On the existence of charged electrostatic black holes in

arbitrary topology

Martín Reiris

<EMAIL_ADDRESS>
Centro de Matemática, Universidad de la República

Montevideo, Uruguay

###### Abstract

The general classification of $3+1$-static black hole solutions of the
Einstein equations, with or without matter, is central in General Relativity
and important in geometry. In the realm of ${\rm S}^{1}$-symmetric vacuum
spacetimes, a recent classification proved that, without restrictions on the
topology or the asymptotic behavior, black hole solutions can be only of three
kinds: (i) Schwarzschild black holes, (ii) Boost black holes, or (iii) Myers-
Korotkin-Nicolai black holes, each one having its distinct asymptotic and
topological type. In contrast to this, very little is known about the general
classification of ${\rm S}^{1}$-symmetric static electrovacuum black holes
although examples show that, on the large picture, there should be striking
differences with respect to the vacuum case. A basic question then is whether
or not there are charged analogs to the static vacuum black holes of types
(i), (ii) and (iii). In this article we prove the remarkable fact that, while
one can ‘charge’ the Schwarzschild solution (resulting in a Reissner-Nordström
spacetime) preserving the asymptotic, one cannot do the same to the Boosts and
to the Myers-Korotkin-Nicolai solutions: the addition of a small or large
electric charge, if possible at all, would transform entirely their asymptotic
behavior. In particular, such vacuum solutions cannot be electromagnetically
perturbed. The results of this paper are consistent but go far beyond the
works of Karlovini and Von Unge on periodic analogs of the Reissner-Nordström
black holes.

The type of result as well as the techniques used are based on comparison
geometry a la Bakry-Émery and appear to be entirely novel in this context. The
findings point to a complex interplay between asymptotic, topology and charge
in spacetime dimension $3+1$, markedly different from what occurs in higher
dimensions.

##  Introduction

Since the early years of General Relativity, the classification of static
solutions has been quite an important problem. Already in 2016, Schwarzschild
presented his (by now) celebrated solution that has been fundamental in our
understanding of black holes. The exterior, static, Schwarzschild solutions
are defined on $\mathbb{R}^{3}$ minus a 3-ball, are asymptotically flat, and
have a horizon at their boundary which is the characteristic feature defining
black holes. Of course, the Schwarzschild solution models also the exterior
gravitational field of spherically symmetric (isolated) material bodies and is
thus of central importance in astrophysics. Now, since already some decades
there has been increasing theoretical interest in the understanding of static
black hole solutions (i.e. having a horizon) with more exotic topology,
matter, and asymptotic behavior. This article investigates various aspects of
the geometry and topology of electro-vacuum static and ${\rm S}^{1}$-symmetric
3+1 black hole solutions, which, as we will see, entangle new phenomena
relative to the vacuum case.

To frame the discussion in the following paragraphs, let us begin recalling
what is known about the relation between the topology and the asymptotic
behavior for static vacuum solutions, (not necessarily black hole solutions).
First, the celebrated uniqueness theorem of the Schwarzschild solutions by
Israel [7], Robinson [20] and Bunting-Masood [2], asserts that a $3+1$
asymptotically flat static vacuum black hole is necessarily Schwarzschild. The
fundamental hypothesis here is asymptotic flatness that ends up constraining
the spatial topology to that of $\mathbb{R}^{3}$ minus a 3-ball. In
particular, it is impossible to achieve static equilibrium between several
black holes when the solution is vacuum and asymptotically flat. Conversely,
if the spatial topology of a static vacuum black hole solution is that of
Schwarzschild and the solution is metrically complete, then it is indeed
asymptotically flat and therefore Schwarzschild again [16], [17]. Thus, in
vacuum at least, the topology and the asymptotic are tightly related.
Similarly, if a static vacuum solution is asymptotically flat and without
horizons (i.e. not a black hole solution) then Lichnerowicz showed that the
solution is the Minkowski spacetime [11]. Conversely, if the spatial topology
is that of the Minkowski spacetime and the vacuum solution is metrically
complete, then Anderson [1] proved that the solution must be the Minkowski
spacetime. More in general, it was shown in [14], [15] that, without
topological and asymptotic assumptions, a metrically complete static vacuum
solution with a non-empty horizon must be either: (i) a Schwarzschild black
hole, (ii) a Boost black hole, or (iii) a black hole of Myers-Korotkin-Nicolai
(MKN) type (see below). This result was refined in [19] but assuming ${\rm
S}^{1}$-symmetry, showing that a ${\rm S}^{1}$-symmetric solution of MKN-type
is indeed a MKN-solution.

The universal MKN solutions [12], [10], [9] are axisymmetric, static
solutions, having a periodic array of infinite coaxial black holes (universal
is because the spatial manifold is simply connected). In Weyl coordinates the
metric is of the form,

$g=-e^{\omega}dt^{2}+e^{-\omega}(e^{2k}(dz^{2}+d\rho^{2})+\rho^{2}d\varphi^{2}).$
(1)

The exponents $\omega$ and $k$ are independent on the axial coordinate
$\varphi$ and $(z,\rho)\in\mathbb{R}_{z}\times\mathbb{R}^{+}_{\rho}$. The axis
and the horizons are located over $\\{\rho=0\\}$ and the exponents $\omega$
and $k$ are suitably singular over the periodic array of segments representing
the horizons (Weyl’s coordinates degenerate on them). As
$\rho\rightarrow\infty$ the metric gets asymptotically Kasner, namely,

$g\sim-c^{2}\rho^{\alpha}dt^{2}+a^{2}\rho^{\alpha^{2}/2-\alpha}(dz^{2}+d\rho^{2})+b^{2}\rho^{2-\alpha}d\varphi^{2}$
(2)

where $0<\alpha<2$. Observe that because $\alpha<2$, the norm of the axial
Killing field $\partial_{\varphi}$ tends to infinity. This information will be
used later. Being periodic (the translation group on the variable $z$ is
$\mathbb{Z}$), the universal MKN solutions can be quotient so that the
quotient possesses only an arbitrary but finite number of black hole horizons.
These are the solutions that we call MKN in this paper. The quotient manifolds
are diffeomorphic to open 3-torus minus a finite number of open 3-balls (see
Figure 1).

Figure 1: The grey region is the spatial manifold of the MKN solutions,
topologically an open solid 3-torus minus the horizons, one in this case. The
only boundary of the manifold is the horizon.

In the presence of electromagnetic matter (electric, magnetic, or both), very
little is known about a general classification of static solutions. The
following well-known examples show that the space of solutions is however more
diverse and more complex than in vacuum, and that there is a more subtle
interplay between topology and asymptotic behavior. First, asymptotically
flat, charged, electrostatic black hole solutions are known to be the
Reissner-Nordström solutions [21]. Roughly speaking, these solutions are the
electrically charged Schwarzschild black holes. However, the magnetostatic
Schwarzschild-Melvin solutions, which are metrically complete and have the
same spatial topology as the Schwarzschild or the Reissner-Nordström solutions
are asymptotic to the magnetized Melvin universe and are therefore not
asymptotically flat. Roughly speaking, these solutions represent an uncharged
Schwarzschild black hole embedded inside a magnetostatic universe. Similarly,
the Melvin and the Bertotti-Robinson solutions are examples of magnetostatic
metrically complete solutions having the same spatial topology of the
Minkowski spacetime but that are not asymptotically flat. Garfinkle and Glass
[5] conjectured that in the presence of magnetic energy only, no other
metrically complete static solution exists in that topology.

We now wonder whether there exist charged electrostatic or magnetostatic black
hole solutions in topologies like those of the MKN black holes. In this
article, we prove the remarkable feature that, while the Schwarzschild black
hole can be perturbed by adding a small electric charge (getting thus a ${\rm
S}^{1}$-symmetric Reissner-Nordström black hole), the Boosts and the Myers-
Korotkin-Nicolai cannot: the addition of any electric charge, small or big,
keeping the ${\rm S}^{1}$-symmetry, would entirely affect the asymptotic
behavior. We conjecture that the Boosts and the Myers-Korotkin-Nicolai
topologies indeed do not accept charged electrostatic or magnetostatic black
hole solutions. We note however that one can quotient the Melvin solution by a
translation along the $z$-axis to obtain indeed a magnetostatic ${\rm
S}^{1}$-symmetric and metrically complete solution whose spatial topology is
that of an open solid $3$-torus. Yet such a solution is uncharged (the end of
the manifold has no magnetic flux).

The main result to be proved in this article, from which the main conclusion
mentioned in the previous paragraph will follow, is a theorem about the
asymptotic behavior of ${\rm S}^{1}$-symmetric electrostatic or magnetostatic
charged solutions on spatial 3-manifolds of the form
$[0,\infty)_{\rho}\times{\rm S}^{1}\times{\rm S}^{1}$, with one of the ${\rm
S}^{1}$-factors being the orbits of the ${\rm S}^{1}$-symmetry (infinity is as
$\rho\rightarrow\infty$). When discussing the applications a few lines below,
this manifold will be thought of as just the asymptotic region of a certain
solution defined on a larger manifold. The main theorem, that we state right
below, says that if an ${\rm S}^{1}$-symmetric electrostatic or magnetostatic
charged solution on such a 3-manifold has the norm of the Killing field
generating the ${\rm S}^{1}$-symmetry bounded away from zero, then the lapse
$N$ must tend to zero at infinity.

######  Theorem 1.

Consider an ${\rm S}^{1}$-symmetric charged electrostatic or magnetostatic
spacetime metric in Weyl coordinates,

${\bf
g}=-e^{2\bar{u}}dt^{2}+e^{-2\bar{u}}(e^{2\bar{k}}(d\rho^{2}+dz^{2})+\rho^{2}d\varphi^{2}),$
(3)

and on the $4$-manifold ${\bf
M}=\mathbb{R}_{t}\times[\rho_{0},\infty)_{\rho}\times{\rm S}^{1}_{z}\times{\rm
S}^{1}_{\varphi}$, ($\rho_{0}>0$). If the spatial Riemannian $3$-manifold is
metrically complete and the norm of the ${\rm S}^{1}$-Killing field
$\partial_{\varphi}$ is bounded below away from zero, then the lapse
$N=e^{\bar{u}}$ tends to zero at infinity (i.e. as $\rho\rightarrow\infty$).

This theorem implies that there are no charged electrostatic (or
magnetostatic) black holes on a 3-manifold with its manifold-end (assume only
one) as in the theorem and with the norm of the $S^{1}$-Killing field bounded
away from zero at infinity. To see this, we argue by contradiction. Suppose
that a solution like that exists. Let $\Sigma$ be its spatial 3-manifold with
compact boundary (the horizon) and with its manifold-end diffeomorphic to
$[0,\infty)\times{\rm S}^{1}\times{\rm S}^{1}$. As we are assuming that the
solution is a black hole, we have $N|_{\partial\Sigma}=0$. On the other hand
by Theorem 1 we have $N\rightarrow 0$ at infinity. Recalling finally that the
lapse equation for an electrostatic solution is,

$\Delta N=|E|^{2}N,$ (4)

then the maximum principle tells us that $N$ must be identically zero,
reaching a contradiction.

This result is consistent but goes far beyond the works of Karlovini and Von
Unge on periodic analogs of the Reissner-Nordström black holes [8], where the
periodic analogs of the Reissner-Nordstróm they construct have a singularity
surrounding the horizon.

Now, as the norm of the ${\rm S}^{1}$-Killing field of the MKN black holes
diverges at infinity, we conclude that it is impossible to perturb (i.e.
barely deform) such vacuum black holes by adding a small charge, because such
a perturbation would keep the norm of the ${\rm S}^{1}$-Killing field also
bounded away from zero. We summarize this second striking consequence below.

######  Corollary 2.

There are no electrostatic ${\rm S}^{1}$-symmetric perturbations of the static
Myers-Korotkin-Nicolai black holes with non-zero charge (no matter how small).

More in general, we conjecture that ${\rm S}^{1}$-symmetric charged
electrostatic solutions do not exist in configurations where the fixed set of
the ${\rm S}^{1}$-symmetry (the ‘axis’) is compact. In particular, we
conjecture that no such type of solution exists on topologies like those of
the MKN-black holes.

######  Conjecture 3.

The are no ${\rm S}^{1}$-symmetric black holes with compact axis and having
non-zero charge at each of its ends.

This conjecture would follow if one could prove that the norm of the ${\rm
S}^{1}$-Killing field of one such solution must bounded below away from zero
at infinity.

Let us say finally a few words about the arguments behind the proof of Theorem
1. The idea is simple and is based on the observation that the lapse function
of charged solutions as (3) but with an extra symmetry in $z$ (i.e. with
metric components $z$-independent) tends to zero as $\rho$ tends to infinity.
This property will be shown in section 4 after solving explicitly the
electrostatic equations with a ${\rm S}^{1}_{z}\times{\rm
S}^{1}_{\varphi}$-symmetry and inspecting the form of the lapse. Then, the
rough steps to prove Theorem 1 are the following: establish first enough a
priori information on the main fields and the geometry at infinity (i.e. as
$\rho\rightarrow\infty$) to show that an additional asymptotic symmetry forms
along the $z$-direction, and then make contact with the previous observation
to conclude that the lapse tends indeed asymptotically to zero. Although it
looks simple, the implementation is technically involved. One of the main
problems one faces is that the structure of the electrostatic equations isn’t
suitable for extracting a priori estimates at infinity. This is not merely a
technical problem. To circumvent this and to extract the desired estimates we
will make a Bonnor-type transformation of the main fields, to obtain new ones
satisfying equations with a much better structure ($f$-harmonic map
equations). Yet, to extract a priori gradient estimates for the main fields
one needs to work with a particular conformal metric which will be shown to be
metrically complete only after using the lower positive bound of the norm of
the ${\rm S}^{1}$-Killing field. All this will be explained in detail in the
next section.

In section 2 we recall the reduced Einstein-Maxwell equations and write down
the mentioned transformation of the main fields as well as the new equations.
After that, we explain how to use the lower bound of the norm of the ${\rm
S}^{1}$-Killing field to guarantee the metric completeness of the new 2-metric
$\hat{\gamma}$ to be used. The decay estimates or gradient estimates are shown
in section 3. Section 4 investigates electrostatic solutions with a ${\rm
S}^{1}_{z}\times{\rm S}^{1}_{\varphi}$ symmetry. The proof of Theorem 1 is
done in section 5.

##  The reduced electrostatic equations and a Bonnor-type transformation.

In this section, we recall the reduced electrostatic equations (see [22]),
explain the mentioned transformation, and explain why we need to work with a
particular conformal metric. The analysis of the resulting equations and the
deduction of the decay estimates is made in the next section.

The electric field $E$, which we assume has no $\partial_{\varphi}$-component
(in the presence of an axis such component is always zero, see [6]), is
locally the gradient of an electrostatic potential $\bar{v}=\bar{v}(z,\rho)$.
Note that such potential does not exist globally in
$[\rho_{0},\infty)_{\rho}\times{\rm S}^{1}_{\varphi}$ but it does in the
universal cover where we will often work. So, in addition to the metric
exponents $\bar{u}$ and $\bar{k}$ in (3), we have $\bar{v}$. We call
$(\bar{u},\bar{v},\bar{k})$ the electrostatic data. The fields $\bar{u}$ and
$\bar{v}$ satisfy the electrostatic equations,

$\displaystyle\Delta_{f}\bar{u}=e^{-2\bar{u}}|\nabla\bar{v}|^{2},$ (5)
$\displaystyle\Delta_{f}\bar{v}=2\langle\nabla\bar{v},\nabla\bar{u}\rangle,$
(6)

where $f=\ln\rho$ and $\Delta_{f}\psi=\Delta\psi+\langle\nabla
f,\nabla\psi\rangle$. Here, the Laplacian and inner product are taken with
respect to the flat metric $d\rho^{2}+dz^{2}$ but because of the conformal
invariance of the equations (5) and (6) they can be taken too with respect to
the metric $\bar{\gamma}=e^{2\bar{k}}(d\rho^{2}+dz^{2})$ or to any other
metric conformally related to the flat metric. Equation (5) is essentially the
lapse equation and (6) is the Gauss equation of electromagnetism. The exponent
$\bar{k}$ satisfies the integrability equations,

$\displaystyle\bar{k}_{\rho}=\rho((\bar{u}_{\rho}^{2}-\bar{u}_{z}^{2})-(\bar{v}_{\rho}^{2}-\bar{v}_{z}^{2})e^{-2\bar{u}}),$
(7)
$\displaystyle\bar{k}_{z}=2\rho(\bar{u}_{\rho}\bar{u}_{z}-\bar{v}_{\rho}\bar{v}_{z}e^{-2\bar{u}}).$
(8)

that can always be solved by line integration.

Now, the Bonnor-type of transformation that we will use,
$(\bar{u},\bar{v},\bar{k})\rightarrow(u,v,k)$ is given by ([22] pg. 524,
Theorem 34.4),

$\displaystyle u$ $\displaystyle=\bar{u}-\ln\rho,$ (9) $\displaystyle\nabla v$
$\displaystyle=\rho e^{-2\bar{u}}*\nabla\bar{v},$ (10) $\displaystyle 4k$
$\displaystyle=\bar{k}-2\bar{u}+\ln\rho.$ (11)

Here $*$ is the Hodge-star with respect to the flat metric $d\rho^{2}+dz^{2}$.
These variables satisfy,

$\displaystyle\Delta_{f}u=e^{2u}|\nabla v|^{2},$ (12)
$\displaystyle\Delta_{f}v=-2\langle\nabla u,\nabla v\rangle.$ (13)

Again, because of the conformal invariance of the equations, the Laplacian and
the inner product can be taken with respect to the flat metric
$d\rho^{2}+dz^{2}$ or with respect to any metric conformally related to it.
The integrability equations for $k$ are,

$\displaystyle
k_{\rho}=\frac{\rho}{4}((u_{\rho}^{2}-u_{z}^{2})+(v_{\rho}^{2}-v_{z}^{2})e^{2u}),$
(14) $\displaystyle k_{z}=\frac{\rho}{2}(u_{\rho}u_{z}+v_{\rho}v_{z}e^{2u}).$
(15)

Crucially, the equations (12) and (13) are now $f$-harmonic map equations into
the hyperbolic plane with $2$-metric $e^{2u}(dv^{2}+du^{2})$ (for the concept
of $f$-harmonic map see [3]). As important as this, the Ricci tensor $Ric$ of
the $2$-metric $\gamma=e^{2k}(d\rho^{2}+dz^{2})$ is given by,

$Ric=\frac{\nabla\nabla\rho}{\rho}+\frac{1}{2}(\nabla u\nabla u+e^{2u}\nabla
v\nabla v).$ (16)

Thus the Bakry-Émery Ricci tensor $Ric_{f}=Ric-\nabla\nabla f$ (called
$f$-Ricci tensor from now on) takes the form,

$Ric_{f}=\frac{1}{2}(\nabla u\nabla u+e^{2u}\nabla v\nabla v)+\nabla f\nabla
f.$ (17)

The $f$-harmonic map equations (12)-(13) for $(u,v)$ and the expression (17)
for the $f$-Ricci tensor of the metric $\gamma=e^{2k}(d\rho^{2}+dz^{2})$ are
very structured and well suited to obtain decay estimates. In fact, (12)-(13))
are the Ernst equations for stationary and axisymmetric solutions of the
Einstein vacuum equations.

We will explain below the nature of such estimates and how to obtain them but
there is a potential problem to notice beforehand and is that the metric
$\gamma$ doesn’t have to be necessarily metrically complete as
$\rho\rightarrow\infty$. So far, the only information available is that the
spatial part of the metric (3),
$g=e^{-2\bar{u}}(e^{2\bar{k}}(d\rho^{2}+dz^{2})+\rho^{2}d\varphi^{2})$, is
complete. This implies that the $2$-metric
$\xi=e^{-2\bar{u}+2\bar{k}}(d\rho^{2}+dz^{2})$ on
$[\rho_{0},\infty)_{\rho}\times{\rm S}^{1}_{z}$ is metrically complete but not
that the metric $\gamma$ is. Even using the lower bound on the norm of the
axial Killing field, $|\partial_{\varphi}|=\rho e^{-\bar{u}}\geq c>0$, one
does not get the necessary metric completeness. But with the bound for
$|\partial_{\varphi}|$ one does get the completeness of the metric
$\hat{\gamma}=e^{8k}(d\rho^{2}+dz^{2})$ because,

$\displaystyle\hat{\gamma}$
$\displaystyle=e^{8k}(d\rho^{2}+dz^{2})=e^{-2\bar{u}+2k-2\bar{u}+2\ln\rho}(d\rho^{2}+dz^{2})=$
(18)
$\displaystyle=|\partial_{\varphi}|^{2}e^{-2\bar{u}+2k}(d\rho^{2}+dz^{2})\geq
c^{2}\xi$ (19)

Remarkably, we will show that the $f$-Ricci tensor of the metric
$\hat{\gamma}$ satisfies,

$\hat{Ric}_{f}=2(\nabla u\nabla u+e^{2u}\nabla v\nabla v)+\nabla f\nabla f$
(20)

This, together with (12)-(13), form a suitable system from which estimates can
be obtained. Indeed, the structure of these equations is such that one can
prove, by what is by now standard technique, the decay estimates,

$|\nabla u|^{2}+e^{2u}|\nabla v|^{2}\leq\frac{c}{\hat{s}^{2}},\quad|\nabla
f|^{2}\leq\frac{c}{\hat{s}^{2}}$ (21)

where $c>0$ is a constant independent on the solutions, and where $\hat{s}$ is
the metric $\hat{\gamma}$-distance function to the boundary $\partial
S=\\{\rho_{0}\\}\times{\rm S}^{1}_{z}$ of the $2$-manifold
$S:=[\rho_{0},\infty)_{\rho}\times{\rm S}^{1}_{z}$,

$\hat{s}(p)={\rm dist}_{\hat{\gamma}}(p,\partial S)$ (22)

The estimates (21) on $u$, $v$ and $\rho$ imply directly the following
important estimate for original variables $\bar{u}$ and $\bar{v}$ that we will
use in the proof of the main theorem,

$|\nabla\bar{u}|^{2}+e^{-2\bar{u}}|\nabla\bar{v}|^{2}\leq\frac{2c}{\hat{s}^{2}}$
(23)

Observe that the Gaussian curvature of $\hat{\gamma}$ is
$\hat{\kappa}=2(|\nabla u|^{2}+e^{2u}|\nabla v|^{2})$ which is non-negative.
Therefore the curvature is non-negative and has scale-invariant quadratic
decay. With all this type of geometry at hand, a standard asymptotic analysis
will permit us to study the geometry at infinity and therefore of the fields
themselves. We do that in the next section.

##  Proof of the decay estimates

From now one we will make $S:=[\rho_{0},\infty)_{\rho}\times{\rm S}^{1}_{z}$.

######  Proposition 4.

Let $\gamma=e^{2k}(d\rho^{2}+dz^{2})$ and
$\hat{\gamma}=e^{2\hat{k}}(d\rho^{2}+dz^{2})$ with $\hat{k}=\alpha k$, and
$\alpha$ a constant. Then,

$(\hat{\kappa}\hat{\gamma}-\frac{\hat{\nabla}\hat{\nabla}\rho}{\rho})=\alpha(\kappa\gamma-\frac{\nabla\nabla\rho}{\rho}).$
(24)

###### Proof.

We will prove that the components of $\kappa\gamma-(\nabla\nabla\rho)/\rho$ in
the base $(\partial_{\rho},\partial_{z})$ are linear in $k$. The proposition
then follows after replacing $k$ by $\hat{k}=\alpha k$. By the standard
formula, we have,

$\kappa\gamma=-(\partial_{\rho}^{2}k+\partial_{z}^{2}k)(d\rho^{2}+dz^{2}),$
(25)

which is linear in $k$. We prove now that
$(\nabla\nabla\rho)(\partial_{z},\partial_{z})$ and
$(\nabla\nabla\rho)(\partial_{\rho},\partial_{z})$ are linear in $k$ too (note
that as $\rho$ is harmonic, we have
$(\nabla\nabla\rho)(\partial_{\rho},\partial_{\rho})=-(\nabla\nabla\rho)(\partial_{z},\partial_{z})$).
Computing we have,

$(\nabla\nabla\rho)(\partial_{z},\partial_{z})=-\Gamma_{zz}^{\rho}=\partial_{\rho}k,$
(26)

and,

$(\nabla\nabla\rho)(\partial_{\rho},\partial_{z})=-\Gamma_{\rho
z}^{\rho}=-\partial_{z}k,$ (27)

as wished. ∎

######  Corollary 5.

Under the notation of Proposition 4. If,

$\kappa\gamma-\frac{\nabla\nabla\rho}{\rho}=\frac{1}{2}(\nabla u\nabla
u+e^{2u}\nabla v\nabla v),$ (28)

then,

$\hat{\kappa}\hat{\gamma}-\frac{\hat{\nabla}\hat{\nabla}\rho}{\rho}=\frac{\alpha}{2}(\nabla
u\nabla u+e^{2u}\nabla v\nabla v).$ (29)

In particular we get that

$\hat{Ric}_{f}:=\hat{\kappa}\hat{\gamma}-\hat{\nabla}\hat{\nabla}f=\frac{\alpha}{2}(\nabla
u\nabla u+e^{2u}\nabla v\nabla v)+\nabla f\nabla f,$ (30)

is non-negative as long as $\alpha\geq 0$.

In the above formulae observe of course that $\nabla h=dh$ for any function,
hence no need to place a hat on it. The following corollary will play a role
in the proof of the main theorem.

######  Corollary 6.

For any $\alpha\geq 0$, the 3-metric $\hat{\gamma}+\rho^{2}d\varphi^{2}$ on
$S\times{\rm S}^{1}_{\varphi}$ has non-negative Ricci curvature.

###### Proof.

Consider any of the totally geodesic hypersurfaces $S\times\\{\varphi\\}$ and
let $e_{1},e_{2}$ be an orthonormal base of tangent vectors at a point $p$ and
$e_{3}$ a perpendicular unit-vector at the same point. By the Gauss-Codazzy we
have,

$Ric(e_{i},e_{3})=0,\ \forall i=1,2,$ (31)

Also, a standard formula for warped product metrics gives,

$Ric(e_{i},e_{j})=(\hat{\kappa}\hat{\gamma}-\frac{\nabla\nabla\rho}{\rho})(e_{i},e_{j}),\
\forall i,j=1,2.$ (32)

Therefore $Ric(e_{i},e_{i})\geq 0$ by Corollary 5. On the other hand, we have,

$2\hat{\kappa}=R-2Ric(e_{3},e_{3})=2\hat{\kappa}-Ric(e_{3},e_{3}),$ (33)

where to obtain the second equality we used (32) and the fact that $\rho$ is
harmonic. We get thus $Ric(e_{3},e_{3})=0$. With these metric components, we
have $Ric\geq 0$ as wished. ∎

The following inequalities are central to obtaining the decay estimates.

######  Proposition 7.

Let $(\gamma;u,v)$ satisfy (12)-(13) and (17), then,

$\displaystyle\Delta_{f}|\nabla f|^{2}\geq 2|\nabla f|^{4},$ (34)
$\displaystyle\Delta_{f}\kappa\geq 4\kappa^{2}.$ (35)

###### Proof.

Recall the Bochner formula,

$\frac{1}{2}\Delta|\nabla\chi|^{2}=|\nabla\nabla\chi|^{2}+\langle\nabla\chi,\nabla(\Delta_{f}\chi)\rangle+Ric_{f}(\nabla\chi,\nabla\chi),$
(36)

which is valid for any function $\chi$. Make now the choice $\chi=f$. Then, as
$\rho$ is harmonic we have $\Delta_{f}f=0$ and recalling (30) we have
$Ric_{f}(\nabla f,\nabla f)\geq|\nabla f|^{4}$. Altogether we obtain (34).

We prove now (35). Now recall the general Bochner formula for a $f$-harmonic
map $h:S\rightarrow X$, [3],

$\displaystyle\frac{1}{2}\Delta_{f}e=$ $\displaystyle|\nabla
dh|^{2}+\sum_{i}\langle dh(Ric_{f}(e_{i})),dh(e_{i})\rangle-$ (37)
$\displaystyle\sum_{i,j}Rm^{X}\langle
dh(e_{i}),dh(e_{j}),dh(e_{i}),dh(e_{j})\rangle,$ (38)

into a target space $X$, (for the definition see for instance [3]), where $e$
is the energy of the map, $Rm^{X}$ is the Riemann curvature of $X$, and
$\\{e_{i}\\}$ is an orthonormal basis of the tangent space of $S$ at the point
where it is being evaluated. We will use this formula for the map
$h=(u,v):S\rightarrow X:=\mathbb{H}^{2}$, from $S$ into the hyperbolic plane
$\mathbb{H}^{2}$ with metric $du^{2}+e^{2u}dv^{2}$. The energy is $e=4\kappa$.
This map is $f=\ln\rho$ harmonic due to equations (12)-(13). As
$\mathbb{H}^{2}$ has negative Riemann curvature, the last term on the r.h.s is
non-negative. The first term in the r.h.s is non-negative too and the second
is greater or equal than $e^{2}/2=8\kappa^{2}$ as the following computation
shows.

Indeed, we have,

$dh(e_{i})=du(e_{i})\partial_{u}+dv(e_{i})\partial_{v},$ (39)

and,

$Ric_{f}(e_{i})=\sum_{j}\frac{1}{2}(du(e_{i})du(e_{j})e_{j}+e^{2u}dv(e_{i})dv(e_{j})e_{j})+df(e_{i})df(e_{j})e_{j}.$
(40)

Thus, using these equations in the second term in the r.h.s of the (37) we
deduce,

$\displaystyle\sum_{i}\langle dh$
$\displaystyle(Ric_{f}(e_{i})),dh(e_{i})\rangle=$ (41) $\displaystyle=|\nabla
u|^{4}+\langle\nabla u,\nabla v\rangle e^{2u}+|\nabla
v|^{4}e^{4u}+\langle\nabla f,\nabla u\rangle^{2}+\langle\nabla f,\nabla
v\rangle^{2}e^{2u}$ (42) $\displaystyle\geq\frac{1}{2}(|\nabla
u|^{2}+e^{2u}|\nabla v|^{2})^{2}=\frac{e^{2}}{2},$ (43)

as wished. ∎

Performing the same computation but with $(\hat{\gamma},u,v)$ we reach the
following.

######  Corollary 8.

For any $\alpha\geq 0$ we have,

$\displaystyle\hat{\Delta}_{f}\hat{\kappa}\geq 4\hat{\kappa}^{2},$ (44)
$\displaystyle\hat{\Delta}_{f}|\nabla f|^{2}\geq 2|\nabla f|^{4},$ (45)

where the norm $|\nabla f|$ is with respect to $\hat{\gamma}$.

The next proposition states finally the decay estimates.

######  Proposition 9.

There is a universal constant $c>0$ such that, for all $\alpha\geq 0$,

$\hat{\kappa}\leq\frac{c^{2}}{\hat{s}^{2}},\qquad\bigg{|}\frac{\nabla\rho}{\rho}\bigg{|}^{2}\leq\frac{c^{2}}{\hat{s}^{2}},$
(46)

###### Proof.

The proof follows from Lemma 3.2 in [18]. ∎

These point-wise estimates imply higher order estimates for the derivatives of
$\hat{\kappa}$ that are useful in the asymptotic analysis as we will do later.
The estimates are,

$|\nabla^{j}\hat{\kappa}|\leq\frac{c_{j}^{2}}{\hat{s}^{2+j}}$ (47)

To obtain them, note that because of the scale invariance of (47) it is enough
to prove them at points $p$ with $\hat{s}(p)=1$. But then note that
$(u^{\prime},v^{\prime},\rho^{\prime})=(u-u(p),e^{-u(p)}v,\rho/\rho(p))$ is a
new solution to (12)-(13) leaving (20) invariant and having
$(u^{\prime}(p),v^{\prime}(0),\rho^{\prime}(0))=(0,0,1)$. From (46) one
obtains point-wise estimates for $(u^{\prime},v^{\prime},\rho^{\prime})$ on
the ball $B_{\hat{\gamma}}(p,1/2)$ and then higher-order estimates on the
universal cover of the ball (where the injectivity radius at $p$ is
controlled) by standard elliptic theory on (12)-(13), which imply the higher-
order estimates for $\hat{\kappa}$.

##  ${\rm S}^{1}_{z}\times{\rm S}^{1}_{\varphi}$-symmetric electrostatic
charged data

Denote $\partial=(\partial_{z},\partial_{\rho})$. In this section we study
charged data $(\bar{u},\bar{v})$ for which $\bar{u}$ and $\partial\bar{v}$ are
$z$-independent. This implies that not only the metric components are
$z$-independent but also the electric field is. We are going to show that if
one data extends to $\rho=+\infty$ then $\bar{u}$ tends to $-\infty$ and
therefore $N=e^{\bar{u}}$ tends to $0$.

We show first that $\partial_{z}\bar{v}$ must be constant. To see this we note
that if $\partial_{\rho}\bar{v}$ is $z$-independent then
$\partial_{z}\partial_{\rho}\bar{v}=0$. Interchanging derivatives and
integrating in $\rho$ between $\rho_{0}$ and $\rho$, leaving $z$ fixed, we
obtain $\partial_{z}\bar{v}(z,\rho)=\partial_{z}\bar{v}(z,\rho_{0})$. But as
$\partial_{z}\bar{v}$ is $z$-independent we conclude that
$\partial_{z}\bar{v}$ is constant, as claimed.

Now, if $\partial_{z}\bar{v}$ is constant, then (6) becomes $(\rho
e^{-\bar{u}}\bar{v}^{\prime})^{\prime}=0$ where ${}^{\prime}=\partial_{\rho}$,
or,

$\rho e^{-2\bar{u}}\bar{v}^{\prime}=\bar{Q},$ (48)

where $Q=2\pi\bar{Q}\neq 0$ is the electric charge. Use (48) and
$\partial_{z}\bar{v}=c$ into (6) to get,

$\bar{u}^{\prime\prime}+\frac{1}{\rho}\bar{u}^{\prime}=(c^{2}e^{-2\bar{u}}+\bar{Q}^{2}\frac{e^{2\bar{u}}}{\rho^{2}})$
(49)

######  Proposition 10.

If $c\neq 0$, then no solution to (49) extends to $\rho=\infty$.

###### Proof.

Arguing by contradiction assume that $u(\rho)$ is a solution defined on
$[\rho_{0},\infty)$. We are going to reach an impossibility. We begin
performing the change of variables

$\bar{u}=\hat{u}+\frac{1}{2}\ln\rho+\frac{1}{4}\ln\frac{c^{2}}{Q^{2}},$ (50)

making equation (49) into

$\hat{u}^{\prime\prime}+\frac{1}{\rho}\hat{u}^{\prime}=|\bar{Q}c|\frac{\cosh
2\hat{u}}{\rho}.$ (51)

Then,

$(\rho\hat{u}^{\prime})^{\prime}=|\bar{Q}c|\cosh 2\hat{u}\geq|Qc|=:c_{1}>0.$
(52)

We deduce therefore $\rho\hat{u}^{\prime}\geq c_{1}\rho+c_{2}$. Thus
$\rho\hat{u}^{\prime}\geq c_{1}\rho/2$ if $\rho\geq\rho_{1}$ for certain
$\rho_{1}$ and hence

$\hat{u}^{\prime}\geq\frac{c_{1}}{2}>0,$ (53)

for all $\rho\geq\rho_{1}$. In particular $\hat{u}\uparrow+\infty$ as
$\rho\rightarrow\infty$. Now assume $\rho\geq\rho_{2}=\max\\{\rho_{1},1\\}$.
Then $1\geq 1/\rho$ and so we get from (51),

$(\rho\hat{u}^{\prime})^{\prime}\geq|\bar{Q}c|\frac{\cosh 2\hat{u}}{\rho}.$
(54)

Then, making the change of variables $x=\ln\rho$, the previous inequality
reads

$\ddot{\hat{u}}\geq|\bar{Q}c|\cosh 2\hat{u}$ (55)

where $\dot{}=d/dx$. We now work assuming $x\geq x_{2}=\ln\rho_{2}$.
Multiplying (55) by $\dot{\hat{u}}$ which is positive by (53), integrating in
$x$, and taking into account that $\hat{u}\rightarrow+\infty$ as
$\rho\rightarrow\infty$ we deduce after some manipulations

$\dot{\hat{u}}\geq c_{3}e^{2\hat{u}},$ (56)

for some $c_{3}>0$ and for $x\geq x_{3}$ for some $x_{3}\geq x_{2}$.
Multiplying by $e^{-2\hat{u}}$ and integrating in $x$ we deduce

$-e^{-\hat{u}}\geq c_{3}(x-x_{3})+e^{2\hat{u}(x_{3})}.$ (57)

But as $x\rightarrow+\infty$ the left hand is negative whereas the right-hand
side tends to $+\infty$. We conclude that the data cannot extend to infinity.
∎

We now assume that $\bar{u}$ is a solution to (49) defined on
$[\rho_{0},\infty)$. We will show that $N=e^{\bar{u}}\rightarrow 0$ as
$\rho\rightarrow\infty$. By the previous proposition we have $c=0$ and thus
$\bar{u}$ satisfies

$\bar{u}^{\prime\prime}+\frac{1}{\rho}\bar{u}^{\prime}=\bar{Q}^{2}\frac{e^{2\bar{u}}}{\rho^{2}}.$
(58)

Then, a simple manipulation after multiplication by $\rho^{2}\bar{u}^{\prime}$
gives

$\big{(}(\rho\bar{u}^{\prime})^{2}\big{)}^{\prime}=\big{(}\bar{Q}^{2}e^{2\bar{u}}\big{)}^{\prime},$
(59)

from which we deduce

$(\rho\bar{u}^{\prime})^{2}=\bar{Q}^{2}e^{2\bar{u}}\pm a^{2},$ (60)

with $a\geq 0$ a constant. Letting $\chi=e^{-\bar{u}}>0$ and defining the new
variable $x=\ln\rho$ we get easily (we use $\partial_{x}\chi=\dot{\chi}$)

$\dot{\chi}^{2}=\bar{Q}^{2}\pm a^{2}\chi^{2},$ (61)

to obtain, after $x$-derivation and cancelation of $\dot{\chi}$ on both sides

$\ddot{\chi}=\pm a^{2}\chi,$ (62)

($\dot{\chi}=0$ or $\chi=\pm\bar{Q}/a$ is not a solution of the original
problem (58) and was introduced during the deduction of (60)). As a
consequence we obtain the following type of solutions for $\chi$ (we take only
those that are positive because $\chi>0$):

  1. 1.

$a=0$,

$\chi=b+|\bar{Q}|\ln x,$ (63)

  2. 2.

$a>0$, case $+a^{2}$ in (62),

$\chi=\frac{|\bar{Q}|}{a}\sinh(\pm ax+b).$ (64)

  3. 3.

$a>0$, case $-a^{2}$ in (62),

$\chi=\frac{|\bar{Q}|}{a}\sin(\pm ax+b).$ (65)

The three expressions have to be taken only in the $x$-domain where they are
positive. The only solutions extending to $x=+\infty$ tend to $+\infty$ as
$x\rightarrow+\infty$. Therefore $N=\chi^{-1}$ tends to $0$ as
$x\rightarrow\infty$.

##  The proof of the main Theorem

From now on we work on $S=[\rho_{0},\infty)_{\rho}\times{\rm S}^{1}_{z}$ with
the metric $\hat{\gamma}$ and $\alpha=4$. All estimates will be done in this
Riemannian manifold. As earlier we let $\hat{s}(p)=d_{\hat{\gamma}}(p,\partial
S)$.

We know by (46) that $|\nabla\ln\rho|\leq c/\hat{s}$. We move first to prove
that there is $b>0$ and a divergent sequence $p_{i}$ such that the opposite
inequality holds,

$\bigg{|}\frac{\nabla\rho}{\rho}\bigg{|}\bigg{|}_{p_{i}}\geq\frac{b}{\hat{s}(p_{i})}.$
(66)

This will turn out to be a fundamental estimate for all that follows. Let us
state this in the following proposition.

######  Proposition 11.

There is $b>0$ and a divergent sequence $p_{i}\in S$ such that,

$\bigg{|}\frac{\nabla\rho}{\rho}\bigg{|}\bigg{|}_{p_{i}}\geq\frac{b}{\hat{s}(p_{i})}.$
(67)

###### Proof.

The idea is to use a technique developed in [15] for harmonic functions on
3-manifolds of non-negative Ricci curvature whose level sets are surfaces of
genus greater than zero. In this instance the harmonic function will be
$U=\ln\rho$ on the 3-manifold $S\times{\rm S}^{1}_{\varphi}$ endowed with the
metric $\hat{h}=\hat{\gamma}+\rho^{2}d\varphi^{2}$ that we know has non-
negative Ricci curvature. Of course, the level sets of $U$ are $2$-tori.

Recall that $S=[\rho_{0},\infty)\times{\rm S}^{1}_{z}$ and that
$\hat{\gamma}=e^{8k}(d\rho^{2}+dz^{2})$. Then, we can write the
$\hat{\gamma}$-length-element of any circle $\\{\rho\\}\times{\rm S}^{1}_{z}$
as $d\ell=e^{4k}dz$. Therefore, taking into account that
$|\nabla\rho|=e^{-4k}$ we deduce that $dz=|\nabla\rho|d\ell$, is constant. We
will use this below.

Let $U_{x}^{-1}$ denote the level set of $U$: $U^{-1}_{x}=\\{(p,\varphi)\in
S\times{\rm S}^{1}_{\varphi}:U(p)=x\\}$. We consider the function of $x$,

$G(x)=\int_{U^{-1}_{x}}|\nabla U|^{2}dA,$ (68)

where $dA=\rho d\ell d\varphi$ is the area element of the level set
$U^{-1}_{x}$. As $dz=|\nabla\rho|d\ell$, then we can write,

$G(x)=\int_{U^{-1}_{x}}|\nabla U|^{2}dA=2\pi\int_{\\{\rho=e^{x}\\}\times{\rm
S}^{1}_{z}}|\nabla U|dz.$ (69)

It turns out, see Proposition 4.2.14 in [15], that
$(G^{\prime}/G)^{\prime}\geq 0$ from which it is deduced that
$G^{\prime}/G\geq(G^{\prime}/G)(x_{0})=:C$. Integrating we reach,

$G(x)\geq\frac{G(x_{0})}{e^{Cx_{0}}}e^{Cx}.$ (70)

If $C>0$ then $G(x)\geq G(x_{0})>0$ for all $x\geq x_{0}$. But, as $dz$ is
constant, then, for every $x$ there is a point $p\in\\{\rho=e^{x}\\}\times{\rm
S}^{1}_{z}$ such that $G(x)=(2\pi)^{2}(|\nabla\rho|/\rho)(p)\geq G(x_{0})>0$.
Therefore $|\nabla\rho|/\rho$ does not go to zero at infinity which is
impossible. Thus $C<0$. Then, using (70) we have,

$\int_{\\{\rho=e^{x}\\}\times{\rm
S}^{1}_{z}}\rho^{-C}\bigg{|}\frac{\nabla\rho}{\rho}\bigg{|}dz\geq d_{1},$ (71)

for some $d_{1}>0$. Therefore, for every $x$ there is
$p_{x}\in\\{\rho=e^{x}\\}\times{\rm S}^{1}_{z}$ such that,

$\big{(}\rho^{-C}\bigg{|}\frac{\nabla\rho}{\rho}\bigg{|}\big{)}\bigg{|}_{p_{x}}\geq
d_{2},$ (72)

for $d_{2}=d_{1}/2\pi>0$. Now, assume that for every $\epsilon=-1/(2C)$ there
is $\hat{s}_{\epsilon}$ such that,

$\bigg{|}\frac{\nabla\rho}{\rho}\bigg{|}\bigg{|}_{q}\leq\frac{\epsilon}{\hat{s}(q)}$
(73)

for all $q$ such that $\hat{s}(q)\geq\hat{s}_{\epsilon}$. Then, a simple
integration gives $\rho(q)\leq d_{3}\hat{s}^{\epsilon}(q)$ and for some
$d_{3}>0$. Then,

$d_{4}\hat{s}(q)^{-C\epsilon-1}\geq\big{(}\rho^{-C}\bigg{|}\frac{\nabla\rho}{\rho}\bigg{|}\big{)}\bigg{|}_{q}$
(74)

for some $d_{4}$ and for all $q$ such that
$\hat{s}(q)\geq\hat{s}_{\epsilon}(q)$. If $q=p_{x}$ then using (72) we get,

$\hat{s}(p_{x})^{-1/2}\geq d_{5}>0.$ (75)

for some $d_{5}>0$. But then the left hand side tends to zero as
$x\rightarrow\infty$ which is impossible. We conclude then that there is a
divergent sequence of points $p_{i}$ such that,

$\bigg{|}\frac{\nabla\rho}{\rho}\bigg{|}\bigg{|}_{p_{i}}\geq\frac{\epsilon}{\hat{s}(p_{i})},$
(76)

where $\epsilon=\frac{-1}{2C}$. It follows (67). ∎

Let us explore now some consequences of what we just proved. To start lets
prove that the area growth on $(S;\hat{\gamma})$ is sub-quadratic. Let
$B(r)=\\{p\in S:d_{\hat{\gamma}}(p,\partial S)<r\\}$ and let $A(B(r))$ be the
$\hat{\gamma}$-area of $B(r)$. As $\hat{\kappa}\geq 0$, then the Bishop-Gromov
monotonicity implies that the quotient $A(B(r))/r^{2}$ is monotonically non-
increasing as $r$ increases. Thus $A(B(r))/r^{2}\rightarrow\mu\geq 0$ as
$r\rightarrow\infty$. We prove now that indeed $\mu=0$, namely, that the area
growth is sub-quadratic.

######  Proposition 12.

The manifold $(S;\hat{\gamma})$ has sub-quadratic area growth, that is
$\mu=0$.

###### Proof.

Assume that $\mu>0$. Then by standard arguments using Bishop-Gromov’s
monotonicity and the scale invariant bound (47) on the Gaussian curvature
$\hat{\kappa}$ and its derivatives, imply that the manifold $(S;\hat{\gamma})$
has a flat cone at infinity. Let us recall this. Consider any divergent
sequence $p_{i}$ and let $r_{i}=\hat{s}(p_{i})$. Given $0<c<d$ let
$\mathcal{A}_{r_{i}}(c,d)=\\{p:c\leq\hat{s}_{r_{i}}(p)\leq d\\}$. Then, there
is a divergent sequence $k_{i}\rightarrow\infty$ such that the pointed annuli
$(\mathcal{A}_{r_{i}}(2^{-k_{i}},2^{k_{i}}),p_{i})$ endowed with the scaled
metric $\hat{\gamma}_{r_{i}}=\hat{\gamma}/r_{i}^{2}$ converge in $C^{\infty}$
and in the pointed sense (see [13]) to the flat cone
$((0,\infty)_{r}\times{\rm
S}^{1}_{\phi};\hat{\gamma}_{\infty}=dr^{2}+(\mu^{2}r^{2}/\pi^{2})d\phi^{2},p_{\infty}=(r,0)$).
Choose now $p_{i}$ to be the sequence in Proposition 11. Define the harmonic
function $\hat{\rho}_{i}=\rho/\rho(p_{i})$. Then, as $\hat{\rho}(p_{i})=1$ and
$|\nabla\hat{\rho}_{i}|_{r_{i}}(p_{i})\geq b$ we can use the scale invariant
bound (46) and standard elliptic estimates to conclude that $\hat{\rho}_{i}$
sub-converges to a limit non-negative and non-constant function
$\hat{\rho}_{\infty}$ on the limit cone. But as $\hat{\kappa}_{\infty}=0$ we
conclude from (20), (that passes to the limit), that
$\hat{\nabla}\hat{\nabla}\hat{\rho}_{\infty}=0$. Take then any infinite
geodesic $\xi(t)$ on the limit cone such that $\xi(0)=p_{\infty}$ and such
that,

$\frac{d}{dt}(\hat{\rho}_{\infty}(\xi(t)))\big{|}_{t=0}=\langle\hat{\nabla}\hat{\rho}_{\infty},\xi^{\prime}(0)\rangle<0,$
(77)

(to see that $\xi(t)$ always exists note that any inextensible geodesic except
the radial ones diverge to infinity in both directions). But then, as
$\hat{\nabla}\hat{\nabla}\hat{\rho}_{\infty}=0$ we deduce that
$d^{2}\hat{\rho}_{\infty}(\xi(t))/dt^{2}=0$ and therefore that
$\hat{\rho}(\xi(t))=(\frac{d}{dt}(\hat{\rho}_{\infty}(\xi(t)))\big{|}_{t=0})t+\hat{\rho}(p_{\infty})$.
But because $\frac{d}{dt}(\hat{\rho}_{\infty}(\xi(t)))\big{|}_{t=0}<0$ we have
$\hat{\rho}_{\infty}(t)<0$ for some $t>0$ which is not possible. ∎

We move now to prove Theorem 1.

###### Proof of Theorem 1.

For $x\geq\rho_{0}$ let $\rho^{-1}_{x}$ be the level set of $\rho$,
$\rho^{-1}_{x}=\\{\rho=x\\}\times{\rm S}^{1}_{z}$. We will show that there is
a sequence of level sets $\rho^{-1}_{x_{i}}$ with $x_{i}\uparrow+\infty$,
($x_{i}$ is strictly increasing), on which $\bar{u}$ tends to $-\infty$, that
is,

$\max\\{\bar{u}(p):p\in\rho^{-1}_{x_{i}}\\}\rightarrow-\infty.$ (78)

Then, as $\hat{\Delta}\bar{u}\geq 0$, it follows by the maximum principle
that,

$\displaystyle\max\\{\bar{u}(p):x_{i-1}$ $\displaystyle\leq\rho(p)\leq
x_{i+1}\\}\leq$ (79)
$\displaystyle\leq\max\\{\max\\{\bar{u}(p):p\in\rho^{-1}_{x_{i-1}}\\},\max\\{\bar{u}(p):p\in\rho^{-1}_{x_{i}}\\}\\}\rightarrow-\infty$
(80)

It follows that $N=e^{\bar{u}}$ tends uniformly to zero at infinity.

Let $p_{i}$ be the sequence of Proposition 11. Again, let
$r_{i}=\hat{s}(p_{i})$ and let $\hat{\gamma}_{r_{i}}=\hat{\gamma}/r^{2}_{i}$.
Scaled quantities constructed out of the metric $\hat{\gamma}_{r_{i}}$ are
denoted with a subindex $r_{i}$. For instance the Gaussian curvature of
$\hat{\gamma}_{r_{i}}$ is $\hat{\kappa}_{r_{i}}:=\hat{\kappa}/r_{i}^{2}$ where
$\hat{\kappa}$ is the Gaussian curvature of $\hat{\gamma}$. We will scale too
the harmonic function $\rho$ as $\rho_{r_{i}}:=\rho/\rho(p_{i})$ and note that
we have,

$\rho_{r_{i}}(p_{i})=1,$ (81)

also,

$\left|\frac{\nabla\rho_{r_{i}}}{\rho_{r_{i}}}\right|^{2}_{r_{i}}\bigg{|}_{p_{i}}\geq
b>0,$ (82)

and

$\left|\frac{\nabla\rho_{r_{i}}}{\rho_{r_{i}}}\right|^{2}_{r_{i}}\bigg{|}_{p}\leq\frac{c^{2}}{\hat{s}_{r_{i}}^{2}(p)},$
(83)

valid on all $p$.

For given integer $k\geq 2$ the sequence of Riemannian annuli
$(\mathcal{A}_{\hat{r}_{i}}(1/2,2^{k}),\hat{\gamma}_{r_{i}})$ collapses in
volume with bounded curvature by Proposition 12. It follows from standard
theory (see [4]) and standard elliptic estimates for the system (12)-(13)-(20)
that there is a sequence of integers $k_{i}\rightarrow\infty$, a sequence of
neighborhoods $\mathcal{U}_{i}$ of $\mathcal{A}_{\hat{r}_{i}}(1/2,2^{k})$ and
of finite coverings (unwrapping)
$\pi_{i}:\tilde{\mathcal{U}}_{i}\rightarrow\mathcal{U}_{i}$, such that
$(\tilde{\mathcal{U}}_{i};\pi^{*}_{i}\hat{\gamma}_{r_{i}})$ sub-converges in
$C^{\infty}$ to a ${\rm S}^{1}$-symmetric surface
$(\tilde{\mathcal{U}},\tilde{\gamma})$ (half an infinite cylinder) of non-
negative Gaussian curvature, with one end and one boundary component. Let
$\tilde{s}$ be the limit of the lifted distance functions $\hat{s}_{r_{i}}$.
We can write $\tilde{\mathcal{U}}=[1/2,\infty)_{\tilde{s}}\times{\rm S}^{1}$
because the distance function $\tilde{s}$ is invariant under the ${\rm
S}^{1}$-action. Observe that because of (81), (82) and (83) the functions
$\rho_{r_{i}}\circ\pi_{i}$ sub-converge to a ${\rm S}^{1}$-symmetric non-
constant ${\rm S}^{1}$-invariant positive harmonic function $\tilde{\rho}$ on
$\tilde{\mathcal{U}}$. We will show now that $\tilde{\rho}$ tends to infinity
at infinity. Following a similar notation as earlier, for $x\geq 1/2$ let
$\tilde{s}^{-1}_{x}:=\\{\tilde{s}=x\\}\times{\rm S}^{1}$, that is, the level
set of $\tilde{s}$ at the value $x$. Now, observe that because $\tilde{\rho}$
is a $\tilde{\gamma}$-harmonic non-constant and ${\rm S}^{1}$-symmetric
function, and because of (82) we have,

$\tilde{\ell}(x)\partial_{x}\tilde{\rho}(x)=b_{1}\neq 0,$ (84)

for all $x\geq 1/2$, where $\tilde{\ell}(x)$ is the $\tilde{\gamma}$-length of
$\tilde{s}^{-1}_{x}$. But because the Gaussian curvature $\tilde{\kappa}$ of
$\tilde{\gamma}$ is non-negative it is easy to see that $\tilde{\ell}(x)$
grows at most linearly, namely, $\tilde{\ell}(x)\leq b_{2}x$ with $b_{2}>0$.
This information together with (84) implies that $\tilde{\rho}$ tends to
infinity as $\tilde{s}$ tends to infinity. The importance of this is that we
can use $\tilde{\rho}$ as a harmonic coordinate extending to infinity. We will
use this fact at the end of the proof.

The task now is to analyze the potentials $\bar{u}$ and $\bar{v}$ over
$\mathcal{U}_{i}$ and by doing so prove that a subsequence of $\bar{u}$ seen
over $\mathcal{U}_{i}$ tends to $-\infty$, and therefore that $N$ tends to $0$
by the argument at the beginning of the proof. We will prove this by
contradiction, assuming that the maximum of $\bar{u}$ over the
$\mathcal{U}_{i}$ does not tend to $-\infty$ on any subsequence. We will be
able to pass to a limit and show that this would contradict Proposition 10
which says that charged ${\rm S}^{1}$-symmetric potentials do not extend to
infinity.

Assume that the maximum of $\bar{u}$ over the $\mathcal{U}_{i}$ does not tend
to $-\infty$ on any subsequence. To start observe that, before passing to any
limit, we have,

$\int_{\rho^{-1}_{x}}\nabla_{n}\rho d\ell=c_{1}>0,$ (85)

for any $x$, where $n$ here is the outgoing normal to the loop
$\rho^{-1}_{x}$. Recall that $\rho_{i}=\rho(p_{i})$. Now, from the previous
identity, we deduce that there is $q_{i}\in\rho^{-1}_{x=\rho_{i}}$ such that,

$\rho_{i}\ell_{i}\left|\frac{\nabla_{n}\rho}{\rho}\right|\bigg{|}_{q_{i}}=c_{1}$
(86)

Scaling $n_{r_{i}}=r_{i}n$ and $\ell_{r_{i}}=\ell_{i}/_{r_{i}}$ preserves the
previous identity, namely,

$\rho_{i}\ell_{r_{i}}\left|\frac{\nabla_{n_{r_{i}}}\rho}{\rho}\right|\bigg{|}_{q_{i}}=c_{1}$
(87)

As,

$\left|\frac{\nabla_{n_{r_{i}}}\rho}{\rho}\right|\bigg{|}_{q_{i}}\rightarrow
b_{2}>0$ (88)

we conclude that,

$\rho_{i}\ell_{r_{i}}\rightarrow c_{2}>0.$ (89)

We will use this information below.

Now, the charge $Q$ has the expression,

$Q=\int_{\rho^{-1}_{x=\rho_{i}}}(\rho e^{-2\bar{u}}\nabla_{n}\bar{v})d\ell$
(90)

So, again, there are $q_{i}\in\rho^{-1}_{x=\rho_{i}}$ such that,

$Q=\rho_{i}\ell_{i}(e^{-2\bar{u}}\nabla_{n}\bar{v})\bigg{|}_{q_{i}}.$ (91)

It is more convenient to write this as,

$Q=\underbrace{(\rho_{i}\ell_{r_{i}})\vphantom{\nabla_{n_{r_{i}}}}}_{\rm(I)}\underbrace{(e^{-\bar{u}})|_{q_{i}}\vphantom{\nabla_{n_{r_{i}}}}}_{\rm(II)}\underbrace{(e^{-\bar{u}}\nabla_{n_{r_{i}}}\bar{v})|_{q_{i}}}_{\rm(III)}.$
(92)

We will think the terms (I), (II) and (III) which depend on $i$ as sequences
(I)i, (II)i and (III)i. We know that (I)i tends to a non-zero constant as
$i\rightarrow\infty$. So, if a subsequence (III)${}_{i_{j}}$ of (III)i tends
to zero then (II)${}_{i_{j}}$ tends to infinity because $Q\neq 0$. Thus we get
a divergent sequence of points $q_{i_{j}}$ on which $\bar{u}(q_{i_{j}})$ tends
to $-\infty$ which is not the case by the initial assumption. On the other
hand as (III)i is bounded above by (23) we deduce that there are
$0<m_{1}<m_{2}$ such that,

$m_{1}\leq{\rm(III)}_{i}\leq m_{2}$ (93)

When we consider the uniform bounds for $|\tilde{u}(q_{i_{j}})|$, the estimate
(93) and the estimates (23), we deduce that one can extract a subsequence of
the variables $\bar{u}$ and $\bar{v}-\bar{v}(q_{i_{j}})$ lifted to
$\tilde{U}_{i_{j}}$, converging in $C^{\infty}$ to a charged ${\rm
S}^{1}$-symmetric solution of (5)-(6) over
$\tilde{U}=[\tilde{\rho}_{0},\infty)_{\tilde{\rho}}\times{\rm S}^{1}$. But we
know from section 4 that such a solution does not exist and we reach thus a
contradiction. ∎

Acknowledgement. I would like to thank Marcus Khuri and Gilbert Weinstein for
having shown me (before this work started) that charged solutions having a
suitable expansion at infinity do not exist.

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  * April 2023

# Bionic Collapsible Wings in Aquatic-aerial Robot

Xiao Xiong & Xinyu Zhang &Huanhao Huang &Kangyao Huang  0000-0002-8308-6270
0000-0003-0034-9037 0000-0001-8190-0390 0000-0002-5708-2620 The School of
Vehicle and Mobility, Tsinghua University, Beijing, P.R.China.
<EMAIL_ADDRESS>
###### Abstract

The concept of aerial-aquatic robots has emerged as an innovative solution
that can operate both in the air and underwater. Previous research on the
design of such robots has been mainly focused on mature technologies such as
fixed-wing and multi-rotor aircraft. Flying fish, a unique aerial-aquatic
animal that can both swim in water and glide over the sea surface, has not
been fully explored as a bionic robot model, especially regarding its motion
patterns with the collapsible pectoral fins. To verify the contribution of the
collapsible wings to the flying fish motion pattern, we have designed a novel
bio-robot with collapsible wings inspired by the flying fish. The bionic
prototype has been successfully designed and fabricated, incorporating
collapsible wings with soft hydraulic actuators, an innovative application of
soft actuators to a micro aquatic-aerial robot. We have analyzed and built a
precise model of dynamics for control, and tested both the soft hydraulic
actuators and detailed aerodynamic coefficients. To further verify the
feasibility of collapsible wings, we conducted simulations in different
situations such as discharge angles, the area of collapsible wings, and the
advantages of using ground effect. The results confirm the control of the
collapsible wings and demonstrate the unique multi-modal motion pattern
between water and air. Overall, our research represents the study of the
collapsible wings in aquatic-aerial robots and significant contributes to the
development of aquatic-aerial robots. The using of the collapsible wings must
a contribution to the future aquatic-aerial robot.

††:

Keywords: Robot, Multi-modal,collapsible wing

##  1 Introduction

In recent years, the aquatic-aerial robot has gained widespread attention as
an emerging concept that offers a convenient and effective solution for water-
air cross-domain tasks. With the ability to move both in air and water. These
robots can operate in a wide range of environments, including oceans, pools,
and complex urban settings. As a result, they hold great potential for
environmental and scientific applications, such as water health monitoring[1]
and animal behavior observation[2].

Mature technologies like fixed-wing robots, multi-rotor robots and multi-modal
robots[3] are used as the flying part of the most aquatic-aerial robots.
Fixed-wing robots use fixed wings to generate lift and can fly at speeds above
20-30 $m/s$, allowing them to be deployed over longer distances. Examples of
fixed-wing aquatic-aerial robots include those developed by North Carolina
State University[4] and Dipper[5]. Multi-rotor robots, on the other hand, use
a structure similar to that of multi-rotor aircraft, with four propellers
controlling the robot’s motion. The main advantages of multi-rotor robots are
their unique hovering and vertical take-off and landing capabilities. Examples
of multi-rotor aquatic-aerial robots include the one developed by Beihang
University[6]. Additionally, some aquatic-aerial robots combine both fixed-
wing and multi-rotor structures, such as Nezha III[7], which has the ability
to perform horizontal flight, vertical flight, and underwater gliding. Each
type of flying method of aquatic-aerial robot has its own unique feature.

In addition, bionic technology provides other methods to design the aquatic-
aerial robot, especially inspired by creatures with the capacity of moving
both in the air and water like sea birds, flying squid and flying fish. For
instance, based on the biological gannet, a kind of sea birds, the robot
developed by Beihang University focuses on wing load investigations during
plunge-diving[8]. Similarly, the AqualMAV is also inspired by plunge-diving
birds and has been tested for performance[9]. Because of the plunge-diving
method to move in the water for sea birds, a common concern of this kind robot
is the huge forces that occur when they impact the water, which may lead to
structural damage. In contrast, marine creature-inspired robots tend to avoid
high-speed impacts with the water and focus more on the motion in the air. For
instance, the robotic jet thruster of AqualMAV was inspired by the flying
squid, which uses pressurized water to achieve high speeds during
conversion[10]. Another squid-like aquatic-aerial vehicle also uses a similar
thrust mechanism but emphasizes bionic mechanisms such as soft morphing fins
and arms[11]. Flying fish have also inspired the design of aquatic-aerial
robots. For example, the Flying Fish ocean surveillance platform developed by
the University of Michigan avoids hydrodynamic drag from ocean waves and
currents while flying, but only has locomotion on the water surface without
underwater movement[12]. MIT has detailed design considerations for a robotic
flying fish prototype[13].

The locomotion capacity in the air of aquatic-aerial robots base on the mature
technologies or inspired by sea birds are active with a power take-off system.
Multi-rotor, propeller and flapping wings all consume the energy in the air,
while aquatic-aerial robots inspired by the marine creatures do not. These
marine based robots usually take off out of water with a fast initial velocity
and finish the air motion consuming the initial kinetic energy. Flying squid
and flying fish are two typical animals move in the air without power output.
However, there are still some difference in air motion between flying squid
and fling fish. Flying squid can only drop back to the water by gravity with a
parabola trajectory without controlling. Flying fish is a kind talented
animal, which can control the gesture and motion state with their pectoral
fins and body center in the air. The flexible locomotion with no power need
contributes to the unique motion pattern of the flying fish. We curious about
this unique motion pattern of flying fish and concentrate on the its
collapsible pectoral fins.

Figure 1: Overview of flying fish motion pattern.

.

The process of flying fish moving in the air consists of four stages,
including taking off, climbing, falling, and gliding stages. During the taking
off stage, flying fishes first accelerate underwater. Once they reach a
certain speed, they change their forward direction upwards and fly out of the
water at maximum speed. Also, they accelerate on the incline water surface and
take off at the top of the wave in this stage. After taking off, flying fishes
open their pectoral fins to create lift, resist gravity, and glide in the air.
Besides, When sliding down the water surface, they only need to swing their
tail fins into the water to accelerate again, regain high speed, and take off
again. In the gliding stage, fully utilizing of the ground effect contributes
to further distance movement[14]. It’s said that flying fishes can continue
this process in the air for a total distance of 400m. In addition, flying
fishes have the capacity to change the motion gesture and statement by
controlling the pectoral fins. For example, they fold their fins and plunge
into the water to escape when they are chased by sea birds and adjust their
attack angle of the fins to generate a higher lift force when drop in the
falling stage.

Fig.1 shows the motion pattern of the flying fish in the air with stages. The
solid line represents the trajectory of the flying fish during the entire
flying process, while the two dotted lines represent the trajectory when the
flying fish folds its wings in the air and the hypothetical situation without
the ground effect. Overall, the unique motion pattern of the flying fish
serves as an excellent model for designing aquatic-aerial robots that can move
efficiently in both air and water.

It’s obviously that the collapsible pectoral fins contribute to all the motion
in the air. Whether in lift generating or motion controlling in the air,
collapsible fin is the main actuator even the only one. The advantage of the
collapsible wings like pectoral fins can contribute to the design of the new
aquatic-aerial robots. However, there are seldom research focus on the
mechanism of the pectoral fins of the flying fish. Thus, further study about
the elements effecting the performance of the collapsible need to be done.

In this study, The novelty of this study include: (1) a new design of the
aerial aquatic robot with collapsible wings inspired by flying fish, (2) the
use and precise control of the soft hydraulic actuator replace the rigid
actuator of the collapsible wings, (3) a precise dynamic modal of the robot
with collapsible wings, and (4) a series of simulation experiments focus on
the collapsible wings, studying the effect of discharge angle, area and ground
effect to the wing performance.

This paper is organized as follows. Section II introduces the design and
fabrication of the prototype of robot and collapsible wings. Section III
analyzes the dynamics of the robot in multi-modal motions. Section IV presents
the experimental results and discussions regarding the locomotion of the
robot, with a focus on the collapsible wings. Finally, Section V concludes the
paper.

##  2 Design and Fabrication of Robot Prototype

###  2.1 General Overview

The unique movement pattern of flying fish is determined by its unique
physiological structure especially the collapsible wings. We design an
aquatic-aerial robot prototype to study the multi-modal motion with
collapsible wings.

The robot was designed with the appearance and physiological mechanism of the
flying fish in mind, with a long and narrow main body and large collapsible
pectoral fins that occupy 70% of the projected area of the body when expanded.
To simplify the complexity of the tail fin, a high-powered motor was used as
the source of power to accelerate. In addition, two pelvic fins were designed
to balance the posture and movement of the robot underwater.

The robot is comprised of a streamlined external structure, collapsible wings
with soft hydraulic actuator, high power density drive system, and precise
control system. The appearance and biological parameters of the flying fish
were both referenced to restore the structure, resulting in a robot prototype
that closely resembles its natural counterpart. Fig. 2 shows an exploded view
and pictures of the robot prototype, while Table 1 provides details on the
physical characteristics of the robot.

Figure 2:  (a) is the exploded view of robot prototype; (b) is the picture of the prototype; (c) is the modal of the prototype.  Table 1: Summary of key design parameters Parameter | Value
---|---
Length | 336.1mm
Wing span | 455mm
Width(wing folded) | 135.4mm
Motor power | 300w
Weight | 353g

###  2.2 Bionic desgin of the main body

Aquatic-aerial robots need the characteristics of both underwater and aerial
robots, such as Waterproof and weight limit. Besides, the streamlined shape
with excellent hydromechanical properties is significant for both types of
robots. The shell frame of the robot is made of basswood, linked with mortise
and tenon joints, reducing the weight without using heavy metallic connectors
like bolts. The outer surface is covered with a heat-shrink film, which shapes
the streamlined surface perfectly by heating and provides great water-proofing
ability. A yellow polyimide film is pasted on the surface to enhance the
hardness for scratch prevention in the uncertain underwater environment. The
vertical flat and elongate body shape similar to fish, along with two upper
and lower tail fins, prevents rotation in the water.

The control system is installed inside the housing, and it is comprised of a
motor with a propeller that generates thrust, a hydraulic system driven by
servos with a connecting rod structure to control the collapsible wings and
the density of the robot for floating or submerging. A receiver receives
signals from the remote control, and IMU sensors, pressure sensors, and other
sensors connecting to the STM32 chip make up the information collection
system. A lithium battery with 11.1V and 1000mAh provides energy for the whole
system, which is distributed to all actuators by an electronic speed
controller.

###  2.3 Collapsible wings with soft actuator

The ability to extend and retract their fins is a critical feature that
enables them to adapt to the different environments of air and water. And the
collapsible wings in the aquatic-aerial robot are a critical feature that
imitates the flying fish mechanism. The use of a soft actuator to control the
pectoral fins’ extending and retracting is a significant improvement over the
rigid rotating structure of most collapsible wings. The soft actuator achieve
the precise control in both air and water and have ability to adapt to
external forces while maintaining control. The flexible membrane used in the
wings of the robot has excellent tensile properties and hydrophobic ability,
which reduces weight gain and improves performance in different environments.
Fig.3 shows the collapsible wings in different open angles.

Figure 3: Different angle of collapsible wings.

.

The use of soft actuators in the collapsible wings of the robot is an
innovative approach to mimic the mechanism of flying fish. Unlike rigid
rotating structures that are heavy and complex, the soft actuator is light and
simple, making it more suitable for use in compact spaces. The injection of
liquid into the actuator controls the bending angle, and the internal pressure
can be directly controlled by the amount of material volume injected. The
flexible capacity of the actuator also allows it to adaptively deform in the
face of external impacts, preventing damage due to huge forces. This design
not only mimics the natural mechanism of flying fish but also provides a
practical and efficient solution when wings of aerial robot get an impact in
the movement.

Soft actuator is designed by the modal of soft hydraulic actuator. Not similar
to the normal straight hydraulic actuator at beginning, a 90 degree bending
actuator was designed to suit the situation that the wings is retracted
without driving. With the injection of the liquid, the actuator formation is
straightened to open the wing. The bending angle is decided by the inner
pressure. Fig.4 is the process of actuator fabrication and the silicone rubber
is Mold Star 30, Smooth-on, Inc, Easton Pa, USA.

Figure 4: Fabrication of the soft actuator. (a) two actuator mold parts (b)
applying mould discharging agent to the inner surface (c) mixing and stirring
the pre-mix silicone rubber evenly (d) pouring uncured silicone rubber into
the molds (e) refrigerating to remove air bubbles (f) curing the silicone
rubber at temperature(140∘F/60∘C) in the oven (g) getting upper and lower
parts of actuator (h) gluing the two parts through secondary pouring

.

To increase the size of the wings without affecting the robot’s propulsion
system, a 170mm carbon fiber tube was used as the arm of the wing to support
the flexible membrane at the end of the actuator. The membrane of the fin was
made of a 0.07mm hydrophobic elastic silicone film, which had excellent
tensile properties. When deployed, the membrane formed a large airfoil, but
when retracted, it was almost negligible. The hydrophobic properties of the
membrane’s surface also reduced weight gain caused by water attachment after
the robot left the water.

##  3 Dynamics Analysis

In order to comprehend the effect of collapsible wings and the motion of the
aquatic-aerial robot inspired by the flying fish, dynamics of the robots
motion must be calculated and analysed. In this part, coordinate system of
robot has been built, and forces and torques in three different stages has
been calculated.

###  3.1 Coordinate System and kinematic

Figure 5: Coordinate system.

The robotic schematic with three coordinate frames were established and showed
in figure 5, including the body frame $O_{body}$ , ground frame $O_{ground}$
and velocity frame $O_{velocity}$. The body frame $O_{body}$ established on
the centroid of the robot. the $x_{b}$-axis points to the front of the robot
and the $z_{b}$-axis point vertically down of the robot body. The $y_{b}$-axis
is determined by the right-hand rule. These three frames rotate to each other
as Eq.1:

$\eqalign{O_{velocity}=R_{vb}O_{body}\\\ O_{body}=R_{bg}O_{ground}}$ (1)

$\eqalign{R_{vb}=\left[\begin{array}[]{ccc}C_{\alpha}C_{\beta}&S_{\beta}&S_{\alpha}C_{\beta}\\\
C_{\alpha}C_{\beta}&C_{\beta}&-S_{\alpha}S_{\beta}\\\
-S_{\alpha}&0&C_{\alpha}\end{array}\right]\\\
R_{bg}=\left[\begin{array}[]{ccc}C_{\theta}C_{\psi}&S_{\theta}S_{\psi}&-S_{\theta}\\\
S_{\theta}C_{\psi}S_{\emptyset}-S_{\psi}C_{\emptyset}&S_{\theta}S_{\psi}S_{\emptyset}+C_{\psi}C_{\emptyset}&C_{\theta}S_{\emptyset}\\\
S_{\theta}C_{\psi}C_{\emptyset}+S_{\psi}S_{\emptyset}&S_{\theta}S_{\psi}C_{\emptyset}-C_{\psi}S_{\emptyset}&C_{\theta}C_{\emptyset}\end{array}\right]}$
(2)

where,$\alpha$ is the angle-of-attack and $\beta$ is the side-slip-angle. The
$\theta$, $\psi$ and $\emptyset$ are the angles of pitch, roll and yaw,
respectively. $C$ and $S$ represent trigonometric functions sine and cosine.

Angular velocity vector and translational velocity vector are recorded as
$\Omega=[p\ q\ r]^{\prime}$ and $V=[u\ v\ w]^{\prime}$. The motion equations
of robot in the body frame are shown as:

$\dot{\Omega}=I^{-1}\left(M-\Omega\times I\Omega\right)$ (3)
$\dot{V}=\frac{F}{m}-\Omega\times V,$ (4)

where $m$ is the mass of robot and $I$ is the inertia matrix. Two vectors F
and M represent force and torque.Then introduce the vectors $\Phi=[\phi\
\theta\ \psi]^{\prime}$ and $X=[x\ y\ z]^{\prime}$, which respectively
represent the Euler angle and the center of gravity position of the flying
fish. Both are representations of the ground coordinate system. The kinematic
equations and navigation equations are:

$\dot{\Phi}={S_{bg}}^{-1}\Omega,\dot{X}={S_{ag}}^{-1}V$ (5)

###  3.2 Forces and Torques in the movement

####  3.2.1 Gliding

When gliding in the air, the center of gravity moment M is the result of
considering the engine thrust, ground effect, and the aerodynamic force of the
pectoral and pelvic fins of the flying fish. It can be expressed as:

$M=M_{m}+M_{p}=\frac{1}{2}\rho
V_{m}^{2}S_{1}\left[\begin{array}[]{ccc}bC_{x}\\\ cC_{y}\\\
bC_{z}\end{array}\right]+\frac{1}{2}\rho
V_{p}^{2}S_{1}\left[\begin{array}[]{ccc}bC_{x}\\\ cC_{y}\\\
bC_{z}\end{array}\right]$ (6)

$M_{m}$, $M_{p}$ are the moments generated by the aerodynamic force of fins.
$C_{x}$, $C_{y}$ and $C_{z}$ are the roll, pitch, and yaw moment coefficients
of the flying fish, respectively. $\rho$ is air density, $V_{m}$, $V_{p}$ are
center of gravity velocity and pelvic fin velocity respectively. The speed of
the pelvic fin is:

$V_{p}=V_{m}+R(\frac{\pi}{2}-\theta_{p})(\dot{\theta_{p}}\times b_{p}),$ (7)

where $R$ is expressed as a matrix rotating around the $Y_{b}$ axis,
$\theta_{p}$is the pitch angle of the pelvic fin, and $b_{p}$ is the average
width of the pelvic fin.

Figure 6: Vertical projected area.

The calculation formula of flying fish pectoral fin area $S_{1}$:

$S_{1}=\int_{0}^{k}\frac{1}{2}l^{2}dk$ (8)

$k$ is the expansion angle of the pectoral fin, and the area of the pelvic fin
is $S_{2}$. $l$ is the length of the pectoral fin. $c$ is the average
aerodynamic chord length of flying fish, $b$ is the fin span of robot:

$b=2lsink+L,$ (9)

where $L$ is is the width of the robot without wings.

In the gliding phase, the force F applied on the flying fish comes from the
gravity G, aerodynamic force on the pectoral fin $F_{m}$ and pelvic fin
$F_{P}$:

$F=G+F_{m}+F_{p}$ (10)

Gravity represents in the body frame as:

$G=R_{bg}\left[\begin{array}[]{ccc}0\\\ 0\\\
mg\end{array}\right]=mg\left[\begin{array}[]{ccc}-S_{\theta}\\\
C_{\theta}S_{\emptyset}\\\ C_{\theta}S_{\emptyset}\end{array}\right]$ (11)

Both aerodynamic forces in the body frame can be showed:

$F_{m}=S_{ab}{}^{-1}\left[\begin{array}[]{ccc}-D_{m}\\\ Y_{m}\\\
-L_{m}\end{array}\right]=\frac{1}{2}S_{ab}{}^{-1}\rho
V_{m}^{2}S_{1}\left[\begin{array}[]{ccc}-C_{D}\\\ C_{Y}\\\
-C_{L}\end{array}\right]$ (12)

$F_{p}=S_{ab}{}^{-1}\left[\begin{array}[]{ccc}-D_{p}\\\ Y_{p}\\\
-L_{p}\end{array}\right]=\frac{1}{2}S_{ab}{}^{-1}\rho
V_{p}^{2}S_{2}\left[\begin{array}[]{ccc}-C_{D}\\\ C_{Y}\\\
-C_{L}\end{array}\right]$ (13)

D, Y, and L are resistance, side force and lift, expressed in the airflow
coordinate system; $C_{D}$, $C_{Y}$, $C_{L}$ are drag, side force, and lift
coefficients, respectively.

####  3.2.2 Conversion

The posture and movements during water exit are key factors affecting the
aerial motion of the robot. A thorough analysis and modeling of the water exit
process can provide sufficient theoretical basis for the subsequent control of
the robot. Before the robot exits the water, the collapsible pectoral fins are
not deployed, and the direction of the body’s velocity is basically the same
as the direction of the body. Here, it is assumed that the velocity coordinate
system and the body coordinate system remain consistent in direction at first
stage during the transition phase of water exit.

Figure 7: Dynamic analysis of water exit.

During the water exit process, the robot as a whole is subject to gravity $G$
and thrust $T$ generated by the propeller rotation showen by Fif.7. The
pectoral fins are deployed, producing aerodynamic lift $F_{L}$. The underwater
part is subject to buoyancy $B$, drag forces $D_{w}$ from horizontal velocity,
and viscous drag forces $F_{w}$ generated by the robot’s surface movement in
the water. At the same time, as the center of force of the underwater part
changes with the degree of water exit, a pitching moment $M_{b}$ is generated,
which affects the pitching angle state.

$F=G+F_{L}+F_{D}+B+T+F_{w}$ (14) $M=M_{m}+M_{B}+M_{T}+M_{w}$ (15)

The buoyancy formula and the buoyancy moment in the body coordinate system
are:

$B=S_{bg}\left[\begin{array}[]{ccc}0\\\ 0\\\
\rho_{w}gV_{w}\end{array}\right],M_{B}=a\cos(\theta)\left[\begin{array}[]{ccc}0\\\
B\\\ 0\end{array}\right]$ (16)

Here, the body is approximated as a cylindrical shape, and the volume of the
body submerged in water is calculated based on the height of the center of
gravity of the body from the water surface.

$V_{m}=\Delta z\pi R^{2}\sin(\theta)$ (17)

where $R$ calculated as 20mm. The hydrodynamic moment and the thrust torture
are

$M_{w}=\frac{1}{2}\rho_{w}v_{w}^{2}AL_{B}\left[\begin{array}[]{ccc}C_{x}\\\
0\\\ C_{z}\end{array}\right],M_{T}=T\times h_{CM}$ (18)

where

$F_{w}=S_{ab}^{-1}\left[\begin{array}[]{ccc}-D\\\ 0\\\
-L\end{array}\right]=\frac{1}{2}\rho_{w}v_{w}^{2}\left[\begin{array}[]{ccc}C_{Dw}\\\
0\\\ C_{Lw}\end{array}\right]$ (19)

Where

$D=\frac{1}{2}\rho_{w}v_{w}^{2}AC_{Dw}$ (20)
$L=\frac{1}{2}\rho_{w}v_{w}^{2}AC_{Lw}$ (21)

$\rho_{w}$ is the density of the fluid, $V_{w}$ is the volume of the flying
fish, $v_{w}$ is the speed of the flying fish in water, $A$ is the lateral
area of the flying fish.

Figure 8: Clearance between the robot and and underlying surface

.

####  3.2.3 Ground effect

Based on previous research on the motion pattern of flying fish, they fully
utilize the ground effect to enhance lift during long-distance gliding[15].
The ground effect refers to the phenomenon where the wing surface near a
underlying surface enhances the ratio of lift and drag coefficient which means
the robot can move longer distance than far away the surface. Lift coefficient
during flight increases rapidly as it approaches the surface. In this process,
the ratio of the distance from an object to the medium surface to the chord
length, $h/b$ , is used to measure the proximity of different objects. The
clearance between the robot and underlying surface is shown in Fig.8. When it
is less than 1/2, the impact of ground effect on lift is extremely
significant, and the calculation formula for lift coefficient will also take
into account the effect of ground effect[16]. $C_{gel}$ and $C_{ged}$
represent the effect coefficients of ground effect to the lift and drag.

$\eqalign{L=\frac{1}{2}\rho_{w}v_{w}^{2}AC_{Lw}C_{gel}\\\
D=\frac{1}{2}\rho_{w}v_{w}^{2}AC_{Dw}C_{ged}}$ (22)

##  4 Experimental Evaluation

To achieve precise control of the collapsible wings for the further study,
several experiments were conducted to test the soft hydraulic actuator. The
relationship between the bending angle and the internal pressure or volume of
injected liquid was examined to establish a direct correlation. Besides,
aerodynamic coefficients were also tested to ensure the locomotion of the
robot can be simulated. Basing on these pretests, three experiments about
discharge angle, area of collapsible wings and ground effect are finished to
explore the performance of collapsible wings in the aquatic-aerial motion.

Figure 9: Experiments for aerodynamic coefficients

.

###  4.1 Soft Actuator

The position of collapsible wings are controlled by the hydraulic system. As
the liquid is injected or sucked out, the deformation angle of the flexible
actuator changes with the internal pressure. Obtaining the correspondence
between the deformation angle and the internal pressure or volume is necessary
for controlling the collapsible wings. However, due to the deformation
characteristics of the material itself of the flexible structure, the
relationship between the two is hard to obtain by numerical calculation or
simulation. Here, the angle of deformation under different pressures is
recorded experimentally to measure the correspondence. Four sets of
experiments were designed using liquid and air as the fillers for the flexible
actuator, respectively. The internal pressure values were recorded at multiple
states during the deformation from 0 to 90 degrees.

Figure 10: (a) Bending angle of soft actuator; (b) Bending angle vary with the
pressure; (c) Bending angle avry with the volume.

.

Based on the data from the four sets of experiments, the relationship between
the internal pressure and the deformation angle was fitted using the least-
squares method, resulting in the graph shown above. It can be seen that the
deformation angle varies linearly with the increase of pressure both in two
medium. The accurate bending angle $\theta$ can be calculated by the equation:

$\theta=1.2*P,$ (23)

where P is the internal pressure. The four sets of experiments had good
consistency without significant differences due to different medium. However,
for the method of bending the actuator by injecting medium, volume is the
direct control parameter. Therefore, it is more important to obtain the
relationship between the injected medium volume and the deformation angle
instead of the pressure. Two sets of experiments were conducted separately,
and the results are shown in Fig.10 (c). The bending angles of the actuator
were measured for two different medium when different volumes injected during
the expansion and contraction processes. Comparing gas and liquid medium, it
was found that only 2 $mL$ of liquid was needed to completely drive the
actuator extension, which was 40% of volume when gas was used as the medium.
The more volume means the larger hydraulic system, which is not suitable in
such a micro robot. For these reasons, liquid is a better choice as the medium
in this situation. In addition, the reset precision and natural affinity to
water environments of the hydraulic soft actuator are more suitable for
aquatic-aerial robot[17].

Figure 11: Relationship between aerodynamic coefficients and atattck angle
.(a) (b) (c) show variations of the lift($C_{L}$), drag($C_{D}$) and
torture($C_{M}$) coefficients with the attack angle respectively. (d) is the
ratio of lift and drag in different attack angles

.

###  4.2 lift coefficient

During stages of takeoff and gliding in the air, lift force is mainly provided
by open collapsible wings. For the motion control of robots in the air, the
relevant aerodynamic coefficients of the robot are crucial, including lift
coefficient, drag coefficient, and torque coefficient. These coefficients are
related to the robot’s motion speed, the vertical projection area of the
robot, and the attitude of the body. To obtain these motion coefficients for
better control of the robot, an experiment was designed as shown in Fig.9. The
robot was installed on a two-dimensional movable workbench, and a wind with a
speed of 10m/s was applied at different angles of attack. The forces acting on
the robot in the upward $F_{z}$ and backward $F_{x}$ in the ground coordinate
system were measured. The lift force $L$ and drag force $D$ at different
angles of attack were then calculated using these two direction forces and the
angle of attack, and the calculation formula is eq.24.

$\eqalign{F_{z}=L*C_{\alpha}+D*S_{\alpha}\\\ F_{x}=L*S_{\alpha}-D*C_{\alpha}}$
(24)

However, since physical experiments cannot measure the rotational torque of
the robot during motion under limited condition, the finite element fluid
simulation method was chosen to measure the rotational torque of the robot at
a speed of 10m/s showed Fig.9(b). Based on the prototype measurement, the
center of gravity of the model was set at a position 13cm away from the head.
After obtaining the lift, drag, and torque at each angle of attack, the
robot’s coefficients are calculated using the following formulas.

$\eqalign{C_{L}=\frac{2L}{\rho SV^{2}}\\\ C_{D}=\frac{2D}{\rho SV^{2}}\\\
C_{M}=\frac{2M}{\rho SV^{2}}}$ (25)

The relationship between lift coefficient and drag coefficient and angle of
attack is shown in Fig.11. At small angles of attack, the lift coefficient
increases linearly with the angle of attack. It tends to stabilize at 0.8
after the angle of attack exceeds 10 degrees. The drag coefficient increases
with the angle of attack when it is positive and decreases with the angle of
attack when it is negative. The drag coefficient is minimum at around -5
degrees angle of attack. Normally, for a typical symmetrical airfoil, the
angle of attack where the lift coefficient is 0 and the angle of attack where
the drag coefficient is minimum are both at 0 degrees angle of attack. The
reason why they are at around -5 degrees angle of attack in this case is a
5-degree angle with respect to the horizontal plane exist in the design,
inspired by the characteristic of flying fish pectoral fins. The torque
coefficient decreases continuously as the angle of attack increases. A
positive torque coefficient indicates that the torque of the robot’s head
tilts upward, while a negative torque tilts it downward. When the angle of
attack is less than 0, the torque coefficient is positive, and the body tilts
upward. When the angle of attack is greater than 0, the torque is negative,
and the body tilts downward. Negative feedback control of the torque ensures
that the body posture of the robot is consistent with its direction of motion,
maintaining the stability of its movement. The lift-to-drag ratio is an
important indicator to measure the flight performance of a winged robot, and a
higher lift-to-drag ratio indicates better flight performance. Fig.11(d) shows
the relationship between the lift-to-drag ratio and the attack angle of the
robot. Starting from 0, as the attack angle increases, the lift-to-drag ratio
increases rapidly, reaches the maximum value 4.37 at $\alpha=5^{\circ}$, and
then slowly decreases.

Figure 12: Simulation of the motion of aquatic-aerial robot in different
incline angles. (a) show the routine in the x-z plane; (b) show the variation
of attack angle with the distance

.

###  4.3 Simulation of the locomotion with collapsible wings

Flying fish evolve a unique motion pattern to escape predators. In the water,
there are many pursuers like sword fish, tuna and other bigger fish. Although
flying fish has a higher maximum speed than the pursuers, the high speed is
hard to sustained for a long time in the fierce chasing. Flying fish evolve
the capacity to break out of the water and hold a long time and distance with
the expanding large pectoral fins over the surface, where the the predators
can not touch. Even if the bigger fish try to jump out of the water, the heavy
body drop it down the water rapidly. Flying fish are able to escape the
predators in the water by pectoral wings in this way. However, the gliding
flying fish turn to the hands-down delicious food for the hunter in the air,
like seabird. For the fish, the gliding process in the air is clumsy to change
the trajectory usually. Fortunately, the adjust of gesture with the pectoral
wings control the flying fish motion in the air. When Flying fish are chased
by the seabird, they fold their fins and drop down to the water speedy. Flying
fish switching in the water and air frequently with the function of
collapisble pectoral wings when chased by predators.

Certainly, the unique motion pattern with collapsible fins is not easy even
for the flying fish. Some flying fish fly too high with a wrong discharge
angle lead to the balance losing and be caught by the seabirds. Appropriate
discharge angle is significant and challenging especially in the roaring
waves. That’s the reason why the first part of simulation focus on the
discharge angle. In addition, the area of opened pectoral fins is also the
significant element effecting the motion of the flying fish. They reduce the
area of the fins to reduce the lift and drop back to the water faster.
Besides, flying fish is an intelligent creature to take full advantage of the
environment. The large area of pectoral fins are used to take advantage of
ground effect, which can generate a higher lift, to extend the time and
distance of gliding. In this section, we also simulate this two motion to
evaluate the effect of the collapsible wings.

Figure 13: (a)is the locomotion routine with different open angles wing in x-z
plane when discharge angle is $15^{\circ}$ and velocity is 10$m/s$.(b) shows
the trajectory of bionic robot with open wings and fold wings at the middle of
process.

####  4.3.1 Discharge angle effect performance of collapsible wings

The base function of collapsible wings is generate lift force to accomplish
the motion in the air. Different discharge angles lead to different
trajectories with the collapsible wings. An exist research introduces the
flying fish can glide for a distance up to 48m and a height up to 10m[18] and
another get the maximum gliding distance is 45.4m with a initial velocity of
20m/s from a series simulations with different discharge angles[19]. It’s hard
to perform perfectly each time especially chased by predator in the water. We
curious about the motion capacity of the bionic robot and simulate the moving
process over the water surface in different discharge angles. Here, 11 groups
of experiments with open wings conducted in intervals of 5 degrees from 10
degrees to 50 degrees with a initial velocity of 10$m/s$. The results are
shown in Fig.12.

Figure 14: Comparison between the motion with and without ground effect.

.

In Fig.12(a), the motion trajectory of the robot with collapsible wings in the
x-z plane after water discharge is shown. As the discharge angle increases,
the gliding distance first increases and then decreases, which is similar to
the truly presentation of the flying fish. When the angle of the water
discharge is 35 degrees, the robot can glide the farthest distance,
approximately 11 meters, at a flight altitude below 2.3 meters. When the water
discharge angle is 50 degrees, the highest altitude is reached, approaching
3.5 meters. In figure (b), the variation of the angle of attack of the robot
in the air at different distances is shown. At the initial stage, the angle of
attack variation is similar for different incline angles, and it begins to
vary after 3 meters. The larger the water discharge angle, the more drastic
the change in the angle of attack after 3 meters, which is due to the faster
change in the velocity incline angle, while the change in body attitude is
relatively stable. As the angle of attack rises to its highest point,
according to the determination of the torque coefficient in experiment two, a
larger reverse torque causes the body to return to the direction close to the
velocity incline angle, which reduces the angle of attack in the latter part
of the gliding process. Not too large attack angle ensure the stability of the
robot with collapsible wings, because once the attack angle is too large, the
stall happens. Stall leads to loss of control easily even to the flying fish.
The outcome explain the robot is able to finish the air motion part like
flying fish, though the distance and height is not better than the real
creature. A more lightweight and miniaturized design or a larger collapsible
wings may the method to enhance the flying ability to the robot.

####  4.3.2 Motion on different angles of collapsible wings

Not like the other fish or flying squid dropping into water immediately
without flexible controlling after breaking out of the water surface, flying
fish’s huge pectoral fins open to glide for a long distance. The main factor
contribute to the motion in the air of flying fish is the collapsible fins. We
curious about the extent angle of the wings affect the movement. Thus, at an
exit angle of 15 degrees and velocity at 10$m/s$, the motion of the robot
under different pectoral fin opening angles was simulated. The four scenarios
were 0 degrees, 30 degrees, 60 degrees, and 90 degrees of fin opening.
Besides, flying fish fold their pectoral fins and adjust their gesture to drop
rapidly when they chases by predator in the air. The agile action also relate
to the use of the pectoral fins. A simulation about the trajectory of robot
folding the collapsible wings when it touch the highest point is set and
compare to the trajectory without folding action.

Fig.13 shows the experimental results. (a) indicates that with a larger area
of the pectoral fins, the flying distance was significantly increased. Without
opening the pectoral fins, the robot could only glide 5.3 meters based on
inertial motion. However, when the pectoral fins were fully open at 90
degrees, the robot glided 7.8 meters, representing a 47.2% increase in gliding
distance. And this was achieved without any control of the robot’s flight
attitude. The fly process with open wings not in a control already induce a
significant improvement, which verify the feasibility to fly far away with
further precise control. (b) shows the perform difference between the open and
fold wings two actions after the highest point. After folding the wings, robot
drop into the water rapidly with a moving distance about 1m. Comparing to the
distance of 3.5m with a perpetual opening wings, nearly two third motion path
are avoided. The less distance means less chance for predators to catch up.
These two simulations verified the further precise control of the collapsible
wings is meaningful and can definitely improve the locomotion capacity of the
robot.

####  4.3.3 The effect of Ground effect

Flying fish is smart to take advantage to the environment especially in the
use of ground effect for a longer gliding distance. The ground effect is the
reason why flying fish evolved to glide at a close distance to the water
surface for long distances with extending pectoral fins. It provides a lift
coefficient by collapsible wings which is more than in normal flight
conditions, and this amplification increases rapidly as the clearance from the
ground decreases. To verify the ground effect for the collapsible wings and
lay the groundwork for future control, simulations were conducted for two
scenarios: one with ground effect and one without, under the condition of
different water discharge angles and 90-degree collapsible wings expansion.
The initial velocity of all experiments is 10$m/s$. Fig.14 shows the results
of the experiment. (a), (b), and (c) show the simulation experiments under
three scenarios with water discharge angles of 10 degrees, 15 degrees, and 30
degrees, respectively. When the water exit angle is 10 degrees, the robot with
ground effect can glide for 6 meters, while the one without can only glide for
4 meters, resulting in a 50% increase in distance. When the water exit angle
is 15 degrees, the ground effect gain decreases to 0.5 meters, resulting in a
6.8% increase, and when the water exit angle is 30 degrees, the ground effect
does not provide an increase in gliding distance, even decreases it, mainly
due to the negative impact of ground effect when exiting the water. At the
same time, the flying height is also of concern. Since the ground effect has a
significant effect only when the clearance $h$ between the collapsible wings
and ground is less than half of the wingspan. For the robot with a wingspan of
0.5, the ground effect only has an evident effect when the clearance above the
ground is below 0.25 meters. For a low water exit angle of 10 degrees, the
gliding height is just within this range, so the ground effect provides a huge
boost to the gliding distance. As the flying height increases, the proportion
of distance traveled below 0.25 meters gradually decreases throughout the
process, and the effect of ground effect to collapsible wings becomes smaller.
The experiments proved the method of collapsible wings to use the ground
effect for a longer moving distance.

###  4.4 Discussion

With the coefficients tested, experiments simulate the moving process of
robots with collapsible wings in the difference discharge angle, explore the
effect by area of collapsible wings and the collapsible wings in ground
effect. In addition, these experiments verify the motion pattern of taking
off, flying and gliding is available for the robot with collapsible wings. As
the outcome of the simulation, bionic robot is able to control the area of the
collapsible wings to adjust the gesture and statement of moving process.
Furthermore, the use of ground effect definitely increase the flying capacity
near the water surface. Taking advantage of the ground effect is an excellent
way to extend the endurance in the future. This motion pattern truly a
outstanding strategy with collapsible wings for the aerial aquatic robot.

##  5 Conclusions

Inspired by the unique motion pattern with collapsible wings, this paper
explore the advantage of the collapsible wings in the aquatic-aerial robots. A
bionic robot prototype with collapsible wings is designed and produced to test
the coefficients of locomotion. After the dynamic modal of the robot is built,
several experiments has been test to verify the advantage of collapsible in
the multi-modal motion pattern.

This article’s main contribution includes designing aquatic-aerial robot with
collapsible wings, which use the soft hydraulic actuator, and introduce the
design and fabrication of the soft actuator structure. Dynamic model of the
robot is built and the study experimentally and computationally analyzed
various dynamic parameters of the robot, including lift coefficient, drag
coefficient, and torque coefficient at different attack angles. Finally, all
the information pretested were used to simulate and analyze the motion state
of the flying fish bio-robot with collapsible wings. The experiments
concentrate to the discharge angle, area of the collapsible wings and the use
of ground effect. Collapsible wings in the aquatic-aerial robots truly
contribute to the unique motion pattern of the flying fish and the future
aquatic-aerial robots. These outcomes of collapsible wings laying a necessary
theoretical foundation for future work.

## Acknowledgments

This work was supported by the National High Technology Research and
Development Program of China under Grant No. 2018YFE0204300, and the National
Natural Science Foundation of China under Grant No. U1964203, and sponsored by
Meituan and Tsinghua University-Didi Joint Research Center for Future
Mobility.

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