i}(s)$ and its complement $\tilde{V}_{i}(s)^{c}$ are not contained
in $\tilde{D}_{j}(t)$. Since $D_{j}(t)\subset U_{j}$ and
$\partial\tilde{D}_{j}(t)\subset\\{f_{j}=t\\}\subset U_{j}$, we have
$\tilde{D}_{j}(t)\subset U_{j}$. Moreover $\tilde{V}_{i}(s)$ contains
$z_{i}\notin U_{j}$, and so $\tilde{V}_{i}(s)\nsubseteq\tilde{D}_{j}(t)$. In
order to get the desired claim we also need to show that
$\tilde{V}_{i}(s)^{c}\nsubseteq\tilde{D}_{j}(t)$. Suppose that
$U_{i}^{c}\subset U_{j_{0}}$ for some
$j_{0}\in\\{1,\cdots,N\\}\setminus\\{i\\}$. Then $\\{U_{i},U_{j_{0}}\\}$ can
cover $M$, and thus $\kappa(\mathcal{U})\leqslant 2$, which contradicts the
assumption $\kappa(\mathcal{U})\geq 3$. Hence $U_{i}^{c}\nsubseteq
U_{j},\forall j\neq i$. Note that $U_{i}^{c}\subset\tilde{V}_{i}(s)^{c}$ and
$\tilde{D}_{j}(t)\subset U_{j}$. We conclude that
$\tilde{V}_{i}(s)^{c}\nsubseteq\tilde{D}_{j}(t)$.

Denote by ${\rm cv}(f_{k})$ the set of critical values of $f_{k}$,
$k=1,\cdots,N$. Define

$\Omega_{ij}:=\\{(s,t)\,:\,s\in(0,1)\setminus{\rm
cv}(f_{i})\;\&\;t\in(0,t^{s}_{j})\setminus{\rm cv}(f_{j})\\}$

and $\Phi_{ij}=(f_{i},f_{j})$ and

$K_{ij}:\mathbb{R}^{2}\to\mathbb{R}\cup\\{\infty\\},\;(s,t)\mapsto\sharp\Phi_{ij}^{-1}(s,t)=\sharp(f_{i}^{-1}(s)\cap
f_{j}^{-1}(t)).$

The above claim implies that $K_{ij}(s,t)\geq 2$ for any
$(s,t)\in\Omega_{ij}$. Applying Lemma 2.2 to $f_{i}$ and $f_{j}$ with
$\Omega:=\Omega_{ij}$ we obtain

$\displaystyle\int_{M}|\\{f_{i},f_{j}\\}|\omega$
$\displaystyle\geq\int_{\Phi_{ij}^{-1}(\Omega_{ij})}|\\{f_{i},f_{j}\\}|\omega$
$\displaystyle=\int_{\Omega_{ij}}K_{ij}(s,t)dsdt$
$\displaystyle\geq\int_{(0,1)\setminus{\rm
cv}(f_{i})}ds\int_{(0,t^{s}_{j})\setminus{\rm
cv}(f_{j})}2dt=2\int_{(0,1)\setminus{\rm cv}(f_{i})}t^{s}_{j}ds.$

Because $t_{j}^{s}=f_{j}(y^{s})$, summing the above inequality over all $j\neq
i$ we get

$\displaystyle\sum_{j=1}^{N}\int_{M}|\\{f_{i},f_{j}\\}|\omega$
$\displaystyle\geq$ $\displaystyle 2\sum_{j\neq i}\int_{(0,1)\setminus{\rm
cv}(f_{i})}t^{s}_{j}ds$ $\displaystyle=$ $\displaystyle
2\int_{(0,1)\setminus{\rm cv}(f_{i})}\sum_{j\neq i}f_{j}(y^{s})ds$
$\displaystyle=$ $\displaystyle 2\int_{(0,1)\setminus{\rm
cv}(f_{i})}(1-f_{i}(y^{s}))ds.$

For every $s\in(0,1)\setminus{\rm cv}(f_{i})$, since
$y^{s}\in\gamma^{s}\subset\\{f_{i}=s\\}$, we have $f_{i}(y^{s})=s$ and thus

$\sum_{j=1}^{N}\int_{M}|\\{f_{i},f_{j}\\}|\geq 2\int_{(0,1)\setminus{\rm
cv}(f_{i})}(1-s)ds=2\int^{1}_{0}(1-s)ds=1.$

Summing over all $i$ satisfying $U_{i}\in\mathcal{J}(\mathcal{U})$, the proof
of Lemma 2.3 is completed. ∎

######  Remark 2.4.

Actually, there is a question about integrality of the function
$(0,1)\setminus{\rm cv}(f_{i})\ni s\mapsto t^{s}_{j}\in\mathbb{R}$ in the
proof above. This can be solved by suitable choices of $y^{s}$. Recall that we
first fix $y^{s}\in\gamma^{s}$ and then define
$t^{s}_{j}:=f_{j}(y^{s})\in\mathbb{R}$ for each $j\neq i$. Since
$(0,1)\setminus{\rm cv}(f_{i})$ is union of at most countable open intervals
contained in $(0,1)$, saying $(0,1)\setminus{\rm
cv}(f_{i})=\cup_{n=1}^{\infty}I_{n}$, for each nonempty $I_{n}$ we fix
$s_{n}\in I_{n}$ and $y^{s_{n}}\in\gamma_{s_{n}}$. Let $\Phi^{s_{n}}$ be the
gradient flow of $f_{i}$ through $y^{s_{n}}$ with respect to some fixed
Riemannian metric on $M$. For each $s\in I_{n}\setminus\\{s_{n}\\}$ let us
choose $y^{s}$ to be the unique intersection point of $f_{i}^{-1}(s)$ with
${\rm Im}(\Phi^{s_{n}})$. Clearly, $y^{s}$ smoothly depends on $s\in
I_{n}\setminus\\{s_{n}\\}$. Hence $I_{n}\setminus\\{s_{n}\\}\ni s\mapsto
t^{s}_{j}:=f_{j}(y^{s})\in\mathbb{R}$ is smooth for each $j\neq i$.

##  3 Proof of Theorem 1.7

We begin with some definitions in [2].

######  Definition 3.1 ([2, Definition 3.3]).

An subset $U$ in a smooth closed surface $M$ is said to have a piecewise
smooth boundary if $\partial U$ is a finite union of disjoint curves
$\Gamma_{1},\cdots,\Gamma_{m}$, such that each $\Gamma_{j}$ is a simple,
closed, piecewise smooth and regular curve.

######  Definition 3.2 ([2, Definition 3.3]).

Let $\gamma_{1},\cdots,\gamma_{m}\subset M$ be a finite collection of smooth
regular curves with a finite number of intersection points. Denote
$\Gamma=\gamma_{1}\cup\cdots\cup\gamma_{m}$. A connected component of the
complement $M\setminus\Gamma$ is called a face of $\Gamma$. A point
$v\in\Gamma$ that lies in the intersection of two (or more) curves is called a
vertex of $\Gamma$.

######  Definition 3.3 (cf. [2, Definition 3.1]).

A cover $\mathcal{U}=\\{U_{i}\\}_{i\in I}$ of $M$ is called good if $\partial
U_{i}$ and $\partial U_{j}$ intersect transversally for all $i,j\in I,i\neq
j$. Two covers $\mathcal{U}=\\{U_{i}\\}_{i\in I},\mathcal{V}=\\{V_{j}\\}_{j\in
J}$ of $M$ are said to be in generic position if

$\partial U_{i}\cap\partial U_{k}\cap\partial
V_{j}=\emptyset\quad\hbox{and}\quad\partial U_{i}\cap\partial
V_{j}\cap\partial V_{l}=\emptyset$

for all $i,k\in I,i\neq k$, and $j,l\in J,j\neq l$.

The following lemma is a variant of [2, Lemmas 3.2 and 3.4].

######  Lemma 3.4.

Let $\mathcal{U}=\\{U_{i}\\}_{i\in I}$ and $\mathcal{V}=\\{V_{j}\\}_{j\in J}$
be good finite open covers of $M$. Assume

(i)

$\mathcal{U}$ and $\mathcal{V}$ are in generic position;

(ii)

there exist $\widetilde{\mathcal{U}}=\\{\widetilde{U}\\}_{i\in I}$ and
$\widetilde{\mathcal{V}}=\\{\widetilde{V}\\}_{j\in J}$ both consisted of
topological discs such that
$\kappa(\widetilde{\mathcal{U}}\cup\widetilde{\mathcal{V}})\geq 3$,
$U_{i}\subset\widetilde{U}_{i}$ and $V_{j}\subset\widetilde{V}_{j}$ for all
$i\in I,j\in J$;

(iii)

there exist $L\in\mathbb{N}$ such that for any point $x\in M$, $\\#\\{i\in
I:x\in U_{i}\\}\geq L$ and $\\#\\{j\in J:x\in V_{j}\\}\geq L$.

Then

$\sharp\cup_{i,j}(\partial U_{i}\cap\partial V_{j})\geq(L+1)^{2}.$

###### Proof.

Let $I=\\{1,2,\cdots,|I|\\}$ and $J=\\{1,2,\cdots,|J|\\}$. Denote by $S_{I}$
and $S_{J}$ the sets of permutations on the elements of $I$ and $J$
respectively. For $\alpha\in S_{I}$ and $\beta\in S_{J}$ define the unions of
curves

$\displaystyle\Gamma_{\alpha}:=\cup_{i\in I}(\partial U_{\alpha(i)}\cap
U^{c}_{\alpha(i-1)}\cap\cdots\cap U^{c}_{\alpha(1)}),$
$\displaystyle\Gamma^{\prime}_{\beta}:=\cup_{j\in J}(\partial V_{\beta(j)}\cap
V^{c}_{\beta(j-1)}\cap\cdots\cap V_{\beta(1)}^{c}).$

Step 1. Given a nonempty connected component $P$ of
$M\setminus\Gamma_{\alpha}$ (resp. $M\setminus\Gamma^{\prime}_{\beta}$), prove
that there exists $i\in I$ (resp. $j\in J$) such that $P\subset U_{i}$ (resp.
$P\subset V_{j}$). We only prove the case for the component of
$M\setminus\Gamma_{\alpha}$. Another case is similar. On the contrary, assume
that $P\nsubseteq U_{i}$ for all $i\in I$. Since $\partial
U_{\alpha(1)}\subset U^{c}_{\alpha(1)}\subset\Gamma_{\alpha}$, we get
$P\cap\partial U_{\alpha(1)}\subset P\cap\Gamma_{\alpha}=\emptyset$. This and
the fact that $P\nsubseteq U_{\alpha(1)}$ imply that $P\subset
U^{c}_{\alpha(1)}$. Assuming $P\subset U^{c}_{\alpha(1)}\cap\cdots\cap
U^{c}_{\alpha(i-1)}$, we conclude that $P\cap\partial U_{\alpha(i)}\subset
P\cap\Gamma_{\alpha}=\emptyset$ since $\partial U_{\alpha(i)}\cap
U^{c}_{\alpha(1)}\cap\cdots\cap U^{c}_{\alpha(i-1)}\subset\Gamma_{\alpha}$. As
above it follows that $P\subset U^{c}_{\alpha(i)}$ because $P\nsubseteq
U_{\alpha(i)}$. By induction on $i\in I$ , we obtain that $P\subset\cap_{i\in
I}U^{c}_{\alpha(i)}=\emptyset$, which contradicts $P\neq\emptyset$.

Step 2. Prove $\\#(\Gamma_{\alpha}\cap\Gamma^{\prime}_{\beta})\geq 1$. First,
let us show that by removing parts from $\Gamma_{\alpha}$ and
$\Gamma^{\prime}_{\beta}$, we can get $\widetilde{\Gamma}_{\alpha}$ and
$\widetilde{\Gamma}^{\prime}_{\beta}$ such that their faces are open
topological discs. By Step 1, for any face $P$ of $\Gamma_{\alpha}$, there
exists $i\in I$ such that $P\subset U_{i}\subset\widetilde{U}_{i}$. Let
$\widetilde{P}$ be the open topological disc of minimal area that contains $P$
and is contained in $\widetilde{U}_{i}$. Then
$\partial{\widetilde{P}}\subset\partial P\subset\Gamma_{\alpha}$, and
$\widetilde{P}$ is a face of
$\Gamma_{\alpha}^{1}:=\Gamma_{\alpha}\setminus(\Gamma_{\alpha}\cap\widetilde{P})$.
For a face ${P}_{1}$ of $\Gamma_{\alpha}^{1}$ which is not an open topological
disc, repeating the above way we finally get $\widetilde{\Gamma}_{\alpha}$
such that each face of $\widetilde{\Gamma}_{\alpha}$ is an open topological
disc. Similarly, from $\Gamma^{\prime}_{\beta}$ we obtain a
$\widetilde{\Gamma}^{\prime}_{\beta}$ with the same properties. It is obvious
that

$\\#(\Gamma_{\alpha}\cap\Gamma^{\prime}_{\beta})\geq\\#(\widetilde{\Gamma}_{\alpha}\cap\widetilde{\Gamma}^{\prime}_{\beta}).$

Call a face $G$ of $\widetilde{\Gamma}_{\alpha}$ (resp.
$\widetilde{\Gamma}^{\prime}_{\beta}$) is maximal if it is not properly
contained in any face of $\widetilde{\Gamma}^{\prime}_{\beta}$ (resp.
$\widetilde{\Gamma}_{\alpha}$). Notice that any non-maximal face of
$\widetilde{\Gamma}_{\alpha}$ is contained in a maximal face of
$\widetilde{\Gamma}^{\prime}_{\beta}$. (Indeed, for a non-maximal face $P$ of
$\widetilde{\Gamma}_{\alpha}$, by the definition there exists a face $Q$ of
$\widetilde{\Gamma}^{\prime}_{\beta}$ such that $P\subsetneq Q$. We claim that
$Q$ is a maximal face of $\widetilde{\Gamma}^{\prime}_{\beta}$. Otherwise,
there exists a face $P^{\prime}$ of $\widetilde{\Gamma}_{\alpha}$ such that
$P\subsetneq Q\subsetneq P^{\prime}$, which contradicts the fact that $P$ is a
face and thus a connected component.) Therefore the union of maximal faces of
both $\widetilde{\Gamma}_{\alpha}$ and $\widetilde{\Gamma}^{\prime}_{\beta}$
covers the complementary of
$\widetilde{\Gamma}_{\alpha}\cap\widetilde{\Gamma}^{\prime}_{\beta}$ in $M$.
Hence either $\widetilde{\Gamma}_{\alpha}$ or
$\widetilde{\Gamma}^{\prime}_{\beta}$ has at least a maximal face. Without
loss of generality, we assume that $\widetilde{\Gamma}_{\alpha}$ has the
maximal faces.

Let $G$ be a maximal face of $\widetilde{\Gamma}_{\alpha}$. Then there exists
a face $G^{\prime}$ of $\widetilde{\Gamma}^{\prime}_{\beta}$ such that

$\partial G\cap G^{\prime}\neq\emptyset.$ (3.1)

(Otherwise, $\partial G\subset\widetilde{\Gamma}^{\prime}_{\beta}$, and hence
$\\#(\widetilde{\Gamma}_{\alpha}\cap\widetilde{\Gamma}^{\prime}_{\beta})=\infty$,
the conclusion is proved.) We claim that

$\partial G\cap(M\setminus G^{\prime})\neq\emptyset.$ (3.2)

In fact, if $\partial G\cap(M\setminus G^{\prime})=\emptyset$, then $\partial
G\subset G^{\prime}$, and we have either $M\setminus G^{\prime}\subset G$ or
$G\subset G^{\prime}$ since $M\setminus G^{\prime}$ is connected. The first
case is impossible since
$\kappa(\widetilde{\mathcal{U}}\cup\widetilde{\mathcal{V}})\geq 3$. In the
second case we get $G\subset G^{\prime}$, which contradicts the maximality of
$G$.

Since the boundary $\partial G$ is a simple closed curve, (3.1) and (3.2)
imply $\\#(\partial G\cap\partial G^{\prime})\geq 1$, and thus

$\\#(\Gamma_{\alpha}\cap\Gamma^{\prime}_{\beta})\geq\\#(\widetilde{\Gamma}_{\alpha}\cap\widetilde{\Gamma}^{\prime}_{\beta})\geq\\#(\partial
G\cap\partial G^{\prime})\geq 1.$

Step 3. Since $\Gamma_{\alpha}\subset\cup_{i\in I}\partial U_{i}$ and
$\Gamma^{\prime}_{\beta}\subset\cup_{j\in J}\partial V_{j}$,
$\Gamma_{\alpha}\cap\Gamma^{\prime}_{\beta}\subset\cup_{i,j}(\partial
U_{i}\cap\partial V_{j})$ and so

$\triangle:=\cup_{\alpha\in S_{I}}\cup_{\beta\in
J}\Gamma_{\alpha}\cap\Gamma^{\prime}_{\beta}\subset\cup_{i,j}(\partial
U_{i}\cap\partial V_{j}).$ (3.3)

Fixing a point $x\in\triangle$ we estimate

$\sharp\\{(\alpha,\beta)\in S_{I}\times
S_{J}\,:\,x\in\Gamma_{\alpha}\cap\Gamma^{\prime}_{\beta}\\}=\sharp\\{\alpha\in
S_{I}\,:\,x\in\Gamma_{\alpha}\\}\times\sharp\\{\alpha\in
S_{J}\,:\,x\in\Gamma^{\prime}_{\beta}\\}.$

Let $i\in I$ such that $x\in\partial U_{i}$. By the definition of
$\Gamma_{\alpha}$, $x\in\Gamma_{\alpha}$ only if
$\alpha^{-1}(i)<\alpha^{-1}(k)$ for any $k\in I$ such that $x\in U_{k}$. By
our assumption, $\sharp\\{k\in I\,:\,x\in U_{k}\\}\geq L$. Because of
symmetry, the number of permutations $\sigma=\alpha^{-1}$ for which
$\sharp\\{k\in I\,:\,\sigma(i)<\sigma(k)\\}\geq L$ is at most $|I|!/(L+1)$. It
follows that $\sharp\\{\alpha\in
S_{I}\,:\,x\in\Gamma_{\alpha}\\}\leq|I|!/(L+1)$. Similarly, we have
$\sharp\\{\alpha\in S_{I}\,:\,x\in\Gamma_{\alpha}\\}\leq|J|!/(L+1)$.

Note that the disjoint union $\coprod_{\alpha\in S_{I}}\coprod_{\beta\in
J}(\Gamma_{\alpha}\cap\Gamma^{\prime}_{\beta})$ has exactly $\sum_{\alpha\in
S_{I}}\sum_{\beta\in S_{J}}\\#(\Gamma_{\alpha}\cap\Gamma^{\prime}_{\beta})$
elements, and that each point $x\in\triangle$ appears
$\sharp\\{(\alpha,\beta)\in S_{I}\times
S_{J}\,:\,x\in\Gamma_{\alpha}\cap\Gamma^{\prime}_{\beta}\\}$ times in
$\coprod_{\alpha\in S_{I}}\coprod_{\beta\in
J}(\Gamma_{\alpha}\cap\Gamma^{\prime}_{\beta})$. Then

$\displaystyle\sharp(\cup_{\alpha\in S_{I}}\cup_{\beta\in
J}\Gamma_{\alpha}\cap\Gamma^{\prime}_{\beta})\times|I|!/(L+1)\times|J|!/(L+1)$
$\displaystyle\geq$ $\displaystyle\sharp(\cup_{\alpha\in S_{I}}\cup_{\beta\in
J}\Gamma_{\alpha}\cap\Gamma^{\prime}_{\beta})\times\max\left\\{\sharp\\{(\alpha,\beta)\in
S_{I}\times
S_{J}\,:\,x\in\Gamma_{\alpha}\cap\Gamma^{\prime}_{\beta}\\}\,:\,x\in\triangle\right\\}$
$\displaystyle\geq$ $\displaystyle\sharp(\coprod_{\alpha\in
S_{I}}\coprod_{\beta\in
J}(\Gamma_{\alpha}\cap\Gamma^{\prime}_{\beta}))=\sum_{\alpha\in
S_{I}}\sum_{\beta\in S_{J}}\sharp(\Gamma_{\alpha}\cap\Gamma^{\prime}_{\beta})$

and hence

$\begin{aligned} \sharp\cup_{i,j}(\partial U_{i}\cap\partial
V_{j})&\geq\sharp(\cup_{\alpha\in S_{I}}\cup_{\beta\in
J}\Gamma_{\alpha}\cap\Gamma^{\prime}_{\beta})\\\
&\geq\frac{L+1}{|I|!}\frac{L+1}{|J|!}\sum_{\alpha\in S_{I}}\sum_{\beta\in
S_{J}}\sharp(\Gamma_{\alpha}\cap\Gamma^{\prime}_{\beta})\\\
&\geq(L+1)^{2}.\end{aligned}.$

∎

###### Proof of Theorem 1.7.

The basic ideas are the same as those of [2, Theorem 1.5]. Fix a large
$L\in\mathbb{N}$ so that $L>|I|$. We shall use the functions
$\\{f_{i}\\}_{i\in I}$ to construct covers satisfying the assumptions of Lemma
3.4. For every $i\in I$ pick $m_{i}\in\mathbb{N}$ such that
$\frac{m_{i}}{L}>\max_{M}f_{i}$, and consider the interval

$\mathcal{I}_{i,k}:=\left[\frac{k-1}{L},\frac{k}{L}\right],\quad
k=1,\cdots,m_{i}.$

Denote by $s_{i,k}\in\mathcal{I}_{i,k}$ an independent variable. We equip the
interval $\mathcal{I}_{i,k}$ with the normalized Lebesgue measure
$\mu_{i,k}:=Lds_{i,k}$. Put $n:=\sum_{i\in I}m_{i}$. Then

$\mathcal{C}:=\Pi_{i\in I}\Pi_{1\leqslant k\leqslant
m_{i}}\mathcal{I}_{i,k}\subset\mathbb{R}^{n}.$

For $s:=(s_{i,k})_{i,k}$, $t:=(t_{j,l})_{j,l}\in\mathcal{C}$, consider the
open sets

$\displaystyle U_{i,k}^{s}:=U_{i,k}(s_{i,k})=\\{f_{i}>s_{i,k}\\},\quad
1\leqslant k\leqslant m_{i},i\in I,$ $\displaystyle
U_{j,l}^{t}:=U_{j,l}(t_{j,l})=\\{f_{j}>t_{j,l}\\},\quad 1\leqslant l\leqslant
m_{j},j\in I.$

Since $\sum_{i\in I}f_{i}=1$, for any $x\in M$ there exists an $i\in I$ such
that $f_{i}(x)\geq 1/|I|>1/L>s_{i,1}$. It follows that
$\mathcal{U}^{s}:=\\{U_{i,k}^{s}\\}_{i,k}$ and
$\mathcal{U}^{t}:=\\{U_{j,l}^{t}\\}_{j,l}$ are open cover of $M$ for any
$s,t\in\mathcal{C}$.

Let us show that for almost all $s,t\in\mathcal{C}$ the covers
$\mathcal{U}^{s}$ and $\mathcal{U}^{t}$ satisfy the assumptions of Lemma 3.4.
The condition (iii) is easily checked. Given a $x\in M$, for every $i\in I$ we
have

$\displaystyle\sharp\\{1\leqslant k\leqslant m_{i}:x\in U_{i,k}^{s}\\}$
$\displaystyle=\sharp\\{1\leqslant k\leqslant m_{i}:f_{i}(x)>s_{i,k}\\}$
$\displaystyle\geq\sharp\\{1\leqslant k\leqslant
m_{i}:f_{i}(x)>\frac{k}{L}\\}$ $\displaystyle\geq Lf_{i}(x)-1.$

This implies that the number of sets in $\mathcal{U}^{s}$ containing $x$ is at
least $\sum_{i\in I}(Lf_{i}(x)-1)=L-|I|$.

Claim. For almost all $(s,t)\in\mathcal{C}\times\mathcal{C}$ the cover
$\mathcal{U}^{s}$ and $\mathcal{U}^{t}$ are good and in generic positions.

Indeed, by Sard’s theorem, for almost all
$(s,t)\in\mathcal{C}\times\mathcal{C}$, $(s_{i,k},t_{j,l})$ is a common
regular value of all maps

$\Phi_{i,j}:M\longrightarrow\mathbb{R}^{2},\;x\mapsto(f_{i}(x),f_{j}(x)),\qquad\forall
i,k,j,l.$

In particular, for such a pair $(s,t)$, the boundaries $\partial U_{i,k}^{s}$
and $\partial U_{j,l}^{t}$ intersect transversely. (Since each $z\in\partial
U_{i,k}^{s}\cap\partial U_{j,l}^{t}$ is a regular point, i.e., ${\rm
det}\Phi_{i,j}(z)\neq 0$, the gradients $\nabla f_{i}(z)$ and $\nabla
f_{j}(z)$ intersect transversely.) Moreover, if $\partial
U_{i,k}^{s}\cap\partial U_{j,l}^{t}\cap\partial U_{q,p}^{t}\neq\emptyset$,
then $(s_{i,k},s_{j,l},s_{q,p})$ is in the image of

$\Phi_{i,j,q}:M\longrightarrow\mathbb{R}^{3},x\mapsto(f_{i}(x),f_{j}(x),f_{q}(x)).$

Since the image ${\rm Im}(\Phi_{i,j,q})$ has the codimension at least $1$ in
$\mathbb{R}^{3}$ and thus the measure of $\cup_{i,j,q}{\rm Im}(\Phi_{i,j,q})$
is equal to zero, we deduce that for almost all
$(s,t)\in\mathcal{C}\times\mathcal{C}$ the cover $\mathcal{U}^{s}$ and
$\mathcal{U}^{t}$ are good and in generic positions.

It remains to prove that (ii) of Lemma 3.4 is satisfied. Since
$\kappa(\mathcal{U})\geq 3$ and

$U_{i,k}^{s}=\\{f_{i}>s_{i,k}\\}\subset{\rm supp}f_{i}\subset U_{i},$

we may take $\widetilde{U_{i,k}^{s}}=U_{i}$ and thus
$\kappa(\\{\widetilde{U_{i,k}^{s}}\,:\,i,k,s\\})\geq 3$.

In summary, for almost all $(s,t)\in\mathcal{C}\times\mathcal{C}$,
$\mathcal{U}^{s}$ and $\mathcal{U}^{t}$ satisfy the assumptions of Lemma3.4.

Applying Lemma 3.4 for $\hat{L}:=L+1-|I|$ and almost every $(s,t)$, we obtain

$\sharp\cup_{i,k,j,l}(\partial U_{i,k}^{s}\cap\partial
U_{j,l}^{t})\geq\hat{L}^{2}.$

Averaging this inequality over $(s,t)\in\mathcal{C}\times\mathcal{C}$ with
respect to the normalized product measure $\mu\times\mu$, where
$\mu:=\Pi_{i,k}\mu_{i,k}$, we obtain

$\displaystyle\hat{L}^{2}$
$\displaystyle\leq\int_{\mathcal{C}\times\mathcal{C}}\sharp\cup_{i,k,j,l}(\partial
U_{i,k}^{s}\cap\partial U_{j,l}^{t})\,d\mu(s)d\mu(t)$
$\displaystyle\leq\int_{\mathcal{C}\times\mathcal{C}}\sum_{i,k,j,l}\sharp(\partial
U_{i,k}^{s}\cap\partial U_{j,l}^{t})d\mu(s)d\mu(t)$
$\displaystyle=\sum_{i,k,j,l}\int_{\frac{k-1}{L}}^{\frac{k}{L}}\int_{\frac{l-1}{L}}^{\frac{l}{L}}\sharp(\partial
U_{i,k}^{s}\cap\partial U_{j,l}^{t})d\mu_{i,k}(s_{i,k})d\mu_{j,l}(t_{j,l})$
$\displaystyle=L^{2}\sum_{i,k,j,l}\int_{\frac{k-1}{L}}^{\frac{k}{L}}\int_{\frac{l-1}{L}}^{\frac{l}{L}}\sharp(\partial
U_{i,k}^{s}\cap\partial U_{j,l}^{t})ds_{i,k}dt_{j,l}$

since $\mu_{i,k}:=Lds_{i,k}$. Moreover, for any values of $s_{i,k}$ and
$t_{j,l}$, we have

$\partial U_{i,k}^{s}=\partial\\{f_{i}>s_{i,k}\\}\subset
f_{i}^{-1}(s_{i,k})\quad\hbox{and}\quad\partial
U_{j,l}^{t}=\partial\\{f_{j}>t_{j,l}\\}\subset f_{j}^{-1}(t_{j,l}).$

Hence

$\hat{L}^{2}\leq
L^{2}\sum_{i,k,j,l}\int_{\frac{k-1}{L}}^{\frac{k}{L}}\int_{\frac{l-1}{L}}^{\frac{l}{L}}\sharp(f_{i}^{-1}(s_{i,k})\cap
f_{j}^{-1}(t_{j,l}))ds_{i,k}dt_{j,l}.$

Denote $\Phi_{i,j}:=(f_{i},f_{j}):M\rightarrow\mathbb{R}^{2}$ and set

$\Omega_{k,l}:=\left(\frac{k-1}{L},\frac{k}{L}\right)\times\left(\frac{l-1}{L},\frac{l}{L}\right)\subset\mathbb{R}^{2}.$

From Lemma 2.2, we deduce

$\displaystyle\hat{L}^{2}$ $\displaystyle\leq
L^{2}\sum_{i,k,j,l}\int_{\Omega_{k,l}}\int_{\frac{l-1}{L}}^{\frac{l}{L}}\sharp(f_{i}^{-1}(s_{i,k})\cap
f_{j}^{-1}(t_{j,l}))ds_{i,k}dt_{j,l}$
$\displaystyle=L^{2}\sum_{i,k,j,l}\int_{\Phi^{-1}_{i,j}(\Omega_{k,l})}|\\{f_{i},f_{j}\\}|\omega$
$\displaystyle\leq L^{2}\sum_{i,j}\int_{M}|\\{f_{i},f_{j}\\}|\omega$

and so

$\sum_{i,j}\int_{M}|\\{f_{i},f_{j}\\}|\omega\geq\frac{\hat{L}^{2}}{L^{2}}=\frac{(L+1-|I|)^{2}}{L^{2}}.$

Letting $L\rightarrow\infty$, the desired result is obtained. ∎

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School of Mathematical Sciences, Beijing Normal University
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Laboratory of Mathematics and Complex Systems, Ministry of Education
Beijing 100875, The People’s Republic of China
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