[ afterskip=-1em, ]subsubsection [section]section

#  The Parametric Approach
to the Willmore Flow

Francesco Palmurella  Tristan Rivière

###### Abstract

We introduce a parametric framework for the study of Willmore gradient flows
which enables to consider a general class of weak, energy-level solutions and
opens the possibility to study energy quantization and finite-time
singularities. We restrict in this first work to a small-energy regime and
prove that, for small-energy weak immersions, the Cauchy problem in this class
admits a unique solution.

MSC 49Q10, 53A05, 58E15, 35J35, 35J48, 35K41, 35K91

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Consequences of DeLellis-Müller’s Theorem
  4. 4 Estimates of Elliptic Type
  5. 5 Conformal Willmore Flows
  6. A DeTurck’s Trick for the Willmore Flow
  7. B The $\overline{\partial}$-operator on Vector Fields

##  1 Introduction

The present paper moves the first steps towards a parametric theory for the
Willmore flow that, we believe, will lead to an effective study of
singularities, bubbling analysis and energy quantization.

We start by recalling what a Willmore surface is.

####  1.1 Willmore Surfaces

The _Willmore energy_ of a surface $\mathcal{S}\subset\mathbb{R}^{n}$ was
initially considered in the work of Poisson [Poi16] and Germain [Ger21] on
elastic plates. It was reconsidered in a purely geometric perspective in the
works of Thomsen [Tho23] and Blaschke [Bla29], in attempt to merge the study
of minimal surfaces with conformal invariance. It was reintroduced in recent
times in the works of Willmore [Wil65], and Bryant [Bry84] in pure mathematics
and by Canham [Can70] and Helfrich [Hel73] in theoretical biology while
modeling the shapes of blood cells and lipid bilayers (and, in fact, it is
sometimes referred to as Canham-Helfrich energy). It comes in the following
three variants:

$\displaystyle\mathcal{W}_{0}(\mathcal{S})$
$\displaystyle=\frac{1}{2}\int_{\mathcal{S}}|A^{\circ}|^{2}\,d\sigma,$
$\displaystyle\mathcal{W}_{1}(\mathcal{S})$
$\displaystyle=\int_{\mathcal{S}}|H|^{2}\,d\sigma,$
$\displaystyle\mathcal{W}_{2}(\mathcal{S})$
$\displaystyle=\frac{1}{4}\int_{\mathcal{S}}|A|^{2}\,d\sigma,$ (1.1)

were $H$ the mean curvature, $A$ is the 2nd fundamental form, $A^{\circ}$ it
tracefree part and $d\sigma$ the area element. If $K$ denotes the Gauss
curvature of $\mathcal{S}$, there holds

$\displaystyle\frac{1}{2}|A^{\circ}|^{2}=|H|^{2}-K=\frac{1}{4}|A|^{2}-\frac{1}{2}K,$
(1.2)

hence, by Gauss-Bonnet theorem, if the topology of $\mathcal{S}$ is fixed, at
least in a smooth setting such energies are all variationally equivalent (in
particular, they have the same Euler-Lagrange operator). Depending on the
context, it may however be more favourable to work with one than another.

The _Willmore operator_ is the associated Euler-Lagrange operator:

$\displaystyle\delta\mathcal{W}=\Delta^{\perp}H+Q(A^{\circ})H,$ (1.3)

where $\Delta^{\perp}$ is the Laplace operator on the normal bundle and

$\displaystyle Q(A^{\circ})H=\big{\langle}A^{\circ},\langle
H,A^{\circ}\rangle\big{\rangle}=g^{\mu\sigma}g^{\nu\tau}\big{\langle}A^{\circ}_{\mu\nu},\langle
A^{\circ}_{\sigma\tau},H\rangle\big{\rangle}.$ (1.4)

Similarly as for the mean curvature, $\delta{\mathcal{W}}$ is a normal-valued
vector field along $\mathcal{S}$. When $n=3$, the expression simplifies
somewhat:

$\displaystyle\delta\mathcal{W}=\Delta^{\perp}H+|A^{\circ}|^{2}H=(\Delta
H_{\text{sc}}+2(H_{\text{sc}}^{2}-K)H_{\text{sc}})N,$ (1.5)

where $N$ is the Gauss map of $\mathcal{S}$ and $H_{\text{sc}}=\langle
H,N\rangle$ is the scalar mean curvature.

_Willmore surfaces_ are those surfaces with vanishing Willmore operator. Any
of the Willmore energies (1.1) is invariant under conformal transformations of
$\mathbb{R}^{n}$ and, in fact, the Lagrangian density $|A^{\circ}|^{2}d\sigma$
is a pointwise conformal invariant, see Chen [Che74]. As a consequence, the
Willmore operator and the notion of Willmore surface are also conformal
invariants.

####  1.2 Willmore Flows

A _Willmore $L^{2}$–gradient flow_ in $\mathbb{R}^{n}$ (_Willmore flow_ for
short) of a closed, abstract surface $\Sigma$ is a 1-parameter family of
immersions $\Phi(t,\cdot):\Sigma\to\mathbb{R}^{n}$, $t\in
I\subseteq\mathbb{R}$ evolving according to the law

$\displaystyle\frac{\partial}{\partial
t}\Phi=-\delta\mathcal{W}+U\quad\text{in }I\times\Sigma,$ (1.6)

where, for each $t$, $\delta\mathcal{W}$ is the Willmore operator of
$\mathcal{S}_{t}=\Phi(t,\Sigma)$ and $U=U^{\mu}\partial_{\mu}\Phi$ is a
tangent tangent vector field, possibly time-dependent.

One good reason to consider Willmore flows is that they satisfy the energy
identity, namely if $I=(0,T)$, then

$\displaystyle\mathcal{W}_{0}(\Phi(t,\cdot))-\mathcal{W}_{0}(\Phi(0))=-\int_{0}^{t}\int_{\Sigma}|\delta\mathcal{W}|^{2}\,d\sigma_{g}\,d\tau,\quad\text{for
}0\leq t<T.$ (1.7)

It may be rephrased by saying that, among all families of immersions whose
velocity vector has normal part with $L^{2}$-norm equal to
$\|\delta\mathcal{W}\|_{{L^{2}}(S^{2})}$, Willmore flows are those with most
rapidly decreasing Willmore energy (in any of the forms given in (1.1)). Thus,
at least in principle, they have the potential to converge efficiently to
Willmore immersions as $t\to+\infty$.

This is a feature common to gradient flows that makes them particularly worth
studying. The first to consider $L^{2}$–gradient flows in a geometric context
were Eells and Sampson [ES64] in the context of harmonic maps. Since then, the
study of parabolic geometric flows has widened to the extent that some of them
constitute research areas on their own right, the mean curvature flow and
Hamilton’s Ricci flow being two of the best-known examples.

It should be noted right away that what is typically called a Willmore flow is
a family solving (1.6) with $U=0$, which we will call here a _normal Willmore
flow_. Since $\Sigma$ is closed, and $\delta\mathcal{W}$ is a tensor, it is
classical fact that there is a bijective correspondence between tangential
components and family of reparametrizations of $\Sigma$, see e.g. Mantegazza
[Man11, Proposition 1.3.4] for the case, entirely analogous in this regard, of
the mean curvature flow. Consequently, if, say, $I$ is a connected interval
containing $0$, for every family solving (1.6) there is a unique family of
diffeomorphisms $\varphi:I\times\Sigma\to\Sigma$ with
$\varphi(0,\cdot)=\operatorname{id}_{\Sigma}$ so that the reparametrized
family $\Phi(t,\varphi(t,\cdot))$, $t\in I$ is a normal Willmore flow, and on
the other hand, every reparametrization of a normal Willmore flow will be a
Willmore flow (1.6) for some $U$.

Thus, in this sense, similarly as for immersions of surfaces, flows can be
regarded as equivalence classes of solutions to (1.6), two of them being
equivalent if one can be reparametrized into another. As for surfaces,
depending on the situation one may choose one parametrization over another,
and in this case this may be done through the choice of the tangential
component. This will be a crucial fact in the present work.

The study of Willmore flows was introduced by Kuwert and Schätzle [KS01, KS02]
and Simonett [Sim01] and is since then subject of a growing number of works.
Particularly useful for us will be the one by Kuwert and Scheuer [KS20]
providing asymptotic estimates on the barycenter along the flow.

Our attention here focuses on the following foundational result. Consider the
Cauchy problem for the normal Willmore flow:

$\displaystyle\left\\{\begin{aligned} \frac{\partial}{\partial
t}\Phi&=-\delta\mathcal{W},&&\text{in }(0,T)\times\Sigma,\\\
\Phi(0,\cdot)&=\Phi_{0}&&\text{on }\Sigma.\\\ \end{aligned}\right.$ (1.8)

######  Theorem ([KS01, KS02]).

There exists $\varepsilon_{0}(n)>0$ so that, for a smooth immersion
${\Phi_{0}:\Sigma\to\mathbb{R}^{n}}$ satisfying
$\mathcal{W}_{0}(\Phi_{0})=\mathcal{W}_{0}(\Phi_{0}(\Sigma))<\varepsilon_{0}$,
then (1.8) has a unique solution in the smooth category, which furthermore
exists for all times and converges to a round sphere.

It should be said immediately that if $\mathcal{W}_{0}(\Phi)$ is sufficiently
small, $\Sigma$ must be a sphere. Indeed, as already noticed in [Wil65], it is
always $\mathcal{W}_{1}(\Phi)\geq 4\pi$ and from (1.2), using Gauss-Bonnet one
sees that

$\displaystyle\chi(\Sigma)=\frac{1}{2\pi}\int_{\Sigma}K\,d\sigma=\frac{1}{2\pi}\big{(}\mathcal{W}_{1}(\Phi)-\mathcal{W}_{0}(\Phi)\big{)}\geq\frac{1}{2\pi}\big{(}4\pi-\varepsilon_{0}\big{)}>1,$
(1.9)

where $\chi(\Sigma)$ is the Euler-Poincaré characteristic of $\Sigma$.

Such theorem was concerned with _smooth_ solutions. As for other geometric
flows however, an effective study of singularities and bubbling analysis
requires eventually to work at the energy level, namely, to consider
appropriate notions of weak solution.

We have in mind as a particular example the classical work on the harmonic map
flow done by Struwe [Str85, Str08] and complemented by the works of others,
the 2nd author [Riv93], Freire [Fre95b, Fre95a], Chang, Ding and Ye [CDY92],
Topping [Top02] Bertsch, Dal Passo and van der Hout [BDvdH02] just to mention
a few.

We believe that the framework introduced by the 2nd author in a series of
works [Riv08, Riv14, Riv16], which led for instance to an effective energy
quantization analysis of Willmore surfaces by Bernard and the 2nd author
[BR14] to be, when suitably adapted, the appropriate one. We want to give in
the present paper an idea of why this should be true by introducing, under
particularly favourable hypotheses, an energy-level class of weak Willmore
flow and prove a uniqueness statement for the corresponding Cauchy problem in
this class for a broad set of weak initial data, which we believe to be
sufficiently close to the largest possible one (among unbranched surfaces).

Let us mention that Lamm and Koch in [KL12] obtained (among other results of
geometric interest) an existence and uniqueness result for the Willmore flow
for entire graphs in a weak framework with Lipschitz initial datum. Such datum
needs to be small in the Lipschitz norm.

####  1.3 Well-Balanced Conformal Willmore Flows

We shall work, in the present paper, always in a low energy regime, namely we
shall arrange things so that the Willmore energy of the surfaces in
consideration $\mathcal{W}_{0}(\mathcal{S})$ is as small as needed;
furthermore, we shall also work in codimension one, namely $n=3$. The first
major consequence of this is that, as already said above, with (1.9) we may
directly assume that the underlying topology is that of the standard sphere
$S^{2}$. The second one is that we can take advantange of results from the
work of De Lellis and Müller [DM05, DM06].

So, from now, it is $\Sigma=S^{2}$, and the underling reference metric and
complex structure are the standard ones.

Central in the theory developed in [Riv08, Riv14, Riv16] and in the present
one is the idea of working with _conformal immersions_. The first advantage of
doing so is that the Willmore operator becomes uniformly elliptic, with
ellipticity constants depending on the conformal factor, and this permits
eventually the regularity bootstrap. The second one is that, exploiting
conservation laws issuing from the conformal invariance (as explained in
Bernard [Ber16]), the Willmore operator of a conformal immersion, which is a
4th order quasilinear elliptic system, can be recast as a 2nd order semilinear
system involving Jacobian-type nonlinearities, which allows regularity
bootstrap by means of integrability by compensation, similarly as in the work
of Hélein [Hél02] on weakly harmonic maps in two dimensions.

The idea is then to consider Willmore flows in conformal gauge, where the
equation becomes uniformly parabolic, if the conformal factor is uniformly
bounded away from zero, and then use a slice-wise in time (elliptic)
integrability by compensation arguments to bootstrap the regularity of the
equation, which – as is often the case when working with parabolic PDEs in
small energy regime – will suffice to get the regularity also in the time
variable. This approach was successfully used by the 2nd author in [Riv93] for
the case of the harmonic map flow.

Indeed, (1.6) is invariant under reparametrizations and thus it is degenerate
parabolic, as is the case for others geometric flows such as the mean
curvature flow or the Ricci flow - and this can be a serious source of
troubles. The celebrated trick of DeTurck [DeT83], originally devised for the
Ricci flow but easily adapted to the present situation, is one way of
overcoming this problem. For the sake of complenetess, we outlined it in
Appendix A. Such method has the advantage of working regardless of the
topology of $\Sigma$, but, as an inspection of the proof reveals, does not
seem to be suitable when working with low degrees of smoothness and moreover
does not give explicitly a control on the parametrization that is chosen by
such gauge.

We are instead going to consider the Cauchy problem

$\displaystyle\left\\{\begin{aligned} \frac{\partial}{\partial
t}\Phi&=-\delta\mathcal{W}+U,&&\text{in }(0,T)\times S^{2},\\\
\Phi(0,\cdot)&=\Phi_{0}&&\text{on }S^{2}.\\\ \end{aligned}\right.$ (1.10)

where the tangential vector field $U$ is chosen so that $\Phi(t,\cdot)$ is
conformal for every $t$.

It is a fairly simple matter to find an explicit characterization for $U$,
which – unsurprisingly, given the relationship between complex and conformal
structures on surfaces – is best expressed in complex notation. To this aim,
we recall that if $\Phi:S^{2}\to\mathbb{R}^{3}$ is a conformal immersion with
metric $g=\mathrm{e}^{2\lambda}g_{S^{2}}$, the second fundamental form may be
written in complex notation as

$\displaystyle A=h_{0}+\overline{h}_{0}+H\otimes g,$ (1.11)

where $H$ is the mean curvature and

$\displaystyle h_{0}=A_{zz}\,dz\otimes
dz=\frac{1}{4}(A_{11}-A_{22}-2iA_{12})dz\otimes dz.$ (1.12)

Similarly, the tracefree second fundamental form is written as

$\displaystyle A^{\circ}=A-Hg=h_{0}+\overline{h}_{0}.$ (1.13)

With this formalism, we have the following.

######  Lemma 1.1.

The tangential component of a conformal Willmore flow satisfies

$\displaystyle\bar{\partial}U^{(1,0)}=-\langle\delta\mathcal{W},\overline{h}_{0}{}\rangle^{\sharp_{g}},$
(1.14)

that is

$\displaystyle\partial_{\bar{z}}(U^{1}+iU^{2})\partial_{z}\otimes
d\bar{z}=2\mathrm{e}^{-2\lambda}\big{\langle}-\delta\mathcal{W},A_{\bar{z}\bar{z}}\big{\rangle}\partial_{z}\otimes
d\bar{z}.$ (1.15)

Basic facts for the $\overline{\partial}$-operator on vector fields is
recalled in Appendix B. It is certainly good news that, at least on the
sphere, it defines an uniformly elliptic, zero-cokernel operator.

However, (1.14) do not suffices per se to guarantee that (1.10) is uniformly
parabolic, not even for short time, since the control on the conformal factor
in time still needs to be addressed. Geometrically this is evident: any
conformal Willmore flow can be composed with any 1-parameter family of
conformal self-maps of $S^{2}$ and remain conformal, and
$\operatorname{Aut}(S^{2})$, the set conformal (i.e. biholomorphic) self-map
of $S^{2}$ is not compact. One can clearly see this also by noting that (1.14)
has a nontrivial kernel given by the conformal Killing (holomorphic, with
complex identification) vector fields on $S^{2}$ which is 6 dimensional, as
$\operatorname{Aut}(S^{2})$.

Moreover, (1.14) is an equation satisfied for every fixed $t$, and does not
give any information on the regularity in time of $U$. Geometrically, this
this means that the 1-parameter family of maps in $\operatorname{Aut}(S^{2})$
which we may compose a conformal Willmore flow may be taken nonsmooth with
respect to $t$.

Precisely because $\operatorname{Aut}(S^{2})$ is 6-dimensional however, a
final choice of a 6-dimensional constraint will be enough to tame the action
of such gauge group. This will be defined by the following.

######  Definition 1.2.

An immersion $\Phi:S^{2}\to\mathbb{R}^{3}$ is called _well-balanced_ if there
holds

$\displaystyle\int_{S^{2}}Id\sigma_{g}=0\quad\text{and}\quad\int_{S^{2}}\Phi\times
I\,d\sigma=0,$ (1.16)

where $I$ denotes standard embedding of $S^{2}$, $d\sigma$ its area element
and $d\sigma_{g}$ the area element for the induced metric
$g=\Phi^{\star}g_{\mathbb{R}^{3}}$.

###### Remark 1.3

Note that being well-balanced is a translation-invariant condition, namely if
$\Phi$ is well-balanced, so is $\Phi+k$ for every $k\in\mathbb{R}^{3}$.

Conditions (1.14) and (1.16) together with a good choice for the
parametrization of the initial datum , which we shall now discuss, will be
sufficient to control the behaviour of the tangential component $U$ in on
(1.10). Moreover, they are meaningful also for the notion of weak conformal
Willmore flow that we are going to define.

####  1.4  Chosing an ad-hoc Parametrization for initial Data with Small
Energy.

From a geometric perspective, both the Cauchy problems (1.8) and (1.10)-(1.14)
possess an obvious “gauge invariance” for the initial datum, namely if
$\Phi_{0}(S^{2})=\mathcal{S}$ is the immersed sphere representing the initial
datum, and $\varphi$ is any diffeomorphism of $S^{2}$, then
$\Phi_{0}\circ\varphi$ is again a parametrization for the same surface
$\mathcal{S}$, and there is no a priori preferred choice – or possibility to
distinguish – between $\Phi_{0}$ and $\Phi_{0}\circ\varphi$. This a relevant
issue for a parametric theory.

The conformal gauge choice helps to reduce this invariance ($\Phi_{0}$ has to
be conformal, and so $\varphi$ must belong to $\operatorname{Aut}(S^{2})$),
but does not break it entirely. To this aim, we shall use the following result
contained in the work of De Lellis and Müller [DM05, DM06].

######  Theorem ([DM05, DM06]).

There exist $\varepsilon_{0},\,C\,>0$ so that, if
$\mathcal{S}\subset\mathbb{R}^{3}$ is an immersed surface with area
$\mathcal{A}(\mathcal{S})=4\pi$ and Willmore energy
$\mathcal{W}_{0}(\mathcal{S})\leq\varepsilon_{0}$, there exists a conformal
parametrization $\Phi:S^{2}\to\mathcal{S}$ satisfying

$\displaystyle\|\Phi-
I-c\|_{W^{2,2}(S^{2})}+\|\mathrm{e}^{\lambda}-1\|_{L^{\infty}(S^{2})}\leq
C\sqrt{\mathcal{W}_{0}(\mathcal{S})},$ (1.17)

where $I:S^{2}\to\mathbb{R}^{3}$ denotes the standard immersion of $S^{2}$ and
$c=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{S^{2}}\Phi\,d\sigma$.

In this theorem the fact that the area of the surface is $4\pi$ can be seen to
a normalization achievable by scaling. We shall need another one, achievable
by translations; to this aim recall that the _barycenter_ of an immersed
surface $\mathcal{S}\subset\mathbb{R}^{3}$ is defined as

$\displaystyle\mathcal{C}(\mathcal{S})=\int_{\mathcal{S}}\operatorname{id}_{\mathbb{R}^{3}}\,d\mathcal{H}^{2}=\int_{\Sigma}\Phi\,d\sigma_{g},$
(1.18)

where $\Phi:\Sigma\to\mathcal{S}$ is any parametrization of $\mathcal{S}$.

The set of initial data for the conformal Willmore flow will consist
geometrically of the set of immersed surfaces
$\mathcal{S}\subset\mathbb{R}^{3}$ with Willmore energy
$\mathcal{W}_{0}(\mathcal{S})\leq\varepsilon$, area $4\pi$ and vanishing
barycenter $\mathcal{C}(\mathcal{S})=0$. Parametrically we shall choose a
parametrization provided by the above theorem, which is in addition well-
balanced as in Definition 1.2. More precisely:

######  Definition 1.4.

For $\varepsilon>0$, $\mathscr{D}^{\varepsilon}(S^{2},\mathbb{R}^{3})$ is the
set of smooth conformal immersions $\Phi:S^{2}\to\mathbb{R}^{3}$ so that the
surface $\mathcal{S}=\Phi(S^{2})$ has Willmore energy
$\mathcal{W}_{0}(\mathcal{S})\leq\varepsilon$, area
$\mathcal{A}(\mathcal{S})=4\pi$, barycenter $\mathcal{C}(\mathcal{S})=0$, is
well-balanced and so that (1.17) holds for $C>0$ given by that estimate.

This is, when restricted to the smooth category, the suitable class of initial
data that shall be considered in this work, for sufficiently small
$\varepsilon>0$. It will be enlarged to its weak $W^{2,2}$-closure when
considering the extension of the theory to the weak framework, which we
discuss below.

We want to stress that the only essential requirement in Definition 1.4 is the
control (smallness) of the Willmore energy. All the others can be seen as
normalizations. More precisely, the first result of this work, which will be
used to prove the main one, is the following extension of the theorem above:

######  Proposition 1.5.

There are $\varepsilon_{0},\,\delta,\,C>0$ with the following properties:

  1. (i)

Any immersed surface $\mathcal{S}\subset\mathbb{R}^{3}$ with area $4\pi$ and
Willmore energy $\mathcal{W}_{0}(\mathcal{S})\leq\varepsilon_{0}$ admits a
conformal parametrization satisfying (1.17) which is also well-balanced.

  2. (ii)

For any well-balanced conformal parametrization $\Psi:S^{2}\to\mathcal{S}$
with conformal factor $\mathrm{e}^{\nu}$ and some vector $c\in\mathbb{R}^{3}$
so that

$\displaystyle\|\Psi-
I-c\|_{W^{2,2}(S^{2})}+\|\mathrm{e}^{\nu}-1\|_{L^{\infty}(S^{2})}\leq\delta,$
(1.19)

there holds

$\displaystyle\|\Psi-
I-c\|_{W^{2,2}(S^{2})}+\|\mathrm{e}^{\nu}-1\|_{L^{\infty}(S^{2})}\leq
C\sqrt{\mathcal{W}_{0}(\mathcal{S})}.$ (1.20)

Furthermore, the following _local uniqueness_ property holds: if
$\Psi^{\prime}$ is another well-balanced conformal immersion satisfying
(1.19), and $\psi\in\operatorname{Aut}(S^{2})$ is the conformal diffeomorphism
so that $\Psi^{\prime}=\Psi\circ\psi$, there is a neighborhood
$\mathcal{O}\subset\operatorname{Aut}(S^{2})$ of the identity $e$ (depending
only on $\delta$) so that, if $\psi\in\mathcal{O}$, then $\psi=e$.

####  1.5 Conformal Weak Flows

We now define an energy-level class of maps where one can consider weak
conformal Willmore flows. We believe it to the a prototype for future works
concerned with Willmore flows at energy level.

Central to the definitions we shall give shortly is that the validity of the
energy identity (1.7) (in fact, a slightly weaker version will suffice). This
should be, broadly speaking, a requirement to avoid the presence of
pathological solutions that invalidate the uniqueness of the solution to the
Cauchy problem, as the examples of Topping [Top02] and Bertsch, Dal Passo and
van der Hout [BDvdH02] show in the case of the harmonic map flow.

From [Riv08, Riv14, Riv16] we recall the notion of weak $W^{2,2}$-Lipschitz
immersion. If $W^{1,\infty}_{\text{imm}}(S^{2},\mathbb{R}^{3})$ denotes the
set of Lipschitz immersions, namely those Lipschitz maps
$\Phi:S^{2}\to\mathbb{R}^{3}$ so that there exists $C=C(\Phi)>0$ with

$\displaystyle\frac{1}{C}g_{S^{2}}\leq g=\Phi^{\star}g_{\mathbb{R}^{3}}\leq
Cg_{S^{2}}$ (1.21)

almost everywhere in the sense of metrics, we let

$\displaystyle\mathscr{E}(S^{2},\mathbb{R}^{3})=W^{1,\infty}_{\text{imm}}(S^{2},\mathbb{R}^{3})\cap
W^{2,2}(S^{2}).$ (1.22)

Every map in such set admits a conformal reparametrization and moreover, it is
possible to define its Willmore operator in the sense of distributions.
Starting from the divergence form of the Willmore operator introduced in
[Riv08]:111 We denote, here and in the sequel:  –=
$\nabla^{*_{g}}(Z\otimes\omega)=\frac{1}{\sqrt{g}}\partial_{\mu}\big{(}\sqrt{g}\,g^{\mu\nu}\omega_{\nu}Z\big{)},$
_minus_ the formal $L^{2}$-adjoint of the covariant derivative induced on the
pull-back bundle $\Phi^{\star}(T\mathbb{R}^{3})$ acting on sections of
$\Phi^{\star}(T\mathbb{R}^{3})\otimes T^{*}S^{2}$, –= $\langle
A,H\rangle^{\sharp_{g}}=g^{\mu\xi}\langle
A_{\xi\nu},H\rangle\partial_{\mu}\otimes dx^{\nu}\simeq g^{\mu\xi}\langle
A_{\xi\nu},H\rangle\partial_{\mu}\Phi\otimes dx^{\nu}$ the the 1st-index
raising of $\langle A,H\rangle$, and similarly for $\langle
A^{\circ},H\rangle$.

$\displaystyle\delta\mathcal{W}=\nabla^{*_{g}}\big{(}\nabla H-2(\nabla
H)^{\top_{\Phi}}-|H|^{2}d\Phi\big{)}=\nabla^{*_{g}}\big{(}\nabla H+\langle
A^{\circ},H\rangle^{\sharp_{g}}+\langle A,H\rangle^{\sharp_{g}}\big{)},$
(1.23)

the Willmore operator of $\Phi\in\mathscr{E}(S^{2},\mathbb{R}^{3})$ is defined
as the distribution-valued two form given by

$\displaystyle\big{(}\delta\mathcal{W}d\sigma_{g},\varphi\big{)}_{\mathcal{D}^{\prime}}$
$\displaystyle=\int_{S^{2}}\Big{(}\big{\langle}H,\Delta_{g}\varphi\big{\rangle}-\big{\langle}\langle
A^{\circ},H\rangle^{\sharp_{g}}+\langle
A,H\rangle^{\sharp_{g}},\nabla\varphi\big{\rangle}_{g}\Big{)}d\sigma_{g},$
(1.24)

for every $\varphi\in C^{\infty}(S^{2},\mathbb{R}^{3})$.

We can now give the following central definitions.

######  Definition 1.6 (Weak Initial Data).

For $\varepsilon>0$, $\mathscr{W}^{\varepsilon}(S^{2},\mathbb{R}^{3})$ is the
closure of $\mathscr{D}^{\varepsilon}(S^{2},\mathbb{R}^{3})$ (from Definition
1.4) with respect to the weak $W^{2,2}(S^{2})$-topology.

We will consider $\mathscr{W}^{\varepsilon}(S^{2},\mathbb{R}^{3})$ only for
$\varepsilon>0$ sufficiently small. As a consequence of the works [Riv08,
Riv14, Riv16], $\mathscr{W}^{\varepsilon}(S^{2},\mathbb{R}^{3})$ is a subset
of the space $\mathscr{E}(S^{2},\mathbb{R}^{3})$, which would be the broadest
possible choice for the theory (among nonbranched surfaces, at least). We do
not know at present whether $\mathscr{W}^{\varepsilon}(S^{2},\mathbb{R}^{3})$
coincides, or strictly contained in, $\mathscr{E}(S^{2},\mathbb{R}^{3})$.

######  Definition 1.7 (Well-Balanced Energy Class).

For $\varepsilon,\,\delta,\,T>0$,
$\mathscr{W}_{[0,T]}^{\varepsilon,\delta}(S^{2},\mathbb{R}^{3})$ is set of
locally integrable maps ${\Phi:(0,T)\times S^{2}\to\mathbb{R}^{3}}$ so that

  1. (i)

For almost every $t$, $\Phi(t,\cdot)$ is in
$\mathscr{E}(S^{2},\mathbb{R}^{3})$ and conformal,

  2. (ii)

There holds

$\displaystyle\|\Phi-
I-c\|_{L^{\infty}((0,T),W^{2,2}(S^{2}))}+\|\mathrm{e}^{\lambda}-1\|_{L^{\infty}((0,T)\times
S^{2})}\leq\delta,$ (1.25)

where $I$ denotes standard embedding of $S^{2}$,
$\mathrm{e}^{\lambda}=\mathrm{e}^{\lambda(t,\cdot)}$ is the conformal factor
of $\Phi(t,\cdot)$ and
$c(t)=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{S^{2}}\Phi(t,\cdot)\,d\sigma$,

  3. (iii)

There holds

$\displaystyle\delta\mathcal{W}\in L^{2}((0,T)\times
S^{2})\quad\text{and}\quad\mathcal{W}_{0}(\Phi(t,\cdot))\leq\varepsilon\;\text{
for a.e. }t,$ (1.26)

  4. (iv)

$\Phi$ is well-balanced for a.e. $t$.

Finally we let, also for $T=+\infty$,

$\displaystyle\mathscr{W}_{[0,T)}^{\varepsilon,\delta}(S^{2},\mathbb{R}^{3})=\bigcap_{\tau\in(0,T)}\mathscr{W}_{[0,\tau]}^{\varepsilon,\delta}(S^{2},\mathbb{R}^{3}).$
(1.27)

Assumption (1.25) is quite natural if we look at Proposition 1.5. In the
energy class, a weak Willmore flow is defined as follows.

######  Definition 1.8 (Weak Willmore Flow).

$\Phi\in\mathscr{W}_{[0,T)}^{\varepsilon,\delta}(S^{2},\mathbb{R}^{3})$ is a
weak solution of the Willmore flow with tangential component
$U=U^{\mu}\partial_{\mu}\Phi$:

$\displaystyle\frac{\partial}{\partial
t}\Phi=-\delta\mathcal{W}+U\quad\text{in }(0,T)\times S^{2},$ (1.28)

if for every $\varphi\in C^{\infty}_{c}((0,T)\times S^{2},\mathbb{R}^{3})$
there holds

$\displaystyle-\int_{0}^{T}\int_{S^{2}}\Big{\langle}\Phi,\frac{\partial}{\partial
t}\varphi\Big{\rangle}\,d\sigma_{g}\,dt=-\int_{0}^{T}\big{(}\delta\mathcal{W}d\sigma_{g},\varphi(t,\cdot)\big{)}_{\mathcal{D}^{\prime}}dt+\int_{0}^{T}\int_{S^{2}}\big{\langle}U,\varphi\big{\rangle}\,d\sigma_{g}\,dt.$
(1.29)

Our main result is the following.

######  Theorem 1.9.

There exists $\varepsilon_{0}>0$ so that the Cauchy problem for the conformal
Willmore flow (1.10)-(1.14) with initial datum in
$\mathscr{W}^{\varepsilon_{0}}(S^{2},\mathbb{R}^{3})$ has a weak solution in
$\mathscr{W}^{\varepsilon_{0},\delta}_{[0,T)}(S^{2},\mathbb{R}^{3})$ for some
$\delta>0$, assuming the initial datum in the sense of traces. Such solution
is smooth, exists for all times and smoothly converges to the standard
embedding $I$ of $S^{2}$ in $\mathbb{R}^{3}$. Furthermore, if the initial
datum is smooth, such weak solution is also unique.

We can compare this result with the above metioned one of Kuwert and Schätzle
[KS01, KS02]. They obtain, in the smooth class, long-time existence,
uniqueness and convergence to a round sphere for the Cauchy problem of the
normal flow (1.8). A central feature our result is that the uniqueness of this
smooth solution is in the broad class of finite energy solutions, and the fact
that it converges exactly to the standard embedding.

We expect the solution to be unique also if the initial datum is nonsmooth; we
plan to address this question in the future.

The proof of the regularity part of Theorem 1.9 shares evident similarities
with the corresponding one for the harmonic map flow obtained in [Riv93]. In
that work, the core estimate that was obtained for weak solutions of the
harmonic map flow was of the form

$\displaystyle\|u(t,\cdot)\|_{W^{2,2}}\leq
C\big{(}\|\partial_{t}u(t,\cdot)\|_{L^{2}}+1\big{)}\quad\text{for a.e. }t,$

which could then be squared and integrated in time to yield higher regularity,
and eventually smoothness by the classical theory by Struwe [Str85, Str08]. We
shall obtain a similar result, namely an inequality of the form

$\displaystyle\|\Phi(t,\cdot)\|_{W^{4,2}}\leq
C\big{(}\|\mathrm{e}^{\lambda}\delta\mathcal{W}(t,\cdot)\|_{L^{2}}+1\big{)}\quad\text{for
a.e. }t,$

for weak solutions of the conformal Willmore flow, and likewise obtain higher
regularity from it. The overall procedure shall be however more technical.

####  1.6 Final Comments

We have intentionally decided to work in a small-energy regime in this first
paper. We plan to consider more general scenarios in future works, where more
technical, localization/energy-concentration arguments will be dealt with.

One has also to take into account that, when the underlying surface is not a
sphere, there is more than one conformal class, so to properly work with a
conformal Willmore flow, one has to take into account the nontriviality of the
corresponding Teichmüller space. The work of Rupflin and Topping [RT16] on the
Teichmüller harmonic map flow also faces the difficulty of “following” the
conformal class along the flow.

In the future we plan to determine whether the class
$\mathscr{W}^{\varepsilon}(S^{2},\mathbb{R}^{3})$ coincides or not with
$\mathscr{E}(S^{2},\mathbb{R}^{3})$, and whether the solution given by Theorem
1.9 is unique also in the case of weak initial data. We shall also seek to
extend the argument for branched weak initial data.

Finally, we plan to carry an accurate study of singularities (blow-up points,
degeneration of conformal factor or conformal class…) in forthcoming works.
For this we will likely build upon some of the work already done on the
subject, Mayer and Simonett [MS02], Blatt [Bla09] and Chill, Fašangová and
Schätzle [CFS09] just to mention a few.

One of the questions relative to the parametric approach of the Willmore flow
is the following: can the conformal class of a conformal Willmore flow –
suitably normalized to remove any obvious gauge invariance – degenerate in
finite time?

##  2 Preliminaries

We state here known results that shall play a key role in the paper.

######  Theorem 2.1.

([Bet92], [Riv93], [Ge99]) Let $\Omega\subseteq\mathbb{R}^{2}$ be a bounded
domain, let $a\in W^{1,2}(\Omega)$ and $f\in L^{p}(\Omega)$ for $1<p<2$. Let
$u\in W^{1,(2,\infty)}(\Omega)$222That is, $u\in W^{1,1}(\Omega)$ and $\nabla
u$ is in the Lorentz space $L^{(2,\infty)}(\Omega)$. be a solution to

$\displaystyle-\Delta u=\langle\nabla^{\perp}a,\nabla u\rangle+f\quad\text{in
}\Omega.$ (2.1)

Then $u\in W^{2,p}_{\text{loc}}(\Omega)$.

######  Theorem 2.2 ([Hél02], [Riv16]).

There exists $\varepsilon_{0}>0$ so that, if
$\Phi\in\mathscr{E}(B_{1},\mathbb{R}^{n})$ is conformal with conformal factor
$e^{\lambda}$ and so that

$\displaystyle\int_{B_{1}}|A|^{2}_{g}\,d\sigma_{g}\leq\varepsilon_{0},$ (2.2)

then, if $C_{(2,\infty)}>0$ is constant so that

$\displaystyle\|d\lambda\|_{L^{(2,\infty)}(B_{1})}\leq C_{(2,\infty)},$ (2.3)

for any $0<r<1$ there holds

$\displaystyle\|d\lambda\|_{L^{2}(B_{r})}+\|\lambda-\ell\|_{L^{\infty}(B_{r})}\leq
C\int_{B_{1}}|A|^{2}_{g}\,d\sigma_{g},$ (2.4)

for some constants $\ell\in\mathbb{R}$ and $C=C(r,C_{(2,\infty)})>0$.

###### Remark 2.3

By the triangle inequalily, without loss of generality we can take
$\ell=\lambda(0)$ in the above estimate.

######  Theorem 2.4 ([DM05, DM06]).

Let $\mathcal{S}\subset\mathbb{R}^{3}$ be an immersed surface with area
$\mathcal{A}(\mathcal{S})=4\pi$ and let $g$ be the induced metric. Then there
holds

$\displaystyle\int_{\mathcal{S}}|A_{\text{sc}}-g|^{2}_{g}\,d\sigma_{g}\leq
C\int_{\mathcal{S}}|A^{\circ}|^{2}_{g}\,d\sigma_{g},$ (2.5)

where $A_{\text{sc}}(\cdot\,,\cdot)=\langle A(\cdot\,,\cdot),N\rangle$ is the
scalar second fundamental form of $\mathcal{S}$. Furthermore, there is
$\varepsilon_{0}>0$ so that if

$\displaystyle\mathcal{W}_{0}(\mathcal{S})=\frac{1}{2}\int_{\mathcal{S}}|A^{\circ}|^{2}_{g}\,d\sigma_{g}\leq\varepsilon_{0},$
(2.6)

there exists a conformal parametrization $\Phi:S^{2}\to\mathbb{R}^{3}$
satisfying

$\displaystyle\|\Phi-(c_{\mathcal{S}}+I)\|_{W^{2,2}(S^{2})}+\|\mathrm{e}^{\lambda}-1\|_{L^{\infty}(S^{2})}\leq
C\sqrt{\mathcal{W}_{0}(\mathcal{S})},$ (2.7)

for some vector $c_{\mathcal{S}}\in\mathbb{R}^{3}$ and for an absolute
constant $C>0$, where $\mathrm{e}^{\lambda}$ is the conformal factor of the
induced metric and $I$ is the standard immersion of $S^{2}$ into
$\mathbb{R}^{3}$.

###### Remark 2.5

From the minimality property of the average:

$\displaystyle\Big{\|}f-\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{S^{2}}f\,d\sigma\Big{\|}_{L^{2}(S^{2})}=\inf_{c\in\mathbb{R}}\|f-c\|_{L^{2}(S^{2})},$
(2.8)

and since $\int_{S^{2}}I\,d\sigma=0$, we may suppose
$c_{\mathcal{S}}=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{S^{2}}\Phi\,d\sigma_{S^{2}}$
in (2.7).

######  Theorem 2.6 ([Riv08], [Riv16], [Ber16]).

The Willmore operator of an immersion of a surface
$\Phi:\Sigma\to\mathbb{R}^{n}$,

$\displaystyle\delta\mathcal{W}=\Delta^{\perp}_{g}H+Q(A^{\circ})H,$ (2.9)

where $\Delta^{\perp}_{g}$ denotes the Laplace operator on the normal bundle
of $\Phi(\Sigma)$ and

$\displaystyle Q(A^{\circ})H=\langle A^{\circ},\langle
H,A^{\circ}\rangle\rangle_{g}=g^{\mu\sigma}g^{\nu\tau}\langle
A^{\circ}_{\mu\nu},\langle A^{\circ}_{\sigma\tau},H\rangle\rangle,$ (2.10)

may be written equivalently in divergence form as

$\displaystyle\delta\mathcal{W}=\nabla^{*_{g}}\big{(}\nabla H+\langle
A^{\circ},H\rangle^{\sharp_{g}}+\langle
A,H\rangle^{\sharp_{g}}\big{)}=\Delta_{g}H+\nabla^{*_{g}}\big{(}\langle
A^{\circ},H\rangle^{\sharp_{g}}+\langle A,H\rangle^{\sharp_{g}}\big{)}.$
(2.11)

When $n=3$, if $w\in\Gamma(\Phi^{\star}(T\mathbb{R}^{3})\otimes T^{*}\Sigma)$
denotes the vector-valued form along $\Phi$ given by

$\displaystyle w=\nabla H+\langle A^{\circ},H\rangle^{\sharp_{g}}+\langle
A,H\rangle^{\sharp_{g}}=\nabla H-2(\nabla H)^{\top}-|H|^{2}d\Phi.$ (2.12)

Then the following formulas hold true:

$\displaystyle\delta\mathcal{W}$ $\displaystyle=\nabla{}^{*_{g}}w,$ (2.13)
$\displaystyle\Phi\times\delta\mathcal{W}$
$\displaystyle=\nabla{}^{*_{g}}\big{(}-d\Phi\times H+\Phi\times w\big{)},$
(2.14) $\displaystyle\big{\langle}\Phi,\delta\mathcal{W}\big{\rangle}$
$\displaystyle=d^{*_{g}}\big{\langle}\Phi,w\big{\rangle},$ (2.15)
$\displaystyle\Phi\times(\Phi\times\delta\mathcal{W})+\Phi\big{\langle}\Phi,\delta\mathcal{W}\big{\rangle}+4H$
$\displaystyle=\nabla{}^{*_{g}}\big{(}-2\Phi\times(d\Phi\times
H)+\Phi\times(\Phi\times w)+\Phi\langle\Phi,w\rangle\big{)}.$ (2.16)

###### Remark 2.7

  1. (i)

Equations (2.13), (2.14), (2.15) and (2.16) follow from Noether’s theorem and
the invariance of the Willmore energy under translations, rotations, dilations
and inversions respectively. To obtain (2.16) (since inversions do not form
per se a 1–parameter family of transformations), one applies Noether’s theorem
to the local family $F(s,y)=\mathcal{I}\,\circ\,\tau_{sv}\circ\mathcal{I}(y)$,
where $\mathcal{I}(y)=\frac{y}{|y|^{2}}$ denotes the inversion with respect to
the unit sphere and $\tau_{v}(y)=y+v$ is the translation of vector
$v\in\mathbb{R}^{3}$. This transformation is generated by the vector field
$\mathcal{X}(y)=v|y|^{2}-2y\langle y,v\rangle$ and gives rise to the law
(2.16).

  2. (ii)

The equivalence between the first and second expressions for $w$ in (2.12) is
obtained by noting that, since $H$ is a normal vector, there holds

$\displaystyle(\nabla H)^{\top}$
$\displaystyle=g^{\mu\nu}\langle\partial_{\alpha}H,\partial_{\mu}\Phi\rangle\partial_{\nu}\Phi\otimes
dx^{\alpha}$ (2.17) $\displaystyle=-g^{\mu\nu}\langle
H,\partial^{2}_{\alpha\mu}\Phi\rangle\partial_{\nu}\Phi\otimes dx^{\alpha}$
(2.18) $\displaystyle=-g^{\mu\nu}\langle
H,A_{\alpha\mu}\rangle\partial_{\nu}\Phi\otimes dx^{\alpha}$ (2.19)
$\displaystyle=-\langle H,A\rangle^{\sharp_{g}},$ (2.20)

and since one similarly has $|H|^{2}d\Phi=\langle H,Hg\rangle^{\sharp_{g}}$,
there holds

$\displaystyle 2(\nabla H)^{\top}+|H|^{2}d\Phi$ $\displaystyle=-2\langle
H,A\rangle^{\sharp_{g}}+\langle H,Hg\rangle^{\sharp_{g}}=-\langle
H,A\rangle^{\sharp_{g}}-\langle H,A^{\circ}\rangle^{\sharp_{g}}.$ (2.21)

######  Theorem 2.8 ([KS01, KS02, KS20]).

There exists an $\varepsilon_{0}=\varepsilon_{0}(n)>0$ so that, if
${\Phi:[0,T)\times S^{2}\to\mathbb{R}^{n}}$ is a smooth normal Willmore flow

$\displaystyle\frac{\partial}{\partial t}\Phi=-\delta\mathcal{W},$ (2.22)

with smooth initial datum $\Phi(0,\cdot)=\Phi_{0}$ and
$\mathcal{W}_{0}(\Phi_{0})\leq\varepsilon_{0}$, then:

  1. (i)

its area satisfies

$\displaystyle|\mathcal{A}(\Phi(t,\cdot))-\mathcal{A}(\Phi_{0})|\leq
C\mathcal{A}_{0}(\Phi_{0})\mathcal{W}_{0}(\mathcal{S}),$ (2.23)

for a constant $C=C(n)>0$.

  2. (ii)

Its barycenter
$\mathcal{C}(\Phi)=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{S^{2}}\Phi\,d\sigma_{g}$
satisfies

$\displaystyle|\mathcal{C}(\Phi(t,\cdot))-\mathcal{C}(\Phi_{0})|\leq
C\mathcal{W}_{0}(\mathcal{S})$ (2.24)

for a constant $C=C(n)>0$.

  3. (iii)

$\Phi$ exists for all times and smoothly converges to a round sphere
$t\to\infty$.

##  3 Consequences of DeLellis-Müller’s Theorem

The proof of Proposition 1.5 will follow from two lemmas.

######  Lemma 3.1.

The function $F:\operatorname{Aut}(S^{2})\to\mathbb{R}^{6}$ given by

$\displaystyle
F(\psi)=(F_{1}(\psi),F_{2}(\psi))=\Big{(}\int_{S^{2}}I\,\psi^{\star}d\sigma,\int_{S^{2}}(I\circ\,\psi)\times
I\,d\sigma\Big{)},$ (3.1)

where $\psi^{\star}d\sigma$ denotes the pullback of area element $d\sigma$ via
$\psi$, is differentiable and $dF(e)$ is an isomorphism.

###### Proof.

Differentiability follows since $F$ is composition of smooth functions and
operations. We shall now use the language of differential forms and so along
this proof it is convenient to temporarily change our notation for the area
element from $d\sigma$ to $\omega_{S^{2}}$.

Recall that $\omega_{S^{2}}=\omega_{\mathbb{R}^{3}}\llcorner
N=\omega_{\mathbb{R}^{3}}(N,\cdot,\cdot)$, where $N$ Gauss map of $S^{2}$.
More explicitly, since $N(y)=y$, we have the formula

$\displaystyle\omega_{S^{2}}=(dy^{1}\wedge dy^{2}\wedge
dy^{3})(y,\cdot,\cdot)=y^{1}dy^{2}\wedge dy^{3}-y^{2}dy^{1}\wedge
dy^{3}+y^{3}dy^{1}\wedge dy^{2}.$ (3.2)

Now $\operatorname{Aut}(S^{2})$ has dimension 6 as a manifold and we consider
the basis for the tangent space $T_{e}\operatorname{Aut}(S^{2})$ given by the
vector fields generating, respectively, rotations and “spherical dilations”
about the coordinate axes:

$\displaystyle Z_{1}(y)$ $\displaystyle=(0,-y^{3},y^{2}),$ $\displaystyle
Z_{2}(y)$ $\displaystyle=(y^{3},0,-y^{1}),$ $\displaystyle Z_{3}(y)$
$\displaystyle=(-y^{2},y^{1},0),$ (3.3) $\displaystyle Z_{4}(y)$
$\displaystyle=e_{1}-y^{1}y,$ $\displaystyle Z_{5}(y)$
$\displaystyle=e_{2}-y^{2}y,$ $\displaystyle Z_{6}(y)$
$\displaystyle=e_{3}-y^{3}y.$ (3.4)

We shall prove that

$\displaystyle\Big{(}\frac{\partial F^{j}}{\partial Z_{a}}(e)\Big{)}_{1\leq
a,j\leq 6}=-\frac{8\pi}{3}\mathbf{1}_{6\times 6}.$ (3.5)

To compute $\partial_{X}F(e)$, if $\Phi^{X}$ denotes the local flow of the
vector field $X$, we have to evaluate

$\displaystyle\frac{\partial}{\partial
X}F(e)=\frac{d}{dt}F(\Phi^{X}(t,\cdot))\bigg{|}_{t=0}.$ (3.6)

Let us look at $F_{1}$. By Cartan’s formula, since $d\omega_{S^{2}}=0$, it is

$\displaystyle\frac{\partial}{\partial
t}\Phi^{X}(t,\cdot)\omega_{S^{2}}\Big{|}_{t=0}=\mathcal{L}_{X}\omega_{S^{2}}=d(X\llcorner\omega_{S^{2}}),$
(3.7)

where $\mathcal{L}_{X}$ denotes the Lie derivative with respect to $X$. Since
$Z_{a}$’s for $a=1,2,3$ generate isometries,
$\mathcal{L}_{Z_{a}}\omega_{S^{2}}=0$ and hence _a fortiori_

$\displaystyle\frac{d}{dt}F_{1}\big{(}\Phi^{Z_{a}}(t,\cdot)\big{)}\bigg{|}_{t=0}=\int_{S^{2}}I\,\mathcal{L}_{Z_{a}}\omega_{S^{2}}=0\quad\quad\text{for
}a=1,2,3.$ (3.8)

As for the $Z_{a}$’s for $a=4,5,6$, we see that

$\displaystyle Z_{4}\llcorner\omega_{S^{2}}$
$\displaystyle=-y^{2}dy^{3}+y^{3}dy^{2},$ $\displaystyle
Z_{5}\llcorner\omega_{S^{2}}$ $\displaystyle=y^{1}dy^{3}-y^{3}dy^{1},$
$\displaystyle Z_{6}\llcorner\omega_{S^{2}}$
$\displaystyle=-y^{1}dy^{2}+y^{2}dy^{1},$ (3.9) $\displaystyle
d(Z_{4}\llcorner\omega_{S^{2}})$ $\displaystyle=-2dy^{2}\wedge dy^{3},$
$\displaystyle d(Z_{5}\llcorner\omega_{S^{2}})$ $\displaystyle=2dy^{1}\wedge
dy^{3},$ $\displaystyle d(Z_{6}\llcorner\omega_{S^{2}})$
$\displaystyle=-2dy^{1}\wedge dy^{2},$ (3.10)

and hence with Stokes’ theorem we get

$\displaystyle\frac{d}{dt}F_{1}\big{(}\Phi^{Z_{4}}(t,\cdot)\big{)}\bigg{|}_{t=0}$
$\displaystyle=\int_{S^{2}}I\mathcal{L}_{Z_{4}}\omega_{S^{2}}$ (3.11)
$\displaystyle=\Big{(}\int_{S^{2}}y^{1}(-2dy^{2}\wedge
dy^{3}),\int_{S^{2}}y^{2}(-2dy^{1}\wedge
dy^{3}),\int_{S^{2}}y^{3}(-2dy^{1}\wedge dy^{2})\Big{)}$ (3.12)
$\displaystyle=-\frac{8\pi}{3}(1,0,0),$ (3.13)

and similarly

$\displaystyle\frac{d}{dt}F_{1}\big{(}\Phi^{Z_{5}}(t,\cdot)\big{)}\bigg{|}_{t=0}=-\frac{8\pi}{3}(0,1,0),\qquad\frac{d}{dt}F_{1}\big{(}\Phi^{Z_{6}}(t,\cdot)\big{)}\bigg{|}_{t=0}=-\frac{8\pi}{3}(0,0,1).$
(3.14)

Now we consider $F_{2}$. Since we can write

$\displaystyle
F_{2}\big{(}\Phi^{X}(t,\cdot)\big{)}=\int_{S^{2}}\Phi^{X}(t,\cdot)\times
I\,d\sigma,$ (3.15)

it is

$\displaystyle\frac{d}{dt}F_{2}\big{(}\Phi^{X}(t,\cdot)\big{)}\bigg{|}_{t=0}=\int_{S^{2}}X\times
I\,d\sigma,$ (3.16)

and thus one directly computes that for $a=1,2,3$ it is

$\displaystyle\int_{S^{2}}Z_{a}\times I\,d\sigma$
$\displaystyle=-\frac{8\pi}{3}e_{a},\qquad\int_{S^{2}}Z_{a+3}\times
I\,d\sigma=0.$ (3.17)

Putting together all these computations yields (3.5) and hence the thesis. ∎

######  Lemma 3.2.

There exists $E>0$ with the following property. For any $\eta_{1}\leq E$ there
exist $\eta_{2}>0$ so that, if $\mathcal{S}\subset\mathbb{R}^{3}$ is an
immersed surface with area $\mathcal{A}(\mathcal{S})=4\pi$ and
$\Phi:S^{2}\to\mathcal{S}$ is a conformal immersion with conformal factor
$\mathrm{e}^{\lambda}$ and $c\in\mathbb{R}^{3}$ is a vector so that

$\displaystyle\|\Phi-
I-c\|_{W^{2,2}(S^{2})}+\|e^{\lambda}-1\|_{L^{\infty}(S^{2})}\leq\eta_{1},$
(3.18)

then there exists a conformal self-map $\psi\in\operatorname{Aut}(S^{2})$ so
that $\Psi=\Phi\circ\psi$ is well-balanced and $\psi$ is the unique self-map
with such property in the Riemannian ball
$B_{\eta_{2}}(e)\subset\operatorname{Aut}(S^{2})$. In addition, if
$\mathrm{e}^{\nu}$ denotes the conformal factor of $\Psi$, there holds

$\displaystyle\|\Psi-
I-c\|_{W^{2,2}(S^{2})}+\|\mathrm{e}^{\nu}-1\|_{L^{\infty}(S^{2})}\leq
2\eta_{1}.$ (3.19)

###### Proof.

Let
$\mathcal{F}:W^{2,2}(S^{2},\mathbb{R}^{3})\times\operatorname{Aut}(S^{2})\to\mathbb{R}^{6}$
be given by

$\displaystyle\quad\mathcal{F}(f,\psi)=\Big{(}\int_{S^{2}}I\,\frac{1}{2}|d(f\circ\psi)|^{2}d\sigma,\int_{S^{2}}(f\circ\psi)\times
I\,d\sigma\Big{)}.$ (3.20)

Note that this definition makes sense for every $f\in W^{2,2}(S^{2})$ and, if
$\Phi$ is a conformal immersion $\mathcal{F}(\Phi,e)=0$ means that $\Phi$ is
well-balanced as in Definition 1.2. Moreover $\mathcal{F}(I,\cdot)=0$
coincides with $F$ given in (3.1). Finally $\mathcal{F}$ is invariant by
translations in its first component:
$\mathcal{F}(f,\cdot)=\mathcal{F}(f+k,\cdot)$ for every $k\in\mathbb{R}^{3}$.

As a consequence of Lemma 3.1,
$d_{\psi}\mathcal{F}(I,e)=d_{\psi}\mathcal{F}(I,\cdot)(e)$ is an isomorphism,
and hence by the implicit function theorem, there exists $E,\eta_{2}>0$ so
that if (3.18) holds for $\eta_{1}\leq E$, there is a unique
$\psi=\psi_{\Phi}$ in the Riemannian ball
$B_{\eta_{2}}(e)\subset\operatorname{Aut}(S^{2})$ so that
$\mathcal{F}(\Phi-c,\psi)=\mathcal{F}(\Phi,\psi)=0$, i.e. so that
$\Psi=\Phi\circ\psi$ is well-balanced (recall also Remark 1.3).

Finally, since $\psi$ is biholomorphic, $\forall N\in\mathbb{N}$ we can
estimate ${\sum_{k=1}^{N}\operatorname{dist}(\nabla^{k}\psi,\nabla^{k}e)\leq
C_{N}\operatorname{dist}(\psi,e)}$ for some $C_{N}>0$ independent of $\psi$.
So (3.18) holds, by the triangle inequality and the continuity of the Lebesgue
integral,

$\displaystyle\phantom{{}={}}\|\Psi-
I-c\|_{W^{2,2}(S^{2})}+\|\mathrm{e}^{\nu}-1\|_{L^{\infty}(S^{2})}$ (3.21)
$\displaystyle=\|\Phi\circ\psi-
I-c\|_{W^{2,2}(S^{2})}+\|\tfrac{1}{\sqrt{2}}|d\psi|\mathrm{e}^{\lambda\circ\psi}-1\|_{L^{\infty}(S^{2})}$
(3.22)
$\displaystyle=\|(\Phi-c)\circ\psi-I\|_{W^{2,2}(S^{2})}+\|\tfrac{1}{\sqrt{2}}|d\psi|\mathrm{e}^{\lambda\circ\psi}-1\|_{L^{\infty}(S^{2})}$
(3.23) $\displaystyle=\eta_{1}+o(1)\quad\text{as
}\operatorname{dist}(\psi,e)\to 0,$ (3.24)

and, since $\|\Psi-c\|_{L^{2}(S^{2})}$ and
$\|e^{\lambda}\|_{L^{\infty}(S^{2})}$ are uniformly bounded, the remainder
$o(1)$ can be taken uniform in $\Phi,\Psi,\psi,c$ and hence, choosing
$\eta_{2}$ sufficiently small we obtain to (3.19). ∎

###### Proof of Proposition 1.5.

It suffices to prove the thesis for
$\mathcal{W}_{0}(\mathcal{S})\leq\varepsilon_{0}$ sufficiently small. For part
(i), combine Theorem 2.4 and Lemma 3.2.

For part (ii), Let $\varepsilon_{0}>0$ be sufficiently small so that

$\displaystyle C\varepsilon_{0}\leq\frac{1}{2}E,$ (3.25)

where $E$ is as in Lemma 3.2 and $C$ is the constant of Theorem 2.4, and let
$\Phi:S^{2}\to\mathcal{S}$ be the conformal parametrization given by that
theorem. By Lemma 3.2 there exists a unique choice of $\alpha=\alpha_{\Phi}$
in $B_{\eta_{2}}(e)\subset\operatorname{Aut}(S^{2})$ with
$\mathcal{F}(\Phi,\alpha)=0$ i.e. $\Phi^{\prime}=\Phi\circ\alpha$ is well-
balanced and

$\displaystyle\|\Phi^{\prime}-I-c\|_{W^{2,2}(S^{2})}+\|e^{\lambda^{\prime}}-1\|_{L^{\infty}(S^{2})}\leq
2C\sqrt{\mathcal{W}_{0}(\Phi^{\prime})}\leq\sqrt{E}.$ (3.26)

So now if $\delta$ is taken so that

$\displaystyle\delta\leq\frac{1}{2}\sqrt{E},$ (3.27)

since $\Psi$ is already well-balanced, by uniqueness it must be
$\Phi^{\prime}=\Psi$, and the thesis follows also for the local uniqueness
part, with $\mathcal{O}=B_{\eta_{2}}(e)$. ∎

The following simple consequence of Theorem 2.4 will also be needed later.

######  Lemma 3.3.

If $\Phi:S^{2}\to\mathbb{R}^{3}$ is a conformal immersion with conformal
factor $e^{\lambda}$ and $B_{r}(x_{0})\subset S^{2}$ is a disk of radius $r$
(in the standard metric of $S^{2}$), there holds

$\displaystyle\int_{B_{r}(x_{0})}|A|^{2}_{g}\,d\sigma_{g}$ $\displaystyle\leq
C\Big{(}\int_{S^{2}}|A^{\circ}|^{2}_{g}\,d\sigma_{g}+\mathrm{e}^{4C_{0}}r^{2}\Big{)},$
(3.28)

where $C_{0}=\|\lambda\|_{L^{\infty}(S^{2})}$ and $C>0$ is an absolute
constant.

###### Proof.

It is a consequence of (2.5) applied to the immersed surface
$\mathcal{S}=a\,\Phi(S^{2})$, where $a=\sqrt{\frac{4\pi}{\mathcal{A}(\Phi)}}$
and $\mathcal{A}(\Phi)=\int_{S^{2}}\mathrm{e}^{2\lambda}\,d\sigma$ is the area
of $\Phi(S^{2})$. Indeed, since

$\displaystyle 4\pi\mathrm{e}^{-2C_{0}}\leq\mathcal{A}(\Phi)\leq
4\pi\mathrm{e}^{2C_{0}},$ (3.29)

it follows that $e^{-C_{0}}\leq a\leq\mathrm{e}^{C_{0}}$ and we can estimate

$\displaystyle\int_{B_{r}(x_{0})}|a\,g|^{2}_{g}d\sigma_{g}=a^{2}\int_{B_{r}(x_{0})}2\mathrm{e}^{2\lambda}d\sigma\leq
C\mathrm{e}^{4C_{0}}r^{2}.$ (3.30)

and thus

$\displaystyle\int_{B_{r}(x_{0})}|A|^{2}_{g}\,d\sigma_{g}$ $\displaystyle\leq
2\int_{B_{r}(x_{0})}|A_{\text{sc}}-ag|^{2}_{g}\,d\sigma_{g}+2\int_{B_{r}(x_{0})}|a\,g|^{2}_{g}\,d\sigma_{g}$
(3.31) $\displaystyle\leq
C\int_{B_{r}(x_{0})}|A^{\circ}|^{2}_{g}\,d\sigma_{g}+C\mathrm{e}^{4C_{0}}r^{2},$
(3.32)

which proves (3.28). ∎

##  4 Estimates of Elliptic Type

This section is dedicated to prove the following.

######  Theorem 4.1.

Let $\Phi\in\mathscr{E}(B_{1},\mathbb{R}^{3})$ be conformal with conformal
factor $\mathrm{e}^{\lambda}$ and Willmore operator $\delta\mathcal{W}$ in
$L^{2}(B_{1})$. Then $\Phi\in W^{4,2}_{\text{loc}}(B_{1})$, and furthermore if
$C_{(2,\infty)}>0$ is constant so that

$\displaystyle\|d\lambda\|_{L^{(2,\infty)}(B_{1})}\leq C_{(2,\infty)},$ (4.1)

there exists an $\varepsilon_{0}>0$ depending only on $C_{(2,\infty)}$ so that
if

$\displaystyle\int_{B_{1}}|A|^{2}_{g}\,d\sigma_{g}\leq\varepsilon_{0},$ (4.2)

then the following estimate holds:

$\displaystyle\|d\Phi\|_{W^{3,2}(B_{1/2})}\leq
C\big{(}\|\mathrm{e}^{4\lambda}\delta\mathcal{W}\|_{L^{2}(B_{1})}+\|\mathrm{e}^{\lambda}\|_{L^{2}(B_{1})}\big{)},$
(4.3)

where $C=C(C_{(2,\infty)})>0$.

###### Remark 4.2

The fact that the estimate (4.3) does not include $\|\Phi\|_{L^{2}(B_{1/2})}$
on the left hand-side is motivated by the translation invariance of all the
quantities on the right-hand side.

###  4.1 Qualitative Estimates

######  Proposition 4.3.

Let $1<q<\infty$ and let $\Phi\in\mathscr{E}(B_{1},\mathbb{R}^{n})$ be a
conformal with Willmore operator $\delta\mathcal{W}$ in $L^{q}(B_{1})$. It
suffices to know that $H\in L^{p}(B_{1})$ for some $p>2$ to deduce that
$\Phi\in W^{4,q}_{\text{loc}}(B_{1})$.

###### Proof.

We may certainly assume $2<p<4$. Since

$\displaystyle\Delta\Phi=2\mathrm{e}^{2\lambda}H\in L^{p},$ (4.4)

elliptic regularity theory gives that $\Phi\in W^{2,p}_{\text{loc}}$, and this
in turn implies

$\displaystyle A=(\nabla^{2}\Phi)^{\perp}\in L^{p}_{\text{loc}}.$ (4.5)

Looking at (2.11):

$\displaystyle-\Delta H=\nabla^{*}(\langle
A^{\circ},H\rangle^{\sharp_{g}}+\langle
A,H\rangle^{\sharp_{g}})-\mathrm{e}^{2\lambda}\delta\mathcal{W},$ (4.6)

we see that

$\displaystyle\langle A^{\circ},H\rangle^{\sharp_{g}}+\langle
A,H\rangle^{\sharp_{g}}=\mathrm{e}^{-2\lambda}\big{(}\langle
A^{\circ},H\rangle^{\sharp}+\langle A,H\rangle^{\sharp}\big{)}\in
L^{\frac{p}{2}}_{\text{loc}},$ (4.7)

and, since $q>1$, we have in particular that $\delta\mathcal{W}\in W^{-1,2}$;
thus $\Delta H\in W^{-1,\frac{p}{2}}_{\text{loc}}$, whence elliptic regularity
and Sobolev embedding give

$\displaystyle H\in W^{1,\frac{p}{2}}_{\text{loc}}\hookrightarrow
L^{\left(\frac{p}{2}\right)^{*}}_{\text{loc}},$ (4.8)

and from this, it follows that $A\in
L^{\left(\frac{p}{2}\right)^{*}}_{\text{loc}}.$ Then

$\displaystyle\langle A^{\circ},H\rangle^{\sharp_{g}}+\langle
A,H\rangle^{\sharp_{g}}\in
L^{\frac{1}{2}\left(\frac{p}{2}\right)^{*}}_{\text{loc}},$ (4.9)

whence $\Delta H\in
W^{-1,\frac{1}{2}\left(\frac{p}{2}\right)^{*}}_{\text{loc}}$, so by elliptic
regularity

$\displaystyle H\in
W^{1,\frac{1}{2}\left(\frac{p}{2}\right)^{*}}_{\text{loc}}.$ (4.10)

This process can be iterated, and since the sequence $p$,
$\big{(}\frac{p}{2}\big{)}^{*}$,
$\big{(}\frac{1}{2}\big{(}\frac{p}{2}\big{)}^{*}\big{)}^{*}$,
$\big{(}\frac{1}{2}\big{(}\frac{1}{2}\big{(}\frac{p}{2}\big{)}^{*}\big{)}^{*}\big{)}^{*}$,$\ldots$
is strictly monotone increasing and unbounded, after a finite number
(depending on $p$) of steps we get that

$\displaystyle\Big{(}\frac{1}{2}\Big{(}\frac{1}{2}\Big{(}\cdots\Big{(}\frac{p}{2}\Big{)}^{*}\cdots\Big{)}^{*}\Big{)}^{*}\Big{)}^{*}\geq
2.$ (4.11)

We then deduce that $-\Delta H\in W^{-1,2}_{\text{loc}}$, and thus that $H\in
W^{1,2}_{\text{loc}}$. By Sobolev embedding, this yields $H\in
L^{r}_{\text{loc}}$ for every $r<\infty$, and in turn elliptic estimates give

$\displaystyle\Phi\in W^{2,r}_{\text{loc}}\quad\forall\,r<\infty,$ (4.12)

hence also $A\in L^{r}_{\text{loc}}$ for every $r<\infty$. From Liouville
equation

$\displaystyle-\Delta\lambda=\mathrm{e}^{2\lambda}K,$ (4.13)

since $|K|\leq C|A|^{2}$, we have $\Delta\lambda\in L^{r}_{\text{loc}}$ and
hence $\lambda\in W^{2,r}_{\text{loc}}$ for every $r<\infty$. With this we
infer that in fact $A,H\in W^{1,r}_{\text{loc}}$ and so

$\displaystyle\langle A^{\circ},H\rangle^{\sharp_{g}}+\langle
A,H\rangle^{\sharp_{g}}\in W^{1,r}_{\text{loc}},$ (4.14)

Thus $-\Delta H\in L^{q}_{\text{loc}}$, which implies $H\in
W^{2,q}_{\text{loc}}$ and hence (again since $\lambda\in
W^{2,r}_{\text{loc}}$) that $\Phi\in W^{4,q}_{\text{loc}}$. ∎

######  Lemma 4.4.

With the same hypotesis as in Theorem 2.6, the vector-valued form (2.12)
satisfies $\langle w,d\Phi\rangle_{g}=0$.

###### Proof.

Since $|d\Phi|_{g}^{2}=2$ and $H$ is a normal vector field, we have

$\displaystyle\left\langle w,d\Phi\right\rangle_{g}$
$\displaystyle=\left\langle\nabla
H,d\Phi\right\rangle_{g}-2\left\langle(\nabla
H)^{\top},d\Phi\right\rangle_{g}-|H|^{2}|d\Phi|_{g}^{2}$ (4.15)
$\displaystyle=-\left\langle\nabla H,d\Phi\right\rangle_{g}-2|H|^{2}$ (4.16)
$\displaystyle=-\frac{1}{2}\left\langle\nabla\Delta_{g}\Phi,d\Phi\right\rangle_{g}-2|H|^{2}$
(4.17)
$\displaystyle=\frac{1}{2}\left\langle\Delta_{g}\Phi,\nabla^{*}d\Phi\right\rangle_{g}-2|H|^{2}$
(4.18)
$\displaystyle=\frac{1}{2}|\Delta_{g}\Phi|^{2}-\frac{1}{2}|\Delta_{g}\Phi|^{2}=0.\qed$
(4.19)

######  Proposition 4.5.

$\Phi:B_{1}\to\mathbb{R}^{3}$ be an immersion. Consider the Hodge
decomposition

$\displaystyle w=d\mathscr{L}+*dL,$ (4.20)

and consider further the Hodge decompositions

$\displaystyle-d\Phi\times H-(*d\Phi)\times L$
$\displaystyle=d\mathscr{R}+*dR,$ (4.21)
$\displaystyle-\langle*d\Phi,L\rangle$ $\displaystyle=d\mathscr{S}+*dS.$
(4.22)

Then the following relations hold:

$\displaystyle\Delta_{g}\Phi$
$\displaystyle=d\Phi\times\big{(}d\mathscr{R}+*dR\big{)}+\big{\langle}d\Phi,d\mathscr{S}+*dS\big{\rangle}$
(4.23) $\displaystyle\Delta_{g}R$
$\displaystyle=dN\times(d\mathscr{R}+*dR)-\langle
dN,d\mathscr{S}+*dS\rangle+(*d\Phi)\times d\mathscr{L},$ (4.24)
$\displaystyle\Delta_{g}S$ $\displaystyle=\langle
dN,d\mathscr{R}+*dR\rangle+\langle*d\Phi,d\mathscr{L}\rangle,$ (4.25)

where $N$ is the Gauss map of $\Phi$.

###### Proof.

Note first that, if the Hodge decompositions (4.20), (4.21), (4.22) hold, we
necessarily have

$\displaystyle\Delta_{g}\mathscr{L}$ $\displaystyle=d^{*}w=\delta\mathcal{W},$
(4.26) $\displaystyle\Delta_{g}\mathscr{R}$ $\displaystyle=-d\Phi\times
d\mathscr{L},$ (4.27) $\displaystyle\Delta_{g}\mathscr{S}$
$\displaystyle=-\langle d\Phi,d\mathscr{L}\rangle.$ (4.28)

With (2.16) and the identity333 here, the notation is: $d\Phi\times
dH=g^{\mu\nu}\partial_{\mu}\Phi\times\partial_{\nu}H$, so that $\Phi\times
d\Phi\times dH=g^{\mu\nu}(\Phi\times\partial_{\mu}\Phi\times\partial_{\nu}H).$
Similarly for $w\times dH$ and $\Phi\times d\Phi\times w$.

$\displaystyle d^{*}(-\Phi\times d\Phi\times H)=2H-\Phi\times d\Phi\times
dH=2H-\Phi\times d\Phi\times w,$ (4.29)

we see that there holds

$\displaystyle\phantom{{}={}}\Phi\times(\Phi\times\delta\mathcal{W})+\Phi\langle\Phi,\delta\mathcal{W}\rangle+4H$
(4.30) $\displaystyle=d^{*}\big{(}-2\Phi\times(d\Phi\times
H)+\Phi\times(\Phi\times w)+\Phi\langle\Phi,w\rangle\big{)}$ (4.31)
$\displaystyle=d^{*}\big{(}-\Phi\times(d\Phi\times H)\big{)}$ (4.32)
$\displaystyle\phantom{{}={}}+d^{*}\Big{(}\Phi\times\big{(}-d\Phi\times
H+\Phi\times(d\mathscr{L}+*dL)\big{)}+\Phi\big{\langle}\Phi,d\mathscr{L}+*dL\big{\rangle}\Big{)}$
(4.33) $\displaystyle=2H-\Phi\times(d\Phi\times w)$ (4.34)
$\displaystyle\phantom{{}={}}+d^{*}\Big{(}\Phi\times\big{(}-d\Phi\times
H+\Phi\times d\mathscr{L}+*d(\Phi\times L)-(*d\Phi)\times L\big{)}\Big{)}$
(4.35)
$\displaystyle\phantom{{}={}}+d^{*}\Big{(}\Phi\big{(}\langle\Phi,d\mathscr{L}\rangle+*d\langle\Phi,L\rangle-\langle*d\Phi,L\rangle\big{)}\Big{)}$
(4.36) $\displaystyle=2H-\Phi\times(d\Phi\times w)$ (4.37)
$\displaystyle\phantom{{}={}}+d^{*}\Big{(}\Phi\times\big{(}\Phi\times
d\mathscr{L}+d\mathscr{R}+*dR+*d(\Phi\times L)\big{)}\Big{)}$ (4.38)
$\displaystyle\phantom{{}={}}+d^{*}\Big{(}\Phi\big{(}\langle\Phi,d\mathscr{L}\rangle+d\mathscr{S}+*dS+*d\langle\Phi,L\rangle\big{)}\Big{)}.$
(4.39)

Now, on the one hand we have, from (4.27),

$\displaystyle\phantom{{}={}}d^{*}\Big{(}\Phi\times\big{(}\Phi\times
d\mathscr{L}+d\mathscr{R}+*dR+*d(\Phi\times L)\big{)}\Big{)}$ (4.40)
$\displaystyle=d\Phi\times\big{(}\Phi\times
d\mathscr{L}+d\mathscr{R}+*dR+*d(\Phi\times
L)\big{)}+\Phi\times\big{(}d\Phi\times
d\mathscr{L}+\Phi\times\Delta_{g}\mathscr{L}+\Delta_{g}\mathscr{R}\big{)}$
(4.41) $\displaystyle=d\Phi\times\big{(}\Phi\times
d\mathscr{L}+d\mathscr{R}+*dR+*d(\Phi\times
L)\big{)}+\Phi\times(\Phi\times\delta\mathcal{W})$ (4.42)

on the other hand from (4.28) it follows that

$\displaystyle\phantom{{}={}}d^{*}\Big{(}\Phi\big{(}\langle\Phi,d\mathscr{L}\rangle+d\mathscr{S}+*dS+*d\langle\Phi,L\rangle\big{)}\Big{)}$
(4.43)
$\displaystyle=\big{\langle}d\Phi,\langle\Phi,d\mathscr{L}\rangle+d\mathscr{S}+*dS+*d\langle\Phi,L\rangle\big{\rangle}+\Phi\big{(}\langle
d\Phi,d\mathscr{L}\rangle+\langle\Phi,\Delta_{g}\mathscr{L}\rangle+\Delta_{g}\mathscr{S}\big{)}$
(4.44)
$\displaystyle=\big{\langle}d\Phi,\langle\Phi,d\mathscr{L}\rangle+d\mathscr{S}+*dS+*d\langle\Phi,L\rangle\big{\rangle}+\Phi\langle\Phi,\delta\mathcal{W}\rangle$
(4.45)

thus we deduce

$\displaystyle 2H$ $\displaystyle=-\Phi\times(d\Phi\times w)$ (4.46)
$\displaystyle\phantom{{}={}}+d\Phi\times\big{(}\Phi\times
d\mathscr{L}+d\mathscr{R}+*dR+*d(\Phi\times L)\big{)}$ (4.47)
$\displaystyle\phantom{{}={}}+\big{\langle}d\Phi,\langle\Phi,d\mathscr{L}\rangle+d\mathscr{S}+*dS+*d\langle\Phi,L\rangle\big{\rangle}$
(4.48) $\displaystyle=-\Phi\times(d\Phi\times
w)+d\Phi\times\big{(}d\mathscr{R}+*dR\big{)}+\big{\langle}d\Phi,d\mathscr{S}+*dS\big{\rangle}$
(4.49) $\displaystyle\phantom{{}={}}+d\Phi\times\big{(}\Phi\times
d\mathscr{L}+*d(\Phi\times
L)\big{)}+\big{\langle}d\Phi,\langle\Phi,d\mathscr{L}\rangle+*\langle\Phi,L\rangle\big{\rangle}$
(4.50)

By definition of $\mathscr{L}$ and $L$, with Lemma 4.4 the last line in the
above expression is

$\displaystyle\phantom{{}={}}d\Phi\times\big{(}\Phi\times
d\mathscr{L}+*d(\Phi\times
L)\big{)}+\big{\langle}d\Phi,\langle\Phi,d\mathscr{L}\rangle+*\langle\Phi,L\rangle\big{\rangle}$
(4.51)
$\displaystyle=d\Phi\times\big{(}\Phi\times(d\mathscr{L}+*dL)+(*d\Phi)\times
L\big{)}+\big{\langle}d\Phi,\langle\Phi,d\mathscr{L}+*dL\rangle+\langle*d\Phi,L\rangle\big{\rangle}$
(4.52) $\displaystyle=d\Phi\times\big{(}\Phi\times w+(*d\Phi)\times
L\big{)}+\big{\langle}d\Phi,\langle\Phi,w\rangle+\langle*d\Phi,L\rangle\big{\rangle}$
(4.53) $\displaystyle=d\Phi\times\big{(}\Phi\times
w\big{)}+\big{\langle}d\Phi,\langle\Phi,w\rangle\big{\rangle}+d\Phi\times\big{(}(*d\Phi)\times
L)+\big{\langle}d\Phi,\langle*d\Phi,L\rangle\big{\rangle}$ (4.54)
$\displaystyle=\Phi\times(d\Phi\times w),$ (4.55)

and this yields (4.23). Next, since $\langle d\Phi\times H,N\rangle=0$, we see
that

$\displaystyle\big{\langle}d\mathscr{R}+*dR,N\big{\rangle}$
$\displaystyle=\big{\langle}-d\Phi\times H-(*d\Phi)\times L,N\big{\rangle}$
(4.56) $\displaystyle=-\big{\langle}N\times(*d\Phi),L\big{\rangle}$ (4.57)
$\displaystyle=-\big{\langle}d\Phi,L\big{\rangle}$ (4.58)
$\displaystyle=-*d\mathscr{S}+dS$ (4.59)

and similarly, using the rules of the vector product, we have

$\displaystyle\big{(}d\mathscr{R}+*dR\big{)}\times N$
$\displaystyle=(-d\Phi\times H-(*d\Phi)\times L\big{)}\times N$ (4.60)
$\displaystyle=N\times\big{(}d\Phi\times H\big{)}+N\times\big{(}(*d\Phi)\times
L\big{)}$ (4.61) $\displaystyle=-H\times(N\times d\Phi)-d\Phi\times(H\times
N)$ (4.62)
$\displaystyle\phantom{{}=}-L\times\big{(}N\times(*d\Phi)\big{)}-(*d\Phi)\times(L\times
N)$ (4.63) $\displaystyle=H\times(*d\Phi)-L\times d\Phi-(*d\Phi)\times(L\times
N)$ (4.64) $\displaystyle=-(*d\Phi)\times H+d\Phi\times
L+N\langle*d\Phi,L\rangle$ (4.65)
$\displaystyle=*d\mathscr{R}-dR-N(d\mathscr{S}+*dS),$ (4.66)

and thus, codifferentiating these identities we have

$\displaystyle\Delta_{g}\mathscr{R}\times
N+\big{(}d\mathscr{R}+*dR\big{)}\times dN$
$\displaystyle=-\Delta_{g}R-\big{\langle}dN,d\mathscr{S}+*dS\big{\rangle}-N\Delta_{g}\mathscr{S},$
(4.67)
$\displaystyle\big{\langle}\Delta_{g}\mathscr{R},N\big{\rangle}+\big{\langle}d\mathscr{R}+*dR,dN\big{\rangle}$
$\displaystyle=\Delta_{g}S.$ (4.68)

It now suffices to notice that, by definition of $\mathscr{R}$ and
$\mathscr{S}$ it is

$\displaystyle N\Delta_{g}\mathscr{S}$ $\displaystyle=-N\langle
d\Phi,d\mathscr{L}\rangle$ (4.69) $\displaystyle\Delta_{g}\mathscr{R}\times N$
$\displaystyle=-(d\Phi\times d\mathscr{L})\times N=N\times(d\Phi\times
d\mathscr{L})=\big{\langle}d\Phi,\langle N,d\mathscr{L}\rangle\big{\rangle},$
(4.70) $\displaystyle\big{\langle}\Delta_{g}\mathscr{R},N\big{\rangle}$
$\displaystyle=-\langle d\Phi\times d\mathscr{L},N\rangle=-\langle N\times
d\Phi,d\mathscr{L}\rangle=\langle*d\Phi,d\mathscr{L}\rangle$ (4.71)

and in particular

$\displaystyle\Delta_{g}\mathscr{R}\times N+N\Delta_{g}\mathscr{S}$
$\displaystyle=\big{\langle}d\Phi,\langle
N,d\mathscr{L}\rangle\big{\rangle}-N\langle d\Phi,d\mathscr{L}\rangle$ (4.72)
$\displaystyle=d\mathscr{L}\times(d\Phi\times N)$ (4.73)
$\displaystyle=d\mathscr{L}\times(*d\Phi).$ (4.74)

Substituting these relations in the ones above then gives (4.24) and (4.25). ∎

######  Proposition 4.6.

Let $\Phi\in\mathscr{E}(B_{1},\mathbb{R}^{3})$ be conformal with conformal
factor $\mathrm{e}^{\lambda}$ and so that $\delta\mathcal{W}\in L^{p}(B_{1})$
for some $p>1$. Then $H\in L^{r}_{\text{loc}}(B_{1})$ for every $r<\infty$.

###### Proof.

We may certainly assume $1<p<2$.

Step 1: there exists $\mathscr{L}$ and $L$ realizing the Hodge decomposition
(4.20) with $\mathscr{L}\in W^{2,p}(B_{1})$ and $L\in
L^{(2,\infty)}_{\text{loc}}(B_{1})$. Indeed, we let $\mathscr{L}$ solve

$\displaystyle\left\\{\begin{aligned}
\Delta\mathscr{L}&=\mathrm{e}^{2\lambda}\delta\mathcal{W}&&\text{in }B_{1},\\\
u&=0&&\text{on }\partial B_{1}.\end{aligned}\right.$ (4.75)

Elliptic regularity theory gives then that $\mathscr{L}\in W^{2,p}$. By
construction we have

$\displaystyle d^{*}(w-d\mathscr{L})=0\quad\text{in }B_{1},$ (4.76)

thus $L$ exists as a distribution in $B_{1}$ thanks to Poincaré’s lemma and it
is determined up to an additive constant. Note now that

$\displaystyle\Delta L$ $\displaystyle=d^{*}(*d\mathscr{L}-*w)$ (4.77)
$\displaystyle=d^{*}\big{(}*d\mathscr{L}-*(\nabla H+\langle
A^{\circ},H\rangle^{\sharp_{g}}+\langle A,H\rangle^{\sharp_{g}})\big{)}$
(4.78) $\displaystyle=-d^{*}\big{(}*(\langle
A^{\circ},H\rangle^{\sharp_{g}}+\langle A,H\rangle^{\sharp_{g}})\big{)},$
(4.79)

so if we let

$\displaystyle L_{0}(x)=-\int_{B_{1}}\big{\langle}dK(x-y),*(\langle
A^{\circ},H\rangle^{\sharp_{g}}+\langle
A,H\rangle^{\sharp_{g}})\big{\rangle}\,dy,$ (4.80)

where $K(x)=-\frac{1}{2\pi}\log|x|$ is the fundamental solution of the Laplace
operator, given that

$\displaystyle|dK(x-y)|\leq\frac{C}{|x-y|}\quad\Longrightarrow\quad\sup_{x\in
B_{1}}\|dK(x-\cdot)\|_{L^{2,\infty}(B_{1})}\leq C,$ (4.81)

we see that $L_{0}\in L^{(2,\infty)}$, and hence, since $L-L_{0}$ is harmonic,
that $L\in L^{(2,\infty)}_{\text{loc}}.$

Step 2: There exist $\mathscr{R}$, $R$, $\mathscr{S}$ and $S$ realizing the
Hodge decompositions (4.21), (4.22) with $\mathscr{R},\mathscr{S}\in
W^{2,p^{*}}(B_{1})$ and $R,S\in W^{1,(2,\infty)}_{\text{loc}}(B_{1})$. Indeed,
define $\mathscr{R}$ and $\mathscr{S}$ by

$\displaystyle\left\\{\begin{aligned} \Delta\mathscr{R}&=-d\Phi\times
d\mathscr{L}&&\text{in }B_{1},\\\ \mathscr{R}&=0&&\text{on }\partial
B_{1},\end{aligned}\right.$ $\displaystyle\left\\{\begin{aligned}
\Delta\mathscr{S}&=-\langle d\Phi,d\mathscr{L}\rangle&&\text{in }B_{1},\\\
\mathscr{S}&=0&&\text{on }\partial B_{1}.\end{aligned}\right.$ (4.82)

Since by Sobolev embedding $d\mathscr{L}\in W^{1,p}\hookrightarrow L^{p^{*}}$,
so elliptic regularity gives $\mathscr{R},\mathscr{S}\in W^{2,p^{*}}$. By
construction we then have

$\displaystyle d^{*}\big{(}-d\Phi\times H-(*d\Phi)\times
L-d\mathscr{R}\big{)}$ $\displaystyle=0,$ (4.83) $\displaystyle
d^{*}\big{(}-\langle*d\Phi,L\rangle-d\mathscr{S}\big{)}$ $\displaystyle=0,$
(4.84)

and thus $R$ and $S$ exist as distributions by Poincaré lemma and are
determined up to additive constants, and, since $L$ is in
$L^{(2,\infty)}_{\text{loc}}$, so are $dR$ and $dS$.

Step 3: conclusion. From relations (4.24), (4.25), we see that $R$ and $S$
satisfy a system Jacobians plus some extra terms, namely

$\displaystyle\left\\{\begin{aligned} \Delta R&=dN\times(*dR)-\langle
dN,*dS\rangle+f_{R},\\\ \Delta S&=\langle
dN,*dR\rangle+f_{S},\end{aligned}\right.$ (4.85)

where, since $d\mathscr{L}\in L^{p^{*}}$ and $d\mathscr{R},d\mathscr{S}\in
W^{1,p^{*}}\hookrightarrow L^{\infty}$, we have

$\displaystyle f_{R}$ $\displaystyle=dN\times d\mathscr{R}-\langle
dN,d\mathscr{S}\rangle+(*d\Phi)\times d\mathscr{L}$ $\displaystyle\in L^{2},$
(4.86) $\displaystyle f_{S}$ $\displaystyle=\langle
dN,d\mathscr{R}\rangle+\langle*d\Phi,d\mathscr{L}\rangle$ $\displaystyle\in
L^{2}.$ (4.87)

Thanks to Theorem 2.1, we get that $R,S\in W^{2,q}_{\text{loc}}$ for every
$q<2$ and hence that

$\displaystyle dR,dS\in W^{1,r}_{\text{loc}}\quad\text{for every }r<\infty.$
(4.88)

Inserting thin information in (4.23) gives that $\Delta\Phi$, and so $H$, is
in $L^{r}_{\text{loc}}$ for every $r<\infty$. ∎

###  4.2 Quantitative Estimates

In the computations that follow, we shall make use of various Gagliardo-
Nirenberg inequalities, namely, of multiplicative Sobolev inequalities such as

$\displaystyle\|u\|_{L^{4}(\Omega)}\leq
C\|u\|_{L^{2}(\Omega)}^{1/2}\|u\|_{W^{1,2}(\Omega)}^{1/2},$

for $u\in W^{1,2}(\Omega)$, $\Omega\subset\mathbb{R}^{2}$ bounded, regular
domain. We refer for instance to [Nir59].

######  Proposition 4.7.

Let $\Omega\subset\mathbb{R}^{2}$ is a bounded, regular domain with
$0\in\Omega$ and $\Phi:\Omega\to\mathbb{R}^{n}$ is a conformal immersion of
class $W^{4,2}$ with conformal factor $\mathrm{e}^{\lambda}$. Let $E>0$ and
$C_{\infty}>0$ be constants so that

$\displaystyle\|\mathrm{e}^{-\lambda}\nabla^{2}\Phi\|_{L^{2}(\Omega)}\leq
E\quad\text{and}\quad\|\lambda-\Lambda\|_{L^{\infty}(\Omega)}\leq C_{\infty},$
(4.89)

where $\Lambda=\lambda(0)$. Then, the following estimate holds:

$\displaystyle\|\Delta^{2}\Phi\|_{L^{2}(\Omega)}\leq
2\|\mathrm{e}^{4\lambda}\delta\mathcal{W}\|_{L^{2}(\Omega)}+C\|\mathrm{e}^{-\lambda}\nabla^{2}\Phi\|_{L^{2}(\Omega)}\|\nabla^{2}\Phi\|_{W^{2,2}(\Omega)},$
(4.90)

for a constant $C=C(\Omega,E,C_{\infty})>0$.

First we point out a few elementary facts.

######  Lemma 4.8.

For an immersion $\Phi:B_{1}\to\mathbb{R}^{n}$ there holds

$\displaystyle\frac{1}{2}\Delta^{2}_{g}\Phi=\Delta_{g}H=\Delta^{\perp}_{g}H-\langle
A,\langle H,A\rangle\rangle_{g}-2\langle\nabla
H,A\rangle^{\sharp_{g}}-\langle\nabla H,H\rangle^{\sharp_{g}},$ (4.91)

and, if $\Phi$ is conformal with conformal factor $\mathrm{e}^{\lambda}$,
there holds

$\displaystyle\Delta^{2}\Phi$
$\displaystyle=\mathrm{e}^{4\lambda}\big{(}\Delta_{g}^{2}\Phi+2\langle
d\lambda,\nabla H\rangle_{g}+(2|d\lambda|^{2}_{g}+\Delta_{g}\lambda)H\big{)}.$
(4.92)

The proof is elementary.

######  Lemma 4.9.

The following pointwise estimates hold for an absolute constant $C>0$ and
$k=0,1,2$:

$\displaystyle\frac{1}{C}|\mathrm{e}^{-\lambda}\nabla^{2}\Phi|$
$\displaystyle\leq|\mathrm{e}^{-\lambda}A|+|d\lambda|\leq
C\,\mathrm{e}^{-\lambda}|\nabla^{2}\Phi|,$ (4.93)
$\displaystyle|\nabla^{k}(\mathrm{e}^{2\lambda}H)|$ $\displaystyle\leq
C|\nabla^{k+2}\Phi|,$ (4.94)
$\displaystyle|\nabla^{k}(\mathrm{e}^{2\lambda}H)|$ $\displaystyle\leq
C|\nabla^{k}A|,$ (4.95)

and the following estimate holds:

$\displaystyle\|A\|_{W^{k,2}(\Omega)}$ $\displaystyle\leq
C\|\nabla^{2}\Phi\|_{W^{k,2}(\Omega)},$ (4.96)

for $C=C(\Omega,E,C_{\infty})>0$ and $k=0,1,2$.

###### Proof.

Since

$\displaystyle
A_{\mu\nu}=\partial^{2}_{\mu\nu}\Phi-\Gamma_{\mu\nu}^{\sigma}\,\partial_{\sigma}\Phi\quad\text{and}\quad
e^{2\lambda}=\frac{1}{2}|d\Phi|^{2},$ (4.97)

and

$\displaystyle\Gamma_{11}^{1}$ $\displaystyle=\partial_{1}\lambda,$
$\displaystyle\Gamma^{1}_{12}$
$\displaystyle=\Gamma^{1}_{21}=\partial_{2}\lambda,$
$\displaystyle\Gamma^{1}_{22}$ $\displaystyle=-\partial_{1}\lambda,$ (4.98)
$\displaystyle\Gamma_{11}^{2}$ $\displaystyle=-\partial_{2}\lambda,$
$\displaystyle\Gamma^{2}_{12}$
$\displaystyle=\Gamma^{1}_{21}=\partial_{1}\lambda,$
$\displaystyle\Gamma^{2}_{22}$ $\displaystyle=\partial_{2}\lambda,$ (4.99)

estimates (4.93) and consequently (4.96) for $k=0$ follow. Next, since

$\displaystyle\mathrm{e}^{2\lambda}H$
$\displaystyle=\frac{1}{2}\Delta\Phi=\frac{1}{2}(\Delta\Phi)^{\perp}=\frac{1}{2}(A_{11}+A_{22}),$
(4.100)

estimates (4.94) and (4.95) follow. Now differentiating of the above
identities:

$\displaystyle\partial_{\xi}A_{\mu\nu}$
$\displaystyle=\partial^{3}_{\xi\mu\nu}\Phi-\partial_{\xi}\Gamma_{\mu\nu}^{\sigma}\,\partial_{\sigma}\Phi-\Gamma_{\mu\nu}^{\sigma}\,\partial^{2}_{\xi\sigma}\Phi,$
(4.101) $\displaystyle\partial^{2}_{\zeta\xi}A_{\mu\nu}$
$\displaystyle=\partial^{4}_{\zeta\xi\mu\nu}\Phi-\partial^{2}_{\zeta\xi}\Gamma_{\mu\nu}^{\sigma}\,\partial_{\sigma}\Phi-\partial_{\xi}\Gamma_{\mu\nu}^{\sigma}\,\partial^{2}_{\zeta\sigma}\Phi-\partial_{\zeta}\Gamma_{\mu\nu}^{\sigma}\,\partial^{2}_{\xi\sigma}\Phi-\Gamma_{\mu\nu}^{\sigma}\,\partial^{3}_{\zeta\xi\sigma}\Phi,$
(4.102)

and

$\displaystyle 2d\lambda\mathrm{e}^{2\lambda}$
$\displaystyle=\langle\nabla^{2}\Phi,d\Phi\rangle,$ (4.103)
$\displaystyle(2\nabla^{2}\lambda+4d\lambda\otimes
d\lambda)\mathrm{e}^{2\lambda}$
$\displaystyle=\langle\nabla^{3}\Phi,d\Phi\rangle+\nabla^{2}\Phi\langle\dot{\otimes}\rangle\nabla^{2}\Phi,$
(4.104) $\displaystyle(2\nabla^{3}\lambda+8\nabla^{2}\lambda\otimes
d\lambda+8d\lambda\otimes d\lambda\otimes d\lambda)\mathrm{e}^{2\lambda}$
$\displaystyle=\langle\nabla^{4}\Phi,d\Phi\rangle+2\nabla^{3}\Phi\langle\dot{\otimes}\rangle\nabla^{2}\Phi,$
(4.105)

(where $\langle\dot{\otimes}\rangle$ means scalar product in the vector part,
inner product in one of the covariant entries and tensor product in the
remaining ones) yields the estimates

$\displaystyle|\Gamma_{\mu\nu}^{\sigma}|$ $\displaystyle\leq C|d\lambda|,$
$\displaystyle|\partial_{\xi}\Gamma_{\mu\nu}^{\sigma}|$ $\displaystyle\leq
C|\nabla^{2}\lambda|,$
$\displaystyle|\partial^{2}_{\zeta\xi}\Gamma_{\mu\nu}^{\sigma}|$
$\displaystyle\leq C|\nabla^{3}\lambda|,$ (4.106)

and

$\displaystyle|d\lambda|$ $\displaystyle\leq
C\mathrm{e}^{-\lambda}|\nabla^{2}\Phi|,$ (4.107)
$\displaystyle|\nabla^{2}\lambda|$ $\displaystyle\leq
C\big{(}|d\lambda|^{2}+\mathrm{e}^{-\lambda}|\nabla^{3}\Phi|+\mathrm{e}^{-2\lambda}|\nabla^{2}\Phi|^{2}\big{)}$
(4.108) $\displaystyle\leq
C\big{(}\mathrm{e}^{-2\lambda}|\nabla^{2}\Phi|^{2}+\mathrm{e}^{-\lambda}|\nabla^{3}\Phi|\big{)},$
(4.109) $\displaystyle|\nabla^{3}\lambda|$ $\displaystyle\leq
C\big{(}|\nabla^{2}\lambda||d\lambda|+|d\lambda|^{3}+\mathrm{e}^{-\lambda}|\nabla^{4}\Phi|+\mathrm{e}^{-2\lambda}|\nabla^{3}\Phi||\nabla^{2}\Phi|\big{)}$
(4.110) $\displaystyle\leq
C\big{(}\mathrm{e}^{-2\lambda}|\nabla^{3}\Phi||\nabla^{2}\Phi|+\mathrm{e}^{-3\lambda}|\nabla^{2}\Phi|^{3}+\mathrm{e}^{-\lambda}|\nabla^{4}\Phi|\big{)},$
(4.111)

thanks to which we estimate in turn

$\displaystyle|\nabla A|$ $\displaystyle\leq
C\big{(}|\nabla^{3}\Phi|+\mathrm{e}^{\lambda}|\nabla^{2}\lambda|+|d\lambda||\nabla^{2}\Phi|\big{)}$
(4.112) $\displaystyle\leq
C\big{(}|\nabla^{3}\Phi|+\mathrm{e}^{-\lambda}|\nabla^{2}\Phi|^{2}\big{)},$
(4.113) $\displaystyle|\nabla^{2}A|$ $\displaystyle\leq
C\big{(}|\nabla^{4}\Phi|+\mathrm{e}^{\lambda}|\nabla^{3}\lambda|+|\nabla\lambda||\nabla^{2}\Phi|+|d\lambda||\nabla^{3}\Phi|\big{)}$
(4.114) $\displaystyle\leq
C\big{(}|\nabla^{4}\Phi|+\mathrm{e}^{-\lambda}|\nabla^{3}\Phi||\nabla^{2}\Phi|+\mathrm{e}^{-2\lambda}|\nabla^{2}\Phi|^{3}\big{)}.$
(4.115)

Thus, with the help of Gagliardo-Nirenberg we can estimate

$\displaystyle\|\nabla A\|_{L^{2}}$ $\displaystyle\leq
C\big{(}\|\nabla^{3}\Phi\|_{L^{2}}+\|\mathrm{e}^{-\lambda}|\nabla^{2}\Phi|^{2}\|_{L^{2}}\big{)}$
(4.116) $\displaystyle\leq
C\big{(}\|\nabla^{3}\Phi\|_{L^{2}}+\mathrm{e}^{-\Lambda}\|\nabla^{2}\Phi\|_{L^{4}}^{2}\big{)}$
(4.117) $\displaystyle\leq
C\big{(}\|\nabla^{3}\Phi\|_{L^{2}}+\mathrm{e}^{-\Lambda}\|\nabla^{2}\Phi\|_{L^{2}}\|\nabla^{2}\Phi\|_{W^{1,2}}\big{)}$
(4.118) $\displaystyle\leq
C\big{(}\|\nabla^{3}\Phi\|_{L^{2}}+\|\mathrm{e}^{-\lambda}\nabla^{2}\Phi\|_{L^{2}}\|\nabla^{2}\Phi\|_{W^{1,2}}\big{)},$
(4.119)

yielding (4.96) for $k=1$. Similarly, since

$\displaystyle\|\nabla^{2}A\|_{L^{2}}$ $\displaystyle\leq
C\big{(}\|\nabla^{4}\Phi\|_{L^{2}}+\|\mathrm{e}^{-\lambda}|\nabla^{3}\Phi||\nabla^{2}\Phi|\|_{L^{2}}+\|\mathrm{e}^{-2\lambda}|\nabla^{2}\Phi|^{3}\|_{L^{2}}\big{)}$
(4.120) $\displaystyle\leq
C\big{(}\|\nabla^{4}\Phi\|_{L^{2}}+\mathrm{e}^{-\Lambda}\||\nabla^{3}\Phi||\nabla^{2}\Phi|\|_{L^{2}}+\mathrm{e}^{-2\Lambda}\|\nabla^{2}\Phi\|_{L^{6}}^{3}\big{)},$
(4.121)

with Hölder and Gagliardo-Nirenberg we estimate

$\displaystyle\||\nabla^{3}\Phi||\nabla^{2}\Phi|\|_{L^{2}}$
$\displaystyle\leq\|\nabla^{3}\Phi\|_{L^{4}}\|\nabla^{2}\Phi\|_{L^{4}}$
(4.122) $\displaystyle\leq
C\|\nabla^{2}\Phi\|_{L^{2}}^{\frac{1}{4}}\|\nabla^{2}\Phi\|_{W^{2,2}}^{\frac{3}{4}}\|\nabla^{2}\Phi\|_{L^{2}}^{\frac{3}{4}}\|\nabla^{2}\Phi\|_{W^{2,2}}^{\frac{1}{4}}$
(4.123) $\displaystyle\leq
C\|\nabla^{2}\Phi\|_{L^{2}}\|\nabla^{2}\Phi\|_{W^{2,2}},$ (4.124)

and

$\displaystyle\|\nabla^{2}\Phi\|_{L^{6}}^{3}\leq
C\|\nabla^{2}\Phi\|_{L^{2}}^{2}\|\nabla^{2}\Phi\|_{W^{2,2}},$ (4.125)

so to obtain

$\displaystyle\|\nabla^{2}A\|_{L^{2}}$ $\displaystyle\leq
C\big{(}\|\nabla^{4}\Phi\|_{L^{2}}+\|\mathrm{e}^{-\lambda}\nabla^{2}\Phi\|_{L^{2}}\|\nabla^{2}\Phi\|_{W^{2,2}}+\|\mathrm{e}^{-\lambda}\nabla^{2}\Phi\|_{L^{2}}^{2}\|\nabla^{2}\Phi\|_{W^{2,2}}\big{)},$
(4.126)

yielding (4.96) also for $k=2$. ∎

###### Proof of Proposition 4.7.

It can be deduced from the following three lemmas and (4.96).

######  Lemma 4.10.

There holds

$\displaystyle\|\mathrm{e}^{4\lambda}\Delta^{\perp}_{g}H\|_{L^{2}(\Omega)}\leq
C\|\mathrm{e}^{-\lambda}A\|_{L^{2}(\Omega)}\|A\|_{W^{2,2}(\Omega)}+\|\mathrm{e}^{4\lambda}\delta\mathcal{W}\|_{L^{2}(\Omega)},$
(4.127)

for a constant $C=C(\Omega,E,C_{\infty})>0$.

######  Lemma 4.11.

There holds

$\displaystyle\|\mathrm{e}^{4\lambda}\Delta_{g}H\|_{L^{2}(\Omega)}\leq\|\mathrm{e}^{4\lambda}\Delta^{\perp}_{g}H\|_{L^{2}(\Omega)}+C\|\mathrm{e}^{-\lambda}A\|_{L^{2}(\Omega)}\|A\|_{W^{2,2}(\Omega)},$
(4.128)

for a constant $C=C(\Omega,E,C_{\infty})>0$.

######  Lemma 4.12.

There holds

$\displaystyle\|\Delta^{2}\Phi\|_{L^{2}(\Omega)}\leq
2\|\mathrm{e}^{4\lambda}\Delta_{g}H\|_{L^{2}(\Omega)}+C\|\mathrm{e}^{-\lambda}\nabla^{2}\Phi\|_{L^{2}(\Omega)}\|\nabla^{2}\Phi\|_{W^{2,2}(\Omega)},$
(4.129)

for a constant $C=C(\Omega,E,C_{\infty})>0$.

###### Proof of Lemma 4.10.

With the classical form of the Willmore operator (2.9) and the pointwise
estimate (4.95), by means of Hölder’s and Gagliardo-Nirenberg inequalities we
estimate

$\displaystyle\|\langle A^{\circ},\langle
H,A^{\circ}\rangle\rangle_{g}\|_{L^{2}}$ $\displaystyle\leq
C\|\mathrm{e}^{-4\lambda}\langle A^{\circ},\langle
H,A^{\circ}\rangle\rangle\|_{L^{2}}$ (4.130) $\displaystyle\leq
C\|\mathrm{e}^{-4\lambda}|A^{\circ}|^{2}H\|_{L^{2}}$ (4.131)
$\displaystyle\leq C\|\mathrm{e}^{-6\lambda}|A|^{3}\|_{L^{2}}$ (4.132)
$\displaystyle\leq C\mathrm{e}^{-6\Lambda}\|A\|_{L^{6}}^{3}$ (4.133)
$\displaystyle\leq C\mathrm{e}^{-6\Lambda}\|A\|_{L^{2}}^{2}\|A\|_{W^{2,2}}$
(4.134) $\displaystyle\leq
C\mathrm{e}^{-4\Lambda}\|\mathrm{e}^{-\lambda}A\|_{L^{2}}^{2}\|A\|_{W^{2,2}},$
(4.135)

which yields (4.127). ∎

###### Proof of Lemma 4.11.

From formula (4.91), we have

$\displaystyle\|\Delta_{g}H\|_{L^{2}}\leq\|\Delta^{\perp}_{g}H\|_{L^{2}}+C\big{(}\|\langle
A,\langle H,A\rangle\rangle_{g}\|_{L^{2}}+\|\langle\nabla
H,A\rangle^{\sharp_{g}}_{g}\|_{L^{2}}+\|\langle\nabla
H,H\rangle^{\sharp_{g}}\|_{L^{2}}\big{)}.$ (4.136)

Similarly as in the proof of Lemma (4.10), we have

$\displaystyle\|\langle A,\langle H,A\rangle\rangle_{g}\|_{L^{2}}\leq
C\mathrm{e}^{-4\Lambda}\|\mathrm{e}^{-\lambda}A\|_{L^{2}}^{2}\|A\|_{W^{2,2}}.$
(4.137)

Next, since

$\displaystyle\nabla H$
$\displaystyle=\mathrm{e}^{-2\lambda}\nabla(\mathrm{e}^{2\lambda}H)-2H\otimes
d\lambda,$ (4.138)

with (4.95) we may pointwise estimate

$\displaystyle|\nabla H|$ $\displaystyle\leq
C\mathrm{e}^{-2\lambda}\big{(}|\nabla A|+|A||d\lambda|\big{)},$ (4.139)

thus allowing to deduce

$\displaystyle\|\langle\nabla H,H\rangle^{\sharp_{g}}\|_{L^{2}}$
$\displaystyle\leq C\|\mathrm{e}^{-\lambda}|\nabla H||H|\|_{L^{2}}$ (4.140)
$\displaystyle\leq C\|\mathrm{e}^{-3\lambda}(|\nabla
A|+|A||d\lambda|)|H|\|_{L^{2}}$ (4.141) $\displaystyle\leq
C\|\mathrm{e}^{-5\lambda}(|\nabla A|+|A||d\lambda|)|A|\|_{L^{2}};$ (4.142)

now with Hölder and Gagliardo-Nirenberg we see that, on the one hand,

$\displaystyle\|\mathrm{e}^{-5\lambda}|\nabla A||A|\|_{L^{2}}$
$\displaystyle\leq C\mathrm{e}^{-5\Lambda}\||\nabla A||A|\|_{L^{2}}$ (4.143)
$\displaystyle\leq C\mathrm{e}^{-5\Lambda}\|\nabla A\|_{L^{4}}\|A\|_{L^{4}}$
(4.144) $\displaystyle\leq
C\mathrm{e}^{-5\Lambda}\|A\|_{L^{2}}\|A\|_{W^{2,2}}$ (4.145)
$\displaystyle\leq
C\mathrm{e}^{-4\Lambda}\|\mathrm{e}^{-\lambda}A\|_{L^{2}}\|A\|_{W^{2,2}},$
(4.146)

and on the other hand

$\displaystyle\|\mathrm{e}^{-5\lambda}|A|^{2}|d\lambda|\|_{L^{2}}$
$\displaystyle\leq
C\mathrm{e}^{-5\Lambda}\|d\lambda\|_{L^{2}}\|A\|_{L^{\infty}}^{2}$ (4.147)
$\displaystyle\leq
C\mathrm{e}^{-5\Lambda}\|d\lambda\|_{L^{2}}\|A\|_{L^{2}}\|A\|_{W^{2,2}}$
(4.148) $\displaystyle\leq
C\mathrm{e}^{-4\Lambda}\|d\lambda\|_{L^{2}}\|\mathrm{e}^{-\lambda}A\|_{L^{2}}\|A\|_{W^{2,2}},$
(4.149)

so we deduce

$\displaystyle\|\langle\nabla H,H\rangle^{\sharp_{g}}\|_{L^{2}}\leq
C\mathrm{e}^{-4\Lambda}\|\mathrm{e}^{-\lambda}A\|_{L^{2}}\|A\|_{W^{2,2}}.$
(4.150)

Similarly, we see that

$\displaystyle\|\langle\nabla H,A\rangle^{\sharp_{g}}_{g}\|_{L^{2}}$
$\displaystyle\leq C\|\mathrm{e}^{-3\lambda}\langle\nabla
H,A\rangle^{\sharp}\|_{L^{2}}$ (4.151) $\displaystyle\leq
C\|\mathrm{e}^{-3\lambda}|\nabla H||A|\|_{L^{2}}$ (4.152) $\displaystyle\leq
C\|\mathrm{e}^{-5\lambda}(|\nabla A|+|A||d\lambda|)|A|\|_{L^{2}},$ (4.153)

and so similarly as before we deduce

$\displaystyle\|\langle\nabla H,A\rangle^{\sharp_{g}}_{g}\|_{L^{2}}\leq
C\mathrm{e}^{-4\Lambda}\|\mathrm{e}^{-\lambda}A\|_{L^{2}}\|A\|_{W^{2,2}},$
(4.154)

yielding (4.128). ∎

###### Proof of Lemma 4.12.

From formula (4.92), it follows that

$\displaystyle\|\Delta\Phi\|_{L^{2}}\leq
2\|\mathrm{e}^{4\lambda}\Delta_{g}H\|_{L^{2}}+C\big{(}\|\mathrm{e}^{4\lambda}\langle
d\lambda,\nabla
H\rangle_{g}\|_{L^{2}}+\|\mathrm{e}^{4\lambda}|d\lambda|^{2}_{g}H\|_{L^{2}}+\|\mathrm{e}^{4\lambda}\Delta_{g}\lambda\,H\|_{L^{2}}\big{)}.$
(4.155)

Now with (4.93), (4.94), and (again) the identity

$\displaystyle\nabla H$
$\displaystyle=\mathrm{e}^{-2\lambda}\nabla(\mathrm{e}^{2\lambda}H)-2H\otimes
d\lambda,$ (4.156)

we estimate with Hölder and Galgliardo-Nirenberg

$\displaystyle\|\mathrm{e}^{4\lambda}\langle d\lambda,\nabla
H\rangle_{g}\|_{L^{2}}$ $\displaystyle\leq C\|\mathrm{e}^{2\lambda}\langle
d\lambda,\nabla H\rangle\|_{L^{2}}$ (4.157) $\displaystyle\leq
C\|\mathrm{e}^{2\lambda}|d\lambda||\nabla H|\|_{L^{2}}$ (4.158)
$\displaystyle\leq
C\big{\|}\mathrm{e}^{2\lambda}\big{(}\mathrm{e}^{-\lambda}|\nabla^{2}\Phi|\big{)}\big{(}\mathrm{e}^{-2\lambda}|\nabla^{3}\Phi|+\mathrm{e}^{-3\lambda}|\nabla^{2}\Phi|^{2}\big{)}\big{\|}_{L^{2}}$
(4.159) $\displaystyle\leq
C\big{(}\mathrm{e}^{-\Lambda}\||\nabla^{3}\Phi|\nabla^{2}\Phi|\|_{L^{2}}+\mathrm{e}^{-2\Lambda}\|\nabla^{2}\Phi\|_{L^{6}}^{3}\big{)}$
(4.160) $\displaystyle\leq
C\big{(}\mathrm{e}^{-\Lambda}\|\nabla^{3}\Phi\|_{L^{4}}\|\nabla^{2}\Phi\|_{L^{4}}+\mathrm{e}^{-2\Lambda}\|\nabla^{2}\Phi\|_{L^{6}}^{3}\big{)}$
(4.161) $\displaystyle\leq
C\big{(}\mathrm{e}^{-\Lambda}\|\nabla^{2}\Phi\|_{L^{2}}\|\nabla^{2}\Phi\|_{W^{2,2}}+\mathrm{e}^{-2\Lambda}\|\nabla^{2}\Phi\|_{L^{2}}^{2}\|\nabla^{2}\Phi\|_{W^{2,2}}\big{)}$
(4.162) $\displaystyle\leq
C\|\mathrm{e}^{-\lambda}\nabla^{2}\Phi\|_{L^{2}}\|\nabla^{2}\Phi\|_{W^{2,2}}.$
(4.163)

Similarly, again with (4.94) and Gagliardo-Nirenberg we estimate

$\displaystyle\|\mathrm{e}^{4\lambda}|d\lambda|^{2}_{g}H\|_{L^{2}}$
$\displaystyle\leq C\|\mathrm{e}^{2\lambda}|d\lambda|^{2}H\|_{L^{2}}$ (4.164)
$\displaystyle\leq
C\|\mathrm{e}^{2\lambda}(\mathrm{e}^{-2\lambda}|\nabla^{2}\Phi|^{2})(\mathrm{e}^{-2\lambda}|\nabla^{2}\Phi|)\|_{L^{2}}$
(4.165) $\displaystyle\leq
C\|\mathrm{e}^{-2\lambda}|\nabla^{2}\Phi|^{3}\|_{L^{2}}$ (4.166)
$\displaystyle\leq
C\|\mathrm{e}^{-\lambda}\nabla^{2}\Phi\|_{L^{2}}\|\nabla^{2}\Phi\|_{W^{2,2}}.$
(4.167)

Finally, with Liouville’s equation

$\displaystyle-\Delta\lambda=\mathrm{e}^{2\lambda}K,$ (4.168)

the pointwise estimate

$\displaystyle|K|\leq|A|^{2}_{g}\leq\mathrm{e}^{-4\lambda}|A|^{2},$ (4.169)

we can estimate with Gagliardo-Nirenberg and (4.96):

$\displaystyle\|\mathrm{e}^{4\lambda}\Delta_{g}\lambda H\|_{L^{2}}$
$\displaystyle\leq C\||A|^{2}H\|_{L^{2}}$ (4.170) $\displaystyle\leq
C\|\mathrm{e}^{-2\lambda}|A|^{3}\|_{L^{2}}$ (4.171) $\displaystyle\leq
C\mathrm{e}^{-2\Lambda}\|A\|_{L^{2}}^{2}\|A\|_{W^{2,2}}$ (4.172)
$\displaystyle\leq C\|\mathrm{e}^{-\lambda}A\|_{L^{2}}^{2}\|A\|_{W^{2,2}}$
(4.173) $\displaystyle\leq
C\|\mathrm{e}^{-\lambda}A\|_{L^{2}}^{2}\|\nabla^{2}\Phi\|_{W^{2,2}}.$ (4.174)

All these estimates together yield (4.129). ∎

The combination of lemmas 4.10, 4.11 and 4.12 immediately gives (4.90), and
concludes the proof of Proposition 4.7. ∎

###### Proof of Theorem 4.1.

The qualitative statement, namely that $\Phi\in W^{4,2}_{\text{loc}}$, is an
immediate consequence of Propositions 4.3 and 4.6. We can therefore apply
Proposition 4.7, so that (4.90) jointly with elliptic estimates for the
bilaplacian give

$\displaystyle\|d\Phi\|_{W^{3,2}(B_{1/2})}\leq
C\big{(}\|\mathrm{e}^{4\lambda}\delta\mathcal{W}\|_{L^{2}(B_{1})}+\|\mathrm{e}^{-\lambda}\nabla^{2}\Phi\|_{L^{2}(B_{1})}\|\nabla^{2}\Phi\|_{W^{2,2}(B_{1})}+\|d\Phi\|_{L^{2}(B_{1})}\big{)}$
(4.175)

for $C=C(E,C_{\infty})>0$. Now we consider rescalings. For $0<r<1$ we let

$\displaystyle\widetilde{\Phi}(x)=\Phi(rx),\quad x\in B_{1},$ (4.176)

and we denote with a tilde all the quantities pertaining to
$\widetilde{\Phi}$. From for $k\in\mathbb{N}$ it follows in particular that
for $\Omega\subseteq B_{1}$ we have

$\displaystyle\nabla^{k}\widetilde{\Phi}(x)$
$\displaystyle=r^{k}\nabla^{k}\Phi(rx),$ (4.177)
$\displaystyle\|\nabla^{k}\widetilde{\Phi}\|_{L^{2}(\Omega)}$
$\displaystyle=r^{k-1}\|\nabla^{k}\Phi\|_{L^{2}(r\Omega)},$ (4.178)
$\displaystyle\mathrm{e}^{\widetilde{\lambda}(x)}$
$\displaystyle=r\mathrm{e}^{\lambda(rx)},$ (4.179)
$\displaystyle\widetilde{\lambda}(x)-\widetilde{\lambda}(0)$
$\displaystyle=\lambda(rx)-\lambda(0),$ (4.180)
$\displaystyle\widetilde{\delta\mathcal{W}}(x)$
$\displaystyle=\delta\mathcal{W}(rx),$ (4.181)

(the last equality follows either by direct inspection or at once recalling
that $\delta\mathcal{W}$ is a vector field) and from these relations, we
deduce in particular that

$\displaystyle\|\mathrm{e}^{4\widetilde{\lambda}}\widetilde{\delta\mathcal{W}}\|_{L^{2}(B_{1})}$
$\displaystyle=r^{3}\|\mathrm{e}^{4\lambda}\delta\mathcal{W}\|_{L^{2}(B_{r})},$
(4.182)
$\displaystyle\|\mathrm{e}^{-2\widetilde{\lambda}}\nabla^{2}\widetilde{\Phi}\|_{L^{2}(B_{1})}$
$\displaystyle=\|\mathrm{e}^{-2\lambda}\nabla^{2}\Phi\|_{L^{2}(B_{r})},$
(4.183)
$\displaystyle\|\widetilde{\lambda}-\widetilde{\Lambda}\|_{L^{\infty}(B_{1})}$
$\displaystyle=\|\lambda-\Lambda\|_{L^{\infty}(B_{r})},$ (4.184)

and the last two relations in particular give that $\widetilde{E}\leq E$ and
$\widetilde{C_{\infty}}\leq C_{\infty}$. Consequently, applying (4.175) to
$\widetilde{\Phi}$ gives

$\displaystyle\|d\Phi\|^{\prime}_{W^{3,2}(B_{r/2})}$ $\displaystyle\leq
C\big{(}r^{3}\|\mathrm{e}^{4\lambda}\delta\mathcal{W}\|_{L^{2}(B_{r})}+\|\mathrm{e}^{-\lambda}\nabla^{2}\Phi\|_{L^{2}(B_{r})}\|\nabla^{2}\Phi\|_{W^{2,2}(B_{r})}^{\prime}+\|d\Phi\|_{L^{2}(B_{r})}\big{)},$
(4.185)

where for $k=1,2$ we denoted
$\|\Phi\|_{W^{k,2}(B_{\rho})}^{\prime}=\big{(}\sum_{h=1}^{k}\rho^{k-1}\|\Phi\|_{W^{h,2}(B_{\rho})}^{2}\big{)}^{1/2}$
the scale-invariant version of the Sobolev norms where, as we said abefore
$C=C(E,C_{\infty})>0$.

Recall now that from (4.93) it is

$\displaystyle\|\mathrm{e}^{-\lambda}\nabla^{2}\Phi\|_{L^{2}(B_{1/2})}\leq
C\big{(}\|\mathrm{e}^{-\lambda}A\|_{L^{2}(B_{1/2})}+\|d\lambda\|_{L^{2}(B_{1/2})}\big{)},$
(4.186)

so by Theorem 2.2 we let $\varepsilon_{0}$ be sufficiently small so to have

$\displaystyle\|d\lambda\|_{L^{2}(B_{1/2})}+\|\lambda-\lambda(0)\|_{L^{\infty}(B_{1/2})}\leq
C\|\mathrm{e}^{-\lambda}A\|_{L^{2}(B_{1})}\leq C\varepsilon_{0},$ (4.187)

for $C=C(C_{(2,\infty)})>0$, whence (4.185) can be improved to

$\displaystyle\|d\Phi\|^{\prime}_{W^{3,2}(B_{r/2})}$ $\displaystyle\leq
C\big{(}r^{3}\|\mathrm{e}^{4\lambda}\delta\mathcal{W}\|_{L^{2}(B_{r})}+\varepsilon_{0}\|\nabla^{2}\Phi\|_{W^{2,2}(B_{r})}^{\prime}+\|d\Phi\|_{L^{2}(B_{r})}\big{)},$
(4.188)

for $C=C(C_{(2,\infty)})>0.$

Choose finally $\varepsilon_{0}>0$ be sufficiently small so that
$C\varepsilon_{0}\leq\frac{1}{2}$ to obtain

$\displaystyle\|d\Phi\|^{\prime}_{W^{3,2}(B_{r/2})}$
$\displaystyle\leq\frac{1}{2}\|d\Phi\|^{\prime}_{W^{3,2}(B_{r})}+C\big{(}r^{3}\|\mathrm{e}^{4\lambda}\delta\mathcal{W}\|_{L^{2}(B_{1})}+\|d\Phi\|_{L^{2}(B_{1})}\big{)},$
(4.189)

for every $0<r\leq 1/2$. A classical iteration/interpolation argument applied
to $\phi(r)=\|d\Phi\|^{\prime}_{W^{3,2}(B_{r})}$ and a covering argument yield
to (4.3). ∎

##  5 Conformal Willmore Flows

###### Proof of Lemma 1.1.

A metric $g$ is conformal if and only if its Hopf differential (computed with
respect to the background complex structure of $S^{2}$) vanishes identically.
In our case it is

$\displaystyle\operatorname{Hopf}(g)=g_{zz}dz\otimes
dz=\langle\partial_{z}\Phi,\partial_{z}\Phi\rangle dz\otimes dz.$ (5.1)

Since $\delta\mathcal{W}$ is a normal vector field, we see that

$\displaystyle\frac{1}{2}\frac{\partial}{\partial
t}\langle\partial_{z}\Phi,\partial_{z}\Phi\rangle$
$\displaystyle=\langle\partial_{z}\partial_{t}\Phi,\partial_{z}\Phi\rangle=\langle-\partial_{z}\delta\mathcal{W}+\partial_{z}U,\partial_{z}\Phi\rangle=\langle\delta\mathcal{W},\partial^{2}_{zz}\Phi\rangle+\langle\partial_{z}U,\partial_{z}\Phi\rangle.$
(5.2)

Since $U=U^{z}\partial_{z}\Phi+U^{\bar{z}}\partial_{\bar{z}}\Phi$ with
$U^{z}=U^{1}+iU^{2}$ and $U^{\bar{z}}=U^{1}-iU^{2}$ we have

$\displaystyle\left\langle\partial_{z}U,\partial_{z}\Phi\right\rangle$
$\displaystyle=\big{\langle}\partial_{z}U^{z}\partial_{z}\Phi+U^{z}\partial^{2}_{zz}\Phi+\partial_{z}U^{\bar{z}}\partial_{\bar{z}}\Phi+U^{\bar{z}}\partial^{2}_{z\bar{z}}\Phi,\partial_{z}\Phi\big{\rangle}$
(5.3)
$\displaystyle=\partial_{z}U^{z}g_{zz}+U^{z}\big{\langle}\partial^{2}_{zz}\Phi,\partial_{z}\Phi\big{\rangle}+\partial_{z}U^{\bar{z}}g_{\bar{z}z}+U^{\bar{z}}\big{\langle}\partial^{2}_{z\bar{z}}\Phi,\partial_{z}\Phi\big{\rangle}$
(5.4)
$\displaystyle=\partial_{z}U^{z}g_{zz}+\partial_{z}U^{\bar{z}}g_{\bar{z}z}+\frac{1}{2}\Big{(}U^{z}\partial_{z}g_{zz}+U^{\bar{z}}\partial_{\bar{z}}g_{zz}\Big{)},$
(5.5)

thus we have

$\displaystyle\frac{1}{2}\frac{\partial}{\partial
t}\langle\partial_{z}\Phi,\partial_{z}\Phi\rangle$
$\displaystyle=\partial_{z}U^{z}g_{zz}+\partial_{z}U^{\bar{z}}g_{\bar{z}z}+\frac{1}{2}\Big{(}U^{z}\partial_{z}g_{zz}+U^{\bar{z}}\partial_{\bar{z}}g_{zz}\Big{)}+\big{\langle}\delta\mathcal{W},\partial^{2}_{zz}\Phi\big{\rangle}.$
(5.6)

If the flow $\Phi$ is conformal, then
$\partial_{t}\operatorname{Hopf}(g)\equiv 0$ and $g_{zz},g_{\bar{z}\bar{z}}$
vanish identically. Since moreover $\delta\mathcal{W}$ is a normal vector we
may replace $\partial^{2}_{zz}\Phi$ with $A_{zz}$ and thus obtain, after
conjugation,

$\displaystyle
g_{\bar{z}z}\partial_{z}U^{\bar{z}}=-\langle\delta\mathcal{W},A_{zz}\rangle,$
(5.7)

and so since $\frac{1}{2}\mathrm{e}^{2\lambda}=g_{\bar{z}{z}}$, this yields
(1.14). ∎

The proof of the following proposition follows directly by direct inspection
of the proof of the original theorems; for the proof of (ii) one uses
additionally that conformal Willmore flows are $W^{4,2}$ for almost every
time, as proved in Proposition 5.2 below.

######  Proposition 5.1.

  1. (i)

If the surface $\mathcal{S}$ is immersed through a $W^{4,2}$-map, then the
same conclusion of Theorem 2.4 and all its consequences obtained in Section 3
still hold.

  2. (ii)

For conformal Willmore flows in
$\mathscr{W}^{\varepsilon,\delta}_{[0,T)}(S^{2},\mathbb{R}^{3})$ the area and
barycenter bounds in Theorem 2.8 still hold if $\varepsilon$ is chosen
sufficiently small.

A first regularity improvement for conformal Willmore flows is the following.

######  Proposition 5.2.

For any $\varepsilon_{0},\,\delta>0$, any conformal Willmore flow
$\Phi\in\mathscr{W}_{[0,T)}^{\varepsilon_{0},\delta}(S^{2},\mathbb{R}^{3})$ is
in $W^{4,2}(S^{2})$ for a.e. $t\in(0,T)$. Futhermore there exist
$\varepsilon_{0},\,\delta,\,C>0$ independent of $\Phi$ so that $\Phi\in
L^{2}((0,T),W^{4,2}(S^{2}))$ with

$\displaystyle\|d\Phi\|_{L^{2}((0,T),W^{3,2}(S^{2}))}\leq
C(\sqrt{T}+\|\mathrm{e}{{}^{\lambda}}\delta\mathcal{W}\|_{L^{2}((0,T)\times
S^{2})}),$ (5.8)

for a constant $C>0$.

###### Proof.

Since $\delta\mathcal{W}\in L^{2}((0,T)\times S^{2})$, Fubini’s theorem
implies $\delta\mathcal{W}(t,\cdot)\in L^{2}(S^{2})$ for a.e. $t$.
Consequently, by Propositions 4.3 and 4.6 it follows that $\Phi(t,\cdot)\in
W^{4,2}(S^{2})$.

From Liouville’s equation

$\displaystyle\Delta_{S^{2}}\lambda=\mathrm{e}^{2\lambda}K-1,$ (5.9)

the pointwise inequality $|K|\leq|A|^{2}_{g}=\mathrm{e}^{-4\lambda}|A|^{2}$
and elliptic estimates

$\displaystyle\|d\lambda\|_{L^{\infty}((0,T),L^{(2,\infty)}(S^{2}))}\leq C.$
(5.10)

Choose now $\varepsilon_{1}>0$ sufficiently small so that, for a.e.
$t\in(0,T)$ so that $\Phi(t,\cdot)$ is $W^{4,2}$, we can fix a value $r>0$
which satisfies, according to Lemma 3.3,

$\displaystyle\int_{B_{r}(x_{0})}|A|^{2}\,d\sigma$ $\displaystyle\leq
C\Big{(}\varepsilon_{1}+\mathrm{e}^{C}r^{2}\Big{)}\leq\varepsilon_{0},$ (5.11)

for every $x_{0}\in S^{2}$, where $\varepsilon_{0}$ is as in Theorem 4.1. With
the estimates (5.10) and (5.11), an application of Theorem 4.1 to a covering
$S^{2}$ with balls of radius $r/2$ gives, for a.e. $t\in(0,T)$,

$\displaystyle\|d\Phi\|_{W^{3,2}(S^{2})}\leq
C\big{(}\|\mathrm{e}^{4\lambda}\delta\mathcal{W}\|_{L^{2}(S^{2})}+\|\mathrm{e}^{\lambda}\|_{L^{2}(S^{2})}\big{)}\leq
C\big{(}\|\mathrm{e}^{\lambda}\delta\mathcal{W}\|_{L^{2}(S^{2})}+1\big{)},$
(5.12)

where the last inequality follows recalling that by assumption the bound
(1.25) holds in the class
$\mathscr{W}_{[0,T)}^{\varepsilon_{0},\delta}(S^{2},\mathbb{R}^{3})$. Now if
we square and integrate in $t$ such inequality, recalling that, since
$\Phi\in\mathscr{W}_{[0,T)}^{\varepsilon_{0},\delta}$, (1.26) holds by
assumption, we obtain (5.8). ∎

One can see that, in fact, along the proof of Proposition 5.2, well-balanced
condition (iv) in Definition 1.7 was not needed, an in fact the proof is
completely independent of the tangential component $U$. Such condition is
however essential to prove the following.

######  Proposition 5.3.

For every $p<2$ there exist $\delta>0$ with the following property. Let
$\Phi\in\mathscr{W}^{\varepsilon,\delta}_{[0,T)}(S^{2},\mathbb{R}^{3})$ be a
weak Willmore flow. Then its tangential component is in
$L^{2}((0,T),L^{p}(S^{2}))$, with

$\displaystyle\|U\|_{L^{2}((0,T),L^{p}(S^{2}))}\leq
C\|\mathrm{e}^{\lambda}\delta\mathcal{W}\|_{L^{2}((0,T)\times S^{2}))},$
(5.13)

for a constant $C=C(p)>0$.

###### Proof.

We may certainly assume $p>1$. In this proof we find it convenient to clearly
distinguish between the non-immersed (pulled-back) tangential component
$U=U^{\mu}\partial_{\mu}$, the immersed one
$d\Phi(U)=U^{\mu}\partial_{\mu}\Phi$, and the associated tangent vector field
on $S^{2}$, $dI(U)=U^{\mu}\partial_{\mu}I$, where $I:S^{2}\to\mathbb{R}^{3}$
denotes is the standard immersion. With this notation $\Phi$ satisfies weakly

$\displaystyle\frac{\partial}{\partial
t}\Phi=-\delta\mathcal{W}+d\Phi(U)\quad\text{in }(0,T)\times S^{2}.$

In what follows, it will be implicitly understood that all the slice-wise
operations are valid for a.e. fixed $t$. Finally we shall make use of the
notation and concepts recalled in Appendix B.

From Lemma 1.1, we deduce that $U$ is given by

$\displaystyle
U^{(1,0)}=-\overline{\partial}{}^{-1}(\langle\delta\mathcal{W},\overline{h}_{0}\rangle^{\sharp_{g}})+\Omega$
(5.14)

for some time-dependent holomorphic vector field
$\Omega=\Omega(t,\cdot)\in\mathfrak{X}^{\omega}(S^{2})$. Classical elliptic
estimates and Hölder’s inequality permit to estimate

$\displaystyle\|\overline{\partial}{}^{-1}(\langle\delta\mathcal{W},\overline{h}_{0}\rangle^{\sharp_{g}})\|_{L^{p}(S^{2})}$
$\displaystyle\leq
C_{p}\|\langle\delta\mathcal{W},\overline{h}_{0}\rangle^{\sharp}_{g}\|_{L^{1}(S^{2})}$
(5.15) $\displaystyle\leq
C_{p}\||\delta\mathcal{W}|\mathrm{e}^{-2\lambda}|A^{\circ}|\|_{L^{1}(S^{2})}$
(5.16) $\displaystyle\leq
C_{p}\|\mathrm{e}^{-\lambda}|A^{\circ}|\|_{L^{2}(S^{2})}\|\mathrm{e}^{-\lambda}\delta\mathcal{W}\|_{L^{2}(S^{2})}$
(5.17) $\displaystyle\leq
C_{p}\sqrt{\mathcal{W}_{0}(\Phi)}\,\|\mathrm{e}^{\lambda}\delta\mathcal{W}\|_{L^{2}(S^{2})}$
(5.18) $\displaystyle\leq
C_{p}\|\mathrm{e}^{\lambda}\delta\mathcal{W}\|_{L^{2}(S^{2})},$ (5.19)

where we estimated
$\|\mathrm{e}^{-\lambda}\delta\mathcal{W}\|_{L^{2}(S^{2})}\leq\|\mathrm{e}^{\lambda}\delta\mathcal{W}\|_{L^{2}(S^{2})}$
thanks to the simple inequality

$\displaystyle
1=\mathrm{e}^{-\lambda}\mathrm{e}^{\lambda}\leq\sup_{S^{2}}(\mathrm{e}^{-\lambda})\mathrm{e}^{\lambda}\leq
C(1+\delta)\mathrm{e}^{\lambda}$ (5.20)

issuing from property (1.25). We now examine $\Omega$. It will be more
practical to look at the associated (time-dependent) conformal Killing vector
field i.e. generating conformal transformations:

$\displaystyle V=\Omega+\overline{\Omega},$ (5.21)

Similarly as in the proof of Lemma 3.1, a basis for the vector space of
conformal Killing fields $T_{e}\operatorname{Aut}(S^{2})$ is given, in its
immersed representative, is given by

$\displaystyle dI(Z_{1})(y)$ $\displaystyle=(0,-y^{3},y^{2}),$ $\displaystyle
dI(Z_{2})(y)$ $\displaystyle=(y^{3},0,-y^{1}),$ $\displaystyle dI(Z_{3})(y)$
$\displaystyle=(-y^{2},y^{1},0),$ (5.22) $\displaystyle dI(Z_{4})(y)$
$\displaystyle=e_{1}-y^{1}y,$ $\displaystyle dI(Z_{5})(y)$
$\displaystyle=e_{2}-y^{2}y,$ $\displaystyle dI(Z_{6})(y)$
$\displaystyle=e_{3}-y^{3}y.$ (5.23)

One checks that this basis is orthogonal with respect the $L^{2}$–scalar
product and each element has the same length. Thus, we may write

$\displaystyle
V=\sum_{a=1}^{6}V^{a}Z_{a}=C\sum_{a=1}^{6}(V,Z_{a})_{L^{2}(S^{2})}Z_{a}=C\sum_{a=1}^{6}(U,Z_{a})_{L^{2}(S^{2})}Z_{a},$
(5.24)

where the last inequality is a consequence of the fact that, by construction,
the normal solution of the $\overline{\partial}$-operator is
$L^{2}$-orthogonal to the space of holomorphic vector fields. So, to estimate
$V$ it suffices to estimate the (time-dependent) coefficients

$\displaystyle(V,Z_{a})_{L^{2}}=\int_{S^{2}}\langle
U,Z_{a}\rangle\,d\sigma\quad\text{for }a=1,\ldots,6.$

Now, we may write the integrand as

$\displaystyle\langle U,Z_{a}\rangle$ $\displaystyle=\langle
dI(U),dI(Z_{a})\rangle=\langle d\Phi(U),dI(Z_{a})\rangle+\langle
dI(U)-d\Phi(U),dI(Z_{a})\rangle$ (5.25)

where the second term can be estimated as

$\displaystyle\big{|}\langle dI(U)-d\Phi(U),dI(Z_{a})\rangle\big{|}\leq
C|dI-d\Phi||U|$ (5.26)

and thus, upon integration, Hölder’s inequality and property (1.25) of the set
$\mathscr{W}^{\varepsilon,\delta}_{[0,T)}(S^{2},\mathbb{R}^{3})$, we see that
the estimate

$\displaystyle\bigg{|}\int_{S^{2}}\langle dI(U)-d\Phi(U),dI(Z_{a})\rangle
d\sigma\bigg{|}$ $\displaystyle\leq C\int_{S^{2}}|dI-d\Phi||U|\,d\sigma$
(5.27) $\displaystyle\leq
C\|dI-d\Phi\|_{L^{p^{\prime}}(S^{2})}\|U\|_{L^{p}(S^{2})}$ (5.28)
$\displaystyle\leq C\|\Phi-I-c\|_{W^{2,2}(S^{2})}\|U\|_{L^{p}(S^{2})}$ (5.29)
$\displaystyle\leq C_{p}\,\delta\,\|U\|_{L^{p}(S^{2})},$ (5.30)

holds, which will suit our purposes. We are left to estimate the terms

$\displaystyle\int_{S^{2}}\langle
d\Phi(U),dI(Z_{a})\rangle\,d\sigma\quad\text{for }a=1,\ldots,6.$ (5.31)

Differentiating the well-balanced conditions (1.16) gives,

$\displaystyle 0$
$\displaystyle=\frac{d}{dt}\int_{S^{2}}I\times\Phi\,d\sigma=\int_{S^{2}}I\times(-\delta\mathcal{W}+d\Phi(U))\,d\sigma,$
(5.32) $\displaystyle 0$
$\displaystyle=\frac{d}{dt}\int_{S^{2}}Id\sigma_{g}=\int_{S^{2}}I\big{(}\langle
2H,\delta\mathcal{W}\rangle+\operatorname{div}_{g}(U)\big{)}d\sigma_{g},$
(5.33)

and thus that

$\displaystyle\int_{S^{2}}I\times d\Phi(U)\,d\sigma$
$\displaystyle=\int_{S^{2}}I\times\delta\mathcal{W}\,d\sigma,$ (5.34)

and, upon integration by parts, that

$\displaystyle\int_{S^{2}}dI(U)\,d\sigma_{g}$
$\displaystyle=\int_{S^{2}}I\langle 2H,\delta\mathcal{W}\rangle\,d\sigma_{g}.$
(5.35)

On the other hand, a direct calculation shows that

$\displaystyle\begin{pmatrix}\langle d\Phi(U),dI(Z_{1})\rangle\\\ \langle
d\Phi(U),dI(Z_{2})\rangle\\\ \langle
d\Phi(U),dI(Z_{3})\rangle\end{pmatrix}=I\times d\Phi(U),$ (5.36)

and similarly that

$\displaystyle\begin{pmatrix}\langle d\Phi(U),dI(Z_{4})\rangle\\\ \langle
d\Phi(U),dI(Z_{5})\rangle\\\ \langle
d\Phi(U),dI(Z_{6})\rangle\end{pmatrix}=d\Phi(U)-\langle d\Phi(U),I\rangle
I=(d\Phi(U))^{\top},$ (5.37)

where $(\cdot)^{\top}$ denotes the orthogonal projection onto the tangent
space of $S^{2}$ in the standard immersion. Integrating (5.36) and using
(5.34) we can estimate for $a=1,2,3$

$\displaystyle\big{|}(d\Phi(U),dI(Z_{a}))_{L^{2}}\big{|}$ $\displaystyle\leq
C\Big{|}\int_{S^{2}}I\times\delta\mathcal{W}\,d\sigma\Big{|}\leq
C\|\mathrm{e}^{\lambda}\delta\mathcal{W}\|_{L^{2}(S^{2})}.$ (5.38)

Integrating (5.37) we get instead

$\displaystyle\sum_{a=1}^{3}C(d\Phi(U),dI(Z_{a}))_{L^{2}}e_{a}=\int_{S^{2}}(d\Phi(U))^{\top}d\sigma,$
(5.39)

and if we write the integrand as

$\displaystyle(d\Phi(U))^{\top}=(dI(U))^{\top}+(d\Phi(U)-dI(U))^{\top}$ (5.40)

and notice that, similarly as for (5.27) we can estimate

$\displaystyle\Big{|}\int_{S^{2}}(d\Phi(U)-dI(U))^{\top}d\sigma\Big{|}$
$\displaystyle\leq C\int_{S^{2}}|d\Phi(U)-dI(U)|\,d\sigma$ (5.41)
$\displaystyle\leq C\|d\Phi-dI\|_{L^{p^{\prime}}(S^{2})}\|U\|_{L^{p}(S^{2})}$
(5.42) $\displaystyle\leq C_{p}\,\delta\,\|U\|_{L^{p}(S^{2})},$ (5.43)

using (5.34) (since $dI(U)=dI(U)^{\top}$), we have for $a=4,5,6$

$\displaystyle|(d\Phi(U),dI(Z_{a}))_{L^{2}}|$ $\displaystyle\leq
C\Big{|}\int_{S^{2}}dI(U)d\sigma\Big{|}+C_{p}\,\delta\,\|U\|_{L^{p}(S^{2})}$
(5.44) $\displaystyle\leq C\Big{|}\int_{S^{2}}I\langle
2H,\delta\mathcal{W}\rangle\,d\sigma_{g}\Big{|}+C_{p}\,\delta\,\|U\|_{L^{p}(S^{2})}$
(5.45) $\displaystyle\leq
C\|H\mathrm{e}^{\lambda}\|_{L^{2}(S^{2})}\|\mathrm{e}^{\lambda}\delta\mathcal{W}\|_{L^{2}(S^{2})}+C_{p}\,\delta\,\|U\|_{L^{p}(S^{2})}$
(5.46) $\displaystyle\leq
C_{p}\big{(}\|\mathrm{e}^{\lambda}\delta\mathcal{W}\|_{L^{2}(S^{2})}+\delta\,\|U\|_{L^{p}(S^{2})}\big{)},$
(5.47)

where we also used that
$\|H\mathrm{e}^{\lambda}\|_{L^{2}(S^{2})}=C\mathcal{W}_{1}(\Phi)$ is bounded
uniformly in $t$.

Estimates (5.27), (5.38) and (5.44) inserted in (5.24) yield

$\displaystyle\|V\|_{L^{\infty}(S^{2})}\leq
C_{p}\big{(}\|\mathrm{e}^{\lambda}\delta\mathcal{W}\|_{L^{2}(S^{2})}+\delta\,\|U\|_{L^{p}(S^{2})}\big{)}.$
(5.48)

and so, in conjunction with (5.15), we get

$\displaystyle\|U\|_{L^{p}(S^{2})}\leq
C_{p}\big{(}\|\mathrm{e}^{\lambda}\delta\mathcal{W}\|_{L^{2}(S^{2})}+\delta\,\|U\|_{L^{p}(S^{2})}\big{)}.$
(5.49)

Taking the $L^{2}$-norm in time of such inequality gives

$\displaystyle\|U\|_{L^{2}((0,T),L^{p}(S^{2}))}\leq
C_{p}\big{(}\|\mathrm{e}^{\lambda}\delta\mathcal{W}\|_{L^{2}((0,T),L^{2}(S^{2}))}+\delta\,\|U\|_{L^{2}((0,T),L^{p}(S^{2}))}\big{)},$
(5.50)

and so, if $\delta$ is chosen sufficiently small, we reach (5.13). ∎

###### Remark 5.4

The $\delta$ of Proposition 5.3 may be smaller than that given by Proposition
1.5. An inspection of the proof however shows however that we may equivalently
have taken the same $\delta$ at the price of choosing $\varepsilon>0$
sufficiently small, because we may apply instead Propositions 5.1 and 1.5 for
a.e. $t$ so that $\Phi(t,\cdot)$ is $W^{4,2}$, and argue in the end similarly
as above. This variant would have been equally fine for our purposes.

######  Corollary 5.5.

There exists $\varepsilon_{0},\delta>0$ so that any Willmore flow in
$\mathscr{W}^{\varepsilon_{0},\delta}_{[0,T)}(S^{2},\mathbb{R}^{3})$ is in
$C^{\infty}((0,T]\times S^{2})$, and, if it is a solution to the Cauchy
problem (1.10) for smooth initial datum $\Phi_{0}$, then it is in
$C^{\infty}([0,T]\times S^{2})$.

###### Proof.

By definition and by Proposition 5.2, $d\Phi$ is in
$L^{\infty}((0,T),W^{1,2}(S^{2}))\cap L^{2}((0,T),W^{3,2}(S^{2}))$, we have
sufficient regularity to expand the Willmore operator in the flow equation by
means of formulas (2.11) and (4.91):

$\displaystyle\frac{\partial}{\partial
t}\Phi+\frac{1}{2}\mathrm{e}^{-4\lambda}\Delta^{2}\Phi$
$\displaystyle=\frac{1}{2}\mathrm{e}^{-4\lambda}\big{(}2\langle
d\lambda,\nabla H\rangle_{g}+(2|d\lambda|^{2}_{g}+\Delta_{g}\lambda)H\big{)}$
$\displaystyle\phantom{{}--{}}-\nabla^{*_{g}}\big{(}\langle
A,H\rangle^{\sharp_{g}}+\langle A^{\circ},H\rangle^{\sharp_{g}}\big{)}+U;$

since $\mathrm{e}^{\lambda}$ is by assumption uniformly bounded, the equation
is uniformly parabolic, and by Proposition 5.3, $U\in
L^{2}(0,T),L^{p}(S^{2}))$ for any $p<2$.

This is enough to start a boostrapping procedure using first $L^{p}$-$L^{q}$
and then Schauder parabolic estimates in a fashion similar to the elliptic
case discussed in Proposition 4.3. To bootstrap the regularity of the
tangential component $U$, one uses higher-regularity variants of Proposition
5.3, whose proofs are similar to the basic case. ∎

###### Proof of Theorem 1.9.

Case of smooth initial datum. By Corollary 5.5, it suffices to prove that
there exists a unique smooth solution in
$\mathscr{W}^{\varepsilon_{0},\delta}_{[0,T)}(S^{2},\mathbb{R}^{3})$ with the
required properties.

An application DeTurck’s trick (see Appendix A) yields existence and
uniqueness of a smooth solution to the Cauchy problem for the normal Willmore
flow:

$\displaystyle\left\\{\begin{aligned} \frac{\partial}{\partial
t}\Phi^{0}&=-\delta\mathcal{W}&&\text{in }(0,T)\times S^{2},\\\
\Phi^{0}(0,\cdot)&=\Phi_{0}&&\text{on }S^{2},\end{aligned}\right.$ (5.51)

and if $\varepsilon_{0}>0$ is small enough, by Theorem 2.8 $\Phi^{0}$ exists
for all and smoothly converges to a round sphere.444Note carefully: the
convergence is to _some_ smooth parametrization, of a round sphere of _some_
center and radius, not to $I$ modulo dilation and translation. For instance,
if $\Phi_{0}$ is _any_ smooth parametrization of a round sphere, then
$\Phi^{0}$ trivially converges to $\Phi_{0}$. We conformalize such flow
composing it with the family $(\phi(t,\cdot))_{t\in[0,+\infty)}$ of canonical
quasi-conformal mappings associated to the family of metrics
$g^{0}(t,\cdot)=\Phi^{0}(t,\cdot)^{\star}g_{\mathbb{R}^{3}}$, see [AB60]. The
fact that it is $\phi(0,\cdot)=e$ that such family is smooth both in the space
and in time follows from the theory of quasi-conformal mappings. Then
$\Phi^{1}(t,\cdot)=\Phi^{0}(t,\phi(t,\cdot))$ is a conformal Willmore flow
defined for all times and converging to a conformal parametrization of a round
sphere.

Let

$\displaystyle a(t)=\sqrt{\frac{4\pi}{\mathcal{A}(\mathcal{S}_{t})}}$ (5.52)

the normalizing function of time so that $a(t)\mathcal{S}_{t}$ has always area
$4\pi$ and let $\varepsilon_{0}>0$ be sufficiently small as in Theorem 2.4 and
also so that $C\varepsilon_{0}\leq\delta$ where $C$ is as in (2.7) and
$\delta$ is sufficiently small as in Propositions 1.5 and 5.3.555See in this
regard Remark 5.4. In this way, there exists a family of conformal
diffeomorphisms
$(\psi(t,\cdot))_{t\in[0,+\infty)}\subset\operatorname{Aut}(S^{2})$ so that
$\Phi(t,\cdot)=\Phi^{1}(t,\psi(t,\cdot))$ is conformal, well-balanced and

$\displaystyle\|a(t)\Phi(t,\cdot)-I-c(t)\|_{W^{2,2}(S^{2})}+\|a(t)\mathrm{e}^{\lambda(t,\cdot)}-1\|_{L^{\infty}(S^{2})}\leq
C\sqrt{\mathcal{W}_{0}(\mathcal{S}_{t})},$ (5.53)

where
$c(t)=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{S^{2}}\Phi(t,\cdot)\,d\sigma$
(see Remark 2.5) and moreover there exist a neighborhood of the identity
$\mathcal{O}\subset\operatorname{Aut}(S^{2})$ where such choice is unique.

As for $a(t)$, thanks the area control of Theorem 2.8 (recall that
$\mathcal{A}(\mathcal{S})=\mathcal{A}(\mathcal{S}_{0})=4\pi$) it is

$\displaystyle|\mathcal{A}(\mathcal{S}_{t})-4\pi|\leq
C\mathcal{W}_{0}(\mathcal{S}_{t})=o(1)\quad\text{as }t\to+\infty,$ (5.54)

and hence

$\displaystyle|a(t)-1|\leq C\mathcal{W}_{0}(\mathcal{S}_{t})=o(1)\quad\text{as
}t\to+\infty,$ (5.55)

which means that we may remove $a(t)$ from the estimate (5.53).

As for $c(t)$, we may write

$\displaystyle
c(t)=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{S^{2}}\Phi(t,\cdot)\,d\sigma$
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{S^{2}}\Phi(t,\cdot)\mathrm{e}^{2\lambda(t,\cdot)}\,d\sigma+\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{S^{2}}\Phi(t,\cdot)(1-\mathrm{e}^{2\lambda(t,\cdot)})d\sigma$
(5.56)
$\displaystyle=\mathcal{C}(\mathcal{S}_{t})+\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{S^{2}}\Phi(t,\cdot)(1-\mathrm{e}^{2\lambda(t,\cdot)})d\sigma,$
(5.57)

where $\mathcal{C}(\mathcal{S}_{t})$ denotes the barycenter as in Theorem 2.8.
Thus, from: the barycenter control of Theorem 2.8, the control on the
conformal factor issuing from (5.53), the fact that
$\psi(t,\cdot)\in\mathcal{O}$, smooth convergence and
$\mathcal{C}(\mathcal{S})=0$, we can estimate

$\displaystyle|c(t)|\leq C\mathcal{W}_{0}(\mathcal{S}_{t}),$ (5.58)

and hence also $c(t)$ can be removed from estimate (5.53) as well and deduce
that

$\displaystyle\|\Phi(t,\cdot)-I\|_{W^{2,2}(S^{2})}+\|\mathrm{e}^{\lambda(t,\cdot)}-1\|_{L^{\infty}(S^{2})}\leq
C\sqrt{\mathcal{W}_{0}(\mathcal{S}_{t})}.$ (5.59)

Since $\mathcal{W}_{0}(\mathcal{S}_{t})=o(1)$ as $t\to+\infty$, we obtain that
$\Phi(t,\cdot)$ converges to $I$ in $W^{2,2}$ and that its conformal factor
converges uniformly to $1$. Convergence of higher order derivatives to $I$
then follow from this and the smooth convergence of $\Phi^{1}$.

Case of weak initial datum. Let
$\Phi_{0,j}\in\mathscr{D}^{\varepsilon_{0}}(S^{2},\mathbb{R}^{3})$ be a
sequence appoximating $\Phi_{0}$ in the weak $W^{2,2}$-topology, i.e.

$\displaystyle\Phi_{0,j}$ $\displaystyle\rightharpoonup\Phi_{0}\quad\text{in
}W^{2,2}(S^{2}).$ (5.60)

If $\varepsilon_{0}$ is taken sufficiently small, it follows from the analysis
in [Riv08, Riv14, Riv16] (see for instance the proof of Theorem 3.36 in
[Riv16]) that also

$\displaystyle\mathrm{e}^{\lambda_{j}}$
$\displaystyle\overset{*}{\rightharpoonup}\mathrm{e}^{\lambda}\quad\text{in
}L^{\infty}(S^{2}).$ (5.61)

For each $j$, we let
$\Phi_{j}\in\mathscr{W}^{\varepsilon_{0},\delta}_{[0,+\infty)}(S^{2},\mathbb{R}^{3})$
be the well-balanced conformal Willmore flow given by Theorem 1.9 with initial
datum $\Phi_{0,j}$, which we know to be smooth. By estimate (5.59) we have
that for every $t>0$ there holds

$\displaystyle\|\Phi_{j}(t,\cdot)-I\|_{W^{2,2}(S^{2})}+\|\mathrm{e}^{\lambda_{j}(t,\cdot)}-1\|_{L^{\infty}(S^{2})}\leq
C\sqrt{\mathcal{W}_{0}(\mathcal{S}_{t})}\leq\delta,$ (5.62)

and by Proposition 5.2 it follows that for every fixed choice of $T>0$ there
holds

$\displaystyle\|\Phi_{j}\|_{L^{2}((0,T),W^{4,2}(S^{2}))}\leq C,$ (5.63)

and finally by Proposition 5.3 we also have, for a fixed choice of $1<p<2$,

$\displaystyle\|\partial_{t}\Phi_{j}\|_{L^{2}((0,T),L^{p}(S^{2}))}=\|-\delta\mathcal{W}_{j}+U_{j}\|_{L^{2}((0,1),L^{p}(S^{2}))}\leq
C,$ (5.64)

where $C$ does not depend on $j$.

Weak sequential compactness properties of Sobolev spaces then imply that, up
to the extraction of a subsequence, there exists measurable functions
${\Phi:(0,+\infty)\times S^{2}\to\mathbb{R}^{3}}$ and
${\lambda:(0,+\infty)\times S^{2}\to\mathbb{R}}$ so that, for every fixed
$T>0$, as $j\to\infty$,

$\displaystyle\Phi_{j}$ $\displaystyle\to\Phi\quad\text{a.e. in
}(0,+\infty)\times S^{2},$ (5.65) $\displaystyle\Phi_{j}$
$\displaystyle\rightharpoonup\Phi\quad\text{in }W^{1,p}((0,T)\times S^{2}),$
(5.66) $\displaystyle\Phi_{j}$ $\displaystyle\rightharpoonup\Phi\quad\text{in
}L^{2}((0,T),W^{4,2}(S^{2})),$ (5.67) $\displaystyle\Phi_{j}$
$\displaystyle\overset{*}{\rightharpoonup}\Phi\quad\text{in
}L^{\infty}((0,T),W^{2,2}(S^{2})),$ (5.68)
$\displaystyle\mathrm{e}^{\lambda_{j}}$
$\displaystyle\overset{*}{\rightharpoonup}\mathrm{e}^{\lambda}\quad\text{in
}L^{\infty}((0,T)\times S^{2}).$ (5.69)

Thus, for every $T>0$, $\Phi$ satisfies the conditions (i), (ii), (iv) of
Definition 1.2 (and $\mathrm{e}^{\lambda}$ is its conformal factor), its
Willmore operator is in $L^{2}((0,T)\times S^{2})$ and finally since also

$\displaystyle\mathcal{W}_{0}(\Phi(t,\cdot))\leq\liminf_{j\to\infty}\mathcal{W}_{0}(\Phi_{j})\leq\varepsilon_{0},$
(5.70)

which means that (ii) is satisfied as well and so
$\Phi\in\mathscr{W}^{\varepsilon_{0},\delta}_{[0,+\infty)}(S^{2},\mathbb{R}^{3})$.
Moreover, since $p>1$ by (5.66) $\Phi$ has a trace at initial time, which, by
uniqueness and continuity of the trace operator, must coincide with
$\Phi_{0}$.

Finally, the convergence properties (5.65)-(5.69) are enough to pass to the
limit as $j\to\infty$ in Definition 1.8 and thus deduce that $\Phi$ is also a
weak Willmore flow. By Corollary 5.5 $\Phi$ is smooth on $(0,+\infty)\times
S^{2}$ as well. ∎

##  Appendix A DeTurck’s Trick for the Willmore Flow

We outline here one way to obtain, in the smooth category, short-time
existence for the Cauchy’s problem (1.8) adapting an idea originally devised
by DeTurck [DeT83] in the context of the Ricci flow. There are other
possibilities, such as the graph Ansatz [HP99] (in codimension 1) or through
the Nash-Moser Implicit Function Theorem [Ham82a, Ham82b].

The key idea is as follows. From the divergence form of the Willmore operator
(2.11), we write

$\displaystyle\delta\mathcal{W}$
$\displaystyle=\Delta_{g}H+\nabla^{*_{g}}\big{(}\langle
A^{\circ},H\rangle^{\sharp_{g}}+\langle
A,H\rangle^{\sharp_{g}}\big{)}=\frac{1}{2}\Delta_{g}^{2}\Phi+F\left(d\Phi,\nabla^{2}\Phi,\nabla^{3}\Phi\right),$
(A.1)

where $F$ is a smooth function and the covariant derivatives are with respect
to a fixed smooth reference metric on $\Sigma$. Now in local coordinates we
may write

$\displaystyle\Delta_{g}\Phi$
$\displaystyle=g^{\mu\nu}\partial^{2}_{\mu\nu}\Phi-g^{\mu\nu}\Gamma_{\mu\nu}^{\sigma}\partial_{\sigma}\Phi,$
(A.2)

and so

$\displaystyle\Delta_{g}^{2}\Phi$ $\displaystyle=\Delta_{g}(\Delta_{g}\Phi)$
(A.3)
$\displaystyle=g^{\mu\nu}\partial^{2}_{\mu\nu}(\Delta_{g}\Phi)-g^{\mu\nu}\Gamma_{\mu\nu}^{\sigma}\partial_{\sigma}(\Delta_{g}\Phi)$
(A.4)
$\displaystyle=g^{\mu\nu}\partial^{2}_{\mu\nu}\left(g^{\alpha\beta}\partial^{2}_{\alpha\beta}\Phi-g^{\alpha\beta}\Gamma_{\alpha\beta}^{\gamma}\partial_{\gamma}\Phi\right)+f\left(\partial_{x}\Phi,\partial^{2}_{xx}\Phi,\partial^{3}_{xxx}\Phi\right),$
(A.5)

for some smooth function $f$. Now, because also the metric depends on $\Phi$:
$g_{\mu\nu}=\langle\partial_{\mu}\Phi,\partial_{\nu}\Phi\rangle$, the term
$g^{\alpha\beta}\left(\partial^{2}_{\mu\nu}\Gamma_{\alpha\beta}^{\gamma}\right)\partial_{\gamma}\Phi$
also contains derivative of order 4 in $\Phi$. This causes the Willmore
operator to be degenerate elliptic, and the corresponding flow to be
degenerate parabolic.

However, writing:

$\displaystyle\Delta^{2}_{g}\Phi$
$\displaystyle=g^{\mu\nu}g^{\alpha\beta}\partial^{4}_{\mu\nu\alpha\beta}\Phi-g^{\mu\nu}\partial^{2}_{\mu\nu}\left(g^{\alpha\beta}\Gamma_{\alpha\beta}^{\gamma}\right)\partial_{\gamma}\Phi+f\left(\partial_{x}\Phi,\partial^{2}_{xx}\Phi,\partial^{3}_{xxx}\Phi\right),$
(A.6)

one guesses that it may be possible to add a tangent vector field to the
Willmore operator which for which the corresponding flow is a uniformly
parabolic one. This motivates the following.

######  Definition A.1.

Let $\Sigma$ be a closed, orientable surface and let
$\Phi_{0}:\Sigma\to\mathbb{R}^{3}$ be a smooth immersion. If
$\Phi:\Sigma\to\mathbb{R}^{3}$ is another smooth immersion, _DeTurck’s vector
field for the Willmore flow_ (for $\Phi$ relative to $\Phi_{0}$) is the vector
field tangent to $\Phi$ given by

$\displaystyle
V=V(\Phi_{0},\Phi)=-\frac{1}{2}\Delta_{g}W=-\frac{1}{2}(\Delta_{g}W)^{\gamma}\partial_{\gamma}\Phi,$
(A.7)

where $W=W^{\gamma}\partial_{\gamma}$ is the vector field on $\Sigma$ given by

$\displaystyle
W^{\gamma}=g^{\alpha\beta}\left(\Gamma_{\alpha\beta}^{\gamma}-\breve{\Gamma}_{\alpha\beta}^{\gamma}\right),$
(A.8)

where $\Gamma_{\alpha\beta}^{\gamma}$ and
$\breve{\Gamma}_{\alpha\beta}^{\gamma}$ denote respectively the Christoffel
symbols of $\Phi$ and $\Phi_{0}$.

Note that this definition makes sense, since it is a well-known fact in
differential geometry that, although the Christoffel symbols are themselves
not tensor, the expression
$\Gamma_{\alpha\beta}^{\gamma}-\breve{\Gamma}_{\alpha\beta}^{\gamma}$ (and
consequently its trace $W$) is.

######  Proposition A.2 (Short-Time Existence for the Smooth DeTurck-Willmore
Flow).

Let $\Sigma$ be a closed, orientable surface and let
$\Phi_{0}:\Sigma\to\mathbb{R}^{3}$ be a smooth immersion. There exists some
$T=T(\Phi_{0})>0$ so that the Cauchy problem

$\displaystyle\left\\{\begin{aligned}
\partial_{t}\Phi&=-\delta\mathcal{W}+V&&\text{in }(0,T)\times\Sigma,\\\
\Phi(0,\cdot)&=\Phi_{0},&&\text{on }\Sigma,\end{aligned}\right.$ (A.9)

has a unique solution in the class
$C^{\infty}([0,T]\times\Sigma,\mathbb{R}^{3})$, where $V=V(\Phi_{0},\Phi)$ is
DeTurck’s vector field for the Willmore flow.

###### Proof.

It is sufficent to prove that (A.9) defines a uniformly parabolic system for
$\Phi$ over $\Sigma$. The existence, uniqueness and smoothness of a solution
follows then from the general theory for such systems in Hölder spaces [Sol65]
(transl. English [MR067]), [LSU68]. We have

$\displaystyle\Delta_{g}W=\operatorname{tr}_{g}\left(\nabla^{(2)}W\right)$
$\displaystyle=g^{\mu\nu}\left(\nabla^{g}_{\partial_{\mu}}\nabla^{g}_{\partial_{\nu}}W-\nabla_{\nabla_{\partial_{c}}^{g}}^{g}W\right)=g^{\mu\nu}\left(\nabla^{g}_{\partial_{\mu}}\nabla^{g}_{\partial_{\nu}}W-\Gamma_{\mu\nu}^{\sigma}\nabla_{\partial_{\sigma}}^{g}W\right),$
(A.10)

so computing directly we see that

$\displaystyle\Delta_{g}W$
$\displaystyle=g^{\mu\nu}\Big{(}\partial^{2}_{\mu\nu}W^{\xi}+\partial_{\nu}W^{\sigma}\Gamma_{\mu\sigma}^{\xi}+\partial_{\mu}W^{\sigma}\Gamma_{\nu\sigma}^{\xi}$
(A.11)
$\displaystyle\phantom{{}----{}}-\Gamma_{\mu\nu}^{\tau}\partial_{\tau}W^{\xi}+W^{\sigma}\partial_{\mu}\Gamma_{\nu\sigma}^{\xi}+W^{\sigma}\Gamma_{\nu\sigma}^{\tau}\Gamma_{\mu\tau}^{\xi}-\Gamma_{\mu\nu}^{\sigma}\Gamma_{\tau\sigma}^{\xi}W^{\sigma}\Big{)}\partial_{\xi}$
(A.12)
$\displaystyle=g^{\mu\nu}\left(\partial^{2}_{\mu\nu}W^{\xi}+f(W,\partial_{x}W,\partial_{x}\Phi,\partial^{2}_{xx}\Phi)\right)\partial_{\xi}$
(A.13)
$\displaystyle=g^{\mu\nu}\left(\partial^{2}_{\mu\nu}W^{\xi}+f\left(\partial_{x}\Phi,\partial^{2}_{xx}\Phi,\partial^{3}_{xxx}\Phi\right)\right)\partial_{\xi},$
(A.14)

thus we have, in _every_ choice of local coordinates, that

$\displaystyle-\delta\mathcal{W}+V$
$\displaystyle=-\frac{1}{2}\Delta^{2}_{g}\Phi+V+f\left(\partial_{x}\Phi,\partial^{2}_{xx}\Phi,\partial^{3}_{xxx}\Phi\right)$
(A.15)
$\displaystyle=-\frac{1}{2}\Big{(}g^{\mu\nu}g^{\alpha\beta}\partial^{4}_{\mu\nu\alpha\beta}\Phi-g^{\mu\nu}\partial^{2}_{\mu\nu}\left(g^{\alpha\beta}\Gamma_{\alpha\beta}^{\gamma}\right)\partial_{\gamma}\Phi$
(A.16)
$\displaystyle\phantom{{}----{}}+g^{\mu\nu}\partial^{2}_{\mu\nu}\left(g^{\alpha\beta}(\Gamma_{\alpha\beta}^{\gamma}-\breve{\Gamma}_{\alpha\beta}^{\gamma})\right)\partial_{\gamma}\vec{\Phi}\Big{)}+f\left(\partial_{x}\Phi,\partial^{2}_{xx}\Phi,\partial^{3}_{xxx}\Phi\right)$
(A.17)
$\displaystyle=-\frac{1}{2}\left(g^{\mu\nu}g^{\alpha\beta}\partial^{4}_{\mu\nu\alpha\beta}\Phi-g^{\mu\nu}\partial^{2}_{\mu\nu}\left(g^{\alpha\beta}(\breve{\Gamma}_{\alpha\beta}^{\gamma})\right)\partial_{\gamma}\vec{\Phi}\right)+f\left(\partial_{x}\Phi,\partial^{2}_{xx}\Phi,\partial^{3}_{xxx}\Phi\right)$
(A.18)
$\displaystyle=-\frac{1}{2}\left(g^{\mu\nu}g^{\alpha\beta}\partial^{4}_{\mu\nu\alpha\beta}\Phi\right)+f\left(\partial_{x}\Phi,\partial^{2}_{xx}\Phi,\partial^{3}_{xxx}\Phi\right),$
(A.19)

so that (A.9) defines, for $T$ sufficiently small and in the smooth category,
a uniformly parabolic system of fourth order in $\Phi$. ∎

One now obtains the analogue short-time existence and uniqueness statement for
the Cauchy problem (1.8) combining Proposition A.2 and the the fact that there
is a bijective correspondence between tangential components an
reparametrizations (see the analogous discussion in [Man11]).

##  Appendix B The $\overline{\partial}$-operator on Vector Fields

Let $\Sigma$ be a Riemann surface. the complexified tangent bundle
$T^{\mathbb{C}}\Sigma=T\Sigma+iT\Sigma$ splits in two sub-bundles:

$\displaystyle T^{\mathbb{C}}\Sigma=T\Sigma^{(1,0)}\oplus T\Sigma^{(0,1)},$
(B.1)

whose sections are respectively $(1,0)$ and $(0,1)$-vector fields:

$\displaystyle\mathfrak{X}^{(1,0)}(\Sigma)$
$\displaystyle=\Gamma(T\Sigma^{(1,0)})\quad\ni\quad
V=(V^{1}+iV^{2})\partial_{z},$ (B.2)
$\displaystyle\mathfrak{X}^{(0,1)}(\Sigma)$
$\displaystyle=\Gamma(T\Sigma^{(0,1)})\quad\ni\quad
W=(W^{1}+iW^{2})\partial_{\bar{z}}.$ (B.3)

One can identify $(1,0)$\- and real vector fields by means of conjugation and
$(1,0)$-projection:

$\displaystyle\mathfrak{X}^{(1,0)}(\Sigma)\simeq\mathfrak{X}(\Sigma)\quad:\quad
V\rightarrow V+\overline{V}\quad\text{and}\quad W^{(1,0)}\leftarrow W.$ (B.4)

With this identification, we can consider holomorphic vector fields as real
vector fields, whose flows consist precisely of families of
$\operatorname{Aut}(\Sigma)$, the space of conformal self-maps (Möbius
transformations) of $\Sigma$. Calling $B=\Gamma(T\Sigma^{(1,0)}\otimes
T^{*}\Sigma^{(0,1)})$, the $\overline{\partial}$-operator over $(1,0)$-vector
fields is

$\displaystyle\overline{\partial}:\mathfrak{X}^{(1,0)}(\Sigma)\to
B,\quad\overline{\partial}V=\frac{1}{2}\partial_{\bar{z}}(V^{1}+iV^{2})\partial_{z}\otimes
d\bar{z},$ (B.5)

whose kernel is the space of holomorphic vector fields
$\ker(\overline{\partial})=\mathfrak{X}^{\omega}(\Sigma).$ Fix now a conformal
metric over $\Sigma$, $g=\mathrm{e}^{2\lambda}|dz|^{2}$. The formal
$L^{2}$-adjoint of $\overline{\partial}$, defined through the formula
$(\overline{\partial}V,F)_{L^{2}}=(V,\overline{\partial}{}^{*}F)_{L^{2}}$ is,
if $F=f\partial_{z}\otimes d\bar{z}$,

$\displaystyle\overline{\partial}{}^{*}:B\to\mathfrak{X}^{(1,0)}(\Sigma),\quad\overline{\partial}{}^{*}F=-2\mathrm{e}^{-4\lambda}\partial_{z}(\mathrm{e}^{2\lambda}f)\partial_{z}.$
(B.6)

We deduce that $\ker(\overline{\partial}{}^{*})$ consists of those tensors
$F=f\partial_{z}\otimes d\bar{z}$ so that $\mathrm{e}^{2\lambda}f$ is
antiholomorphic. If we lower the fist index of $F$:

$\displaystyle F{}^{\flat}=fg_{\bar{z}z}\,d\overline{z}\otimes
d\overline{z}=\frac{1}{2}e^{2\lambda}f\,d\overline{z}\otimes d\overline{z},$
(B.7)

then $F^{\flat}$ is an antiholomorphic quadratic differential, and so its
conjugate is a holomorphic quadratic differential. With these identifications,
we have

$\displaystyle\ker(\overline{\partial}{}^{*})\simeq Q^{\omega}(\Sigma).$ (B.8)

In particular, we note that even though $\overline{\partial}{}^{*}$ does
depend on the chosen metric, $Q^{\omega}(\Sigma)$ does not.

For given $F\in B$, we consider the equation

$\displaystyle\overline{\partial}V=F\quad\text{on }\Sigma.$ (B.9)

Then, (B.9) has a solution if and only if

$\displaystyle
F\in\overline{\partial}(\mathfrak{X}^{(1,0)}(\Sigma))=\ker(\overline{\partial}{}^{*})^{\perp}$
(B.10)

In such case, if $V_{0}$ is one such solution, every other one is of the form
$V=V_{0}+v$ for $v\in\mathfrak{X}^{\omega}(\Sigma)$.

The _normal solution_ to (B.9) is the only one in
$\ker(\overline{\partial})^{\perp}$, and we denote it by
$\overline{\partial}{}^{-1}F$. Normal solutions satisfy the typical elliptic
estimates, such as for instance

$\displaystyle\|\overline{\partial}^{-1}F\|_{W^{1,2}(\Sigma)}\leq
C\|F\|_{L^{2}(\Sigma)}\quad\forall\,F\in\ker(\overline{\partial}{}^{*})^{\perp}.$
(B.11)

for a constant $C=C(\Sigma,g)>0$.

As a consequence of the Riemann-Roch formula, if $\gamma$ is the genus of
$\Sigma$, we have

$\displaystyle\dim_{\mathbb{C}}Q^{\omega}(\Sigma)=\begin{cases}0&\text{if
}\gamma=0\\\ 1&\text{if }\gamma=1,\\\ 3\gamma-3&\text{if }\gamma\geq
2,\end{cases}\quad\text{and}\quad\dim_{\mathbb{C}}\mathfrak{X}^{\omega}(\Sigma)=\begin{cases}3&\text{if
}\gamma=0,\\\ 1&\text{if }\gamma=1,\\\ 0&\text{if }\gamma\geq 2.\end{cases}$
(B.12)

In particular, (B.9) can be solved for any $F$ when $\Sigma=S^{2}$.

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