# Non-uniqueness of bubbling for wave maps

Max Engelstein  and  Dana Mendelson  Department of Mathematics
University of Minnesota-Twin Cities
Minneapolis, MN, 55455<EMAIL_ADDRESS>Department of Mathematics
University of Chicago
Chicago, IL, 60637<EMAIL_ADDRESS>
###### Abstract.

We consider wave maps from $\mathbb{R}^{2+1}$ to a $C^{\infty}$-smooth
Riemannian manifold, $\mathcal{N}$. Such maps can exhibit energy
concentration, and at points of concentration, it is known that the map
(suitably rescaled and translated) converges weakly to a harmonic map, known
as a bubble. We give an example of a wave map which exhibits a type of non-
uniqueness of bubbling. In particular, we exhibit a continuum of different
bubbles at the origin, each of which arise as the weak limit along a different
sequence of times approaching the blow-up time.

This is the first known example of non-uniqueness of bubbling for dispersive
equations. Our construction is inspired by the work of Peter Topping [Top04b],
who demonstrated a similar phenomena can occur in the setting of harmonic map
heat flow, and our mechanism of non-uniqueness is the same ’winding’ behavior
exhibited in that work.

ME was partially supported by an NSF Postdoc, NSF DMS-1703306. DM was
partially supported by NSF DMS-180069.

##  1\. Introduction

We consider wave maps from $(1+2)$-dimensional Minkowski space into a compact
Riemannian manifold $({\mathcal{N}},g)$, which are defined formally as
critical points of the Lagrangian

$\mathcal{L}(U,\partial
U)=\frac{1}{2}\int_{{\mathbb{R}}^{1+2}}\eta^{\alpha\beta}\,\langle\partial_{\alpha}U,\partial_{\beta}U\rangle_{g}dtdx,$

where $\eta$ is the Minkowski metric and g is the Riemannian manifold on
${\mathcal{N}}$. In local coordinates on ${\mathcal{N}}$, wave maps are
solutions $u$ to the system

(1.1) $\left\\{\begin{aligned} &\Box u^{i}=\Gamma_{k\ell}^{i}(u)\nabla
u^{k}\cdot\nabla u^{\ell},\\\
&(u,\partial_{t}u)\big{|}_{t=0}=(u_{0},u_{1}),\end{aligned}\right.$

where $\Gamma_{k\ell}^{i}$ are the Christoffel symbols for ${\mathcal{N}}$.

There is a conserved energy associated to (1.1) given by

(1.2) $\mathcal{E}(u,\partial_{t}u)(t)=\int_{\mathbb{R}^{2}}|\nabla
u|_{g}^{2}+|\partial_{t}u|_{g}^{2}\equiv\mathcal{E}(u_{0},u_{1}),$

whenever the right-hand side is finite. The coercivity of this energy implies
that that the $\dot{H}^{1}$-norm of solutions remains bounded for all time.
However, it is still possible for energy to concentrate producing “solitons"
or “bubbles" in the suitably rescaled weak limit. Bubbling is a well-studied
phenomena in nonlinear evolution equations, particularly in the parabolic
setting, dating back to the work of Struwe [Str85], which built on the work of
Sacks and Uhlenbeck [SU81]. We will not attempt to do justice to the vast
literature here, particularly pertaining to parabolic flows, but we refer to
[dP17] and references therein for an overview on the history. The purpose of
this paper is to provide the first example, in the context of (1.1) where this
weak limit is non-unique, see Theorem 1.7 below. In particular, we construct a
solution where, after rescaling around a particular point in space-time,
different solitons are obtained by considering weak limits along different
sequences of times.

To more precisely state our results, let us recall some background on wave
maps. We consider (1.1) for smooth, finite energy initial data belonging to a
certain symmetry class and satisfying certain quantitative higher regularity
bounds; we will make these assumptions precise below. For such initial data,
classical energy methods show that (1.1) admits a unique smooth solution for
small times, see e.g. [SS98]. When the domain is $\mathbb{R}^{2+1}$, the
energy (1.2) is scale invariant, thus solutions to (1.1) can exhibit energy
concentration even if the initial data is smooth or highly symmetric, see,
e.g. [KST08, RR12, RS10]. This concentration can be ruled in certain cases
under additional assumptions, say on the smallness of the initial energy, see
[Tao01, Tat05, Kri04], or by restricting the topology of the target, see, e.g.
[Str03a, KS12, Tao].

For general targets, Sterbenz and Tataru [ST10b] showed that inside any light
cone a dichotomy holds: either energy does not concentrate too much, in which
case solution can be smoothly extended to the whole space with quantitative
control, or the energy concentrates and the solution blows up via bubbling of
harmonic maps. To state their result (in the infinite time setting), we follow
the notation in [ST10b], and fix

$C_{[t_{0},t_{1}]}=\\{t_{0}\leq t\leq t_{1},r\leq t\\}$

for the truncated light cone and

$S_{t}=\\{(x,t)\mid|x|\leq t\\}$

for the time sections of the light cone. We set

$\mathcal{E}_{S_{t}}=\frac{1}{2}\int_{S_{t}}|\nabla
u|_{g}^{2}+|\partial_{t}u|_{g}^{2}.$

######  Theorem 1.1 ([ST10b, Theorem 1.5]).

Let $u:C_{[1,\infty)}\to\mathcal{N}$ be a $C^{\infty}$ wave map such that

$\lim_{t\to\infty}\mathcal{E}_{S_{t}}[u]<\infty.$

Then exactly one of the following possibilities must hold:

  * (a)

There exists a sequence of points $(t_{n},x_{n})\in C_{[1,\infty)}$ and scales
$r_{n}$ with

$t_{n}\to\infty,\quad\limsup_{n\to\infty}\frac{|x_{n}|}{t_{n}}<1,\quad\lim_{n\to\infty}\frac{r_{n}}{t_{n}}=0.$

so that the rescaled sequence of maps

$u^{(n)}(t,x)=u(t_{n}+r_{n}t,x_{n}+r_{n}x)$

converges strongly in $H^{1}_{loc}$ to a Lorentz transform of an entire (time-
independent) harmonic map

$u^{(\infty)}:\mathbb{R}^{2}\to\mathcal{N}$

of nontrivial energy:

$u^{(\infty)}:\mathbb{R}^{2}\to\mathcal{N},\quad
0<\|u^{(\infty)}\|_{\dot{H}^{1}({\mathbb{R}}^{2})}\leq\lim_{t\to\infty}\mathcal{E}_{S_{t}}[u].$

  * (b)

For each $\varepsilon>0$, there exists $t_{0}>1$ and a wave map extension

$u:\mathbb{R}^{2}\times[t_{0},\infty)\to\mathcal{N}$

with bounded energy:

$\mathcal{E}(u)\leq(1+\varepsilon^{8})\lim_{t\to\infty}\mathcal{E}_{S_{t}}[u],$

which satisfies

$\sup_{t\in[t_{0},\infty)}\sup_{k\in\mathbb{Z}}(\|P_{k}u(t)\|_{L^{\infty}_{x}}+2^{-k}\|P_{k}\partial_{t}u(t)\|_{L^{\infty}_{x}})\leq\varepsilon,$

where $P_{k}$ are the Littlewood-Paley projections to frequency $2^{k}$.

An analogous dichotomy holds for finite time blowup, see [ST10b, Theorem 1.3].

######  Remark 1.2.

In the previous theorem and throughout, we use $\dot{H}^{s}({\mathbb{R}}^{n})$
to denote the usual homogeneous Sobolev space, with norm defined as

$\|u\|_{\dot{H}^{s}({\mathbb{R}}^{n})}=\|(-\Delta)^{s/2}u\|_{L^{2}({\mathbb{R}}^{n})}.$

######  Remark 1.3.

It was observed in [ST10b] that part (b) in Theorem 1.1 implies that a certain
controlling norm for $u$ is finite, and in [ST10a] it was proved that this
implies that $u$ converges to a linear wave after applying a suitable gauge
transformation. We note that this is called scattering in the terminology of
[ST10b], but is different from the usual use of that term in nonlinear
dispersive equations, see [LO16, Remark 1] for some discussion. We adopt this
terminology for brevity when referring to the behavior in part (b) of this
theorem.

Instead of working with the modified definition of scattering in [ST10a], we
instead adopt a more direct approach to demonstrate that our flow exhibits
energy concentration via degree considerations. We believe this approach
illuminates certain features of scattering in the current setting. This
argument is inspired by arguments in [CTZ93, Str13, CKLS15b, LO16], see
Section 4 for more details.

The first case in Theorem 1.1 describes singularity formation via “bubbling”
(a.k.a. energy concentration) to non-constant harmonic maps. Note that the
bubbling phenomenon described in Theorem 1.1 leaves open the question of
whether the convergence along a discrete sequence of times and scales can be
strengthened to convergence as $t\rightarrow\infty$ after rescaling,
translating and applying Lorentz transformations by some continuous functions,
$r(t)$, $x(t)$ and $\gamma(t)$.

To reduce the number of parameters, and to provide an example of non-
uniqueness in what we speculate is the simplest possible setting, we will
impose an additional symmetry on our solution, which will allow us to ignore
translations and Lorentz transformations:

######  Definition 1.4.

We say that a wave map $u(x,t):{\mathbb{R}}^{1+2}\rightarrow\mathcal{M}$ is
quasi-equivariant if there exists a smooth one-parameter family of isometries,
$\Phi_{s}\in\mathrm{Isom}(\mathcal{M})$. with
$\Phi_{0}=\mathrm{Id}_{\mathcal{M}}$, such that (using polar coordinates) one
has $u(r,\theta+s,t)=\Psi_{s}\circ u(r,\theta,t)$ for all
$r>0,s,\theta\in\mathbb{S}^{1}$ and $t\in[0,T_{\max})$.

######  Remark 1.5.

Many of Topping’s constructions for harmonic map heat flow (see, e.g. [Top04b,
Top97]) exhibit quasi-equivariant symmetries. While, Topping does not give
this symmetry a name, we introduce the term “quasi-equivariance" to stress its
relation to the well-studied notion of equivariant wave maps. In the language
of Definition 1.4, if $\mathcal{M}$ is a surface of rotation and $\Phi_{s}$ is
a rotation of $\mathcal{M}$ by $s$ radians around the same axis, then $u$ is
equivariant.

We can now define of uniqueness of a bubbling singularities for quasi-
equivariant wave maps:

######  Definition 1.6.

We say that a quasi-equivariant wave map,
$u:\mathbb{R}^{1+2}\rightarrow\mathcal{M}$, has a unique bubble at a point
$0\in\mathbb{R}^{2}$ and time $T_{max}>0$ if there exists a continuous
function, $r(t):[0,T_{\max})\rightarrow(0,+\infty)$, with

(1.3) $r(t)=\begin{cases}o(T_{max}-t)&T_{max}<\infty\\\
o(t)&T_{max}=\infty\end{cases}$

such that $u(t,r(t)x)\rightarrow\omega(x)$ in
$C_{\mathrm{loc}}^{0}(\mathbb{R}^{2}\backslash\\{0\\};\mathcal{M})$ for some
non-trivial harmonic map $\omega:\mathbb{R}^{2}\rightarrow\mathcal{M}$.

In contrast to part $(a)$ of Sterbenz-Tataru’s Theorem 1.1, which yields
convergence to a harmonic map along a sequence of times, the previous
definition describes convergence (after application of the relevant symmetries
in the quasi-equivariant class) to a unique harmonic map along _all times_. In
general, it is difficult to prove that uniqueness in the sense of Definition
1.6 fails. However, our Theorem 6.6 shows that if a singularity is “winding"
(see Definition 1.10), then the bubble is not unique.

Using this approach, our main theorem gives the first example of a bubbling
solution to a non-linear wave equation for which uniqueness in the sense of
Definition 1.6 is known not to hold.

######  Theorem 1.7 (Main Theorem).

There exists a compact smooth Riemannian manifold $(\mathcal{N},g)$ given by

$\mathcal{N}=\mathbb{T}^{2}\times_{f}\mathcal{S}^{2}$

for a certain $C^{\infty}({\mathbb{T}}^{2})$ warping function, $f$, and
$C^{\infty}$-smooth, finite energy, quasi-equivariant initial data
$(u_{0},u_{1})$, which satisfy

$\|(u_{0},u_{1})\|_{\dot{H}^{3}\times\dot{H}^{2}}<\infty,\qquad\mathcal{E}(u_{0},u_{1})<\mathcal{E}_{\mathrm{quasi}}(\mathcal{N})+\varepsilon_{1},$

such that the corresponding solution $(u,u_{t})$ to (1.1) has a bubbling
singularity as $t\to T_{\max}$ which fails to be unique in the sense of
Definition 1.6. Above, $\varepsilon_{1}>0$ is a constant which depends only on
$\mathcal{N}$, and $\mathcal{E}_{\mathrm{quasi}}(\mathcal{N})$ denotes the
smallest energy of a non-trivial quasi-equivariant harmonic map,
$\omega:\mathbb{S}^{2}\rightarrow\mathcal{N}$.

The energy assumption above is reminiscent of work which considers wave maps
with energies slightly above the “ground state" (that is, the lowest energy of
a non-trivial harmonic map into the target manifold). We do not show that
$\mathcal{E}_{\mathrm{quasi}}(\mathcal{N})$ is the energy of the ground state,
but it plays that role within our considered symmetry class and we will
sometimes abuse terminology and refer to it as the energy of the ground state
(see Lemma 5.4 and the remark after).

######  Remark 1.8.

A few clarifying remarks on Theorem 1.7:

  * •

We note that blow-up for radially symmetric wave maps into general targets has
been ruled out under a number of different weak assumptions [Str03b, Nah13,
CKL18]. Thus it is perhaps unreasonable to expect that Theorem 1.7 could hold
under stronger symmetry assumptions on the initial data.

  * •

The fact that $f\in C^{\infty}$, as opposed to analytic, allows $f(p)-f(q)$ to
vanish to arbitrarily high order as $p\rightarrow q$. Much previous work on
wave maps has assumed non-degeneracy conditions (e.g. conditions (A1)-(A3) in
Section 3 of [JK17]) which rule out this behavior.

  * •

In addition to $C^{\infty}$ smoothness and finite energy, we require some
additional quantitative regularity of the initial data. While
$\dot{H}^{1+\varepsilon}\times\dot{H}^{\varepsilon}$ regularity (for any
$\varepsilon>0$) would suffice, we assume
$\dot{H}^{3}\times\dot{H}^{2}$-regularity of the initial data since the proof
of local wellposedness proceeds by classical energy methods and is simpler
than the techniques required to get the almost critical result. This
additional regularity is used in Lemma 4.1.

  * •

The proof of Theorem 1.7 actually establishes the existence of an infinite
family of quasi-equivariant initial data $(u_{0},u_{1})$ for which the
corresponding solution has non-unique bubbling, as opposed to a specific
construction of the initial data.

######  Remark 1.9.

We do not, currently have more precise control on the rates $\\{r_{n}\\}$ for
the bubbling in Theorem 1.7 than what is provided in [ST10b]. In the case of
equivariant wave maps into $\mathbb{S}^{2}$, it is remarked in [CKLS15b] that
various possibilities exist, and we refer to [Jen17, Pil19] for the
construction of an infinite-time blow-up in this setting. In [Top04b], Topping
establishes a lower bound on the rate of blow-up in the context of harmonic
map heat flow. A key ingredient in Topping’s proof of this lower bound is a
quantitative “neck estimate”, see [Top04b, Lemma 4.4], due to Qing and Tian
[QT97], see also Lemma 2.9 in [Top04a]. While neck estimates have been
recently made available for wave maps into spheres in the work of Grinis
[Gri17], those estimates are non-quantitative, and thus it remains unclear how
to use them to gain control on the rate of bubbling for wave maps, even if we
were to adapt them to the case of general targets.

###  1.1. Soliton Resolution Conjecture and Prior Work

One motivation for establishing Theorem 1.7 stems from the large amount of
recent activity in establishing the soliton resolution conjecture for
nonlinear wave and wave map equations. The soliton resolution conjecture
posits that solutions for a broad class of nonlinear dispersive equations
should decompose asymptotically as a sum of “bubbles" and radiation. In the
setting of (quasi-)equivariant wave maps,
$\psi:\mathbb{R}^{2+1}\rightarrow\mathcal{M}$, the conjecture states that if
$\psi$ develops at least one bubble at $t=\infty$, then there exists a
collection of finite energy harmonic maps,
$\\{Q_{j}\\}_{j=1}^{J}:\mathbb{R}^{2}\rightarrow\mathcal{M}$, continuous
scaling parameters,
$0<\lambda_{1}(t)\ll\lambda_{2}(t)\ll\ldots\ll\lambda_{J}(t)\ll t$, and a
finite energy linear wave, $\phi_{L}$, such that

(1.4)
$\psi(-,t)=\phi_{L}(-,t)+\sum_{j=1}^{J}Q_{j}\left(\frac{-}{\lambda_{j}(t)}\right)+\varepsilon(t),$

where $\varepsilon(t)\rightarrow 0$ in the appropriate function space as
$t\rightarrow\infty$ (see the introduction of [Côt15, Gri17] for a more
detailed discussion of the soliton resolution conjecture for wave maps into
the sphere). In particular, (1.4) implies uniqueness in the sense of
Definition 1.6 for each of the bubbles, $Q_{j}$. We note that such a
description goes beyond Theorem 1.1 which, together with uniqueness in the
sense of Definition 1.6, only describes the dynamics of energy concentration
at one scale. We refer to the examples of [Jen19, JL18] which demonstrate that
such multi-scale concentration is in fact possible.

Much progress has been made on the soliton resolution conjecture a variety of
settings. Thus far, the full conjecture has been proved in the work of
Duyckaerts, Kenig and Merle [DKM13] and [DKM19] for the radial, focusing
energy-critical nonlinear wave equation in odd space dimensions, and wave maps
into $\mathbb{S}^{2}$ under various symmetry and energy assumptions, see, for
instance, [CKLS15a, CKLS15b, DJKM18, JL18] and references therein.
Nonetheless, in many cases the conjecture in its full strength remains open.
Often, one can show that an asymptotic decomposition in the vein of (1.4)
holds along a sequence of times $t_{n}\rightarrow T_{\max}$, see [Côt15, JK17,
DJKM17]. In these instances, the difficulty becomes proving that the
decomposition is independent of the sequence $\\{t_{n}\\}$. Our main theorem,
Theorem 1.7, demonstrates that in some settings, it may be impossible to move
beyond the soliton resolution along a sequence of times to the full
conjecture.

We hope that our theorem may shed some light on the specific difficulties in
proving continuous time soliton resolution. Indeed, for quasi-equivariant non-
linear wave equations there are two main ways in which the decomposition could
depend on the sequence of times. The first is that the bubbles can “switch
places", i.e. $Q_{j}$ may develop at a smaller scale than $Q_{k}$ along one
sequence of times but at a larger scale along another sequence of times. The
second is that the bubbles, $Q_{j}$, themselves could depend on the sequence
of times considered. In both cases, a careful understanding of the potential
bubbles, i.e. non-trivial harmonic maps at specified energy levels, and
possible interactions between bubbles separated in scale or in space has
appeared to be a crucial ingredient in proofs which ultimately establish
uniqueness for such problems (see, e.g. [JL18, DKM19] respectively).

A complete understanding of the possible bubbles and their interactions is
often achieved by imposing symmetry assumptions; for example, there is one (up
to rescaling) radially symmetric stationary solution to the non-linear wave
equation considered in [DKM13]. In other situations, energy constraints can
rule out multi-soliton configurations, see, for instance, [CKLS15a, CKLS15b]
which establishes continuous-time soliton resolution for equivariant wave maps
into $\mathbb{S}^{2}$ with energy less than three times the energy of the
lowest energy harmonic map into the sphere, and [DJKM18] for the result
without the equivariance assumption when the energy is restricted to just
above the energy of the ground state. In contrast to these cases, the target
manifold in Theorem 1.7 admits a continuum of quasi-equivariant harmonic maps
$\omega:\mathbb{R}^{2}\rightarrow\mathcal{N}$ at the lowest admissible non-
zero energy level, and the richness of this family plays an essential role in
the proof of non-uniqueness for Theorem 1.7.

Finally, we note one further aspect of our setting which contributes to non-
uniqueness, specifically compared to the setting considered in [DJKM18]. While
in both cases wave maps with energy just above the ground state are
considered, in [DJKM18] the authors exploit the fact that for wave maps into
spheres, the energy is coercive near the traveling waves, which traps the wave
map in increasingly small neighborhoods of the traveling wave, yielding
uniqueness. In contrast, while the second component of our target manifold is
the sphere, the first component has an infinite length geodesic which, using
standard coordinates $\mathbb{T}^{2}=(w,z)$, wraps around the torus infinitely
many times as it approaches the circle $\\{w=0\\}$. This winding behavior
allows the first coordinate of the wave map to exit a small neighborhood of a
soliton infinitely many times, even though the behavior of the second
component of the map will be well controlled (i.e. the second component may be
almost constant).

###  1.2. Comparison with non-uniqueness in elliptic and parabolic problems

Our construction is heavily inspired by Topping’s construction of a harmonic
map heat flow which develops a non-unique bubble in finite time [Top04b]. Our
target manifold, including the warping function, is essentially equivalent to
Topping’s (more on this in Section 3). We also use the same mechanism as
Topping to ensure non-uniqueness, what he calls “winding". Note that we record
only the definition of winding as it applies in the quasi-equivariant setting,
and a more general definition would take into account translational and
Lorentz symmetries.

######  Definition 1.10 ([Top04b, Definition 1.12]).

A quasi-equivariant wave map $u:\mathbb{R}^{1+2}\rightarrow\mathcal{M}$ has a
winding singularity at time $T_{\max}$ and the origin $0\in\mathbb{R}^{2}$ if
there exists sequences $\\{r_{n}\\}$, and $\\{t_{n}\\}$, satisfying

(1.5) $t_{n}\uparrow T_{\max},\qquad
r_{n}=\begin{cases}o(T_{max}-t_{n})&T_{max}<\infty\\\
o(t_{n})&T_{max}=\infty\end{cases}$

such that $u(t_{n},r_{n}x)\rightarrow\omega(x)$ in
$C^{0}_{\mathrm{loc}}({\mathbb{R}}^{2}\backslash\\{x_{0}\\};\mathcal{M})$,
where $\omega$ is a non-constant harmonic map, the lifts
$\hat{u}(t_{n},r_{n}x)$ have no convergent subsequence in
$C^{0}_{\mathrm{loc}}({\mathbb{R}}^{2}\backslash\\{x_{0}\\};\widehat{\mathcal{M}})$.

Despite the similarities in set-up, the execution of our proof differs
substantially from Topping’s. This is due to fundamental differences between
the parabolic and Hamiltonian settings, even at the level of ODEs, which we
had to overcome. To elucidate these issues, we elaborate here on two examples.

For analytic functions, $f$, bounded gradient flows, $\dot{x}(t)=-\nabla
f(x(t))$, have a unique limit as $t\rightarrow\infty$, due to the Łojasiewicz
inequalities [Łoj65]. It is a beautiful observation of L. Simon [Sim83] that
this fact about ODEs can be applied to study the long time behavior of the
gradient flows of many elliptic functionals which arise naturally in geometry.
These “Łojasiewicz-Simon" inequalities have found subsequent use in a huge
range of geometric and variational problems. For example, to prove uniqueness
of tangent objects for variational problems (e.g. minimal surfaces, [Sim83]
and mean curvature flow [CM15]) and to show the uniqueness of long time limits
of geometric flows (e.g. Yamabe flows [Bre05], harmonic map heat flow [FM19]).
However, this phenomena does not hold in the Hamiltonian setting. Indeed, if
$f(x,y)=\frac{1}{2}(x^{2}+y^{2})$ then the equation $\ddot{x}(t)=-\nabla
f(x(t))$ becomes $\ddot{x}(t)=-x(t)$. One solution to this ODE is the bounded
flow $x(t)=(\cos(t),\sin(t))$, which clearly does not have a unique limit as
$x(t)\rightarrow\infty$.

When $f$ is not analytic, but is $C^{\infty}$, then the classic “goat tracks"
example

(1.6) $f(r,\theta)=\begin{cases}1&\quad r\leq 1,\\\
1+e^{-\frac{1}{r-1}}\left(\sin(\frac{1}{r-1}+\theta)+2\right)&\quad
r>1,\end{cases}$

generates a gradient flow $\dot{x}(t)=-\nabla f(x(t))$ which is bounded but
doesn’t have a unique limit as $t\rightarrow\infty$, in fact, every point on
the circle $r=1$ is a limit point. This example is at the heart of Topping’s
construction [Top04b] (see also the examples of non-uniqueness for
singularities of harmonic maps, [Whi92], and the long term behavior of
harmonic map heat flow, [Top97]). In contrast, the corresponding flow given by
$\ddot{x}(t)=-\nabla f(x(t))$ cannot exhibit the same asymptotic behavior as
the gradient flow does. The Hamiltonian flow stays bounded as long as $|x(0)|$
is close to one and $\dot{x}(0)$ is small enough, however, by working in polar
coordinates, one can see that if $|x(t)|\rightarrow 1$ as $t\rightarrow\infty$
it must also be the case that $|\dot{x}(t)|\rightarrow 0$ as
$t\rightarrow\infty$. This would violate the conservation of the energy,
$|\dot{x}|^{2}(t)+f(x(t))$ (provided $\dot{x}(0)\neq 0$).

As these examples show, one needs caution when using long-term behavior of
non-linear parabolic flows to provide insight into the Hamiltonian setting. On
a practical level, while energy conservation provides some control for
Hamiltonian flows, it is not a substitute for the maximum principle and energy
dissipation, which hold in the parabolic setting. For example, as in [Top04b]
we show that the image of the flow is contained in a geodesic in the target
manifold, see Lemma 4.1. However, we are faced with the additional difficulty
of showing that the flow cannot leave this geodesic before the blow-up time.
We note that this issue is not present in the parabolic setting, where
stationary solutions act as a barrier to constrain the flow. Additionally,
energy dissipation allows Topping to determine that his flow blows up in
finite time via topological considerations and Lemaire’s theorem. On the other
hand, we must leave open the possibility that the winding singularity may
occur at either finite or infinite time (we speculate either situation can
occur).

Nonetheless, we believe that our proof exhibits, morally, a phenomenon
exploited in work of Grinis [Gri17]: that nonlinear wave equations start to
exhibit elliptic behavior in the (strict) interior of a light-cone in which
energy is concentrating.

###  1.3. Structure of the Paper

Here we briefly outline the structure of the paper. In Section 2 we record
some preliminaries about wave maps and harmonic maps into general Riemannian
manifolds.

In Section 3 we construct the target manifold $\mathcal{N}$ from Theorem 1.7.
This follows much as in [Top04b, Section 3], with additional complications
caused by the fact that we are unable to use the maximum principle to
constrain the image of the flow. We overcome this difficulty by working with a
compact target manifold and carefully defining our metric twist globally on
${\mathbb{T}}^{2}$.

In Section 4 we establish some preliminary results on the Hamiltonian flow. In
subsection 4.1 we also rule out “scattering" to a solution of the underlying
linear wave equation. In [Top04b] the analogous parabolic phenomena,
relaxation to a stationary solution, can be quickly ruled out using topology.
The wave map setting requires a more involved estimate of the energy flux
through the wall of the light cone.

In Section 5 we study the harmonic maps into $\mathcal{N}$ which can arise as
bubbles in the flow. Here the analysis is complicated by the fact that the
metric twist had to be defined globally on ${\mathbb{T}}^{2}$. In particular,
we use energy arguments to rule out bubbles which wrap “the wrong way" around
the ${\mathbb{T}}^{2}$ component, see Lemma 5.4.

Finally, in Section 6 we establish properties of the singularity, in
particular winding, proving Theorem 1.7.

###  1.4. Acknowledgements

This work was done while ME was visiting the University of Chicago for the AY
2019-2020; he thanks the department and especially Carlos Kenig for their
hospitality. He also thanks Carlos Kenig for suggesting he investigate the
phenomena of bubbling for harmonic map heat flow. DM learned about Topping’s
work on non-uniqueness for the harmonic map heat flow bubbling, and the open
question of non-uniqueness for wave maps bubbling, in a topics course given by
Carlos Kenig at the University of Chicago. Both ME and DM benefited from
several discussions with Andrew Lawrie, including discussions about Topping’s
work on harmonic map heat flow and about scattering for wave maps. The authors
thank both Carlos Kenig and Andrew Lawrie for helpful comments on an earlier
version of this manuscript.

##  2\. Preliminaries for Wave Maps and Harmonic Maps

In this section we collect some basic facts about the regularity of (quasi-
equivariant) wave maps and harmonic maps. As mentioned in the introduction, we
will consider wave maps

$u:\mathbb{R}^{1+2}\to\mathbb{T}^{2}\times_{f}\mathbb{S}^{2}$

for a certain $C^{\infty}({\mathbb{T}}^{2})$ warping function, $f$. Using
polar coordinates in the domain of $u$, we suppose that the initial condition
has the form

(2.1) $u_{0}(r,\theta)=(0,y_{0},\alpha_{0}(r),\theta),\qquad
u_{1}(r,\theta)=(0,y_{1}(r),\alpha_{1}(r),0).$

We will rely on the following local existence and persistence of regularity
result, which follows via energy estimates. Note too that the optimal local
theory is known, see for instance [KS97].

######  Proposition 2.1 (Classical local existence and persistence of
regularity).

Let $s_{0}>1$ and let
$(u_{0},u_{1})\in\dot{H}^{s_{0}+1}\times\dot{H}^{s_{0}}$. There exists a
$T_{max}\equiv T_{max}(u)>0$ such that for every $T<T_{max}$, there exists a
unique solution

$u:[0,T]\times\mathbb{R}^{1+2}\to\mathcal{N}$

of (1.1) such that

(2.2) $\displaystyle\sup_{0<t\leq T}\|u(t)\|_{\dot{H}^{s_{0}+1}}<\infty.$

Moreover, if $(u_{0},u_{1})\in H^{s+1}\times H^{s}$ for any $s>s_{0}$, then

$\sup_{0<t\leq T}\|u(t)\|_{\dot{H}^{s+1}}<\infty.$

The uniqueness conclusion of Proposition 2.1 implies that the wave map retains
rotational and quasi-equivarient symmetry. Therefore, the solution will have
the form

(2.3) $\displaystyle u(t,r,\theta)=(X(t,r),Y(t,r),\alpha(t,r),\theta).$

In the coordinates of (2.3), the energy has the form

(2.4)
$\mathcal{E}(t)=2\pi\int_{0}^{\infty}\left(\frac{1}{2}|\nabla(X,Y)|^{2}+\frac{1}{2}|\partial_{t}(X,Y)|^{2}+f(X,Y)e(\alpha)\right)rdr,$

where $e(\alpha)$ is the “spherical" part of the energy

$e(\alpha):=\frac{1}{2}\left[\left(\frac{\partial\alpha}{\partial
r}\right)^{2}+\left(\frac{\partial\alpha}{\partial
t}\right)^{2}+\frac{\sin^{2}\alpha}{r^{2}}\right].$

If the initial conditions (2.1) have finite energy,
$\mathcal{E}(u_{0},u_{1})<\infty$, then
$\mathcal{E}(t)\equiv\mathcal{E}(u_{0},u_{1})$ for all $t<T_{\max}$.
Furthermore, if $u$ bubbles to $\omega$ at $T_{\max}$ in the sense of Theorem
1.1 or the analogous finite time result, then as stated in case (a) of that
theorem, $\int|D\omega|^{2}\leq\mathcal{E}(u_{0},u_{1})$.

In order to satisfy

$\int_{0}^{\infty}f(X,Y)e(\alpha)rdr<\infty$

for all $t$ it must be the case that $\alpha(t,0)=m\pi$ and
$\alpha(t,\infty)=n\pi$ for all $t\in I_{max}(u)$ for some integers $n,m$.
That the integers must be constant for all $t$ follows from the continuity in
time of the flow. We will take $m=0$ in the sequel, and we define the _degree_
of the wave map to be the integer $n$.

###  2.1. Regularity for Weakly Harmonic Maps from ${\mathbb{R}}^{2}$

Let $(\mathcal{N},g)$ be a closed smooth Riemannian manifold. Throughout we
will assume that $\mathcal{N}$ is smoothly embedded into $\mathbb{R}^{n}$.

Weakly harmonic maps
$\omega:{\mathbb{R}}^{2}\rightarrow\mathcal{M}\subset\mathbb{R}^{n}$ are
critical points of the energy $E(u)\equiv\int_{{\mathbb{R}}^{2}}|Du|^{2}_{g}$
under perturbations of the form
$u_{\varepsilon}(x)=\pi(u+\varepsilon\varphi(x))$ where $\pi$ is the nearest
point projection of $\mathbb{R}^{n}$ onto $\mathcal{N}$, which is well defined
and smooth in a small neighborhood of $\mathcal{N}$, and $\varphi\in
C^{1}_{c}({\mathbb{R}}^{2};{\mathbb{R}}^{n})$.

Equivalently, these maps (weakly) satisfy the semi-linear PDE:

$-\Delta\omega+A(\omega)(\nabla\omega,\nabla\omega)=0$

where $A(\omega)$ is the second fundamental form of $\mathcal{N}$. In general,
weakly harmonic maps need not be continuous (see, e.g. [Riv95]), but when the
domain is two-dimensional all weakly harmonic maps are $C^{\infty}$ by the
fundamental result of [Hél91]. As such, we will refer to weakly harmonic maps
from $\mathbb{R}^{2}$ as simply harmonic maps. Finally, if
$\omega:{\mathbb{R}}^{2}\rightarrow{\mathcal{N}}$ is harmonic, then (composing
with a stereographic projection) we get a harmonic map
$\widetilde{\omega}:\mathbb{S}^{2}\backslash\\{\infty\\}\rightarrow{\mathcal{N}}$,
which we can smoothly extend to all of $\mathbb{S}^{2}$ by the work of Sachs-
Uhlenbeck [SU81]. We will often abuse notation and identify the harmonic maps
$\omega,\widetilde{\omega}$.

Our first theorem quantifies the regularity of two-dimensional harmonic maps
(the precise statement non-minimizing maps on ${\mathbb{R}}^{2}$ is hard to
track down. However, one can argue as in [SU83] or consider the stationary
case of the parabolic regularity proven in [Str88]):

######  Theorem 2.2.

Let $\omega:\mathbb{R}^{2}\rightarrow\mathcal{N}$ be a harmonic map. There
exists an $\bar{\varepsilon}>0$ small, and depending on $\mathcal{N}$, such
that if

(2.5) $\int_{B_{R}(x_{0})}|D\omega|^{2}\leq\bar{\varepsilon},$

then,

(2.6) $R^{2}\sup_{B_{R/2}(x_{0})}|D\omega|^{2}\leq
C\int_{B_{R}(x_{0})}|D\omega|^{2}dx,$

where $C>0$ depends on $\mathcal{N}$ but not on $x_{0},R$ or $\omega$.
Equivalently, given the small energy condition (2.5)

(2.7) $\sup_{x,y\in B_{R/2}(x_{0})}d_{\mathcal{N}}(\omega(x),\omega(y))\leq
C\left(\int_{B_{R}(x_{0})}|D\omega|^{2}dx\right)^{1/2},$

where $d_{\mathcal{N}}(p,q)$ is the geodesic distance between
$p,q\in\mathcal{N}$ and $C>0$ is as above.

There are two standard corollaries of Theorem 2.2 which will be important to
us. The first states that there is a least energy non-trivial harmonic map
into any target.

######  Corollary 2.3.

For any weakly harmonic map $\omega:{\mathbb{R}}^{2}\rightarrow\mathcal{N}$,
one of following two hold:

  * •

$\mathcal{E}(\omega)\equiv\int_{{\mathbb{R}}^{2}}|D\omega|^{2}=0$ and $\omega$
is almost everywhere equal to a constant $p\in\mathcal{N}$.

  * •

$\mathcal{E}(\omega)\geq\bar{\varepsilon}>0$, where $\bar{\varepsilon}$ is the
constant from Theorem 2.2

The second corollary states that the shortest path between two points in the
image of a harmonic map cannot have infinite length:

######  Lemma 2.4.

Let $\omega:{\mathbb{R}}^{2}\rightarrow\mathcal{N}$ be a weakly harmonic map.
There cannot be two points $p_{1},p_{2}\in\mathrm{Im}\,\omega$ such that the
shortest path between the two points in $\mathcal{N}$ has infinite length.

###### Proof.

Let $x,y\in{\mathbb{R}}^{2}$ and let $L_{x,y}$ be the line segment connecting
the two. Let $E$ be the energy of the harmonic map. Since $\omega\in
C^{\infty}({\mathbb{R}}^{2})$, for all $z\in L_{x,y}$ there exists an
$r_{z}>0$ such that

$\int_{B_{r_{z}}(z)}|D\omega|^{2}<\bar{\varepsilon}.$

By compactness there are finitely many $z_{i}\in L_{x,y}$ such that

$L_{x,y}\subset\bigcup_{i=1}^{M}B_{r_{z_{i}}}(z_{i}).$

Using the oscillation estimate (2.7), we have that
$d_{\mathcal{N}}(\omega(x),\omega(y))\leq CM\sqrt{\bar{\varepsilon}}<\infty$.
∎

We end this section with some elementary facts about harmonic maps
$\omega:\mathbb{S}^{2}\rightarrow\mathbb{S}^{2}$.

######  Lemma 2.5.

Let $\omega$ be a harmonic map $\mathbb{S}^{2}\rightarrow\mathbb{S}^{2}$. Let
$\mathcal{E}_{\mathbb{S}^{2}}$ denote the lowest energy level of a non-trivial
harmonic map between spheres, which is guaranteed to exist by Corollary 2.3.
There is a unique (up to a conformal transformation of $\mathbb{S}^{2}$)
equivariant harmonic map $\omega(r,\theta)=(\alpha(r),\theta)$ such that
$\mathcal{E}(\omega)=\mathcal{E}_{\mathbb{S}^{2}}$. Furthermore, if $\omega$
is equivariant and $\mathcal{E}(\omega)>\mathcal{E}_{\mathbb{S}^{2}}$ then it
must be that $\mathcal{E}(\omega)\geq 2\mathcal{E}_{\mathbb{S}^{2}}$.

##  3\. Construction of the target

We now construct the target manifold ${\mathcal{N}}$ using a modification of
Topping’s construction in [Top04b]. Recall that we will construct
${\mathcal{N}}$ as a twisted product of the torus $\mathbb{T}^{2}$ with
$S^{2}$, i.e.

${\mathcal{N}}=\mathbb{T}^{2}\times_{f}S^{2},$

where $f\in C^{\infty}({\mathbb{T}}^{2})$ denotes the warping function. We
introduce coordinates $(w,z)$ on ${\mathbb{T}}^{2}$ where
$w,z\in[0,1]\cong\mathbb{S}^{1}$.

We define the curve

$\gamma(s)=\left(\frac{1}{\pi}\cot^{-1}(s),\,s\mod 1\right),\quad
s\in(-\infty,\infty),$

where inverse cotangent is defined so that
$\cot^{-1}(s):(-\infty,\infty)\rightarrow[0,\pi]$. Observe that $\cot^{-1}$ is
$C^{\infty}$ with uniform control on all derivatives in any
$[-K,K]\subset(-\infty,\infty)$. We want to define a metric on
${\mathbb{T}}^{2}$ such that $\gamma$ is a geodesic. We claim that

$h(w,z)=\begin{pmatrix}\pi^{2}&\pi\sin(\pi w)^{2}\\\ \pi\sin(\pi
w)^{2}&1+\sin(\pi w)^{4}\end{pmatrix}$

gives such a metric. To see that this is the case, and to simplify our
analysis of the wave maps equation, it will be useful to make the following
coordinate change:

$\Phi:(w,z)\mapsto(x,y)=(\cot(\pi w)-z,z),\qquad
D\Phi=\begin{pmatrix}\frac{-\pi}{\sin^{2}(\pi w)}&-1\\\ 0&1\end{pmatrix}$

We note that $D\Phi$ is well defined and invertible away from $w=0\equiv 1$,
and we will show in Lemma 4.1 that when $t<T_{\max}$ the flow stays away from
this curve, hence the change of variables remains valid up until the first
blow-up time. We further note that $\Phi$ respects the symmetry of
${\mathbb{T}}^{2}$ given by $w\cong w+1$ and that the image of $\Phi$ will
have the symmetry $(x,y)\cong(x+1,y-1)$, and thus will be a cylinder. We
shouldn’t expect the image of $\Phi$ to be a topological torus as $\Phi$ is
not well defined at $w=0,1$.

In the $(x,y)$ coordinates

$\gamma(s)=(0,s),\quad s\in(-\infty,\infty)$

and the pushforward of the metric, $h$, is given by

$\textbf{h}(x,y)=\begin{pmatrix}\frac{1}{(1+(x+y)^{2})^{2}}&0\\\
0&1\end{pmatrix}.$

It is now straightforward to compute that $\gamma$ is a geodesic. As a check,
we observe that $(x,y)\cong(x+1,y-1)$ is an isometry of $\mathrm{Im}\,\Phi$
equipped with the metric h.

We now turn to the construction of the warping function $f(w,z)$. Here we face
a technical difficulty not present in [Top04b] which is our inability to
constrain the flow using the maximum principle. Thus we need to carefully
define the warping function on all of $\mathcal{N}$ and not just in a
neighborhood of $w=0$. We do this by means of a smooth cutoff function, the
existence of which is guaranteed by the following lemma.

######  Lemma 3.1.

There exists a $C^{\infty}$ bump function
$\chi:{\mathbb{T}}^{2}\rightarrow[0,1]$ with the following properties:

(3.1) $\begin{cases}\chi\equiv 1&w\in[0,1/4]\cup[3/4,1]\\\ \chi\equiv
0&w\in[7/16,9/16]\\\
\nabla\chi|_{\gamma}\parallel\dot{\gamma}&w\in(1/4,3/4).\end{cases}$

###### Proof.

We provide only a sketch of this proof. To ease notation, let $X=\\{(w,z)\mid
w\in(1/4,3/4)\\}\subset{\mathbb{T}}^{2}$. To define the cut-off function
$\chi$, we smoothly connect a cut-off defined in a neighborhood of the
geodesic $\gamma$, with another depending only on $w$ outside this
neighborhood. The latter construction is simpler, so we only describe the
former. Let
$\gamma_{\delta}=\\{p\in{\mathbb{T}}^{2}\mid\mathrm{dist}(p,\gamma)<\delta\\}$.
Inside of $X$, $\gamma$ is parameterized by a $C^{\infty}$ function with
uniform bounds on all the derivatives, therefore, there exists a
$\delta_{0}>0$ such that we can smoothly parameterize $\gamma_{\delta_{0}}$ by
$(t,n)$ where the $t$ coordinates are parallel to $\gamma$ and the $n$
coordinates are normal to $\gamma$. In $X\cap\gamma_{\delta_{0}/2}$, we define
$\chi$ to depend only on $t$ (working in $t,n$ coordinates) and to
$C^{\infty}$ interpolate between $0$ and $1$ so that the first two conditions
of (3.1) hold. ∎

With $\chi$ defined we can now define the metric twist $\tilde{f}$ in a
neighborhood of $w=0\cong 1$:

$\tilde{f}(w,z)\equiv e^{-2\pi\cot(\pi w)}\left(\sin 2\pi\bigl{(}\cot(\pi
w)-z-1/8\bigr{)}+\sqrt{2}\right)+1,$

and then globally define the twist by

(3.2) $f(w,z)=\begin{cases}1&\>\>w=0\\\ \tilde{f}(w,z)&\>\>0<w\leq 1/4\\\
\chi(w,z)\tilde{f}(w,z)+(1-\chi(w,z))M&\>\>1/4<w\leq 1/2\\\
\chi(w,z)\tilde{f}(1-w,1-z)+(1-\chi(w,z))M&\>\>1/2<w\leq 3/4\\\
\tilde{f}(1-w,1-z)&\>\>3/2<w<1\end{cases}$

where $M>1$ is a constant to be chosen later. Note that $f$ is $C^{\infty}$
away from $w=0,1$ (as it is the sum of products of $C^{\infty}$ functions). At
$w=0$ we observe that $\tilde{f}-1$ vanishes to infinite order and similarly
$\tilde{f}(1-w,1-z)$ at $w=1$. Thus $f\in C^{\infty}({\mathbb{T}}^{2})$, and
furthermore, $f$ is invariant under the isometries of the space, i.e.
$(w,z)\cong(w+n,z+m)$ for $(n,m)\in\mathbb{Z}\times\mathbb{Z}$.

In $(x,y)$ coordinates, we have

$\tilde{f}(x,y)=e^{-2\pi(x+y)}\left(\sin
2\pi\bigl{(}x-1/8\bigr{)}+\sqrt{2}\right)+1.$

It is a straightforward calculation to see the following properties of $f$:

  * (i)

$\partial_{x}\tilde{f}(0,y)=0$

  * (ii)

$\partial_{y}\tilde{f}(0,y)<0$.

Combined with what we know about $\chi$, this implies that $\nabla
f|_{\gamma}$ is parallel to $\gamma$ in all of
${\mathbb{T}}^{2}\backslash\\{w=0\\}$, allowing for the fact that $\nabla f$
is zero away from the support of $\chi$.

We can define $f$ globally in $(x,y)$ coordinates by

(3.3) $f(x,y)=\begin{cases}\tilde{f}(-x,-y)&\>\>x+y<-\cot^{-1}(\pi/4)\\\
\chi(x,y)\tilde{f}(-x,-y)+(1-\chi(x,y))M&\>\>x+y\in[-\cot^{-1}(\pi/4),0)\\\
\chi(x,y)\tilde{f}(x,y)+(1-\chi(x,y))M&\>\>x+y\in[0,\cot^{-1}(\pi/4))\\\
\tilde{f}(x,y)&\>\>x+y>\cot^{-1}(\pi/4)\end{cases}$

Furthermore, by picking $M>2\sup_{{\mathbb{T}}^{2}}\tilde{f}$ we can guarantee
that

$\mathrm{sgn}(y)\partial_{y}f(0,y)\leq 0\textup{ with equality if and only if
}f\equiv M$

(i.e. $y\notin\mathrm{supp}\chi$). This can be seen through a chain rule
computation and the fact that $\partial_{y}\tilde{f}(0,y)<0$.

We end this section by summarizing the properties of $f$ and $\mathcal{N}$
which are important to us.

######  Lemma 3.2 (Properties of the target Manifold).

The function $f$, manifold
$\mathcal{N}={\mathbb{T}}^{2}\times_{f}{\mathbb{S}}^{2}$ and curve $\gamma$,
described above, have the following properties:

  1. (1)

$f\in C^{\infty}({\mathbb{T}}^{2})$ and $f\geq 1$ always with $f=1$ iff $w=0$.

  2. (2)

The curve $\gamma(s):(-\infty,\infty)\rightarrow{\mathbb{T}}^{2}$ is a
geodesic with the following properties:

    1. (a)

For any $s\in\mathbb{R}$,
$\ell(\gamma((-\infty,s))=\infty=\ell(\gamma((s,\infty)))$.

    2. (b)

$\mathrm{sgn}(s)\frac{d}{ds}f(\gamma(s))>0$ except in a neighborhood of $0$,
in which $f\equiv M\gg 1$

    3. (c)

$\\{w=0\\}\subset\overline{\gamma((-\infty,\infty))}$ but
$\\{w=0\\}\cap\gamma(-\infty,\infty)=\emptyset$.

    4. (d)

For any $s\in{\mathbb{R}}$, $\nabla f(\gamma(s))\parallel\gamma^{\prime}(s)$.

Let us quickly comment on some of these conditions:

######  Remark 3.3.

It is not so important that $f,\mathcal{N}$ satisfy the conditions 1, 2c
precisely. The arguments here work for any $f$ which is globally bounded away
from zero and which achieves its minimum on a topological circle that is in
the closure of (but does not intersect) a geodesic $\gamma$. These facts will
help us control the image of possible bubbles; cf. Lemma 5.4.

Second, conditions 2a,2b 2d imply that the gradient flow generated by $f$
starting at a point along the curve $\gamma$ will be bounded but will not have
a unique limit as $t\rightarrow\infty$. Rather, each point in $\\{w=0\\}$ will
be an accumulation point of the flow. As we mentioned in Section 1.2, this
property would not possible if $f$ were analytic.

##  4\. Analysis of the Hamiltonian Flow

We now turn to the setting of wave maps into $\mathcal{N}$. Our first result
of this section establishes that for wave maps with initial conditions of the
form (2.1), the flow stays inside $\mathrm{Im}\gamma\times_{f}\mathbb{S}^{2}$
for all $t<T_{max}(u)$. Throughout this section, we will denote by $P_{1}$ and
$P_{2}$ the projection of ${\mathcal{N}}$ onto its two-dimensional components:

$P_{1}:{\mathcal{N}}\to{\mathbb{T}}^{2},\quad
P_{2}:{\mathcal{N}}\to{\mathbb{S}}.$

######  Lemma 4.1.

Let $u:\mathbb{R}^{2+1}\rightarrow{\mathcal{N}}$ be a wave map with initial
conditions (2.1), which also satisfy
$\|(u_{0},u_{1})\|_{\dot{H}^{3}\times\dot{H}^{2}}<\infty$. Then for all
$t<T_{max}(u)$, using the notation of (2.3), $X(t,\cdot)\equiv 0$, and

(4.1) $\displaystyle-\infty<Y(t,\cdot)<\infty.$

###### Proof.

We prove at as long as the initial conditions lay in $\gamma\times T\gamma$ we
have $X(t,r)\equiv 0$. This follows from the fact that
$\gamma\times\mathbb{S}^{2}$ lies totally geodesically within ${\mathcal{N}}$
(cf. [Top04b] Section 3), and then the classical fact that wave maps with
initial conditions in totally geodesic submanifolds stay in that submanifold.

Let us briefly sketch how this works: $\gamma$ is a geodesic in
${\mathbb{T}}^{2}$ which implies that the image of
$(\Delta_{{\mathbb{T}}^{2}})|_{\gamma}$ is contained in $T\gamma\subset
T{\mathbb{T}}^{2}$. The first component of the wave map satisfies the equation

$\partial^{2}_{tt}P_{1}\circ u=-\Delta_{{\mathbb{T}}^{2}}P_{1}\circ u-\nabla
f(P_{1}\circ u)e(\alpha).$

Since $\nabla f(P_{1}\circ u)$ and $\Delta_{{\mathbb{T}}^{2}}P_{1}\circ u$ lie
tangent to $\gamma$, the flow stays in $\gamma$ for the whole time of
existence.

Finally, to obtain boundedness of $Y$, we note that by assumption
$(0,y_{0})\in\gamma$, so, to fix notation, suppose that $s_{0}\in\mathbb{R}$
is such that $\gamma(s_{0})=(0,y_{0})$. Note that

$\ell\bigl{(}\gamma\bigl{(}(s_{0},\infty)\bigr{)}\bigr{)}=\infty=\ell\bigl{(}\gamma\bigl{(}(-\infty,s_{0})\bigr{)}\bigr{)},\qquad\forall
s_{0}\in(-\infty,\infty),$

so we will conclude by establishing that the image of a finite energy wave map
with $\dot{H}^{3}\times\dot{H}^{2}$ bounds cannot contain an infinite length
path, which will establish that $|Y(\cdot,t)|<\infty$ for all $t<T_{\max}$
proving (4.1). This is where we rely on the persistence of regularity result
from Proposition 2.1. We note again that $\dot{H}^{1+}\times\dot{H}^{0+}$
bounds would suffice for this argument.

For any $r_{0}>0$ and $r_{1},r_{2}>r_{0}$, letting $b=(0,Y(t,r_{2}))$ and
$a=(0,Y(t,r_{1}))$, we have

$\ell(a,b)\leq\int_{r_{1}}^{r_{2}}|DY|dr\leq\mathcal{E}(u)\frac{1}{r_{0}},$

Hence for any given time, the only point in the domain at which $Y(t,r)$ can
be infinite is $r=0$. However, for $p>1$ and for any $r_{1}\leq 1$, letting
$c=Y(t,r_{1})$ and $d=Y(t,0)$, we further have that

$\displaystyle\ell(c,d)\leq\left(\int_{0}^{r_{1}}|DY|^{p}rdr\right)^{1/p}$
$\displaystyle\leq\left(\int_{0}^{\infty}|DY|^{2}rdr\right)^{1-\theta}\left(\int_{0}^{\infty}|D^{3}Y|^{2}rdr\right)^{\theta}$
$\displaystyle\leq
C\bigl{(}t,\|(u_{0},u_{1})\|_{\dot{H}^{3}\times\dot{H}^{2}},\mathcal{E}(u_{0},u_{1})\bigr{)}$

where the second inequality holds for some $0<\theta<1$ by Sobolev embedding.
Hence $Y(t,0)$ cannot pass through $\\{\pm\infty\\}$. ∎

###  4.1. Energy Concentration at $T_{\max}=\infty$

In this subsection we assume that $T_{\max}=+\infty$. We want to show that
scattering (i.e. outcome (b) in Theorem 1.1) cannot occur unless the wave map
is degree zero.

We begin by rewriting the system of equations for $Y$ and $\alpha$, which, in
light of Lemma 4.1, is given by

(4.2) $\begin{split}\frac{\partial^{2}Y}{\partial
t^{2}}&=\frac{\partial^{2}Y}{\partial r^{2}}+\frac{1}{r}\frac{\partial
Y}{\partial r}-\frac{\partial f}{\partial
y}(0,Y)e(\alpha),\,\,\,\,\qquad\qquad\qquad\qquad\qquad(Y,Y_{t})\big{|}_{t=0}=(y_{0},y_{1}),\\\
\frac{\partial^{2}\alpha}{\partial t^{2}}&=\frac{\partial^{2}\alpha}{\partial
r^{2}}+\frac{1}{r}\frac{\partial\alpha}{\partial
r}-\frac{\sin(2\alpha)}{2r^{2}}+\frac{1}{f(0,Y)}\frac{\partial f}{\partial
y}(0,Y)\frac{\partial Y}{\partial r}\frac{\partial\alpha}{\partial
r},\qquad(\alpha,\alpha_{t})\big{|}_{t=0}=(\alpha_{0},\alpha_{1}).\end{split}$

We plan on showing that energy concentrates inside the light cone $|x|<t$ as
$t\rightarrow\infty$ (i.e. that outcome (a) of Theorem 1.1 holds). We start by
observing that the norms of derivatives of quasi-equivariant functions have
radial symmetry:

######  Proposition 4.2.

Let $u:\mathbb{R}^{1+2}\rightarrow(\mathcal{M},g)$ be a quasi-equivariant wave
map, then $|\nabla_{x}u|_{g},|\nabla_{t}u|_{g}$ and
$\left\langle\nabla_{x}u,\nabla_{t}u\right\rangle_{g}$ are all radially
symmetric functions.

###### Proof.

Recall that $u(r,\theta+s,t)=\Phi_{s}\circ u(r,\theta,t)$ for
$\Phi_{s}\in\mathrm{Isom}(\mathcal{M})$. In particular this implies that
$D\Phi_{s}$ satisfies

$\left\langle D\Phi_{s}v,D\Phi_{s}w\right\rangle_{g(\Phi_{s}(p))}=\left\langle
v,w\right\rangle_{g(p)},\qquad\forall p\in\mathcal{M},v,w\in
T_{p}\mathcal{M}.$

We can then compute,

$\displaystyle\left\langle\nabla_{x}u(r,\theta+s,t),\nabla_{t}u(r,\theta+s,t)\right\rangle_{g}=$
$\displaystyle\left\langle
D\Phi_{s}\nabla_{x}u(r,\theta,t),D\Phi_{s}\nabla_{t}u(r,\theta,t)\right\rangle_{g}$
$\displaystyle=$
$\displaystyle\left\langle\nabla_{x}u(r,\theta,t),\nabla_{t}u(r,\theta,t)\right\rangle_{g}.$

The same argument applies for $|\nabla_{x}u|_{g},|\nabla_{t}u|_{g}$. ∎

We also record a standard Hölder regularity estimate for quasi-equivariant
wave maps. Note that we abuse notation and use $|\cdot|$ to denote the
distance within the manifold.

######  Lemma 4.3.

Let $u:\mathbb{R}^{1+2}\rightarrow\mathcal{M}$ be a quasi-equivariant finite
energy wave map into a smooth manifolds $\mathcal{M}$ and
$\\{\Phi_{s}\\}_{s\in\mathbb{S}^{1}}$ the associated smoothly parameterized
one parameter family of isometries (see Definition 1.4). Then, for any
$r_{0}>0$ there exists $C\equiv
C(\mathcal{E}(u_{0},u_{1}),r_{0},\sup_{s\in{\mathbb{S}}^{1}}\|\partial_{s}\Phi_{s}\|_{C^{\infty}(T\mathcal{N})})>0$
such that for any $t\in\mathbb{R},\theta_{1},\theta_{2}\in{\mathbb{S}}^{1}$
and any $r,s\in\mathbb{R}$ with $r_{0}<r,s$ we have

$|u(t,r,\theta_{1})-u(t,s,\theta_{2})|\leq
C\bigl{(}|r-s|^{1/2}+|\theta_{1}-\theta_{2}|\bigr{)}.$

###### Proof.

First, fix $r>0$. For any $\phi,\theta\in{\mathbb{S}}^{1},t\in{\mathbb{R}}$ we
have

(4.3)
$\displaystyle|u(t,r,\theta)-u(t,r,\phi)|=|u(t,r,\theta)-\mathrm{\Phi}_{\phi-\theta}(u(t,r,\theta))|<\|\mathrm{Id}-\mathrm{\Phi}_{\phi-\theta}\|_{C^{\infty}(\mathcal{N})}<C|\phi-\theta|.$

Note, a similar argument shows that for $r,s,t>0$
$|u(t,r,\theta)-u(t,s,\theta)|$ is independent of $\theta$.

Fix $\theta\in{\mathbb{S}}^{1}$, $s>r>r_{0}>0$. Then, by Hölder’s inequality,

(4.4) $\displaystyle|u(t,r,\theta)-u(t,s,\theta)|^{2}$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}\left(\int_{r}^{s}\partial_{r}u(t,r^{\prime},\theta)dr^{\prime}\right)^{2}d\theta$
(4.5)
$\displaystyle\leq\left(\frac{1}{2\pi}\int_{0}^{2\pi}\int_{r}^{s}|\partial_{r}u(t,r^{\prime},\theta)|^{2}r^{\prime}dr^{\prime}d\theta\right)\frac{|r-s|}{r_{0}}.$

Putting (4.3) and (4.4) together, the conclusion follows. ∎

For $0<T<\infty$ and define, for any $A\geq 0$

(4.6)
$\mathrm{Flux}(u,T,A):=\int_{r=T-A}|\nabla_{x}u(r,\theta,T)+\nabla_{t}u(r,\theta,T)|_{g}^{2}d\sigma(\theta).$

As suggested by its name, the flux measures the energy entering the
(translated inwards by $A$) light cone at time $T$. We can see this in the
following energy identity:

(4.7)
$\int_{|x|<T_{1}-A}|\nabla_{x,t}u(x,T_{1})|_{g}^{2}dx-\int_{|x|<T_{2}-A}|\nabla_{x,t}u(x,T_{2})|_{g}^{2}dx=\int_{T_{2}}^{T_{1}}\mathrm{Flux}(u,t,A)dt.$

It follows from (4.7) that

$T\mapsto\int_{|x|<T-A}|\nabla_{x,t}u(x,T)|_{g}^{2}dx$

is monotone increasing for any $A\geq 0$. Since the integral is monotone
increasing in $T$ and bounded by $E(u_{0},u_{1})$, it follows that

$\lim_{T\rightarrow\infty}\int_{r<T-A}|\nabla_{x,t}u(x,T)|_{g}^{2}dx$

exists and, therefore,

(4.8) $\lim_{T\uparrow\infty}\mathrm{Flux}(u,T,A)=0.$

Our next proposition shows that the energy in a linear neighborhood of the
boundary of the light cone still goes to zero in infinite time; this
observation was first made in the setting of radial wave maps, c.f. [CTZ93],
see also Proposition 2.1 in [CKLS15b] and Lemma 4.1 in [Str13].

######  Proposition 4.4.

Let $u$ be finite energy quasi-equivariant wave map, with $T_{\max}=+\infty$.
Then for any $\lambda\in(0,1)$ we have

(4.9) $\int_{\lambda
T<|x|<T-A}|\nabla_{x}u(x,T)|_{g}^{2}+|\nabla_{t}u(x,T)|^{2}_{g}dx\to
0\qquad\textup{as $T,A\to\infty$ for $A\leq(1-\lambda)T$}.$

###### Proof.

We follow the notation from [Str13], see also [SS98]. Let

$e:=\frac{1}{2}\left(|\nabla_{x}u|_{g}^{2}+|\nabla_{t}u|_{g}^{2}\right),\>\>m:=\left\langle\nabla_{x}u,\nabla_{t}u\right\rangle,\>\>L:=\frac{1}{2}\left(|\nabla_{x}u|_{g}^{2}-|\nabla_{t}u|_{g}^{2}\right).$

From these we have the algebraic relations,

(4.10) $\displaystyle\partial_{t}(re)-\partial_{r}(rm)=$ $\displaystyle 0$
$\displaystyle\partial_{t}(rm)-\partial_{r}(re)=$ $\displaystyle L.$

Note that these identities follow from the wave map equation (1.1) and
Proposition 4.2, which, in this context, implies that
$\partial_{r}e=\partial_{x}e$ and similarly with $m$.

It will be convenient to reparameterize $(t,r)$ space by the coordinates
$\xi=t-r$ and $\eta=t+r$. We also introduce the quantities
$\mathcal{A}^{2}=r(e+m)$ and $\mathcal{B}^{2}=r(e-m)$. Then
$\partial_{\xi}\mathcal{A}^{2}=L=-\partial_{\eta}\mathcal{B}^{2}$. From this
equation and the algebraic observation that $8r^{2}(e^{2}-m^{2})\geq L^{2}$,
we see that

(4.11) $\displaystyle|\partial_{\xi}\mathcal{A}^{2}|\leq$
$\displaystyle\frac{C}{r}\mathcal{B}$
$\displaystyle|\partial_{\eta}\mathcal{B}^{2}|\leq$
$\displaystyle\frac{C}{r}\mathcal{A}.$

Let $Q$ denote the quadrilateral in $(\xi,\eta)$ space with vertices

$((1-\lambda)T,(1+\lambda)T),\quad(A,2T-A),\quad(A,2s-(1-\lambda)T),\quad((1-\lambda)T,2s-(1-\lambda)T)$

where $s\gg T>0$, see Figure 1 (the ordering above is the order of the
vertices in the Figure, starting from the lower left and moving counter-
clockwise).

Figure 1. The quadrilateral $Q$

By (4.10) the vector $(re,-rm)$ is divergence free, so we can conclude

(4.12) $\displaystyle 0=$ $\displaystyle\int_{\partial Q}(re,-rm)\cdot\hat{n}$
$\displaystyle=$ $\displaystyle\underbrace{-\int_{\lambda
T}^{T-A}erdr}_{I}-\underbrace{\int_{2T-A}^{2s-(1-\lambda)T}\mathcal{A}^{2}(A,\eta^{\prime})d\eta^{\prime}}_{II}$
$\displaystyle+$
$\displaystyle\underbrace{\int_{A}^{(1-\lambda)T}\mathcal{B}^{2}(\xi^{\prime},2s-(1-\lambda)T)d\xi^{\prime}}_{III}-\underbrace{\int_{(1+\lambda)T}^{2s-(1-\lambda)T}\mathcal{A}^{2}((1-\lambda)T,\eta^{\prime})d\eta^{\prime}}_{IV}.$

The terms $I-IV$ correspond to integrating along the sides of the
quadrilateral $Q$ in Figure 1, beginning at $I$ and moving counter-clockwise.

In (4.12), integral $I$ is exactly the one we want to show goes to zero as
$T,A$ go to infinity, and hence we have

$|I|\leq|II|+|III|+|IV|.$

We will handle each term on the right separately. Note that

$II\leq\int_{T}^{\infty}\mathrm{Flux}(u,t,A)dt,$

and hence letting $T\uparrow\infty$, (4.7) implies that $II$ goes to zero.

For the next terms, we introduce some notation: let

$\mathcal{E}_{\lambda}(\xi)=\int_{\frac{1+\lambda}{1-\lambda}\xi}^{\infty}\mathcal{A}^{2}(\xi,\eta^{\prime})d\eta^{\prime}.$

Figure 2. The shaded region is created by taking a lightcone with vertex
$(0,a)$, removing the lightcone with vertex $(0,b)$ and intersecting that
region with the backwards lightcone with vertex $(0,\eta_{0})$

Now we handle term III. Integrating between two light cones, i.e. the shaded
region in Figure 2, we see that

$\int_{a}^{b}\mathcal{B}^{2}(\xi^{\prime},\eta_{0})d\xi^{\prime}=\int_{a}^{\eta_{0}}\mathcal{A}^{2}(a,\eta^{\prime})d\eta^{\prime}-\int_{b}^{\eta_{0}}\mathcal{A}^{2}(b,\eta^{\prime})d\eta^{\prime}.$

Define

$\mathcal{F}(a,b)=\lim_{\eta\rightarrow\infty}\int_{a}^{b}\mathcal{B}^{2}(\xi^{\prime},\eta)d\xi^{\prime},$

and note that the limit in this definition exists since

(4.13) $\displaystyle\mathcal{F}(a,b)=\mathcal{E}_{0}(a)-\mathcal{E}_{0}(b).$

The identity (4.13) also implies that $\mathcal{E}_{0}(-)$ is a decreasing
function since $\mathcal{F}(a,b)\geq 0$. Therefore, since
$\mathcal{F}(a,b)\leq\mathcal{E}_{0}(a)<\mathcal{E}(u_{0},u_{1})$, we can
define

$\mathcal{F}(a)\equiv\lim_{b\rightarrow\infty}\mathcal{F}(a,b).$

Now, noting that

$III\leq\mathcal{F}(A),$

we need to establish the limit of the right-hand side is zero. But since we
know that $\mathcal{E}_{0}(-)$ is decreasing and non-negative,
$\lim_{b\rightarrow\infty}\mathcal{E}_{0}(b)$ exists, and, consequently,
$\lim_{a\rightarrow\infty}\mathcal{F}(a)=0$, which concludes the proof for
term $III$.

Finally, it is clear that $IV\leq\mathcal{E}_{\lambda}((1-\lambda)T)$, and
hence we are done if we can show that

(4.14) $\lim_{T\uparrow\infty}\mathcal{E}_{\lambda}((1-\lambda)T)=0.$

Here we argue exactly as in [CTZ93, Lemma 1]. While that lemma is written in
the setting of radially symmetric wave maps (with arbitrary target), we note
that all the relevant quantities in our setting are radial by Proposition 4.2.
Therefore, (4.14) holds and the proof is complete. ∎

We can then quickly conclude that the kinetic energy of a quasi-equivariant
wave map vanishes inside of the light cone (this corresponds to Corollary 2.2
in [CKLS15b]):

######  Corollary 4.5.

Let $u$ be a finite energy quasi-equivariant wave map with $T_{\max}=\infty$.
Then

(4.15)
$\lim_{A\rightarrow\infty}\limsup_{T\rightarrow\infty}\frac{1}{T}\int_{A}^{T}\int_{0}^{t-A}|\partial_{t}u|^{2}(r,t)rdrdt=0.$

###### Proof.

From (4.10), it follows that

$\partial_{t}(r^{2}m)=\partial_{r}(r^{2}e)-r|\partial_{t}u|^{2}_{g}.$

The proof then follows as in [CKLS15b, Corollary 2.2]. ∎

Now we narrow our focus to solutions to the system (4.2). We aim to prove the
following:

######  Proposition 4.6.

For wave maps, $u$, given by (4.2), scattering cannot occur when $\alpha$ is
not of degree zero. In particular, if $T_{\max}(u)=+\infty$ then there exists
$t_{n}\uparrow\infty$ and $\lambda(t_{n})\ll t_{n}$ such that
$u(t_{n},\lambda(t_{n})r)\rightarrow\omega$ in $H^{1}_{\mathrm{loc}}$, where
$\omega:{\mathbb{R}}^{2}\rightarrow\mathcal{N}$ is a non-trivial harmonic map.

Our arguments will follow very closely those in [CKLS15b, Sections 2 and 3],
where an analogous proposition is proven for equivariant wave maps into
spheres, see [CKLS15b, Theorem 3.2]. The $\alpha$ component of our wave map is
not equivariant wave map itself (it satisfies a different equation), however,
we will be able to use, essentially unchanged, any arguments in [CKLS15b]
which are purely energy theoretic, since $E(\alpha)\leq E(u)$.

The first step is to show that the $\alpha$ component converges to a constant
when $r\geq\lambda t$. Here we can argue exactly as in [CKLS15b] with our
Proposition 4.4 and Corollary 4.5 taking the place of [CKLS15b, Proposition
2.1 and Corollary 2.2].

######  Corollary 4.7 (Cf. [CKLS15b, Corollary 2.3]).

Let $\lambda>0$ and $u=(Y,\alpha)$ be a finite energy wave map which solves
the system (4.2). Then
$\alpha(\infty,t):=\lim_{r\rightarrow\infty}\alpha(r,t)$ exists and,
furthermore,

$\lim_{t\uparrow\infty}\|\alpha(r,t)-\alpha(\infty,t)\|_{L^{\infty}(r\geq\lambda
t)}=0.$

From Corollary 4.7 we can construct our sequence $\lambda(t_{n})$:

######  Lemma 4.8.

Let $u=(Y,\alpha)$ be a finite energy wave map which solves (4.2). Further
assume that $\alpha$ is not of zero degree. For each $t>T_{0}$, let
$\lambda(t)$ be such that

(4.16) $1\leq 2\pi\int_{0}^{2\lambda(t)}e(r,t)rdr\leq 2,$

then $\lambda(t)\ll t$.

We first note that if $\alpha$ has degree not equal to zero then
$E(u)>\mathcal{E}_{\mathbb{S}^{2}}=2\pi$. So, by continuity of the energy,
there exist $\lambda(t)$ which satisfy (4.16).

###### Proof.

Imagine that there are $t_{n}\uparrow\infty$ and a $\lambda>0$ such that
$\lambda(t_{n})>\lambda t_{n}$. It would then follow that

$\displaystyle\int_{0}^{2\lambda(t_{n})}e(r,t_{n})rdr\geq$
$\displaystyle\int_{0}^{\lambda t_{n}}e(r,t_{n})rdr\geq\int_{0}^{\lambda
t_{n}}\left((\partial_{r}\alpha)^{2}+\frac{\sin^{2}(\alpha)}{2r^{2}}\right)rdr$
$\displaystyle\geq$ $\displaystyle\int_{0}^{\lambda
t_{n}}\left|\partial_{r}\alpha\right|\left|\sin(\alpha)\right|dr\geq\int_{0}^{\alpha(\lambda
t_{n})}\sin(\rho)d\rho\stackrel{{\scriptstyle\mathrm{Corollary}\;\ref{c:constantoutsideofcone}}}{{\rightarrow}}\int_{0}^{\pi}\sin(\rho)d\rho=2.$

This contradicts the defintion of $\lambda(t)$ (i.e. (4.16)). As such
$\lambda(t_{n})\ll\lambda t_{n}$ for any $t_{n}\uparrow\infty$ and
$\lambda>0$. ∎

We can now finish just as in the proof of [CKLS15b, Theorem 3.2], we sketch
the argument below:

###### Sketch of Proof of Proposition 4.6.

Let $\lambda(t)$ be defined as in Lemma 4.8 and let $A(t)$ be such that
$A(t)\rightarrow\infty$ as $t\rightarrow\infty$ but also $\lambda(t)\leq
A(t)\ll t$. Arguing from Corollary 4.5 (see [CKLS15b, Lemma 3.3]) there exists
$t_{n}\uparrow\infty$ and $\lambda_{n}:=\lambda(t_{n}),A_{n}:=A(t_{n})$ such
that

$\lim_{n\rightarrow\infty}\frac{1}{\lambda_{n}}\int_{t_{n}}^{t_{n}+\lambda_{n}}\int_{0}^{t-A_{n}}|\partial_{t}u|^{2}rdrdt=0.$

Rescaling so that $u_{n}(x,t)=u(\lambda_{n}x,\lambda_{n}t+t_{n})$ gives that

(4.17)
$\int_{0}^{1}\int_{0}^{r_{n}}|\partial_{t}u_{n}|_{g}^{2}rdrdt\rightarrow 0,$

where $r_{n}=(t_{n}-A_{n})/\lambda_{n}\rightarrow\infty$.

This scaling also preserves the energy. Thus the $u_{n}\rightharpoonup
u_{\infty}$ in $\dot{H}^{1}_{\mathrm{loc}}$. Furthermore, $u_{\infty}$ is time
independent by (4.17) and therefore is a finite energy harmonic map, which can
be assumed to be smooth and globally defined by the work of [Hél91] and
[SU81]. We now want to show that this convergence is strong, which also
implies that $u_{\infty}$ is non-trivial by (4.16).

Away from $0$, we may assume the sequence converges locally uniformly by local
uniform Hölder continuity of Lemma 4.3. Near $0$ we use (4.16) to see that
$u_{n}(B_{1}(0))$ is contained in a coordinate chart in $\mathcal{N}$ (and the
chart can be taken uniformly for $n$ large enough). Then one can argue exactly
as in [Str03a, Theorem 2.1] (see in particular (3.17) onwards) to conclude
that the convergence is strong near $0$. ∎

##  5\. An Energy Gap for Bubbles arising in the flow

In this section we study the space of harmonic maps which can possible arise
as bubbles in the flow (4.2). We begin with the following consequence of Lemma
4.3.

######  Proposition 5.1.

Let $u:\mathbb{R}^{1+2}\rightarrow\mathcal{M}$ be a quasi-equivariant wave-map
to the smooth manifold $\mathcal{M}$. If $u^{(n)}$ (defined above) converges
in $H^{1}_{loc}$ to a non-trivial harmonic map,
$\omega:{\mathbb{S}}^{2}\rightarrow\mathcal{M}$ then, up to passing to a
susbsequence,

$u^{(n)}(t,x)\to\omega:\mathbb{R}^{2}\to\mathcal{M},$

in $C^{0}_{loc}(\mathbb{R}^{2}\setminus\\{0\\},\,\mathcal{M})$ as
$n\to\infty$.

###### Proof.

By Lemma 4.3, the sequence $u^{(n)}=u(t_{n},r_{n}x)$ is uniformly Hölder
continuous on any compact region
$K\subset\subset\mathbb{R}^{2}\setminus\\{0\\}$. By Arzelá-Ascoli, to extract
a uniformly convergent subsequence, hence by a diagonal argument and passing
to a further subsequence, we can product a subsequence which converges
uniformly on any compact subset of $\mathbb{R}^{2}\setminus\\{0\\}$. By
uniqueness of limits, we have convergence to $\omega$. ∎

From this convergence, we can conclude first that the image of the bubble is
topologically connected to the image of the rest of the flow.

######  Proposition 5.2 (Connectedness of the flow).

Let $u:\mathbb{R}^{1+2}\rightarrow\mathcal{M}$ be a quasi-equivariant wave map
into a smooth manifold $\mathcal{M}$, such that bubbling (as above) occurs at
time $T_{\max}(u)$ and that the rescaled wave maps converge to the non-trivial
harmonic map $\omega:\mathbb{S}^{2}\rightarrow\mathcal{M}$. Then

$\bigcup_{t<T_{max}(u)}u(t,\mathbb{R}^{2})\cup\omega(\mathbb{S}^{2})$

is a connected subset of $\mathcal{M}$.

###### Proof.

The fact that union of the images of the wave map components

$\mathcal{U}:=\bigcup_{t<T_{max}(u)}u(t,\mathbb{R}^{2})$

and the image of the harmonic map $\omega(\mathbb{S}^{2})$ are each connected
follows from the continuity of the involved maps (in both space and time in
the case of the wave maps). To see that the union of these two components is
connected, we note that by Proposition 5.1, the set
$\omega(\mathbb{R}^{2}\setminus\\{0\\})$ contains limit points of
$\mathcal{U}$, which implies the result. ∎

The following immediate corollary of Propositions 5.1 and 5.2 constrains the
form and image of any bubble arising in the flow (4.2):

######  Corollary 5.3.

Any harmonic map $\omega:{\mathbb{S}}^{2}\rightarrow{\mathcal{N}}$ arising as
a non-trivial bubble in the flow (4.2) must be of the form

(5.1) $\omega(r,\theta)=(0,Y(r),\alpha(r),\theta),$

and satisfy
$\omega({\mathbb{S}}^{2})\subset\overline{\gamma}\times_{f}{\mathbb{S}}^{2}$.

We are now ready to state and prove the main result of this section: a
classification of the bubbles arising in the flow (4.2) at low energies.
Before we do, let us recall from Lemma 2.5 that $\mathcal{E}_{\mathbb{S}^{2}}$
is the smallest possible energy of a non-trivial harmonic map
$\omega:{\mathbb{S}}^{2}\rightarrow{\mathbb{S}}^{2}$. Also recall, from above,
that $P_{1}:{\mathcal{N}}\rightarrow{\mathbb{T}}^{2}$ is the projection map
onto the first coordinate and $P_{2}:{\mathcal{N}}\rightarrow{\mathbb{S}}^{2}$
is the projection map onto the second coordinate.

######  Lemma 5.4.

Let $\omega:{\mathbb{S}}^{2}\rightarrow{\mathcal{N}}$ be a non-trivial
harmonic map arising as a bubble in the flow (4.2). Let
$\varepsilon_{0}=\min\\{\bar{\varepsilon},\cot(7\pi/16)/2\\}>0$, where
$\bar{\varepsilon}>0$ is the constant given by Theorem 2.2. Then, either
$\mathcal{E}(\omega)=\mathcal{E}_{{\mathbb{S}}^{2}}$, or
$\mathcal{E}(\omega)>\mathcal{E}_{{\mathbb{S}}^{2}}+\varepsilon_{0}/2$. In the
former case, $P_{1}\omega$ is a constant map into the set $\\{w=0\\}$ and
$P_{2}\omega$ is a degree one harmonic map from
${\mathbb{S}}^{2}\rightarrow{\mathbb{S}}^{2}$.

In this proof we will use $Y,\alpha$ interchangeably with
$P_{1}\omega,P_{2}\omega$ respectively.

###### Proof.

By Lemma 2.4, $\mathrm{Im}\,P_{1}\omega$ is either a subset of $\gamma$ or of
$\\{w=0\\}\equiv\overline{\gamma}\,\setminus\,\gamma$. In the latter case, $f$
is constant on $\\{w=0\\}$ so $P_{2}\omega$ is a harmonic map between spheres
and $P_{1}\omega$ is a harmonic map from
$\mathbb{S}^{2}\rightarrow\mathbb{S}^{1}$. The maximum principle implies that
$P_{1}\omega$ is a constant, so $P_{2}\omega$ must be non-trivial, as $\omega$
is non-trivial. Therefore, $P_{2}\omega$ is either a degree one harmonic map
or $\mathcal{E}(P_{2}\omega)\geq 2\mathcal{E}_{{\mathbb{S}}^{2}}$, and the
conclusion of the lemma holds in both cases.

Thus, it suffices to rule out the presence of harmonic maps where
$\mathrm{Im}\,P_{1}\omega\subset\gamma$ satisfying our energy constraint. We
first claim that $P_{2}\omega$ must be non-trivial. Indeed, if $P_{2}\omega$
is trivial, then $P_{1}\omega$ is a bounded harmonic map from
${\mathbb{R}}^{2}$ to $\gamma(-\infty,\infty)\cong{\mathbb{R}}$. Such a map
must be trivial which contradicts the non-triviality of $\omega$. Moreover,
since $P_{2}\omega$ is non-trivial and equivariant, the maximum principle
implies that the image of $P_{2}\omega$ cannot be contained in a hemisphere,
and thus $\mathcal{E}(P_{2}\omega)\geq\mathcal{E}_{{\mathbb{S}}^{2}}$.

Let

$S=\\{r\in{\mathbb{R}}^{2}\mid f(0,Y(r))<2\\},$

and assume that
$\mathcal{E}(\omega)<\mathcal{E}_{{\mathbb{S}}^{2}}+\varepsilon_{0}/2$. Since

$\int_{\mathbb{R}^{2}\backslash
S}f(0,Y)e(\alpha)rdr>2\int_{\mathbb{R}^{2}\backslash S}e(\alpha)rdr,$

it must be the case that

(5.2) $\int_{S}e(\alpha)rdr>\mathcal{E}_{\mathbb{S}^{2}}-\varepsilon_{0}/2,$

and therefore

$\int_{{\mathbb{R}}^{2}\backslash S}|D\omega|_{g}^{2}<\varepsilon_{0}.$

Thus the hypothesis of Theorem 2.2 are fulfilled inside of any ball
$B\subset{\mathbb{R}}^{2}\backslash S$. $P_{1}\omega=Y$ is radial, so the
oscillation of $Y$ inside any component of ${\mathbb{R}}^{2}\backslash S$ is
bounded by $2\varepsilon_{0}$. Note that $M>2$ so $f(0,Y(r))<2$ implies that
$|Y|>\cot(7\pi/16)$ (see (3.2), (3.1)). As $2\varepsilon_{0}<\cot(7\pi/16)$,
we conclude that $Y$ does not change sign on $\mathbb{R}^{2}$.

Composing with the diffeomorphism
$\Psi_{s}:(x,y,\alpha,\theta)\mapsto(x,y+s,\alpha,\theta)$ we see that

(5.3)
$\frac{d}{ds}\mathcal{E}(\Psi_{s}\circ\omega)|_{s=0}=\int_{{\mathbb{R}}^{2}}\partial_{y}f(0,Y)e(\alpha)rdr\neq
0,$

where the last “non-equality" follows from (5.2) and the fact that $Y$ does
not change sign on ${\mathbb{R}}^{2}$ (and, consequently, $\partial_{y}f(0,Y)$
has a sign). That the deformation in (5.3) changes the energy to first order
contradicts the fact that $\omega$ is harmonic. Thus there are no harmonic
maps $\omega$ with $P_{1}\omega\subset\gamma$ satisfying our energy
constraint, and our proof is complete. ∎

######  Remark 5.5.

It is not hard to show that $\mathcal{E}_{\mathbb{S}^{2}}$ is exactly the
energy of the lowest energy non-trivial quasi-equivariant harmonic map,
$\omega:\mathbb{S}^{2}\rightarrow\mathcal{N}$, what we call
$\mathcal{E}_{\mathrm{quasi}}(\mathcal{N})$ in Theorem 1.7.

##  6\. Properties of the singularity

We now turn to the proof of our main theorem, which we recall here:

######  Theorem (Main theorem).

There exists a compact smooth Riemannian manifold $(\mathcal{N},g)$ given by

$\mathcal{N}=\mathbb{T}^{2}\times_{f}\mathcal{S}^{2}$

for a certain $C^{\infty}({\mathbb{T}}^{2})$ warping function, $f$, and
$C^{\infty}$-smooth, finite energy, quasi-equivariant initial data
$(u_{0},u_{1})$, which satisfy

$\|(u_{0},u_{1})\|_{\dot{H}^{3}\times\dot{H}^{2}}<\infty,\qquad\mathcal{E}(u_{0},u_{1})<\mathcal{E}_{\mathrm{quasi}}(\mathcal{N})+\varepsilon_{1},$

such that the corresponding solution $(u,u_{t})$ to (1.1) has a bubbling
singularity as $t\to T_{\max}$ which fails to be unique in the sense of
Definition 1.6. Above, $\varepsilon_{1}>0$ is a constant which depends only on
$\mathcal{N}$, and $\mathcal{E}_{\mathrm{quasi}}(\mathcal{N})$ denotes the
smallest energy of a non-trivial quasi-equivariant harmonic map,
$\omega:\mathbb{S}^{2}\rightarrow\mathcal{N}$.

Before proceeding with the proof of the main result we summarize what we know
so far about solutions to (1.1), or more precisely to solutions of (4.2). In
light of Sterbenz and Tataru’s dichotomy, Theorem 1.1, at every finite time
$t_{0}$, the solution either concentrates energy at a point and bubbles off (a
Lorentz transform of) a finite energy harmonic map or it can be continued
smoothly past $t_{0}$. If the solution exists as $t\rightarrow\infty$ then it
either scatters in the limit or bubbles off (a Lorentz transform of) a finite
energy harmonic map. If the “spherical part" of the initial data has degree
one, Proposition 4.6 implies that blow-up in the form of bubbling off a
harmonic map must occur (either in finite time or as $t\rightarrow\infty$).

Now note that quasi-equivariance implies that there is no Lorentz transform
symmetry in the solutions, and hence the convergence of the bubbling sequence
(as described in part (a) of Theorem 1.1) must be to an entire, non-constant
harmonic map. Moreover, singularity formation at any point in the domain
requires the bubbling of at least as much energy as the lowest energy non-
trivial harmonic map. Hence by additivity of the energy, bubbling can occur
only at finitely many points, and so in the quasi-equivariant setting the only
possible blow-up point in the domain is the origin $x=0$.

Finally, if the initial conditions (2.1) have energy less than
$\mathcal{E}_{\mathbb{S}^{2}}+\varepsilon_{0}/2$, then it must be the case
that the bubble also has energy less than this. By Lemma 5.4, this implies
that the harmonic map, $\omega$, decomposes into a constant map into
$\\{w=0\\}\subset{\mathbb{T}}^{2}$ and a degree one equivariant harmonic map
between spheres.

Summarizing the previous discussion, we have, thus far, established that:

######  Theorem 6.1.

Consider the wave map equation (1.1) with smooth, quasi-equivariant symmetric,
finite energy initial data of the form (2.1). Further assume that $\alpha_{0}$
(the “spherical part" of the initial conditions) is a degree one map between
2-spheres. Then there exists a unique solution to (1.1) on the time interval
$[0,T_{\max})$, and a sequence of times $t_{n}\uparrow T_{\max}$ and scales
$r_{n}$ with

(6.1) $r_{n}=\begin{cases}o(T_{max}-t_{n})&T_{max}<\infty\\\
o(t_{n})&T_{max}=\infty\end{cases}$

so that the rescaled sequence of maps

$u^{(n)}(t,x)=u(t_{n},r_{n}x)$

converges strongly in $H^{1}_{loc}$ to an entire Harmonic map

$\omega:\mathbb{R}^{2}\to\mathcal{N}$

of nontrivial energy. Furthermore, if

$\mathcal{E}(u_{0},u_{1})<\mathcal{E}_{\mathbb{S}^{2}}+\varepsilon_{0}/2,$

then $\omega=(P_{1}\omega,P_{2}\omega)$ satisfies that $P_{1}\omega=(0,z_{0})$
for some $z_{0}\in[0,1]$ and $P_{2}\omega$ is a degree one equivariant
harmonic maps between 2-spheres.

We note that it is easy to produce initial conditions which satisfy the
hypothesis of Theorem 6.1. Using the notation of (2.1), one such example is
letting $(\alpha_{0}(r),\theta)$ be a degree one equivariant harmonic map,
$\alpha_{1}(r)\equiv 0\equiv Y_{1}(r)$ and $Y_{0}(r)\equiv c$ where $c$ is a
constant large enough such that
$f(0,c)<1+\varepsilon_{0}/(2\mathcal{E}_{\mathbb{S}^{2}})$.

###  6.1. Winding Singularities

In this subsection we prove that the singularity guaranteed by Theorem 6.1 is
winding in the sense of [Top04b], and in fact enjoys a stronger form of
winding which we introduce below, which we call “strongly winding”. We will
then prove that a strongly winding singularity implies non-uniqueness in the
sense of Definition 1.6. Our stronger notion of winding requires winding along
_all_ sequences, as opposed to just one. We find this definition to be
slightly easier to work with, and, in the situation where only one bubble
develops at a given point and time, equivalent to the standard definition of
winding.

Before we introduce the following stronger notion of winding recall that if
$\mathcal{M}$ is a Riemannian manifold, then $\widehat{\mathcal{M}}$ its its
universal cover and, given a function $u:\mathbb{R}^{1+d}\to\mathcal{M}$, we
let $\hat{u}:\mathbb{R}^{1+d}\rightarrow\widehat{\mathcal{M}}$ be its lift to
the universal cover.

######  Definition 6.2 (Strongly winding singularity).

A quasi-equivariant wave map $u:\mathbb{R}^{1+2}\rightarrow\mathcal{M}$ has a
strongly winding singularity at time $T_{\max}$ and the origin
$0\in\mathbb{R}^{2}$ if for any sequences $\\{r_{n}\\}$, and $\\{t_{n}\\}$,
satisfying

(6.2) $t_{n}\uparrow T_{\max},\qquad
r_{n}=\begin{cases}o(T_{max}-t_{n})&T_{max}<\infty\\\
o(t_{n})&T_{max}=\infty\end{cases}$

such that $u(t_{n},r_{n}x)\rightarrow\omega(x)$ in
$C^{0}_{\mathrm{loc}}({\mathbb{R}}^{2}\backslash\\{0\\};\mathcal{M})$, where
$\omega$ is a non-constant harmonic map, the lifts $\hat{u}(t_{n},r_{n}x)$
have no convergent subsequence in
$C^{0}_{\mathrm{loc}}({\mathbb{R}}^{2}\backslash\\{0\\};\widehat{\mathcal{M}})$.

For a solution $u$ with winding singularity, we say that $u$ has a winding
singularity with respect to sequences $\\{t_{n}\\}$ and $\\{r_{n}\\}$, when
$\\{t_{n}\\}$ and $\\{r_{n}\\}$ are the sequences guaranteed by Definition
1.10. The following proposition provides an equivalent definition of
(strongly) winding.

######  Proposition 6.3.

If $u:\mathbb{R}^{1+d}\rightarrow\mathcal{M}$ has a singularity at
$(T_{\max},0)$, then the singularity is winding with respect to sequences
$\\{t_{n}\\}$ and $\\{r_{n}\\}$ if and only if there exists a compact
$K\subset\subset{\mathbb{R}}^{d}\backslash\\{0\\}$ such that every subsequence

$\\{\hat{u}(t_{n_{j}},r_{n_{j}}x)\big{|}_{K}\\}$

is unbounded.

###### Proof.

This proposition follows readily from the equicontinuity of the lifts and
compactness, but we include the details for completeness. Suppose first that
the singularity is winding in the sense of Definition 1.10. Since the sequence
$\\{u^{(n)}\\}$ is equicontinuous on every compact set, the family given by
the lifts $\hat{u}^{(n)}:=\hat{u}(t_{n},r_{n}x)$ is still equicontinuous on
every compact set. Indeed, fix a compact set
$K\subset\subset{\mathbb{R}}^{d}\backslash\\{0\\}$, $x\in K$ and
$\varepsilon>0$, and let $\delta>0$ be as in the definition of equicontinuity
of the family $\\{u^{(n)}\\}$. Making $\delta$ smaller if necessary, we can
restrict to a sufficiently small neighborhood so that the covering map
trivializes. This yields the equicontinuity of the lifts. Hence, the only way
the lifted sequence can fail to be precompact is if there exists a compact set
on which the family is unbounded.

Conversely, if there exists such a compact subset $K$, then unboundedness
implies that there is no convergent subsequence, and hence the singularity is
winding. ∎

######  Remark 6.4.

The previous proposition adapts readily to strongly winding singularities.

We are now prepared to establish the following main proposition about the
nature of the singularity.

######  Proposition 6.5.

The flow described (4.2) with initial conditions as in Theorem 6.1 has a
strongly winding singularity at $(T_{max},0)$.

###### Proof.

We know the flow needs to converge to a bubble as described in Lemma 5.4. Let
$\gamma$ be the geodesic described in Section 3. By the path lifting property,
$\gamma$ may be lifted to a unique path $\hat{\gamma}$ in the universal cover,
which is necessarily an unbounded curve since $\gamma$ is not null-homotopic
in ${\mathcal{N}}$ and $\pi_{1}({\mathcal{N}})={\mathbb{Z}}^{2}$.

Let now $\\{t_{n}\\}$ and $\\{r_{n}\\}$ be any sequences along which

$u^{(n)}(t,x)=u(t_{n},r_{n}x)$

converge to $\omega$, see Theorem 6.1. Lemma 5.4 tells us that
$P_{1}\omega\subset\\{w=0\\}$ which implies that for any $t_{0}\gg 1$ and any
compact set $K$, there exists an $N=N(t_{0},K)$ such that if $n\geq N$, then
$u^{n}(K)\subset\gamma((-\infty,-t_{0}]\cup[t_{0},\infty))$. Lifting to the
universal cover, the unboundedness of $\hat{\gamma}$ implies that the sequence
$\hat{u}^{(n)}|_{K}:=\hat{u}(t_{n},r_{n}x)|_{K}$ must be unbounded, and hence
by Proposition 6.3, the singularity is winding. ∎

Our final result shows that a strongly winding singularity cannot give rise to
a unique bubble in the sense of Definition 1.6.

######  Theorem 6.6.

If $u$ has a strongly winding singularity at $x_{0}$ and time $T_{\max}$, then
it does not have a unique bubble at that point and time.

###### Proof.

Suppose for contradiction that $u$ has a winding singularity at the origin at
time $T_{max}$, and a unique bubble $\omega$ to which it converges under
rescaling by $r(t)$. and translation by $x(t)$.

We will show that for every compact
$K\subset\subset{\mathbb{R}}^{d}\backslash\\{0\\}$, there exists a subsequence

$\\{\hat{u}(t_{n_{j}},r_{n_{j}}x)\big{|}_{K}\\}$

which is bounded, contradicting Proposition 6.3. Fix a compact set
$K\subset\subset{\mathbb{R}}^{d}\backslash\\{0\\}$ and note that
$\widehat{\omega}(K)$ is compact. Define the $\varepsilon$-neighbourhood of a
set $A$ by

$N_{\varepsilon}(A):=\\{x\mid\mathrm{dist}(x,A)<\varepsilon\\}.$

Fix $\varepsilon=1$, since $u(t,r(t)-)\rightarrow\omega$ in
$C^{0}_{\mathrm{loc}}$, there exists $t_{0}$, such that for all $t>t_{0}$ we
have

$u(t,r(t)K)\subset N_{1}(\omega(K)).$

But we then have that $\\{\hat{u}(t_{n},r(t_{n})K)\\}_{\\{n\mid
t_{n}>t_{0}\\}}\subset N_{1}(\widehat{\omega}(K))$, and, in particular, the
sequence is bounded, which contradicts Proposition 6.3, and concludes the
proof. ∎

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