Introduction

Let (Mg) be an asymptotically flat Riemannian three-manifold with non-negative scalar curvature. Under suitable decay conditions on the metric, such a manifold possesses a global non-negative invariant called the ADM mass and is denoted by \(m_{\text {ADM}}\) (see [1, 2, 32]). On the other hand, finding the right notion of quasi-local mass corresponding to this global invariant remains an interesting open problem (see [28]). A promising candidate is the so-called Hawking mass \(m_H\) defined by

$$\begin{aligned} m_H(\Sigma ):=\frac{|\Sigma |^{\frac{1}{2}}}{(16\pi )^{\frac{3}{2}}}\bigg (16\pi -\int \nolimits _\Sigma H^2\mathrm{{d}}\mu \bigg ), \end{aligned}$$

where \(\Sigma \) is a compact surface bounding a region \(\Omega \) whose mass is to be determined. With the help of the Hawking mass, the ADM mass can be quantified in terms of the local geometry: in a celebrated work, Huisken and Illmanen used a weak version of the inverse mean curvature flow to prove the Riemannian Penrose inequality which states that the ADM mass of an asymptotically flat manifold is bounded from below by the Hawking mass of any connected outward minimizing surface (see [16]). A different version of the Penrose inequality, where the comparison surface is required to be minimal but not necessarily connected, was later on shown by Bray using a quasistatic flow (see [4]). More recently, Huisken introduced a concept of isoperimetric mass which only relies on the \(C^0\)-data of the metric and provides a notion of quasi-local as well as global mass. The global mass can be shown to agree with the ADM mass in case the latter is well defined. It turns out that the isoperimetric mass can be characterized in terms of the Hawking mass of outward minimizing surfaces (see [17] or [20] for a more detailed discussion).

While the Hawking mass enjoys such desirable connections to the global geometry, there are unfortunately many surfaces with negative Hawking mass. This is a contrast to some other concepts of quasi-local mass such as the Brown-York mass (see [31]). It was therefore a crucial insight by Christodolou and Yau that the Hawking mass of a closed stable constant mean curvature surface is non-negative (see [12]). This suggested that such surfaces are suitable to test the gravitational field of an asymptotically flat manifold and motivated the study of the isoperimetric problem in such spaces. As some of the following results require stronger decay conditions on the metric than asymptotical flatness, we make the following definition: the metric g is said to be \(C^k\)-close to Schwarzschild with decay coefficient \(\eta \) and ADM mass m if in the chart at infinity, there holds \(g=g_S+h\), where h is a symmetric two tensor satisfying

$$\begin{aligned} |\partial ^j h|\le \eta r^{-2-j},\ \end{aligned}$$
(1)

for any \(0\le j\le k\). Here, \(g_S\) is the Schwarzschild metric with mass m, \(\partial \) the Euclidean derivative and r the radial parameter in the chart at infinity. The Schwarzschild space models a static, single black hole and a space which is \(C^k\)-close can be understood to be a small perturbation. The first breakthrough in the study of the isoperimetric problem was accomplished by Huisken and Yau, who used a volume-preserving version of the mean curvature flow to show that an asymptotic region of an asymptotically flat manifold which is \(C^4\)-close to Schwarzschild and has non-negative scalar curvature is foliated by embedded stable constant mean curvature spheres. Such a foliation induces a natural coordinate system and also gives rise to a geometric centre of mass. This result was later on refined by Qing and Tian who showed uniqueness of this foliation (see [29]). Using an ingenious argument, Bray showed in his PhD-thesis that the centred spheres are the unique non-null-homologous isoperimetric surfaces in the Schwarzschild manifold (see [5]). This provided evidence that the leaves of the foliation in [18] might actually be isoperimetric. In another breakthrough, Eichmair and Metzger extended the idea of Bray and showed in [15] that a foliation as in [18] exists even if the manifold is only \(C^2\)-close to Schwarzschild. Furthermore, they proved that in the asymptotic region the leaves are in fact the unique isoperimetric surfaces enclosing a sufficiently large volume. It was later on shown by Chodosh et al. that a unique minimizer of the isoperimetric problem exists even if the manifold is only asymptotically flat and satisfies a certain decay condition on the scalar curvature (see [11]). On the other hand, studying the uniqueness of stable constant mean curvature spheres which are not necessarily isoperimetric turned out to be a more difficult problem. As a first step in this direction, Brendle showed a Heintze–Karcher type inequality and used a conformal flow in an elegant way to show that the centred spheres are the only constant mean curvature surfaces contained in one half of the Schwarzschild manifold (see [6]). Finally, in a series of crucial results, Chodosh and Eichmair obtained the unconditional characterization of stable constant mean curvature surfaces in asymptotically flat manifolds which are \(C^6\)-close to Schwarzschild and whose scalar curvature is non-negative and satisfies a certain decay condition. By comparing certain mass flux integrals ([9]) and using a Lyapunov–Schmidt analysis to study null-homologous surfaces ([10]), they showed that the leaves of the foliation are the only stable compact constant mean curvature surfaces without any assumption on their homology class. In the proof, the result [8] by Carlotto, Chodosh and Eichmair played an important part where they showed among other things that any asymptotically flat manifold with non-negative scalar curvature admitting an unbounded area minimizing minimal surface must be isometric to the flat Euclidean space. The results in [9] seem to be optimal in some sense (see also [3]). Moreover, it should be noted that they stand in stark contrast to the situation in the Euclidean space. The presence of positive mass seems to rule out all but one isoperimetric surface.

For any concept of quasi-local mass, it is natural to look for regions which contain a maximal amount of mass. Usually, one can only hope to find such regions if one fixes a certain geometric quantity such as the volume of the region \(\Omega \) or the area of its boundary \(\Sigma \). In the case of the Hawking mass, the latter seems to be the more natural quantity. While isoperimetric surfaces enjoy non-negative Hawking mass, they in fact maximize Huisken’s quasi-local isoperimetric mass when fixing the volume of \(\Omega \). Hence, when studying the Hawking mass, it might be a more natural problem to directly look for maximizers of the Hawking mass when fixing the area. This approach is equivalent to finding area-constrained minimizers of the Willmore functional \({\mathcal {W}}\) which is defined to be

$$\begin{aligned} {\mathcal {W}}(\Sigma ):=\frac{1}{4}\int \nolimits _\Sigma H^2 \mathrm{{d}}\mu . \end{aligned}$$

While the isoperimetric problem can be formulated solely in terms of \(C^0\)-data, the Hawking mass depends on higher order quantities and therefore seems to be more complicated to investigate. In fact, the Euler-Lagrange equation for the Willmore functional is a fourth-order elliptic equation and cannot be studied with the same techniques as the second order problem of finding constant mean curvature surfaces. Nevertheless, using a continuity method and integral curvature estimates, Lamm, Metzger and Schulze showed the following result (see [24] and Sect. 2 for a more precise statement).

Theorem 1.1

Let (Mg) be an asymptotically flat manifold which is \(C^3\)-close to Schwarzschild, with mass \(m>0\) and decay coefficient \(0<\eta \le \eta _0\) for some constant \(\eta _0\) depending only on m and satisfies \(|\text {Sc}|\le \eta r^{-5}\). Then there exists a constant \(\lambda _0>0\) and a compact set K depending only on m and \(\eta _0\) such that \(M\setminus K\) is foliated by surfaces of Willmore type \(\Sigma _\lambda \) where \(\lambda \in (0,\lambda _0)\). Moreover, every sufficiently centred, strictly mean-convex sphere \(\Sigma \subset M\setminus K\) which is of Willmore type belongs to this foliation.

Here, a surface of Willmore type is a critical point of the area prescribed Willmore energy. More precisely, every \(\Sigma _\lambda \) satisfies the equation

$$\begin{aligned} \Delta H+H(\text {Rc}(\nu ,\nu )+|\mathring{A}|^2+\lambda )=0. \end{aligned}$$

As for the isoperimetric problem, the positivity of the ADM mass is related to uniqueness which is evidently violated in the Euclidean space. The leaves of the foliation enjoy various desirable properties: if the scalar curvature is non-negative, the Hawking mass is positive and non-decreasing along the foliation and approaches the ADM mass as \(\lambda \rightarrow 0\). Given the results obtained for the isoperimetric problem, one is tempted to believe that in an asymptotic region, the leaves \(\Sigma _\lambda \) are the global maximizers of the Hawking mass and perhaps the only surfaces of Willmore type with non-negative Hawking mass and a sufficiently large area. Up to now, this has not even been known in Schwarzschild. In fact, a result comparable to the one obtained in [4] cannot be expected as one can easily construct spheres which are close and homologous to the horizon, but have arbitrarily large Hawking mass. On the other hand, it is possible to construct off-centre surfaces whose Hawking mass is arbitrarily close to the ADM-mass which in turn equals the Hawking mass of the centred spheres (see the remark below Corollary 5.4). Hence, the problem of maximizing the Hawking mass even with fixed area seems particularly challenging from a variational point of view.

In this work we make partial progress in understanding the role of the centred spheres in the Schwarzschild space or more generally of the leaves \(\Sigma _\lambda \) in the foliation constructed in [24] in asymptotically Schwarzschild spaces regarding the maximization of the Hawking mass. In fact, we show the following:

Theorem 1.2

Let (Mg) be \(C^3\)-close to Schwarzschild with decay coefficient \(0<\eta \le \eta _0(m)\) and mass \(m>0\) and let \(\{\Sigma _\lambda | \lambda \in (0,\lambda _0)\}\) be the foliation from the previous theorem. Then there is a constant \(\Lambda <\lambda _0\) depending only on m and \(\eta _0\) such that any immersed sphere \(\Sigma \subset M\) enclosing \(\Sigma _{\Lambda }\) which is sufficiently centred satisfies \(m_H(\Sigma )\le m_H(\Sigma _\lambda )\), where \(\lambda \) is chosen such that \(|\Sigma |=|\Sigma _\lambda |\). If equality holds, then \(\Sigma =\Sigma _\lambda \). Moreover, if the scalar curvature of (Mg) is non-negative, there holds \(m_H(\Sigma )\le m\) with equality if and only if \(\Sigma \) is a centred sphere in the spatial Schwarzschild manifold.

A more local version of this result in the deSitter-Schwarzschild space was shown by Maximo and Nunes, see [27]. They considered graphical surfaces with respect to the centred spheres and computed the second variation of the Hawking mass. Our approach relies instead on a stability result for the area preserving Willmore flow which we will discuss below.

Theorem 1.3

Let (Mg) be \(C^3\)-close to Schwarzschild with decay coefficient \(0<\eta \le \eta _0(m)\) and mass \(m>0\) and let \(\Sigma \) be an embedded sphere which is obtained as a small area-preserving \(W^{2,2}\)-perturbation of a leave of the foliation \(\{\Sigma _\lambda \}\) from Theorem 1.1. Then the area preserving Willmore flow starting at \(\Sigma \) exists for all times and converges smoothly to one of the leaves in the foliation \(\{\Sigma _\lambda \}\).

For a more precise statement of Theorems 1.2 and  1.3, we refer to Theorem 5.3 and Corollary 5.4. The Willmore flow was introduced by Kuwert and Schätzle in [21, 22] as the \(L^2\)-gradient flow for the Willmore energy in the Euclidean space and it has been studied in various contexts ever since. The area preserving Willmore flow is the \(L^2\)-projection of the Willmore flow onto area preserving immersions and was introduced by Jachan in his PhD-thesis, see [19]. It is a smooth one-parameter family of surfaces leaving the area constant while decreasing the Willmore energy and hence increases the Hawking mass. Using methods similar to Kuwert and Schätlze in [21,22,23], Jachan showed long time existence and subsequential convergence for topological spheres to a surface of Willmore type requiring a specific bound on the initial energy and assuming that the flow avoids a sufficiently large compact set for all times. One might therefore expect that the area-preserving Willmore flow can be used to produce area-prescribed critical points of the Hawking mass. It is, however, in general not clear under which initial conditions this assumption can be verified.

In order to prove Theorem 1.3, we verify the constraints of the long time existence result of [19]. By a result of Müller and deLellis, surfaces with small traceless part of the second fundamental form are \(W^{2,2}\)-close to a round sphere. From this, it follows that proving long time existence eventually reduces to controlling the evolution of the barycentre. In order to do this, we derive a differential inequality which we calculate in terms of the approximating round sphere. It turns out that this inequality is governed by the positivity of mass. The crucial part is then to control the error terms which require precise estimates of the evolving geometric quantities. Then, this argument can be used to show that the barycentre does not move too much if it is initially not too far away from the origin. Finally, we can use the uniqueness statement in Theorem 1.1 to identify the limit of the flow and deduce smooth convergence. Theorem 1.2 then follows from Theorem 1.3 and the fact that sufficiently centred surfaces can either be flown back to a leave of the foliation or have negative Hawking mass.

The rest of this paper is organized as follows. In Sect. 2, we fix some notation, collect results about asymptotically flat manifolds which are \(C^3\)-close to Schwarzschild and show that the area preserving Willmore flow of a small \(W^{2,2}\)-perturbation of a leave of the canonical foliation remains round and avoids a large compact set if its barycenter does not move too much. In Sect. 3, we prove general integral curvature estimates in the spirit of [21] for asymptotically Schwarzschild manifolds. In Sect. 4, we combine these estimates with a careful analysis of the evolution equation for the Willmore energy to obtain precise a-priori estimates for certain geometric quantities under the area preserving Willmore flow. Finally, in Sect. 5, we derive a differential inequality for the barycentre to find that the evolution is governed by the translation sensitivity of the Willmore energy in the Schwarzschild space. We then proceed to prove Theorem 1.3. For the convenience of the reader, we have included a summary of the argument used by Jachan in [19] in the appendix.

Preliminaries

Let (Mg) be an a three-dimensional, complete and asymptotically flat Riemannian manifold which is \(C^3\)-close to Schwarzschild with mass \(m>0\) and decay coefficient \(\eta >0\). More precisely, we assume that there is a compact set K such that \(M\setminus K\) is diffeomorphic to \(R^{3}\setminus \overline{B_{\sigma }(0)}\) for some \(\sigma >m/2\) and that the following estimate holds on \(M\setminus K\):

$$\begin{aligned} r^2|g-g_S|+r^3|{{\overline{\nabla }}}_g-{{\overline{\nabla }}}_S|+r^4|\text {Rc}_g-\text {Rc}_S|+r^5|{\overline{\nabla }}_g\text {Rc}_g-{\overline{\nabla }}_S\text {Rc}_S|\le \eta . \end{aligned}$$
(2)

Here, r denotes the radial function of the asymptotic chart \({\mathbb {R}}^3\setminus \overline{B_{\sigma }(0)}\), \({{\overline{\nabla }}}_g\) the gradient of the ambient space and \(\text {Rm}_g\) and \(\text {Rc}_g\) the Riemann curvature tensor and the Ricci curvature of the ambient space, respectively. \(g_S\) denotes the Schwarzschild metric with mass m, which is defined by

$$\begin{aligned} g_S:=\bigg (1+\frac{m}{2r}\bigg )^4g_e=\phi ^4 g_e. \end{aligned}$$

The subscripts S and e indicate that the geometric quantity is computed with respect to the Schwarzschild and the Euclidean metric, respectively. On the other hand, we will usually omit the subscript g. The definition of being \(C^{3}\)-close to Schwarzschild is equivalent to the definition given in the introduction and the mass parameter m is of course equal to the ADM-mass of (Mg). Finally, we assume that the scalar curvature of (Mg), denoted by \(\text {Sc}_g\), satisfies

$$\begin{aligned} |r^5\text {Sc}_g|\le \eta . \end{aligned}$$
(3)

Such manifolds were called asymptotically Schwarzschild in [24] and we will adopt this terminology from now on. The Schwarzschild manifold \((M_S,g_S)=({\mathbb {R}}^3\setminus \{0\},g_S)\) is the model space for the problem studied in this paper. It models a single static black hole with mass \(m>0\) and the metric being static is expressed in the following equation:

$$\begin{aligned} ({{\overline{\Delta }}}_S f) g_S -{{\overline{\nabla }}}^2_Sf+f\text {Rc}_S= -{{\overline{\nabla }}}^2_Sf+f\text {Rc}_S=0, \end{aligned}$$
(4)

where \(f=(2-\phi )/\phi \) is the so-called potential function. It follows that \((M_S,g_S)\) has vanishing scalar curvature and that the Ricci curvature is given by

$$\begin{aligned} \text {Rc}_S(\cdot ,\cdot )=mr^{-3}\phi ^{-2}(g_e(\cdot ,\cdot )-3g_e(\partial _r,\cdot )g_e(\partial _r,\cdot )). \end{aligned}$$
(5)

We consider an immersed, closed and orientable surface \(\Sigma \subset M\) and denote its first fundamental form by \(\gamma \), its connection by \(\nabla \), its outward normal by \(\nu \), its second fundamental form by A, the traceless part by \(\mathring{A}\), the mean curvature by H and the area element by \(\mathrm{{d}}\mu \). Moreover, we denote the induced curvature by \(\text {Rc}^\Sigma \) and \(\text {Rm}^\Sigma \), respectively. \(\Sigma \) can also be regarded as an embedded surface in \(({\mathbb {R}}^3\setminus \overline{B_{\sigma }(0)},g_e)\) or \(({\mathbb {R}}^3\setminus \overline{B_{\sigma }(0)},g_S)\). We indicate the corresponding geometric quantities by the subscripts e and S. If we want to emphasize the correspondence to g, we sometimes use the subscript g. We use the letter c for any constant that only depends on \(m,\eta \) in a non-decreasing way. The meaning of such a constant will be different in most of the following inequalities. If a constant has a geometric dependency, we will explicitly state it. We fix a chart at infinity and extend the radial parameter r in a smooth way to all of M. We define \(r_{\min }\) and \(r_{\max }\) to be the minimal and maximal value of the radial function on \(\Sigma \), respectively. As all of our results concern surfaces which are contained in the asymptotic region, we will always assume that \(r_{\min }\ge R_0\) for some positive constant \(R_0(\eta ,m)\) which is to be determined. We let x be the position vector in the asymptotic region. If \(\Sigma \) is contained in the asymptotic region, we define the Euclidean barycentre, approximate radius and centring parameter by

$$\begin{aligned} a_e=\frac{\int \nolimits _\Sigma x\mathrm{{d}}\mu _e }{|\Sigma |_g}, \qquad R_e:=\frac{|\Sigma |^{\frac{1}{2}}_e}{\sqrt{4\pi }}, \qquad \tau _e=\frac{|a_e|}{R_e}, \end{aligned}$$

respectively, and reiterate these definitions for the metrics g and \(g_S\).Footnote 1 The Hawking mass \(m_H\) of a surface \(\Sigma \) is defined by

$$\begin{aligned} m_H(\Sigma )=\frac{|\Sigma |^{1/2}}{(16\pi )^{3/2}}\bigg (16\pi -\int \nolimits _\Sigma H^2\mathrm{{d}}\mu \bigg )=\frac{|\Sigma |^{1/2}}{(16\pi )^{3/2}}\bigg (16\pi -4{\mathcal {W}}(\Sigma )\bigg ), \end{aligned}$$
(6)

where \({\mathcal {W}}\) denotes the Willmore energy, that is,

$$\begin{aligned} {\mathcal {W}}(\Sigma )=\frac{1}{4}\int \nolimits _\Sigma H^2\mathrm{{d}}\mu . \end{aligned}$$

In [24], Lamm, Metzger and Schulze studied area-prescribed critical points of the Willmore energy and called them surfaces of Willmore type. Such surfaces satisfy the equation

$$\begin{aligned} \Delta H+H(\text {Rc}(\nu ,\nu )+|\mathring{A}|^2+\lambda )=0 \end{aligned}$$
(7)

for some scalar parameter \(\lambda \). As Lamm, Metzger and Schulze showed by using a continuity method, an asymptotic region of an asymptotically Schwarzschild manifold is foliated by such surfaces. More precisely, they showed the following theorem.

Theorem 2.1

Let (Mg) be an asymptotically Schwarzschild manifold with mass m and decay coefficient \(\eta \le \eta _0\) for some \(\eta _0>0\) depending only on m. Then there exists a constant \(\lambda _0>0\) and a compact set K depending only on \(m,\eta _0\) such that \(M\setminus K\) is foliated by embedded spheres of Willmore type \(\Sigma _\lambda \) where \(\lambda \in (0,\lambda _0)\). Moreover, there are constants \(\chi , {{\tilde{\tau }}}>0\) which only depend on \(m,\eta _0\) such that any strictly mean convex sphere \(\Sigma \subset M\setminus K\) of Willmore type satisfying \(\tau _e\le {{\tilde{\tau }}}\) and \(R_e\le \chi r_{\min }^2\) belongs to the foliation.

Another tool to find surfaces of Willmore type is the so-called area preserving Willmore flow which was introduced by Jachan in [19]. In this paper, we will study the evolution of spherical surfaces under the area preserving Willmore flow which is defined as follows. We say that a smooth family of surfaces \(\{\Sigma _t|0\le t\le T\}\), \(\Sigma _0=\Sigma \) flows by the area preserving Willmore flow with initial data \(\Sigma \) if it satisfies the following evolution equation:

$$\begin{aligned} \frac{\mathrm{{d}}}{\mathrm{{d}}t}x=(\Delta H + H(\text {Rc}(\nu ,\nu )+|\mathring{A}|^2+\lambda ))\nu =:W\nu +\lambda H\nu , \end{aligned}$$
(8)

where the Lagrange parameter \(\lambda \) is given by

$$\begin{aligned} \lambda (t)=|H|^{-2}_{L^2(\Sigma _t)}\int \nolimits _{\Sigma _t}(|\nabla H|^2-H^2\text {Rc}(\nu ,\nu )-H^2|\mathring{A}|^2)\mathrm{{d}}\mu . \end{aligned}$$
(9)

The area preserving Willmore flow is the \(L^2\)-projection of the Willmore flow onto the class of area-preserving flows. It is easy to see that this evolution leaves the area constant, decreases the Willmore energy and consequently increases the Hawking mass. In fact, for a flow with normal speed \(\zeta \), there holds

$$\begin{aligned} \frac{\mathrm{{d}}}{\mathrm{{d}}t}H=-\Delta \zeta -\zeta |A|^2-\zeta \text {Rc}(\nu ,\nu ),\qquad \frac{\mathrm{{d}}}{\mathrm{{d}}t}\mathrm{{d}}\mu =\zeta H\mathrm{{d}}\mu . \end{aligned}$$
(10)

On the other hand, integration by parts reveals that

$$\begin{aligned} \lambda (t)\int \nolimits _{\Sigma }H^2\mathrm{{d}}\mu =\int \nolimits _{\Sigma } WH\mathrm{{d}}\mu . \end{aligned}$$
(11)

Plugging in \(\zeta =W+\lambda H\), integrating by parts and using the identity \(2|A|^2=H^2+2|\mathring{A}|^2\), we find

$$\begin{aligned} \frac{\mathrm{{d}}}{\mathrm{{d}}t}\int \nolimits _{\Sigma } H^2\mathrm{{d}}\mu&=\int \nolimits _{\Sigma }\bigg (2H(-\Delta (W+H\lambda )-|A|^2(W+H\lambda )\nonumber \\&\quad -\text {Rc}(\nu ,\nu )(W+H\lambda ))+H^3(W+\lambda H)\bigg )\mathrm{{d}}\mu \nonumber \\&=\int \nolimits _{\Sigma } \bigg (2W(-\Delta H-|\mathring{A}|^2-\text {Rc}(\nu ,\nu ))+\lambda (|\nabla H|^2-|\mathring{A}|^2-\text {Rc}(\nu ,\nu ))\bigg )\mathrm{{d}}\mu \nonumber \\&=-2\int \nolimits _{\Sigma } W^2\mathrm{{d}}\mu +2\bigg (\int \nolimits _{\Sigma } H^2\bigg )^{-1}\mathrm{{d}}\mu \bigg (\int \nolimits _\Sigma WH\mathrm{{d}}\mu \bigg )^2 \nonumber \\&\le 0, \end{aligned}$$
(12)

where we used Hölder’s inequality in the last step. Similarly,

$$\begin{aligned} \frac{\mathrm{{d}}}{\mathrm{{d}}t}|\Sigma |=\int \nolimits _{\Sigma } WH\mathrm{{d}}\mu +\lambda (t)\int \nolimits _{\Sigma }H^2\mathrm{{d}}\mu =0, \end{aligned}$$
(13)

where we used (11). In [19, Theorems 1.6, 1.7], Jachan showed the following:

Theorem 2.2

Let \(\Sigma \) be an immersed surface homologous to a coordinate sphere. Then a solution to the equation (8) exists on a maximal time interval [0, T). Moreover, if \({\mathcal {W}}(\Sigma )\le 8\pi -\rho \) for some constant \(\rho >0\), there exists \(R_0=R_0(m,\eta ,\rho ^{-1})\) such that if \(r_{\min }(t)\ge R_0\) for all \(t\in [0,T)\), then \(T=\infty \). If this is true, the flow subsequentially converges smoothly to a surface of Willmore type.

For the convenience of the reader, we have included an outline of the proof in the appendix. We also remark that a similar problem has been studied by Link in [26]. From (12) and (13), it follows that the Hawking mass is non-decreasing along the area preserving Willmore flow. It is consequently a natural flow to find local maximizers of the Hawking mass. In this paper, we will study the stability of the foliation constructed in [24] under small \(W^{2,2}\)-perturbations with respect to the area preserving Willmore flow. To this end, we say that a surface \(\Sigma \) is admissible if it is an embedded sphere satisfying the following conditions:

$$\begin{aligned} |\mathring{A}|_{L^2(\Sigma )}^2 \le \epsilon , \qquad \tau _e \le \delta , \qquad r_{\min } \ge R_0, \end{aligned}$$
(14)

for some constants \(\epsilon , \delta >0\). Moreover, we say that an area preserving Willmore flow is admissible up to time \(T>0\) if every \(\Sigma _t\) with \(0\le t\le T\) is admissible. Before we proceed, we need the following lemma to relate geometric quantities with regard to the different background metrics. The lemma is a straight-forward consequence of the asymptotic behaviour of the metric (2), c.f. [24, Sect. 1].

Lemma 2.3

Let \(\Sigma \) be an embedded sphere satisfying \(r_{\min }\ge R_0\) for some \(R_0(m,\eta )\) sufficiently large. There is a universal constant c such that

$$\begin{aligned} \nu _S=\phi ^{-2}\nu _e, \qquad&|\nu _S-\nu |\le c\eta r^{-2}, \\ \nabla _S=\nabla _e+\phi ^{-1}\nabla _e\phi , \qquad&|\nabla _S-\nabla |\le c\eta r^{-3}, \\ H_S=\phi ^{-2}H_e-2mr^{-2}\phi ^{-3}\nu _e\cdot \partial _r, \qquad&|H_S-H|\le c\eta r^{-3}+c\eta r^{-2}|A| , \\ \mathring{A}_S=\phi ^{2}\mathring{A}_e, \qquad&|\mathring{A}_S-\mathring{A}|\le c\eta r^{-3}+c\eta r^{-2}|A|, \\ |A_S-A_e|\le cmr^{-2}+cmr^{-1}A_S, \qquad&|A_S-A|\le c\eta r^{-3}+c\eta r^{-2}|A|, \\ \mathrm{{d}}\mu _S=\phi ^4 \mathrm{{d}}\mu _e, \qquad&|\mathrm{{d}}\mu _S -\mathrm{{d}}\mu |\le c\eta r^{-2}. \end{aligned}$$

as well as

$$\begin{aligned} |\nabla _SA_S-\nabla _e A_e|&\le cmr^{-3}+cmr^{-2}|A_S|+cmr^{-1}|\nabla _S A_S|, \\ |\nabla _SA_S-\nabla A|&\le c\eta r^{-4}+c\eta r^{-3}|A|+c\eta r^{-2}|\nabla A|, \\ |\nabla _SH_S-\nabla _e H_e|&\le cmr^{-3}+cmr^{-2}|A_S|+cmr^{-1}|\nabla _S A_S|, \\ |\nabla _SH_S-\nabla H|&\le c\eta r^{-4}+c\eta r^{-3}|A|+c\eta r^{-2}|\nabla A|. \end{aligned}$$

By conformal invariance, there holds

$$\begin{aligned} |\mathring{A}_S|_{L^2(\Sigma )}&=|\mathring{A}_e|_{L^2(\Sigma )}, \\ ||H_S|^2_{L^2(\Sigma )}-|H_e|^2_{L^2(\Sigma )}|&\le cmr_{\min }^{-1}|H_S|^2_{L^2(\Sigma )}. \end{aligned}$$

Finally, there holds

$$\begin{aligned} |\mathring{A}|^2_{L^2(\Sigma )}&\le (1+c\eta r^{-2}_{\min })|\mathring{A}_e|^2_{L^2(\Sigma )}+c\eta r_{\min }^{-3} |H|^2_{L^2(\Sigma )}, \\ ||H|^2_{L^2(\Sigma )}-|H_S|^2_{L^2(\Sigma )}|&\le c\eta r_{\min }^{-2}+ c\eta r_{\min }^{-2} |H|_{L^2(\Sigma )}^2. \end{aligned}$$

Admissible surfaces enjoy various properties: Müller and de Lellis showed (see [13, 14]) that in the Euclidean space, a surface \(\Sigma \) with small traceless part of the second fundamental form in the \(L^2\)-sense is \(W^{2,2}\)-close to a round sphere S and that there is a conformal parametrization mapping S onto \(\Sigma \). Using the conformal parametrization, geometric quantities on S and \(\Sigma \) can be related. The corresponding quantities on S will be indicated by a tilde. The exact statement of the result in [13, 14] is as follows.

Lemma 2.4

Let \(\Sigma \subset {\mathbb {R}}^3\) be a surface satisfying \(|\mathring{A}_e|^2_{L^2(\Sigma )}<8\pi \). Then \(\Sigma \) is a sphere and there exists a universal constant c independent of \(\Sigma \) and a conformal parametrization \(\psi :S:=S^2_{a_e}(R_e)\rightarrow {\mathbb {R}}^3\) such that

$$\begin{aligned} |\psi -\text {id}|_{L^2(S)}&\le c R_e^2 |\mathring{A}_e|_{L^2(\Sigma )}, \\ |\nabla _e \psi -{{\tilde{\nabla }}}_e\text {id}|_{L^2(S)}&\le c R_e |\mathring{A}_e|_{L^2(\Sigma )}, \\ |\nabla _e ^2 \psi -{{\tilde{\nabla }}}_e ^2 \text {id}|_{L^2(S)}&\le c |\mathring{A}_e|_{L^2(\Sigma )}, \\ |\nu _e\circ \psi - {{\tilde{\nu }}}_e\circ \text {id}|_{L^2(S)}&\le c R_e |\mathring{A}_e|_{L^2(\Sigma )}, \\ |\nabla _e \nu _e\circ \psi - {{\tilde{\nabla }}}_e {{\tilde{\nu }}}_e\circ \text {id}|_{L^2(S)}&\le c |\mathring{A}_e|_{L^2(\Sigma )}, \\ |E^2-1|_{L^\infty (S)}&\le c |\mathring{A}_e|_{L^2(\Sigma )}, \end{aligned}$$

where \(\text {id}:S\rightarrow {\mathbb {R}}^3\) is the identity, E the conformal factor of \(\psi \) and \(\nu _e,{{\tilde{\nu }}}_e\) the outward normals of \(\psi \) and \(\text {id}\), respectively. Moreover, the Sobolev embedding theorem implies

$$\begin{aligned} |\psi -\text {id}|_{L^\infty (S)}\le c R_e |\mathring{A}_e|_{L^2(\Sigma )} \\ \end{aligned}$$

and one easily obtains

$$\begin{aligned} |{{\tilde{A}}}_e-A_e|_{L^2(S)}\le c |\mathring{A}_e|_{L^2(\Sigma )}. \end{aligned}$$

Using the previous two lemmas, we can relate \(r_{\min },r_{\max },R_e\) provided \(\epsilon ,\delta \) are chosen sufficiently small in (14):

Lemma 2.5

Let \(|\mathring{A}_e|^2_{L^2}<8\pi \). There holds

$$\begin{aligned} (1-\tau _e-c|\mathring{A}_e|_{L^2})R_e\le r_{\min } \le r_{\max } \le (1+\tau _e+c|\mathring{A}_e|_{L^2})R_e. \end{aligned}$$

There are constants \(\upsilon ,\tau _0>0\) such that

$$\begin{aligned} c^{-1}R_e\le r_{\min } \le r_{\max } \le cR_e, \end{aligned}$$

provided \(|\mathring{A}_e|_{L^2}<\upsilon \) and \(\tau _e \le \tau _0\). In particular, \(\epsilon ,\delta \) in (14) can be chosen such that the previous estimate is true for any admissible surface.

We also need to compare the quantities \(|\Sigma |,R,\tau ,a\) with respect to the different background metrics.

Lemma 2.6

Let \(\Sigma \) be an admissible surface. Then we have the following estimates

$$\begin{aligned} |\Sigma |_g c^{-1}\le&|\Sigma |_e \le |\Sigma |_g c, \\ |R_g| c^{-1}\le&|R_e| \le |R_g| c, \\ |R_g-R_S|\le c\eta R_e^{-1},&\qquad |R_S-R_e|\le cm, \\ |a_g-a_e|\le c\eta R_e^{-1}+cm,&\qquad |\tau _g-\tau _e|\le c\eta R_e^{-2}+cmR_e^{-1}. \end{aligned}$$

Moreover, let \(\overline{R_S}:=\phi ^2(R_e)R_e\), then

$$\begin{aligned} |\overline{R_S}-R_S|\le cm(\tau _e+|\mathring{A}_e|_{L^2(\Sigma )}). \end{aligned}$$

Another useful tool is an integrated version of the Gauss equation. More precisely, any sphere \(\Sigma \) satisfies

$$\begin{aligned} 16\pi -\int \nolimits _\Sigma H^2=-2\int \nolimits _\Sigma |\mathring{A}|^2\mathrm{{d}}\mu -4\int \nolimits _\Sigma \text {G}(\nu ,\nu )\mathrm{{d}}\mu , \end{aligned}$$
(15)

where \(G=\text {Rc}-\frac{1}{2} g \text {Sc}\) is the Einstein tensor. It follows from (15), Lemma 2.5 and (5) that any admissible surface enjoys uniform bounds on \(|H|_{L^2(\Sigma )}\) and \(|A|_{L^2(\Sigma )}\), that is,

$$\begin{aligned} |H|_{L^2(\Sigma )}, |A|_{L^2(\Sigma )}\le c(\epsilon ,\delta ). \end{aligned}$$
(16)

Finally, we need the Michael–Simon–Sobolev inequality which can, for instance, be found as Proposition 5.4 in [18].

Lemma 2.7

If \(r_{\min }\ge R_0\) for some \(R_0(\eta ,m)\) sufficiently large, then any smooth function \(\omega \) satisfies

$$\begin{aligned} \int \nolimits _\Sigma |u|^2\mathrm{{d}}\mu \le c \bigg (\int \nolimits _\Sigma (|\nabla u|+|u H|)\mathrm{{d}}\mu \bigg )^2. \end{aligned}$$

We finish this section by showing that an area preserving Willmore flow can only cease to be admissible if its barycentre moves outwards. We also prove an estimate for the excess Willmore energy.

Lemma 2.8

Let \(\Sigma _t, t\in [0,T),\) be an admissible Willmore flow starting at a surface \(\Sigma _0=\Sigma \) which satisfies \(|\mathring{A}|_{L^2(\Sigma )}<\epsilon /2, \tau _e<\delta /2\) and \(r_{\min }\ge \Xi R_0\) for some constant \(\Xi >1\). If \(R_0\) and \(\Xi \) are sufficiently large, then the following estimates hold:

$$\begin{aligned} |\mathring{A}|^2_{L^2(\Sigma _t)}(t)\le \frac{3}{4} \epsilon , \qquad {\mathcal {W}}(t)-{\mathcal {W}}(0)\le \epsilon , \qquad r_{\min }\ge 2R_0. \end{aligned}$$

Proof

Lemma 2.5 and Lemma 2.6 imply the crude estimate \(\int \nolimits _{\Sigma } G(\nu ,\nu )\mathrm{{d}}\mu \le cR_e^{-1}\). Using (15) and the fact that the flow decreases the Willmore energy we obtain

$$\begin{aligned} 0&\ge |\mathring{A}|_{L^2(\Sigma _t)}^2-|\mathring{A}|_{L^2(\Sigma _0)}^2-cR_e^{-1}. \end{aligned}$$

Hence, the claim follows if \(R_0\) is sufficiently large such that \(cR_e^{-1}\le cR_0^{-1}<\epsilon /4\). For the second claim, we trivially estimate

$$\begin{aligned} {\mathcal {W}}(0)-{\mathcal {W}}(t)\le&|\mathring{A}|_{L^2(\Sigma _0)}^2+cR_e^{-1}<\epsilon . \end{aligned}$$

For the last claim, we note that according to Lemmas 2.5 and  2.6, there holds \(r_{\min }(t)\ge c^{-1} R_g(t)\) as well as \(R_g(0)\ge c^{-1} r_{\min }(0)\ge c^{-1}\Xi R_0\). Since \(R_g(t)=R_g(0)\), the claim now follows if \(\Xi \) is chosen sufficiently large. \(\square \)

Remark

According to a classical inequality by Li and Yau ([25]), embeddedness is automatically implied by the smallness of \(|\mathring{A}|^2_{L^2(\Sigma _t)}\).

It follows from the previous lemma that a flow can only cease to be admissible if \(\tau _e\) reaches \(\delta \). In order to study the evolution of \(\tau _e\), we need to establish precise curvature estimates.

Integral Curvature Estimates

In this section, we prove general curvature estimates for embedded spheres with small traceless part of the second fundamental form in the \(L^2\)-sense. In the next section, we will use these to obtain precise estimates for the evolution of certain geometric quantities. However, this section might be of independent interest, too. Unless otherwise stated, we only assume that \(\Sigma \) is spherical, that the conclusion of Lemma 2.5 holds, that \(r_{\min }\ge R_0\) and that the estimate (16) is valid. We will then state all estimates in terms of \(R_e\). As before, we denote the connection of \(\Sigma \) by \(\nabla \) and the connection of the ambient space by \({\overline{\nabla }}\). We also use the common \(*\)-notation to summarize geometric terms. In order to obtain estimates for the different components of the second fundamental form, we follow the approach of [21, Sect. 2]. However, we need to take the effect of the ambient curvature into account. We need the following lemma which follows from a straight-forward computation:

Lemma 3.1

Let \(\omega \in \Omega ^k(\Sigma )\) for some \(k\in {\mathbb {N}}\), \(p\in \Sigma \) and \(e_1,e_2\) be and orthonormal frame of \(\Sigma \) at p. Let \(X_i\in \{e_1,e_2\}\), \(1\le i\le k\). There holds

$$\begin{aligned}&(\nabla \nabla ^*\omega -\nabla ^*\nabla \omega )(X_1,...,X_k)\\&\quad =\omega (\text {Rm}^\Sigma (X_1,e_i,e_i),X_2,...,X_k) +\sum _{j=2}^{k}\omega (e_i,\dots ,\text {Rm}^\Sigma (X_1,e_i,X_j),...,X_k)\\&\quad \quad -\nabla ^*Q_\omega (X_1,...,X_k), \end{aligned}$$

where \(\nabla ^*=-\text {div}\) is the adjoint of \(\nabla \) and

$$\begin{aligned} Q_\omega (X_0,..X_k)=\nabla _{X_0}\omega (X_1,..,X_k)-\nabla _{X_1}\omega (X_0,..,X_k). \end{aligned}$$

We would now like to use the previous lemma to express certain geometric quantities in terms of the Willmore operator W defined by \(W=\Delta H+H\text {Rc}(\nu ,\nu )+H|\mathring{A}|^2\). To this end, we first need the following lemma.

Lemma 3.2

The following identities hold:

$$\begin{aligned} \Delta \mathring{A}=&(\nabla ^2 H-\frac{1}{2}\gamma \Delta H)+\frac{1}{2} H^2\mathring{A}+\mathring{A}*\mathring{A}*\mathring{A}+\text {Rm}*A+{\overline{\nabla }}{\text {Rm}}*1, \end{aligned}$$
(17)
$$\begin{aligned} \nabla ^*\nabla ^2 H =&-\nabla (\Delta H)+{\text {Rm}}*\nabla H+\mathring{A}*\mathring{A}*\nabla \mathring{A}+{\text {Rm}}*\mathring{A}*\mathring{A}-\frac{1}{4} H^2\nabla H, \end{aligned}$$
(18)
$$\begin{aligned} \nabla ^*\nabla ^2 H=&-\nabla (\Delta H)+{\text {Rm}}*\nabla H+\mathring{A}*\mathring{A}*\nabla H-\frac{1}{4}H^2\nabla H, \end{aligned}$$
(19)
$$\begin{aligned} \nabla ^*(\nabla ^2\mathring{A})=&\nabla (\nabla ^*\nabla \mathring{A})+\nabla \mathring{A}*A*A+{\overline{\nabla }}{\text {Rm}}* \mathring{A}+{\text {Rm}}*\mathring{A}*A+\text {Rm}*\nabla \mathring{A}. \end{aligned}$$
(20)

Proof

With the convention

$$\begin{aligned} A_{ij}=A(X_i,X_j)=g({\overline{\nabla }}_{X_i}X_j,\nu ) \end{aligned}$$

the orthonormal frame satisfies

$$\begin{aligned} {\overline{\nabla }}_{X_i}X_j=\nabla _{X_i}X_j+A_{ij}\nu =A_{ij}\nu . \end{aligned}$$
(21)

Chosing \(\omega =A\) in Lemma 3.1 we obtain

$$\begin{aligned} Q_A(X_1,X_2,X_3)=X_1(g({\bar{\nabla }}_{X_2}X_3,\nu ))-X_2(g({\bar{\nabla }}_{X_1}X_3,\nu ))={\text {Rm}}(X_1,X_2,X_3,\nu ) \end{aligned}$$

as \({\overline{\nabla }}\nu \) is tangential. We clearly have \(\nabla ^*Q(X_1,X_2)=X_i({\text {Rm}}(X_i,X_1,X_2,\nu ))\) which implies

$$\begin{aligned} \nabla ^*Q_A={\overline{\nabla }}{\text {Rm}}*1+{\text {Rm}}*A. \end{aligned}$$

A straight forward calculation gives

$$\begin{aligned} \nabla ^*A=-\nabla H +{\text {Rm}}*1. \end{aligned}$$

Evaluated at \((X_1,X_2)\), this yields (with slight abuse of notation)

$$\begin{aligned} \Delta A=&\nabla ^2 H+A(\text {Rm}^\Sigma (X_1,e_i,e_i),X_2)+A(e_i,\text {Rm}^\Sigma (X_1,e_i,X_2))\nonumber \\&+{\overline{\nabla }}{\text {Rm}}*1+{\text {Rm}}*A. \end{aligned}$$
(22)

If we furthermore assume that the \(e_i\) are principal directions, we obtain

$$\begin{aligned}&A(\text {Rm}^\Sigma (X_1,e_i,e_i),X_2)+A(e_i,\text {Rm}^\Sigma (X_1,e_i,X_2)) \\&\quad = \text {Rm}^\Sigma (X_1,e_i,e_i,X_2)A(X_2,X_2)+\text {Rm}^\Sigma (X_1,e_i,X_2,e_i) A(e_i,e_i). \end{aligned}$$

If \(X_1, X_2\) are distinct, both terms vanish. Otherwise, we can assume that \(X_1=X_2=e_1\). Then, using the Gauss equation

$$\begin{aligned} \text {Rm}^\Sigma (a,b,c,d)={\text {Rm}}(a,b,c,d)+A(a,d)A(b,c)-A(a,c)A(b,d) \end{aligned}$$

the right-hand side becomes (again with abuse of notation)

$$\begin{aligned}&\text {Rm}^\Sigma (e_1,e_2,e_2,e_1)(A(e_1,e_1)-A(e_2,e_2)) \\&\quad ={\text {Rm}}*A+A_{11}A_{22}(A_{11}-A_{22}). \\&\quad ={\text {Rm}}*A+\left( \mathring{A}_{11}+\frac{1}{2}H\right) \left( \mathring{A}_{22}+\frac{1}{2}H\right) (\mathring{A}_{11}-\mathring{A}_{22}) \\&\quad ={\text {Rm}}*A-2\left( \mathring{A}_{11}+\frac{1}{2} H\right) \left( \mathring{A}_{11}-\frac{1}{2} H\right) \mathring{A}_{11}, \end{aligned}$$

where we used that \(\mathring{A}\) is trace free. Together with (22) this clearly implies

$$\begin{aligned} \Delta A=\nabla ^2 H+\frac{1}{2} H^2\mathring{A}+\mathring{A}*\mathring{A}*\mathring{A}+{\text {Rm}}*A+{\overline{\nabla }}{\text {Rm}}*1. \end{aligned}$$

As \(\nabla \gamma =0\), there holds \(\Delta (\gamma H)=\gamma \Delta H\) and we obtain the first claim. Choosing \(\omega =\nabla H\), we find \(Q_{\nabla H}=0\) by the symmetry of second derivatives and evaluating at \(X_1=e_1\), we obtain (again, with abuse of notation)

$$\begin{aligned} \nabla ^*\nabla ^2 H=&\nabla \nabla ^* \nabla H-\text {Rm}^\Sigma (X_1,e_i,e_i,e_j)\nabla H(e_j) \\ =&\nabla \nabla ^* \nabla H- \text {Rm}^\Sigma (e_1,e_2,e_2,e_1)e_1(H) \\ =&-\nabla (\Delta H)+{\text {Rm}}*\nabla H+(\mathring{A}_{11}^2-\frac{1}{4} H^2)e_1(H) \\ =&-\nabla (\Delta H)+{\text {Rm}}*\nabla H+\mathring{A}*\mathring{A}*\nabla ^*\mathring{A}+{\text {Rm}}*\mathring{A}*\mathring{A}-\frac{1}{4} H^2\nabla H, \end{aligned}$$

as \(\nabla ^* \mathring{A}=-\frac{1}{2} \nabla H +{\text {Rm}}*1\). This implies the second and third claim. Finally, if \(\omega =\nabla \mathring{A}\), then at \((X_1,X_2,X_3,X_4)\), we have

$$\begin{aligned} Q_{\nabla \mathring{A}}&=\mathring{A}(\text {Rm}^\Sigma (X_1,X_2,X_3),X_4)+\mathring{A}(X_3,\text {Rm}^\Sigma (X_1,X_2,X_4))\\&={\text {Rm}}*\mathring{A}+\mathring{A}*A*A, \end{aligned}$$

hence

$$\begin{aligned} \nabla ^*Q_{\nabla \mathring{A}}={\overline{\nabla }}{\text {Rm}}* \mathring{A}+{\text {Rm}}*\mathring{A}*A+\text {Rm}*\nabla \mathring{A}+\nabla \mathring{A}* A * A. \end{aligned}$$

This gives the very rough identity

$$\begin{aligned} \nabla ^*(\nabla ^2\mathring{A})=\nabla (\nabla ^*\nabla \mathring{A})+\nabla \mathring{A}*A*A+{\overline{\nabla }}{\text {Rm}}* \mathring{A}+{\text {Rm}}*\mathring{A}*A+\text {Rm}*\nabla \mathring{A}. \end{aligned}$$

\(\square \)

Remark

In fact, one can actually show the more precise identity

$$\begin{aligned} \mathring{A}\cdot \nabla ^2 \mathring{A}= \mathring{A}\cdot \nabla ^2 H+\frac{1}{2} H^2 |\mathring{A}|^2-|\mathring{A}|^4+\mathring{A}*\mathring{A}*\text {Rm}+2\mathring{A}\cdot \nabla \text {Rc}(\nu ,\cdot )^T, \end{aligned}$$
(23)

see (1.7) in [24]. This can be seen by using the identity \(\nabla ^* \mathring{A}=-\frac{1}{2} \nabla H +\text {Rc}^T(\cdot ,\nu )\).

For the following proof, we remark that \(|\text {Rc}|\le c\eta R_e^{-4}+cmR_e^{-3}\le cR_e^{-3}\) and \(|{\overline{\nabla }}\text {Rc}|\le c\eta R_e^{-5}+cmR_e^{-4}\le cR_e^{-4}\). Moreover, since \(|H_e|^2_{L^2(\Sigma )}\ge 16\pi \), there evidently holds \(|A|^2_{L^2(\Sigma )}\ge 8\pi \), provided \(r_{\min }\ge R_0\) (c.f. Lemma 2.3).

Lemma 3.3

Let \(\Sigma \) be a spherical surface satisfying \(r_{\min }\ge R_0\), the conclusion of Lemma 2.5 and (16). Then, there holds

$$\begin{aligned} \int \nolimits _{\Sigma } |\nabla A|^2 \mathrm{{d}}\mu \le -c\int \nolimits _{\Sigma } HW\mathrm{{d}}\mu +\frac{c}{R_e^3}\int \nolimits _{\Sigma } |A|^2\mathrm{{d}}\mu +c\int \nolimits _\Sigma |\mathring{A}|^4 \mathrm{{d}}\mu , \end{aligned}$$

provided \(R_0\) is chosen to be sufficiently large. In this estimate, we may replace \(\nabla A\) by \(\nabla \mathring{A}\) or \(\nabla H\).

Proof

Multiplying (17) by \(\mathring{A}\), integrating and using that \(\mathring{A}\) is trace free we obtain

$$\begin{aligned} \int \nolimits _{\Sigma } |\nabla \mathring{A}|^2 \mathrm{{d}}\mu +\frac{1}{2}\int \nolimits _{\Sigma } H^2 |\mathring{A}|^2\mathrm{{d}}\mu&\le - \int \nolimits _{\Sigma } \nabla H\cdot \nabla ^* \mathring{A}\mathrm{{d}}\mu \\&\quad +c\int \nolimits _{\Sigma } |\mathring{A}|^4 \mathrm{{d}}\mu +\frac{c}{R_e^3} \int \nolimits _{\Sigma } |A||\mathring{A}| \mathrm{{d}}\mu +\frac{c}{R_e^4}\int \nolimits _{\Sigma } |\mathring{A}|\mathrm{{d}}\mu \nonumber \\&\le \frac{1}{2} \int \nolimits _{\Sigma } |\nabla H|^2\mathrm{{d}}\mu +\frac{c}{R_e^3}\int \nolimits _{\Sigma } |\nabla H|\mathrm{{d}}\mu \\&\quad +c\int \nolimits _{\Sigma } |\mathring{A}|^4\mathrm{{d}}\mu +\frac{c}{R_e^3} \int \nolimits _{\Sigma } |A|^2\mathrm{{d}}\mu , \end{aligned}$$

where we used \(\nabla H=-\nabla ^* \mathring{A}+\text {Rm}*1\), \(|\mathring{A}||A|\le R_e^{-1}|A|^2+R_e|\mathring{A}|^2\) and estimated

$$\begin{aligned} \int \nolimits _\Sigma |\mathring{A}|\mathrm{{d}}\mu&\le c\int \nolimits _{\Sigma } R_e|\mathring{A}|^2\mathrm{{d}}\mu +cR_e^{-1}\int \nolimits _\Sigma 1\mathrm{{d}}\mu \le c R_e^3 \int \nolimits _\Sigma |\mathring{A}|^4+cR_e \\&\le c R_e^3 \int \nolimits _\Sigma |\mathring{A}|^4\mathrm{{d}}\mu + c R_e \int \nolimits _{\Sigma } |A|^2\mathrm{{d}}\mu . \end{aligned}$$

Integrating by parts and using the definition of W, we obtain

$$\begin{aligned} \frac{1}{2}\int \nolimits _{\Sigma } |\nabla H|^2\mathrm{{d}}\mu =&\frac{1}{2} \int \nolimits _{\Sigma } ( -HW+H^2{\text {Rc}}(\nu ,\nu )+H^2|\mathring{A}|^2)\mathrm{{d}}\mu \\ \le&\frac{1}{2} \int \nolimits _{\Sigma } ( -HW+H^2|\mathring{A}|^2)\mathrm{{d}}\mu +cR_e^{-3}\int \nolimits _{\Sigma } |A|^2 \mathrm{{d}}\mu . \end{aligned}$$

Next, \(|\nabla H|\le c|\nabla \mathring{A}|+cR_e^{-3}\) and hence

$$\begin{aligned} \int \nolimits _{\Sigma } |\nabla H|R_e^{-3}\mathrm{{d}}\mu \le c R_e^{-1} \int \nolimits _{\Sigma } |\nabla \mathring{A}|^2\mathrm{{d}}\mu +cR_e^{-3}\int \nolimits _{\Sigma } |A|^2\mathrm{{d}}\mu . \end{aligned}$$

The claimed inequality now follows from combining these estimates, absorbing the \(|\nabla \mathring{A}|^2\) term and using \(|\nabla A|^2\le c |\nabla \mathring{A}|^2+cR_e^{-6}\). Finally, we clearly have \(|\nabla H|\le c |\nabla A|\). \(\square \)

Lemma 3.4

Under the assumptions of the previous lemma, there holds

$$\begin{aligned}&\int \nolimits _{\Sigma } |\nabla ^2H|^2\mathrm{{d}}\mu +\int \nolimits _{\Sigma } |A|^2|\nabla A|^2\mathrm{{d}}\mu +\int \nolimits _{\Sigma } |A|^4|\mathring{A}|^2\mathrm{{d}}\mu \\&\quad \le c\int \nolimits _{\Sigma } |W|^2 \mathrm{{d}}\mu -cR_e^{-3} \int \nolimits _{\Sigma } WH\mathrm{{d}}\mu +c\int \nolimits _{\Sigma } (|\mathring{A}|^2|\nabla \mathring{A}|^2+|\mathring{A}|^6) \mathrm{{d}}\mu +cR_e^{-6} \int \nolimits _{\Sigma } |A|^2 \mathrm{{d}}\mu . \end{aligned}$$

Proof

Multiplying (18) with \(\nabla H\) and integrating, we obtain

$$\begin{aligned} \int \nolimits _{\Sigma } |\nabla ^2H|^2\mathrm{{d}}\mu +\frac{1}{4}\int \nolimits _{\Sigma } H^2|\nabla H|^2 \mathrm{{d}}\mu \le&\int \nolimits _{\Sigma } |\Delta H|^2\mathrm{{d}}\mu + cR_e^{-3}\int \nolimits _{\Sigma } |\nabla H|^2\mathrm{{d}}\mu \\&+c\int \nolimits _{\Sigma } |\mathring{A}|^2|\nabla \mathring{A}||\nabla H|\mathrm{{d}}\mu \\&+cR_e^{-3}\int \nolimits _{\Sigma } |\mathring{A}|^2|\nabla H|\mathrm{{d}}\mu . \end{aligned}$$

Given a positive number \(\kappa ,\) the last term can be estimated by

$$\begin{aligned} c(\kappa )R_e^{-6} \int \nolimits _{\Sigma } |A|^2 \mathrm{{d}}\mu +\kappa /2 \int \nolimits _{\Sigma } |\nabla A|^2|A|^2|\mathrm{{d}}\mu . \end{aligned}$$

Next, we have

$$\begin{aligned} \int \nolimits _{\Sigma } |\Delta H|^2 \mathrm{{d}}\mu \le c \int \nolimits _{\Sigma } |W|^2 \mathrm{{d}}\mu +cR_e^{-6} \int \nolimits _{\Sigma } |A|^2 \mathrm{{d}}\mu + c \int \nolimits _{\Sigma } |\mathring{A}|^4|A|^2 \mathrm{{d}}\mu \end{aligned}$$

and \(\int \nolimits _{\Sigma } |\mathring{A}|^4|A|^2\mathrm{{d}}\mu \le c(\kappa ) \int \nolimits _{\Sigma } |\mathring{A}|^6\mathrm{{d}}\mu +\kappa \int \nolimits _{\Sigma } |A|^4|\mathring{A}|^2\mathrm{{d}}\mu \). There also holds

$$\begin{aligned} \int \nolimits _{\Sigma } |\mathring{A}|^2|\nabla \mathring{A}||\nabla H|\mathrm{{d}}\mu \le \kappa /2 \int \nolimits _{\Sigma } |A|^2|\nabla A|^2 \mathrm{{d}}\mu +c(\kappa )\int \nolimits _{\Sigma } |\nabla \mathring{A}|^2|\mathring{A}|^2\mathrm{{d}}\mu \end{aligned}$$

and the term \(R_e^{-3}\int \nolimits _\Sigma |\nabla H|^2 \mathrm{{d}}\mu \) can be estimated with the previous lemma and the trivial estimate \(\int \nolimits _{\Sigma } |\mathring{A}|^4\mathrm{{d}}\mu \le 2 R_e^3 \int \nolimits _{\Sigma } |\mathring{A}|^6\mathrm{{d}}\mu +2R_e^{-3}\int \nolimits _{\Sigma } |A|^2\mathrm{{d}}\mu \). Hence, so far we have shown that

$$\begin{aligned} \int \nolimits _{\Sigma } |\nabla ^2H|^2\mathrm{{d}}\mu +\frac{1}{4}\int \nolimits _{\Sigma } H^2|\nabla H|^2 \mathrm{{d}}\mu&\le c \int \nolimits _{\Sigma } |W|^2 \mathrm{{d}}\mu +c(\kappa )R_e^{-6} \int \nolimits _{\Sigma } |A|^2 \mathrm{{d}}\mu \\&\quad + \kappa \int \nolimits _{\Sigma } |\mathring{A}|^2|A|^4 \mathrm{{d}}\mu \\&\quad +\kappa \int \nolimits _{\Sigma } |A|^2|\nabla A|^2 \mathrm{{d}}\mu +c(\kappa )\int \nolimits _{\Sigma } |\nabla \mathring{A}|^2|\mathring{A}|^2\mathrm{{d}}\mu \\&\quad -cR_e^{-3}\int \nolimits _{\Sigma } WH\mathrm{{d}}\mu +c(\kappa )\int \nolimits _\Sigma |\mathring{A}|^6 \mathrm{{d}}\mu . \end{aligned}$$

From the estimate \(|\nabla A|^2\le c |\nabla \mathring{A}|^2+cR_e^{-6}\) and (17) it follows that

$$\begin{aligned}&\frac{1}{c} \int \nolimits _{\Sigma } H^2 |\nabla A|^2\mathrm{{d}}\mu -cR_e^{-6}\int \nolimits _{\Sigma }|A|^2\mathrm{{d}}\mu \\&\quad \le \int H^2|\nabla \mathring{A}|^2\mathrm{{d}}\mu \\&\quad =-\int \nolimits _{\Sigma } H^2 \mathring{A}\cdot \nabla ^2H\mathrm{{d}}\mu -\frac{1}{2} \int H^4|\mathring{A}|^2 \mathrm{{d}}\mu \\&\quad \quad +\int \nolimits _{\Sigma } (H^2*\mathring{A}*\mathring{A}*\mathring{A}*\mathring{A}+H^2*\mathring{A}*({\text {Rm}}*A+{\overline{\nabla }}{\text {Rm}})\mathrm{{d}}\mu \\&\quad \quad +\int \nolimits _{\Sigma } H*\nabla H*\mathring{A}*\nabla \mathring{A})\mathrm{{d}}\mu . \end{aligned}$$

The second and third row can be estimated by

$$\begin{aligned}&\kappa \int \nolimits _{\Sigma } |A|^4 |\mathring{A}|^2\mathrm{{d}}\mu +c(\kappa ) \int \nolimits _{\Sigma } |\mathring{A}|^6 \mathrm{{d}}\mu +c(\kappa )R_e^{-6}\int \nolimits _{\Sigma } |A|^2\mathrm{{d}}\mu +\kappa \int \nolimits _{\Sigma } |A|^2|\nabla A|^2\mathrm{{d}}\mu \\&\quad +\int \nolimits _{\Sigma } c(\kappa )|\mathring{A}|^2|\nabla \mathring{A}|^2\mathrm{{d}}\mu . \end{aligned}$$

Using partial integration and the identity \(-\nabla ^*\mathring{A}=\frac{1}{2} \nabla H+\text {Rm}*1\), the first term in the first row can be computed to be

$$\begin{aligned} -\int \nolimits _{\Sigma } H^2\mathring{A}\nabla ^2H\mathrm{{d}}\mu= & {} \frac{1}{2} \int \nolimits _{\Sigma } H^2|\nabla H|^2 \mathrm{{d}}\mu + \int \nolimits _{\Sigma } H^2*\nabla H *{\text {Rm}}\mathrm{{d}}\mu \\&+ \int \nolimits _{\Sigma } \nabla H*\nabla H * \mathring{A}* H\mathrm{{d}}\mu . \end{aligned}$$

In the last equation, the third term can be estimated as before and the second term can be estimated by

$$\begin{aligned} \frac{c(\kappa )}{R_e^6} \int \nolimits _{\Sigma } |A|^2\mathrm{{d}}\mu +\kappa \int \nolimits _{\Sigma }|A|^2|\nabla A|^2\mathrm{{d}}\mu . \end{aligned}$$

Finally, we note that

$$\begin{aligned} \int \nolimits _{\Sigma } A^2|\nabla A|^2\mathrm{{d}}\mu \le c\int \nolimits _{\Sigma } H^2|\nabla A|^2\mathrm{{d}}\mu +c\int \nolimits _{\Sigma } |\mathring{A}|^2|\nabla \mathring{A}|^2\mathrm{{d}}\mu +\frac{c}{R_e^6}\int \nolimits _{\Sigma } |A|^2\mathrm{{d}}\mu . \end{aligned}$$

Combining all these inequalities, choosing \(\kappa >0\) sufficiently small, absorbing all terms when possible and noting that

$$\begin{aligned} \int \nolimits _{\Sigma } |A|^4|\mathring{A}|^2\mathrm{{d}}\mu \le \int \nolimits _{\Sigma } H^4|\mathring{A}|^2\mathrm{{d}}\mu +\int \nolimits _{\Sigma }|\mathring{A}|^6\mathrm{{d}}\mu \end{aligned}$$

we obtain the desired statement. \(\square \)

Lemma 3.5

Under the assumptions of Lemma 3.3 there holds

$$\begin{aligned}&\int \nolimits _{\Sigma } |\nabla ^2 \mathring{A}|^2\mathrm{{d}}\mu +\int \nolimits _{\Sigma } |A|^4|\mathring{A}|^2\mathrm{{d}}\mu +\int \nolimits _{\Sigma } |\nabla A|^2|A|^2\mathrm{{d}}\mu \\&\quad \le c \int \nolimits _{\Sigma } |W|^2\mathrm{{d}}\mu +c\int \nolimits _{\Sigma } (|\nabla \mathring{A}|^2|\mathring{A}|^2+|\mathring{A}|^6)\mathrm{{d}}\mu +cR_e^{-6}\int \nolimits _{\Sigma } |A|^2\mathrm{{d}}\mu , \end{aligned}$$

provided \(R_0\) is sufficiently large.

Proof

Multiplying (20) by \(\nabla \mathring{A}\) and applying (17) we obtain

$$\begin{aligned} \int \nolimits _{\Sigma } |\nabla ^2 \mathring{A}|^2\mathrm{{d}}\mu \le&\int \nolimits _{\Sigma } |\Delta \mathring{A}|^2+c\int \nolimits _{\Sigma } |\nabla \mathring{A}|^2|A|^2\mathrm{{d}}\mu +cR_e^{-4}\int \nolimits _{\Sigma } |\mathring{A}||\nabla \mathring{A}|\mathrm{{d}}\mu \\ {}&+cR_e^{-3}\int \nolimits _{\Sigma } |\mathring{A}||A||\nabla \mathring{A}|\mathrm{{d}}\mu +cR_e^{-3}\int \nolimits _{\Sigma } |\nabla \mathring{A}|^2\mathrm{{d}}\mu \\ \le&c\int \nolimits _{\Sigma } |\nabla ^2 H|^2\mathrm{{d}}\mu +c\int \nolimits _{\Sigma } |A|^4|\mathring{A}|^2 \mathrm{{d}}\mu +c\int |\mathring{A}|^6\mathrm{{d}}\mu +c\int \nolimits _{\Sigma } |\nabla A|^2|A|^2\mathrm{{d}}\mu \\&+\frac{c}{R_e^6}\int \nolimits _{\Sigma } |A|^2\mathrm{{d}}\mu +\frac{c}{R_e^3}\int \nolimits _{\Sigma } |\nabla A|^2\mathrm{{d}}\mu , \end{aligned}$$

where we used \(|\mathring{A}||\nabla \mathring{A}|\le c|A||\nabla A|\). The claim now follows from the two previous lemmas and from estimating

$$\begin{aligned} R_e^{-3}\int \nolimits _{\Sigma } |HW|\mathrm{{d}}\mu \le 2R_e^{-6}\int \nolimits _{\Sigma } |A|^2\mathrm{{d}}\mu +2\int \nolimits _{\Sigma } |W|^2\mathrm{{d}}\mu . \end{aligned}$$

\(\square \)

We now need the following Sobolev-type interpolation inequality.

Lemma 3.6

Let \(\Sigma \) be a compact surface satisfying \(r_{\min }\ge R_0\). If \(R_0\) is chosen sufficiently large, then any smooth k-form \(\omega \) satisfies

$$\begin{aligned} |\omega |^4_{L^\infty (\Sigma )}\le c|\omega |^2_{L^2(\Sigma )}\bigg (\int \nolimits _{\Sigma } (|\nabla ^2\omega |^2+|\omega |^2H^4)\mathrm{{d}}\mu \bigg ). \end{aligned}$$

Proof

The proof of Theorem 5.6. in [22] caries over to our setting as \(\Sigma \) is compact and since the Michael–Simon–Sobolev inequality holds in an asymptotic region of an asymptotically Schwarzschild manifold, see Lemma 2.7. One then easily adapts the proof of Lemma 2.8 in [21]. \(\square \)

We also need the following multiplicative Sobolev inequality.

Lemma 3.7

Under the assumptions of the previous lemma we have

$$\begin{aligned} \int \nolimits _{\Sigma } (|\mathring{A}|^2|\nabla \mathring{A}|^2+|\mathring{A}|^6)\mathrm{{d}}\mu \le c \int \nolimits _{\Sigma } |\mathring{A}|^2\mathrm{{d}}\mu \int \nolimits _{\Sigma } (|\nabla ^2 \mathring{A}|^2 +|A|^2|\nabla A|^2+|A|^4|\mathring{A}|^2)\mathrm{{d}}\mu . \end{aligned}$$

Proof

This is a straight forward adaption of Lemma 2.5 in [21]. Again, the proof only relies on the Michael–Simon–Sobolev inequality, Young’s inequality and Hölder’s inequality. \(\square \)

At this point, a small curvature assumption allows us to absorb the term on the left-hand side of the previous equation in Lemma 3.5. This finally yields an \(L^\infty \)-estimate for \(\mathring{A}\).

Lemma 3.8

Under the assumptions of Lemma 3.3 there exists a constant \(\epsilon (m,\eta , R_0)>0\) such that if

$$\begin{aligned} \int \nolimits _{\Sigma } |{\mathring{A}}|^2\mathrm{{d}}\mu \le \epsilon , \end{aligned}$$

then

$$\begin{aligned} \int \nolimits _{\Sigma } |\nabla ^2\mathring{A}|^2\mathrm{{d}}\mu&+\int \nolimits _{\Sigma }|A|^2|\nabla A|^2\mathrm{{d}}\mu +\int \nolimits _{\Sigma } |A|^4|\mathring{A}|^2\mathrm{{d}}\mu \le c\int \nolimits _{\Sigma } |W|^2\mathrm{{d}}\mu \nonumber \\ {}&+cR_{e}^{-6}\int \nolimits _{\Sigma } |A|^2\mathrm{{d}}\mu . \end{aligned}$$
(24)

In particular, we have

$$\begin{aligned} |\mathring{A}|_{L^\infty (\Sigma )}^4\le c|\mathring{A}|^2_{L^2(\Sigma )}\bigg ( \int \nolimits _{\Sigma } |W|^2\mathrm{{d}}\mu +R_e^{-6}\int \nolimits _\Sigma |A|^2 \mathrm{{d}}\mu \bigg ). \end{aligned}$$

Proof

This is a direct consequence of the previous three lemmas. \(\square \)

It turns out that the integral curvature estimates also imply an improved estimate for \(|\mathring{A}|_{L^2(\Sigma )}\).

Lemma 3.9

Under the assumptions of Lemma 3.3 there exists a constant \(\epsilon (m,\eta ,R_0)>0\) such that if

$$\begin{aligned} \int \nolimits _{\Sigma } |{\mathring{A}}|^2\mathrm{{d}}\mu \le \epsilon , \end{aligned}$$

then

$$\begin{aligned} \int \nolimits _\Sigma |\mathring{A}|^2 \mathrm{{d}}\mu \le cR_e^2 \int \nolimits _\Sigma (|\nabla H|^2+|\text {Rc}(\nu ,\cdot )^T|^2) \mathrm{{d}}\mu . \end{aligned}$$

Proof

Integrating (23) and using integration by parts as well as the identity \(\nabla ^* \mathring{A}=-\frac{1}{2} \nabla H +\text {Rc}(\nu ,\cdot )^T\) yields

$$\begin{aligned} \int \nolimits _\Sigma (|\nabla \mathring{A}|^2+H^2|\mathring{A}|^2) \mathrm{{d}}\mu \le c \int \nolimits _\Sigma (|\nabla H|^2 + |\mathring{A}|^4+|\text {Rc}(\nu ,\cdot )^T|^2+R_e^{-3}|\mathring{A}|^2)\mathrm{{d}}\mu . \end{aligned}$$

Next, using Lemma 2.7 and Hölder’s inequality, we obtain

$$\begin{aligned} \int \nolimits _\Sigma |\mathring{A}|^4 \mathrm{{d}}\mu \le |\mathring{A}|_{L^2(\Sigma )}^2\bigg ( \int \nolimits _\Sigma (|\nabla \mathring{A}|^2+H^2|\mathring{A}|^2)\mathrm{{d}}\mu \bigg ). \end{aligned}$$

Hence, this term can be absorbed. Again with the Michael–Simon–Sobolev inequality and Hölder’s inequality, we get

$$\begin{aligned} \int \nolimits _\Sigma |\mathring{A}|^2 \mathrm{{d}}\mu \le c |\Sigma |\int \nolimits _\Sigma (|\nabla \mathring{A}|^2+H^2|\mathring{A}|^2) \mathrm{{d}}\mu . \end{aligned}$$

From this the claim follows. \(\square \)

Finally, we prove two useful \(W^{2,2}\)- and \(L^\infty \)-estimates for the mean curvature.

Lemma 3.10

Under the assumptions of Lemma 3.3 there holds

$$\begin{aligned} \int \nolimits _{\Sigma _t} |\nabla ^2 H|^2\mathrm{{d}}\mu +\frac{1}{4} \int \nolimits _{\Sigma _t} H^2|\nabla H|^2\mathrm{{d}}\mu \le c(|\Delta H|_{L^2(\Sigma _t)}^2+|\mathring{A}|^4_{L^\infty (\Sigma _t)}+R_e^{-6}). \end{aligned}$$

Proof

We multiply (19) by \(\nabla H\) to obtain the Bochner-type identity

$$\begin{aligned}&\int \nolimits _{\Sigma _t} |\nabla ^2 H|^2\mathrm{{d}}\mu +\frac{1}{4} \int \nolimits _{\Sigma _t} H^2|\nabla H|^2\mathrm{{d}}\mu \\&\quad \le \int \nolimits _{\Sigma _t} |\Delta H|^2\mathrm{{d}}\mu +\int \nolimits _{\Sigma _t} |\overline{\text {Rm}}*\nabla H * \nabla H| \mathrm{{d}}\mu +\int \nolimits _{\Sigma _t} |\mathring{A}|^2|\nabla H|^2\mathrm{{d}}\mu \\&\quad \le |\Delta H|_{L^2(\Sigma _t)}^2+c\epsilon (R_e^{-3}+|\mathring{A}|^2_{L^\infty (\Sigma _t)})|\Delta H|_{L^2(\Sigma _t)} \\&\quad \le c(|\Delta H|_{L^2(\Sigma _t)}^2+|\mathring{A}|^4_{L^\infty (\Sigma _t)}+R_e^{-6}). \end{aligned}$$

\(\square \)

In order to state the \(L^\infty \)-estimate, we introduce the quantity \(\overline{H_S}=2R_e^{-1}\phi ^{-2}(R_e)-2mR_e^{-2}\phi ^{-3}(R_e)\) to be the mean curvature, with respect to the Schwarzschild metric, of a centred sphere with Euclidean radius \(R_e\).

Lemma 3.11

Under the assumptions of Lemma 3.3 and provided \(R_0^{-1}, |\mathring{A}|^2_{L^2(\Sigma ))}<\epsilon \) there holds

$$\begin{aligned} |H-\overline{ H_S}|^4_{L^\infty (\Sigma _t)}\le c(|\mathring{A}|^2_{L^2(\Sigma _t)}+R_e^{-2})(|\Delta H|_{L^2(\Sigma _t)}^2+|\mathring{A}|^4_{L^\infty (\Sigma _t)}+R_e^{-6}). \end{aligned}$$

Proof

Using Lemma 4.1 below we obtain \(|H-\overline{ H_S}|^2_{L^2(\Sigma _t)}\le c(|\mathring{A}|^2_{L^2(\Sigma _t)}+R_e^{-2})\le c\epsilon \). Now, Lemma 3.6 and Lemma 3.10 as well as \(|\overline{ H_S}|\le cR_e^{-1}\) imply that

$$\begin{aligned}&|H-\overline{ H_S}|^4_{L^\infty (\Sigma _t)}\\&\quad \le c|H-\overline{ H_S}|^2_{L^2}\bigg (\int \nolimits _{\Sigma _t} |\nabla ^2 H|^2\mathrm{{d}}\mu +\int \nolimits _{\Sigma _t} H^4|H-\overline{ H_S}|^2\mathrm{{d}}\mu \bigg ) \\&\quad \le c(|\mathring{A}|^2_{L^2(\Sigma _t)}+R_e^{-2})(|\Delta H|_{L^2(\Sigma _t)}^2+|\mathring{A}|^4_{L^\infty (\Sigma _t)}+R_e^{-6})\\&\quad \quad +c(|\mathring{A}|^2_{L^2(\Sigma _t)}+R_e^{-2})\bigg (R_e^{-4}\int \nolimits _{\Sigma _t} |H-\overline{ H_S}|^2\mathrm{{d}}\mu +|H-\overline{ H_S}|^4_{L^\infty (\Sigma _t)}\int \nolimits _{\Sigma _t} |H-\overline{ H_S}|^2\mathrm{{d}}\mu \bigg ). \end{aligned}$$

The third term can be absorbed using (26) and estimating \(|\mathring{A}|^2_{L^2(\Sigma _t)}\le \epsilon \). Another application of (26) yields

$$\begin{aligned} |H-\overline{ H_S}|^2_{L^2(\Sigma _t)}\le |\mathring{A}|^2_{L^2(\Sigma _t)}+cR_e^{-2} \le c R_e^{4 }|\mathring{A}|^4_{L^\infty (\Sigma _t)}+cR_e^{-2}. \end{aligned}$$

Using this to estimate the second term, the claim follows. \(\square \)

A-Priori Estimates

In this section, we specify the estimates from the previous section to the situation of an area-preserving Willmore flow. We first prove the following useful lemma.

Lemma 4.1

Let \(\Sigma _t\) be an admissible area preserving Willmore flow. Then every \(\Sigma _t\) satisfies

$$\begin{aligned} \int \nolimits _{\Sigma _t} |\text {Rc}(\nu ,\cdot )^T|^2 \mathrm{{d}}\mu \le c R_e^{-4}(R_e|\mathring{A}|^2_{L^2(\Sigma _t)}+\tau _e^2+ \eta R_e^{-1}), \end{aligned}$$
(25)
$$\begin{aligned} |H-\overline{ H_S}|_{L^2(\Sigma _t)} \le c (|\mathring{A}|_{L^2(\Sigma _t)}+m\tau _e R_e^{-1}+\eta R_e^{-2}). \end{aligned}$$
(26)

Here, \({\overline{H}}_S:=2R_e^{-1}\phi ^{-2}(R_e)-2mR_e^{-2}\phi ^{-3}(R_e)\).

Proof

Using Lemma 2.3, we may assume that \(\eta =0\). Let \(e_i\) be an orthonormal frame of \(T_p\Sigma _t\) at a point p. There holds

$$\begin{aligned} m^2r^6\phi ^4 |\text {Rc}(\nu ,\cdot )^T|^2=9 (\partial _r\cdot \nu _e)^2 \partial _r \cdot e_i \partial _r \cdot e_i =9 (\partial _r\cdot \nu _e)^2 \partial _r\cdot (\partial _r-\nu _e\cdot \partial _r \nu _e). \end{aligned}$$

We would like to express the right-hand side in terms of the approximating sphere S introduced in Lemma 2.4. Denoting the respective quantities regarding S with a tilde we find using Lemma 2.4

$$\begin{aligned}&\bigg |\int \nolimits _S {{\tilde{r}}}^{-6}({{\tilde{\partial }}}_r\cdot {{\tilde{\nu }}}_e)^2 {{\tilde{\partial }}}_r\cdot ({{\tilde{\partial }}}_r-{{\tilde{\nu }}}_e\cdot {{\tilde{\partial }}}_r {{\tilde{\nu }}}_e)d{{\tilde{\mu }}}_e-\int \nolimits _\Sigma r^{-6}(\partial _r\cdot \nu _e)^2 \partial _r\cdot (\partial _r-\nu _e\cdot \partial _r \nu _e) \mathrm{{d}}\mu _e \bigg |\nonumber \\&\quad \le cR_e^{-4}|\mathring{A}_e|^2_{L^2(\Sigma _t)}. \end{aligned}$$
(27)

By Young’s inequality and Lemma 2.3, this error can be further estimated via

$$\begin{aligned} R_e^{-4}|\mathring{A}_e|_{L^2(\Sigma _t)} \le cR_e^{-5}+cR_e^{-3}|\mathring{A}|^2_{L^2(\Sigma _t)}. \end{aligned}$$

Now, for a round sphere, there holds \({{\tilde{\partial }}}_{ r}=\tilde{r}^{-1}(a_e+R_e{{\tilde{\nu }}}_e)\). Hence,

$$\begin{aligned} ({{\tilde{\partial }}}_{ r}-{{\tilde{\partial }}}_{ r}\cdot {{\tilde{\nu }}}_e \tilde{\nu }_e)\cdot \partial _{{{\tilde{r}}}}= & {} {{\tilde{r}}}^{-2}(a_e+R_e\tilde{\nu }_e-a_e\cdot {{\tilde{\nu }}}_e{{\tilde{\nu }}}_e-R_e{{\tilde{\nu }}}_e)\cdot (a_e+R_e{{\tilde{\nu }}}_e) \\= & {} {{\tilde{r}}}^{-2}(a_e\cdot a_e-(a_e\cdot \tilde{\nu }_e)^2). \end{aligned}$$

As this term can be estimated by \(c\tau _e^2\), the first claim follows. The second claim is a straight-forward application of Lemmas 2.4 and  2.6. \(\square \)

The next lemma provides some basic control on the evolution of many geometric quantities. However, it will turn out later on that these estimates can be sharpened in a considerable way.

Lemma 4.2

Let \(\Sigma _t\) be an admissible area preserving Willmore flow and let \(R_0^{-1}<\epsilon \). The following estimates hold:

$$\begin{aligned} |\mathring{A}|^4_{L^\infty (\Sigma _t)}&\le c|\mathring{A}|^2_{L^2(\Sigma _t)}(R_e^{-6} - \partial _t {\mathcal {W}}(\Sigma _t)), \\ |\mathring{A}|^2_{L^2(\Sigma _t)}&\le c R_e^{-2}-cR_e^{4} \partial _t{\mathcal {W}}(\Sigma _t), \\ |\nabla H|_{L^2(\Sigma _t)}^4&\le c|\mathring{A}|_{L^2(\Sigma _t)}(R_e^{-6}-\partial _t {\mathcal {W}}(\Sigma _t)), \\ |\Delta H|_{L^2(\Sigma _t)}^2&\le cR_e^{-6}-c \partial _t {\mathcal {W}}(\Sigma _t), \\ |W|_{L^2(\Sigma _t)}^2&\le cR_e^{-6}-c \partial _t {\mathcal {W}}(\Sigma _t), \\ \lambda ^2&\le cR_e^{-6}-c\epsilon \partial _t {\mathcal {W}}(\Sigma _t), \\ |\nabla ^2 H|_{L^2(\Sigma _t)}^4&\le cR_e^{-6}-c \partial _t {\mathcal {W}}(\Sigma _t), \\ |H-{\overline{H}}_S|^4_{L^\infty (\Sigma _t)}&\le (|\mathring{A}|^2_{L^2(\Sigma _t)}+cR_e^{-2})(R_e^{-6}-\partial _t{\mathcal {W}}(\Sigma _t)). \end{aligned}$$

Proof

Lemma 3.8 implies that

$$\begin{aligned} |\mathring{A}|_{L^\infty (\Sigma _t)}^4\le c|\mathring{A}|_{L^2(\Sigma _t)}^2(|W|^2_{L^2(\Sigma _t)}+R_e^{-6}) \le c\epsilon (|W|^2_{L^2(\Sigma _t)}+R_e^{-6}). \end{aligned}$$
(28)

Next, using (26) and \(R_0^{-1}\le \epsilon \) we obtain

$$\begin{aligned} |H-\overline{ H_S}|_{L^2(\Sigma _t)}\le c(\eta R_e^{-2}+\tau _em R_e^{-1}+c|\mathring{A}_e|_{L^2})\le c\sqrt{\epsilon }. \end{aligned}$$

Integrating by parts and using Young’s inequality, we conclude that

$$\begin{aligned} \int \nolimits _{\Sigma _t} |\nabla H|^2\mathrm{{d}}\mu =\int \nolimits _{\Sigma _t} ({\overline{H}}_S-H)\Delta H\mathrm{{d}}\mu \le c\sqrt{\epsilon }|\Delta H|_{L^2(\Sigma _t)}. \end{aligned}$$
(29)

Now we use these estimates to obtain an estimate for \(|\Delta H|_{L^2(\Sigma _t)}\). Recall that (c.f. (12))

$$\begin{aligned} 2\partial _t {\mathcal {W}}(\Sigma _t)=-|W|^2_{L^2}(\Sigma _t)+\lambda \int \nolimits _{\Sigma _t} HW\mathrm{{d}}\mu =-|W|^2_{L^2(\Sigma _t)}+|H|_{L^2(\Sigma _t)}^{-2}\bigg (\int \nolimits _{\Sigma _t} WH\mathrm{{d}}\mu \bigg )^2. \end{aligned}$$
(30)

Integrating by parts, using (29), (16), \(|\text {Rc}|\le cR_e^{-3}\) and (28), we obtain

$$\begin{aligned} \bigg (\int \nolimits _{\Sigma _t} WH\mathrm{{d}}\mu \bigg )^2=&\bigg (\int \nolimits _{\Sigma _t} \bigg (|\nabla H|^2-{\overline{Rc}}(\nu ,\nu )H^2-|\mathring{A}|^2H^2\bigg )\mathrm{{d}}\mu \bigg )^2 \nonumber \\ \le&c\epsilon |\Delta H|^2_{L^2(\Sigma _t)}+c\epsilon |W|_{L^2(\Sigma _t)}^2 +cR_e^{-6}. \end{aligned}$$
(31)

Again by (28), \(|\text {Rc}|\le cR_e^{-3}\) and the definition of W we find

$$\begin{aligned} |W|_{L^2(\Sigma _t)}^2 \ge |\Delta H|^2_{L^2(\Sigma _t)} -cR_e^{-6}-c\epsilon |W|_{L^2(\Sigma _t)}^2 . \end{aligned}$$
(32)

Combining (30), (31) and (32), we find

$$\begin{aligned} |\Delta H|^2_{L^2(\Sigma _t)}\le -c\partial _t{\mathcal {W}}(\Sigma _t)+cR_e^{-6}, \end{aligned}$$
(33)

provided \(\epsilon \) is sufficiently small. Returning to (30) and (31) we conclude

$$\begin{aligned} |W|_{L^2(\Sigma _t)}^2 \le -c\partial _t{\mathcal {W}}(\Sigma _t)+cR_e^{-6}. \end{aligned}$$

This implies the third, fourth and fifth estimate. Returning to (28), we then find that the first estimate holds. Now we can use (9), (16), \(|\text {Rc}|\le cR_e^{-3}\) and the first and third estimate to conclude that

$$\begin{aligned} \lambda ^2 \le c(|\nabla H|^4_{L^2}+|H|\mathring{A}||_{L^2}^4+\bigg (\int \nolimits _\Sigma H^2\overline{\text {Rc}}(\nu ,\nu )\mathrm{{d}}\mu \bigg )^2)\le -c\epsilon \partial _t{\mathcal {W}}(\Sigma _t)+cR_e^{-6}. \end{aligned}$$

Next, in the situation of Lemma 3.9 we apply (25) and the estimate

$$\begin{aligned} |\nabla H|^2_{L^2(\Sigma _t)}\le & {} |H-\overline{ H_S}|_{L^2(\Sigma _t)}|\Delta H|_{L^2(\Sigma _t)} \le \kappa R_e^{-2} |H-\overline{ H_S}|^2_{L^2(\Sigma _t)}\\&+C(\kappa )R_e^4|\Delta H|^2_{L^2(\Sigma _t)}, \end{aligned}$$

valid for any \(\kappa >0\), together with (26). Absorbing the \(|\mathring{A}|_{L^2(\Sigma _t)}^2\) terms and using the estimate (33) then implies the second estimate. The two missing estimates are now straight-forward consequences of Lemmas 3.10 and  3.11. \(\square \)

As promised, we now prove the refined a-priori estimates.

Lemma 4.3

If \(\epsilon , R_0^{-1}\) are chosen sufficiently small, then an admissible area preserving Willmore flow satisfies the improved estimates

$$\begin{aligned} |\Delta H|_{L^2(\Sigma _t)}^2&\le -c\partial _t{\mathcal {W}}(\Sigma _t)+c(\tau _e^2+R_e^{-1})R_e^{-6},\\ |\nabla H|_{L^2(\Sigma _t)}^4&\le -c\epsilon \partial _t{\mathcal {W}}(\Sigma _t)+c(\tau _e^2+R_e^{-2}) R_e^{-8}, \\ |\mathring{A}|_{L^2(\Sigma _t)}^2&\le -cR_e^{4}\partial _t{\mathcal {W}}(\Sigma _t)+ c(\tau _e^2+R_e^{-1})R_e^{-2}, \\ (\lambda -2mR_e^{-3})^2&\le -c\epsilon \partial _t{\mathcal {W}}(\Sigma _t)+ c(\tau _e^2+R_e^{-1})R_e^{-6}. \end{aligned}$$

Proof

First, let us recall that the asymptotic behaviour of the metric implies that \(|\text {Rc}|\le c R_e^{-3}\) as well as \(|\text {Rc}-\text {Rc}_S|\le cR_e^{-4}\). Moreover, according to Lemma 2.3, there holds \(|\nu -\nu _S|\le cR_e^{-2}\) as well as \(|\mathrm{{d}}\mu -d\mu _S|\le cR_e^{-4}\) while (16) states that \(|H|_{L^2(\Sigma _t)}\le c\). We will use these estimates at various points without explicitly stating them. There holds

$$\begin{aligned} -2\partial _t{\mathcal {W}}(\Sigma _t)=&|W|^2_{L^2(\Sigma _t)}-\lambda ^2|H|^2_{L^2(\Sigma _t)} \nonumber \\ =&|\Delta H|^2_{L^2(\Sigma _t)}+2|H\text {Rc}(\nu ,\nu )|^2_{L^2(\Sigma _t)} +2||\mathring{A}|^2H|^2_{L^2(\Sigma _t)}\nonumber \\&+2\langle \Delta H H,\text {Rc}(\nu ,\nu )\rangle _{L^2(\Sigma _t)}+2\langle \Delta H H,|\mathring{A}|^2\rangle _{L^2(\Sigma _t)}\nonumber \\&+2\langle H^2|\mathring{A}|^2,\text {Rc}(\nu ,\nu )\rangle _{L^2(\Sigma _t)} \nonumber \\&-|H|^{-2}_{L^2(\Sigma _t)}(|\nabla H|_{L^2(\Sigma _t)}^4+\langle H^2,\text {Rc}(\nu ,\nu )\rangle ^2_{L^2(\Sigma _t)}+|H|\mathring{A}||_{L^2(\Sigma _t)}^4) \nonumber \\&+2|H|^{-2}_{L^2(\Sigma _t)}(|\nabla H|^2_{L^2(\Sigma _t)} \langle H^2, \text {Rc}(\nu ,\nu )\rangle _{L^2(\Sigma _t)}+|\nabla H|^2_{L^2(\Sigma _t)}|\mathring{A}H|_{L^2(\Sigma _t)}^2 \nonumber \\&-|\mathring{A}H|_{L^2(\Sigma _t)}^2\langle H^2, \text {Rc}(\nu ,\nu )\rangle _{L^2(\Sigma _t)}). \end{aligned}$$
(34)

We denote the 12 terms in the last equation by the Latin numbers \(I-XII\). There holds

$$\begin{aligned}&|III+V+IX+XI|\nonumber \\&\quad \le c |\mathring{A}|_{L^\infty }^4+c|\mathring{A}|_{L^\infty (\Sigma _t)}^2(|\Delta H|^2_{L^2(\Sigma _t)}+|\nabla H|^2_{L^2(\Sigma _t)}) \nonumber \\&\quad \le c|\mathring{A}|_{L^\infty (\Sigma _t)}^4+\frac{1}{8}|\Delta H|_{L^2(\Sigma _t)}^2+c|\nabla H|_{L^2(\Sigma _t)}^4 \nonumber \\&\quad \le c|\mathring{A}|^2_{L^2(\Sigma _t)}(R_e^{-6}-\partial _t{\mathcal {W}}(\Sigma _t))+\frac{1}{8}|\Delta H|_{L^2(\Sigma _t)}^2+c|\nabla H|_{L^2(\Sigma _t)}^4, \end{aligned}$$
(35)

where we used Lemma 4.2 in the last inequality. We now focus on IV and X. We replace \(\text {Rc}\) by \(\text {Rc}_S\), \(\nu \) by \(\nu _S\) and \(\mathrm{{d}}\mu \) by \(\mathrm{{d}}\mu _S\). This results in error terms that can be estimated by

$$\begin{aligned} c\eta R_e^{-4}(|\Delta H|_{L^2(\Sigma _t)}+|\nabla H|^2_{L^2(\Sigma _t)}) \le c R_e^{-8}+\frac{1}{8}|\Delta H|_{L^2(\Sigma _t)}^2. \end{aligned}$$
(36)

In the last inequality, we used the crude estimate \(|\nabla H|_{L^2(\Sigma _t)}^2\le c|\Delta H|_{L^2(\Sigma _t)}\), see (29). In order to estimate these terms further, we express them in terms of the approximate sphere S from Lemma 2.4. To this end, we denote the conformal parametrization \(S\rightarrow \Sigma _t\) by \(\psi \) and indicate the respective geometric quantities of S by a tilde. According to Lemma 2.4 we have \(|r^{-3}-R_e^{-3}|\le R_e^{-3}(|\mathring{A}_e|_{L^2(\Sigma _t)}+\tau _e)\) as well as \(|\partial _r-{{\tilde{\partial }}}_r|\le |\mathring{A}_e|_{L^2(\Sigma _t)}\) while Lemma 2.5 and Taylor’s theorem imply that \(|\phi ^{-6}-1|\le cR_e^{-1}\). Since \(|{{\tilde{\nu }}}_e-\nu _e|\le c|\nabla _e \psi - {{\tilde{\nabla }}}_e\text {Id}|\) we find

$$\begin{aligned}&|\text {Rc}_S(\nu _S,\nu _S)\mathrm{{d}}\mu _S-mR_e^{-3}(1-3(\tilde{\partial }_r\cdot {{\tilde{\nu }}}_e)^2)d \mu _e|\\&\quad \le cR_e^{-3}(\tau _e+|\mathring{A}_e|_{L^2(\Sigma _t)})+cR_e^{-3}|\nabla _e \psi -\nabla _e \text {id}|. \end{aligned}$$

Consequently, it follows from Hölder’s inequality and Lemma 2.4 that replacing \(\text {Rc}_S(\nu _S,\nu _S)\mathrm{{d}}\mu _S\) by \(mR_e^{-3}(1-3(\tilde{\partial }_r\cdot {{\tilde{\nu }}}_e)^2)\mathrm{{d}}\mu _e\) in IV and X results in error terms that can be estimated by

$$\begin{aligned}&cR_e^{-3}(|\Delta HH|_{L^2(\Sigma _t)}R_e|\mathring{A}_e|_{L^2(\Sigma _t)}+(|\Delta H|_{L^2(\Sigma _t)}+|\nabla H|^2_{L^2(\Sigma _t)}) (\tau _e+|\mathring{A}_e|_{L^2(\Sigma _t)})\nonumber \\&\quad \quad +|\nabla H|^2_{L^2(\Sigma _t)}|H^2|_{L^2(\Sigma _t)}R_e|\mathring{A}_e|_{L^2(\Sigma _t)}) \nonumber \\&\quad \le c\tau _e^2 R_e^{-6}+c|\mathring{A}_e|^2_{L^2(\Sigma _t)}R_e^{-6}+1/8 |\Delta H|^2_{L^2(\Sigma _t)}+c|\nabla H|^4_{L^2(\Sigma _t)}\nonumber \\&\quad \quad +cR_e^{-4}|\mathring{A}_e|^2_{L^2(\Sigma _t)}(R_e^{-2}+R_e^2|H|^4_{L^\infty }) \nonumber \\&\quad \le c\tau _e^2R_e^{-6}+c|\mathring{A}|^2_{L^2(\Sigma _t)} R_e^{-6}+cR_e^{-8}-c\epsilon \partial _t{\mathcal {W}}(\Sigma _t)+1/8 |\Delta H|^2_{L^2(\Sigma _t)}+c|\nabla H|^4_{L^2(\Sigma _t)}. \end{aligned}$$
(37)

In the first inequality, we used Young’s inequality, Hölder’s inequality and the estimate \(|H^2|^2_{L^2(\Sigma _t)}\le cR_e^2|H|_{L^\infty (\Sigma _t)}^4\). In the second inequality we used the estimate \(|H|^4_{L^\infty (\Sigma _t)}\le |H-\overline{ H_S}|^4_{L^\infty (\Sigma _t)}+cR_e^{-4}\), Lemma 4.2 to estimate

$$\begin{aligned} |H-\overline{ H_S}|^4_{L^\infty (\Sigma _t)}\le cR_e^{-6}|\mathring{A}|^2_{L^2(\Sigma _t)}+cR_e^{-8}-c\epsilon \partial _t{\mathcal {W}}(\Sigma _t) \end{aligned}$$

as well as Lemma 2.3 to replace \(|\mathring{A}_e|_{L^2(\Sigma _t)}\) by \(|\mathring{A}|_{L^2(\Sigma _t)}\). Performing the same two procedures with VI yields an error term that can be estimated by

$$\begin{aligned}&cR_e^{-2}|H^2\mathring{A}^2|_{L^2(\Sigma _t)}(|\mathring{A}|_{L^2(\Sigma _t)}+\tau _e+R_e^{-1})\nonumber \nonumber \\&\quad \le c(R_e^{-1}|H-\overline{ H_S}|^2_{L^\infty (\Sigma _t)}|\mathring{A}|^2_{L^\infty (\Sigma _t)}+R_e^{-3}|\mathring{A}|^2_{L^\infty (\Sigma _t)}) (|\mathring{A}|_{L^2(\Sigma _t)}+\tau _e+R_e^{-1}) \nonumber \\&\quad \le c(|\mathring{A}|^4_{L^\infty (\Sigma _t)}+R_e^{-2}|H-\overline{ H_S}|^4_{L^\infty (\Sigma _t)})+cR_e^{-3}|\mathring{A}|^2_{L^\infty (\Sigma _t)}(|\mathring{A}|_{L^2(\Sigma _t)}+\tau _e+R_e^{-1}) \nonumber \\&\quad \le cR_e^{-6}|\mathring{A}|_{L^2(\Sigma _t)}^2+cR_e^{-6}(\tau _e^2+R_e^{-2})-c\epsilon \partial _t{\mathcal {W}}(\Sigma _t). \end{aligned}$$
(38)

In the first inequality we used Young’s inequality as well as the estimate \(|H|_{L^\infty (\Sigma _t)}^2\le c |H-\overline{H_S}|_{L^\infty (\Sigma _t)}^2+cR_e^{-2}\). In the second inequality we used Young’s inequality again and the fact that \(|\mathring{A}|_{L^2(\Sigma _t)}+\tau _e+R_e^{-1}\) is bounded. In the third inequality we used Young’s inequality one more time and Lemma 4.2 to estimate \(|\mathring{A}|^4_{L^\infty (\Sigma _t)}\) and \(|H-\overline{ H_S}|^4_{L^\infty (\Sigma _t)}\). Performing this procedure on XII yields a similar error term. On the other hand, there holds \( {{\tilde{\partial }}}_r \cdot {{\tilde{\nu }}}_e = r^{-1}(R_e{{\tilde{\nu }}}_e +a_e)\cdot {{\tilde{\nu }}}_e.\) Again, \(\tilde{r}^{-1}\) can be replaced by \(R_e^{-1}\) and since \(\tau _e=|a_e|/R_e\), we find that

$$\begin{aligned} |mR_e^{-3}(1-3({{\tilde{\partial }}}_r\cdot {{\tilde{\nu }}}_e)^2)+2mR_e^{-3}|\le c{\tau _eR_e^{-3}}. \end{aligned}$$
(39)

Integrating by parts, we find that

$$\begin{aligned} -2mR_e^{-3}\langle \Delta H H,1\rangle _{L^2(\Sigma _t)}-2mR_e^{-3}|H|^{-2}_{L^2(\Sigma _t)}|\nabla H|^2_{L^2(\Sigma _t)}\langle H^2,1\rangle _{L^2(\Sigma _t)}=0. \end{aligned}$$

Combining this with (36), (37) and (39) we obtain

$$\begin{aligned} |IV+X|&\le c\tau _eR_e^{-3}|\Delta H|_{L^2(\Sigma _t)}+c\tau _e^2R_e^{-6}+cR_e^{-8}+c|\mathring{A}|^2_{L^2(\Sigma _t)} R_e^{-6}-c\epsilon \partial _t{\mathcal {W}}(\Sigma _t)\nonumber \\&\quad +1/8 |\Delta H|^2_{L^2(\Sigma _t)}+c|\nabla H|^4_{L^2(\Sigma _t)} \nonumber \\&\le c\tau _e^2R_e^{-6}+cR_e^{-8}+c|\mathring{A}|^2_{L^2(\Sigma _t)} R_e^{-6}-c\epsilon \partial _t{\mathcal {W}}(\Sigma _t)+1/4 |\Delta H|^2_{L^2(\Sigma _t)}\nonumber \\&\quad +c|\nabla H|^4_{L^2(\Sigma _t)}. \end{aligned}$$
(40)

In a similar way we can use (38) to find

$$\begin{aligned} |VI+XII|&\le c\tau _e R_e^{-3} |H\mathring{A}|_{L^2(\Sigma _t)}^2+cR_e^{-6}|\mathring{A}|_{L^2(\Sigma _t)}^2+cR_e^{-6}(\tau _e^2+R_e^{-2})-c\epsilon \partial _t{\mathcal {W}}(\Sigma _t) \nonumber \\&\le cR_e^{-6}|\mathring{A}|_{L^2(\Sigma _t)}^2+cR_e^{-6}(\tau _e^2+R_e^{-2})-c\epsilon \partial _t{\mathcal {W}}(\Sigma _t). \end{aligned}$$
(41)

Here, we estimated \(\tau _eR_e^{-3}|H\mathring{A}|^2_{L^2(\Sigma _t)}\le c\tau _e^2R_e^{-6}+c|\mathring{A}|^4_{L^\infty (\Sigma _t)}\) and then used Lemma 4.2. In II and VIII we first replace every H by \(\overline{ H_S}\). In light of Lemma 4.2, the error can be estimated by

$$\begin{aligned} c|H-\overline{ H_S}|_{L^\infty (\Sigma _t)}R_e^{-5}&\le c|H-\overline{ H_S}|^4_{L^\infty (\Sigma _t)}R_e^{-2}+cR_e^{-8}\le cR_e^{-6}|\mathring{A}|_{L^2(\Sigma _t)}^2\\&\quad +cR_e^{-8}-c\epsilon \partial _t{\mathcal {W}}(\Sigma _t). \end{aligned}$$

From this we find that

$$\begin{aligned}&|II+VIII| \nonumber \\&\quad \le cR_e^{-2} \bigg |\int \nolimits _{\Sigma _t}(\text {Rc}(\nu ,\nu ))^2\mathrm{{d}}\mu -|\Sigma _t|^{-1}\bigg (\int \nolimits _{\Sigma _t} \text {Rc}(\nu ,\nu )\mathrm{{d}}\mu \bigg )^2 \bigg |\nonumber \\&\quad \quad + c|\mathring{A}|^2_{L^2(\Sigma _t)}R_e^{-6}+cR_e^{-8} -c\epsilon \partial _t{\mathcal {W}}(\Sigma _t) \nonumber \\&\quad \le c|\nabla \text {Rc}(\nu ,\nu )|^2_{L^2(\Sigma _t)}+ c|\mathring{A}|^2_{L^2(\Sigma _t)}R_e^{-6}+c R_e^{-8}-c\epsilon \partial _t{\mathcal {W}}(\Sigma _t) \nonumber \\&\quad \le c|{{\overline{\nabla }}} \text {Rc}(\nu ,\nu )|^2_{L^2(\Sigma _t)}+c|A|^2_{L^\infty (\Sigma _t)}|\text {Rc}(\nu ,\cdot )^T|_{L^2(\Sigma _t)}^2+ c|\mathring{A}|^2_{L^2(\Sigma _t)}R_e^{-6}\nonumber \\&\quad \quad +c R_e^{-8}-c\epsilon \partial _t{\mathcal {W}}(\Sigma _t) \nonumber \\&\quad \le c|\mathring{A}|^2_{L^2(\Sigma _t)}R_e^{-5}+c\tau _e^2R_e^{-6}+c R_e^{-7}-c\epsilon \partial _t{\mathcal {W}}(\Sigma _t). \end{aligned}$$
(42)

In the second inequality, we used the Poincare inequality with zero mean. In the third inequality, we expressed \(\nabla \) in terms of \({{\overline{\nabla }}}\) and A, see (21). In the fourth inequality, we used (25) and the estimate \(|({{\overline{\nabla }}} \text {Rc})(\nu ,\nu )|^2_{L^2(\Sigma _t)} \le c|\mathring{A}|^2_{L^2(\Sigma _t)} R_e^{-5}+c\tau _e^2R_e^{-6}+cR_e^{-7}\), which can be shown in the same fashion as (25). Finally, we used \(|A|^2_{L^\infty (\Sigma _t)}\le 2|\mathring{A}|^2_{L^\infty (\Sigma _t)}+ 2|H|^2_{L^\infty (\Sigma _t)}\) and estimated these terms in the usual way. Combining (34),(35),(40),(41) and (42), we finally obtain

$$\begin{aligned} |\Delta H|_{L^2(\Sigma _t)}^2 \le -c\partial _t{\mathcal {W}}(\Sigma _t)+ c|\nabla H|_{L^2(\Sigma _t)}^4+c(\tau ^2_e+ R_e^{-1})R_e^{-6}+cR_e^{-5}|\mathring{A}|_{L^2(\Sigma _t)}^2.\nonumber \\ \end{aligned}$$
(43)

In light of the inequality

$$\begin{aligned} |\nabla H|_{L^2(\Sigma _t)}^4&\le |H-\overline{H_S}|_{L^2(\Sigma _t)}^2|\Delta H|_{L^2(\Sigma _t)}^2\nonumber \\&\le c(\tau _e^2R_e^{-2}+|\mathring{A}|_{L^2(\Sigma _t)}^2+CR_e^{-4})|\Delta H|^2_{L^2(\Sigma _t)}, \end{aligned}$$
(44)

which follows from (26) and the divergence theorem, we can eventually absorb the second term on the right-hand side in (43) to obtain

$$\begin{aligned} |\Delta H|_{L^2(\Sigma _t)}^2 \le -c\partial _t{\mathcal {W}}(\Sigma _t)+c(\tau ^2_e+ R_e^{-1})R_e^{-6}+cR_e^{-5}|\mathring{A}|_{L^2(\Sigma _t)}^2. \end{aligned}$$
(45)

Next, using Lemma 3.9, (25) and Young’s inequality we find for any \(\kappa >0\)

$$\begin{aligned} |\mathring{A}|^2_{L^2(\Sigma _t)}\le&c|\mathring{A}|^2_{L^2(\Sigma _t)}R_e^{-1}+c\tau _e^2 R_e^{-2}+cR_e^{-3}+\kappa |H-\overline{ H_S}|^2_{L^2(\Sigma _t)} \\&+C(\kappa )R_e^{4}|H-\overline{ H_S}|^{-2}_{L^2(\Sigma _t)}|\nabla H|^4_{L^2(\Sigma _t)} \\ \le&|\mathring{A}|^2_{L^2(\Sigma _t)}(\kappa +cR_e^{-1})+c\tau _e^2 R_e^{-2}+cR_e^{-3}-cR_e^{4}\partial _t{\mathcal {W}}. \end{aligned}$$

In the last step, we have used (26), \(|\nabla H|^4_{L^2(\Sigma _t)}\le |H-\overline{H_S}|_{L^2(\Sigma _t)}^2|\Delta H|_{L^2(\Sigma _t)}^2\) and (45). Absorbing yields the claimed estimate for \(|\mathring{A}|^2_{L^2(\Sigma _t)}\). Reinserting into (45) gives the claimed estimate for \(|\Delta H|^2_{L^2(\Sigma _t)}\). This then implies

$$\begin{aligned} |\nabla H|_{L^2(\Sigma _t)}^4\le&|H-\overline{H_S}|_{L^2(\Sigma _t)}^2|\Delta H|_{L^2(\Sigma _t)}^2\\&\le c(\tau _e^2R_e^{-2} +|\mathring{A}|^2_{L^2(\Sigma _t)}+cR_e^{-4})(-\partial _t{\mathcal {W}}(\Sigma _t)+\tau _e^2 R_e^{-6}+cR_e^{-7})\\&\le -c\epsilon \partial _t{\mathcal {W}}+c\tau _e^2 R_e^{-8}+cR_e^{-10}, \end{aligned}$$

as claimed. Finally, we recall the definition of \(\lambda \), see (9). We have

$$\begin{aligned} \lambda \le c|\nabla H|_{L^2(\Sigma _t)}^2+c|\mathring{A}|_{L^\infty (\Sigma _t)}^2-|H|_{L^2(\Sigma _t)}^{-2}\int \nolimits _{\Sigma _t} H^2 \text {Rc}(\nu ,\nu )\mathrm{{d}}\mu . \end{aligned}$$

Using the same methods as before, the last term can be computed explicitly to give

$$\begin{aligned} (\lambda -2mR_e^{-3})^2\le & {} c|H-\overline{ H_S}|^2_{L^\infty (\Sigma _t)}R_e^{-4}+c\tau _e^2R_e^{-6}+c R_e^{-8}+c|\mathring{A}|^2_{L^2(\Sigma _t)}R_e^{-4}\\&+c|\mathring{A}|^4_{L^\infty (\Sigma _t)}+|\nabla H|^4_{L^2(\Sigma _t)}. \end{aligned}$$

The claim follows. \(\square \)

The Evolution of the Barycenter

In this section, we proof Theorems 1.3 and 1.2. To this end, we derive a differential inequality for \(\tau _g\). In the next Lemma, we show that the evolution of \(\tau _g\) is linked to the translation sensitivity of \({\mathcal {W}}\) in terms of the Schwarzschild background. We note that the symbol \(``\cdot ''\) indicates the Euclidean inner product.

Lemma 5.1

Let \(\Sigma _t\) be an admissible area preserving Willmore flow. Then the following holds

$$\begin{aligned} \partial _t \tau _g=&\frac{1}{|\Sigma _t|_gR_g(t)}b_g\cdot \bigg (\int \nolimits _{\Sigma _t} \nu (\Delta H +H{\text {Rc}}(\nu ,\nu )+H|\mathring{A}|^2+\lambda H)\mathrm{{d}}\mu \nonumber \\&+\int \nolimits _{\Sigma _t} (x-a_e)(H\Delta H+H^2{\text {Rc}}(\nu ,\nu )+H^2|\mathring{A}|^2+\lambda H^2)\mathrm{{d}}\mu \bigg ), \end{aligned}$$
(46)

where \(b_g=a_g/|a_g|\). Moreover, we have

$$\begin{aligned} \partial _t \tau _g=&\frac{3-2mR_g^{-1}}{|\Sigma _t|_gR_g(t)}\bigg (b_g\cdot \int \nolimits _{\Sigma _t} \nu _S(\Delta _S H_S +H_S{\text {Rc}_\mathrm{S}}(\nu _S,\nu _S)+H_S|\mathring{A}_S|^2+\lambda H_S)\mathrm{{d}}\mu _S \nonumber \\&+{\mathcal {O}}((\tau ^2_e+\eta +R_e^{-1})R_e^{-3}-R_e^3\partial _t{\mathcal {W}}(\Sigma _t)) \bigg ). \end{aligned}$$
(47)

Proof

The first identity is a straight forward computation using the flow equation (8) and the fact that \(HW+\lambda H^2\) has zero mean. We first show that the second line of (46) is \(2(1-m/R_g)\) times the first line of (46) up to an error term by replacing \((x-a_e)H\) by \((2(1-m/R_g))\nu \). To this end, we first replace H by \(\overline{ H_S}\) in the second line of (46) which according to (26) yields an error term that can be estimated by

$$\begin{aligned} cR_e(|\mathring{A}|_{L^2(\Sigma _t)}+\tau _eR_e^{-1}+\eta R_e^{-2})|W+\lambda H|_{L^2(\Sigma _t)}. \end{aligned}$$
(48)

Similarly, Lemmas 2.4 and  2.3 imply that replacing \((x-a_e)\) by \(({{\tilde{x}}}-a_e)=R_e{{\tilde{\nu }}}_e\) and then \({{\tilde{\nu }}}_e\) by \(\nu _e\) results in error terms that can be estimated by

$$\begin{aligned} cR_e(|\mathring{A}|_{L^2(\Sigma _t)}+\eta R_e^{-2})|W+\lambda H|_{L^2(\Sigma _t)}. \end{aligned}$$
(49)

Finally, replacing \(R_e\overline{ H_S}=2\phi ^{-2}(R_e)-2m R_e^{-1}\phi ^{-3}(R_e)\) by \(2\phi ^{-2}(r)(1-m/R_g)\) and then \(\nu _S=\phi ^{-2}\nu _e\) by \(\nu \) leads to an error that can be estimated by

$$\begin{aligned} cR_e(|\mathring{A}|_{L^2(\Sigma _t)}+\tau _e R_e^{-1}+\nu R_e^{-2})|W+\lambda H|_{L^2(\Sigma _t)}. \end{aligned}$$
(50)

Here, we also used Lemma 2.6. We observe that

$$\begin{aligned} |W+\lambda H|_{L^2(\Sigma _t)}^2=|W|_{L^2(\Sigma _t)}^2-\lambda \int \nolimits _{\Sigma _t}WH\mathrm{{d}}\mu =-\partial _t{\mathcal {W}}(\Sigma _t). \end{aligned}$$
(51)

Therefore, we obtain

$$\begin{aligned} (c\tau _e+cR_e^{-1})|W+\lambda H|_{L^2(\Sigma _t)}\le c(\tau _e^2+R_e^{-2})R_e^{-3}-cR_e^{3}\partial _t{\mathcal {W}} \end{aligned}$$
(52)

Moreover, using Lemma 4.3 and Young’s inequality, we estimate

$$\begin{aligned} cR_e|\mathring{A}|_{L^2(\Sigma _t)}|W+\lambda H|_{L^2(\Sigma _t)} \le c(\tau _e^2+R_e^{-1})R_e^{-3}-cR_e^{3}\partial _t{\mathcal {W}}. \end{aligned}$$
(53)

Combining (48)–(53) shows that

$$\begin{aligned} \partial _t \tau _g=&\frac{3-2mR_g^{-1}}{|\Sigma _t|_gR_g(t)}\bigg (b_g\cdot \int \nolimits _{\Sigma _t} \nu (\Delta H +H{\text {Rc}}(\nu ,\nu )+H|\mathring{A}|^2+\lambda H)\mathrm{{d}}\mu \\&+{\mathcal {O}}(\tau ^2_eR_e^{-3}+R_e^{-4}-R_e^3\partial _t{\mathcal {W}}(\Sigma _t)) \bigg ). \end{aligned}$$

Next, we would like to express this quantity in terms of the Schwarzschild geometry. To this end, we will make implicit use of Lemmas 2.3 and  4.3. Recalling \(|\mathrm{{d}}\mu -d\mu _S|\le c\eta R_e^{-2} \mathrm{{d}}\mu \), \(|\nu -\nu _S|\le c\eta R_e^{-2}\) as well as \(|H-H_S|\le c\eta R_e^{-3}\) we find

$$\begin{aligned} \lambda \int \nolimits _{\Sigma _t}\nu H\mathrm{{d}}\mu =\lambda \int \nolimits _{\Sigma _t}\nu _S H_S\mathrm{{d}}\mu _S+\lambda {\mathcal {O}}(\eta R_e^{-1}). \end{aligned}$$
(54)

Using Lemma 4.3, this error can be estimated via

$$\begin{aligned} \eta \lambda R_e^{-1}\le 2\eta ^2 R_e^{-4}+2\lambda ^2R_e^2 \le cR_e^{-4}-c R_e^{2}\partial _t {\mathcal {W}}(\Sigma _t). \end{aligned}$$
(55)

Next, we have

$$\begin{aligned} \int \nolimits _{\Sigma _t}\nu H|\mathring{A}|^2\mathrm{{d}}\mu =\int \nolimits _{\Sigma _t} \nu _S H_S|\mathring{A}|^2\mathrm{{d}}\mu _S+{\mathcal {O}}(\eta |\mathring{A}|^2_{L^\infty (\Sigma _t)}R_e^{-2}). \end{aligned}$$

According to Lemma 2.3 there holds \(||\mathring{A}|^2-|\mathring{A}_S|^2|\le c\eta (|\mathring{A}|+R_e^{-3}+|A|R_e^{-2})(R_e^{-3}+|A|R_e^{-2})\). From this, it follows that replacing \(|\mathring{A}|^2\) by \(|\mathring{A}_S|^2\) in the previous equation yields an error that can be estimated by

$$\begin{aligned}&c\eta \int \nolimits _{\Sigma _t} |H_S|\bigg (|\mathring{A}|+R_e^{-3}+|A|R_e^{-2}\bigg )\bigg (R_e^{-3}+|A|R_e^{-2}\bigg )\mathrm{{d}}\mu \\&\quad \le c\eta (R_e^{-4}+|\mathring{A}|^2_{L^2(\Sigma _t)}R_e^{-2}+|\mathring{A}|_{L^\infty (\Sigma _t)}|H|_{L^\infty (\Sigma _t)}R_e^{-1}+|H|^2_{L^\infty (\Sigma _t)}R_e^{-3}) \\&\quad \le c\eta R_e^{-3}-cR_e^2\partial _t{\mathcal {W}}(\Sigma _t) \end{aligned}$$

In the first inequality, we used the crude estimate \(|H_S|\le cR_e^{-1}+|H|\), applied Young’s inequality several times and used (16). In the second inequality, we used Young’s inequality, Lemma 4.2 as well as Lemma 4.3. Hence,

$$\begin{aligned} \int \nolimits _{\Sigma _t} \nu H|\mathring{A}|^2\mathrm{{d}}\mu =\int \nolimits _{\Sigma _t} \nu _S H_S|\mathring{A}_S|^2\mathrm{{d}}\mu _S +{\mathcal {O}}(\eta (R_e^{-3}-R_e^{2}\partial _t{\mathcal {W}}(\Sigma _t))). \end{aligned}$$
(56)

Using the asymptotic behaviour of \({\text {Rc}},\nu ,\mathrm{{d}}\mu \), it is easy to see that

$$\begin{aligned} \int \nolimits _{\Sigma _t} \nu H{\text {Rc}}(\nu ,\nu )\mathrm{{d}}\mu = \int \nolimits _{\Sigma _t} \nu _S H_S{\text {Rc}}_S(\nu _S,\nu _S)\mathrm{{d}}\mu _S + {\mathcal {O}}(\eta R_e^{-3}). \end{aligned}$$
(57)

Regarding the last term, we note

$$\begin{aligned} \int \nu \Delta H\mathrm{{d}}\mu =\int \nu _S \Delta H\mathrm{{d}}\mu _S +{\mathcal {O}}(\eta R_e^{-3}+\eta R_e|\Delta H|^2_{L^2}). \end{aligned}$$

It is then straightforward to see that any smooth function u satisfies the estimate \(|\Delta _S u-\Delta u|\le c\eta (|\nabla ^2 u|R_e^{-2}+|\nabla u|R_e^{-3})\). Hence replacing \(\Delta \) by \(\Delta _S\) results in an error that can be estimated by

$$\begin{aligned} c\eta \int \nolimits _{\Sigma _t} (R_e^{-2}|\nabla ^2H|+R_e^{-3}|\nabla H|)\mathrm{{d}}\mu _S&\le c\eta R_e^{-3}+c\eta R_e|\nabla ^2 H|^2_{L^2(\Sigma _t)}+c\eta R_e^{-1}|\nabla H|^2_{L^2(\Sigma _t)} \\&\le c\eta R_e^{-3} +c\eta R_e|\nabla ^2H|^2_{L^2(\Sigma _t)}+c\eta R_e |\nabla H|^4_{L^2(\Sigma _t)} \\&\le c\eta R_e^{-3} -c\eta R_e\partial _t{\mathcal {W}}(\Sigma _t), \end{aligned}$$

where we used Lemmas 4.2 and  4.3. Integrating by parts, we obtain

$$\begin{aligned} \int \nolimits _{\Sigma _t} \nu \Delta H\mathrm{{d}}\mu = -\int \nolimits _{\Sigma _t} \nabla _S \nu _S\nabla _S H\mathrm{{d}}\mu _S+{\mathcal {O}}(\eta R_e^{-3}-c\eta R_e\partial _t{\mathcal {W}}(\Sigma _t)). \end{aligned}$$

There holds \(\nabla _S \nu _S\le cR_e^{-2}+|A_S|\le cR_e^{-2}+c|A|\). Hence, using Lemma 2.3 to replace \(\nabla _S H\) by \(\nabla _S H_S\) gives an error that can be estimated by

$$\begin{aligned} c\eta \int \nolimits _{\Sigma _t} (R_e^{-2}+|A|)(|\nabla A|R_e^{-2}+|A|R_e^{-3}+R_e^{-4})&\le c\eta R_e^{-3}+c\eta R_e^{-1}|\nabla A|^2_{L^2(\Sigma _t)}\\&\le c\eta R_e^{-3}+\eta R_e |\nabla A|^4_{L^2(\Sigma _t)}. \end{aligned}$$

According to Lemmas 3.3 and 4.3 there holds

$$\begin{aligned} |\nabla A|^4_{L^2(\Sigma _t)}\le c\lambda ^2+cR_e^{-6}+R_e^2|\mathring{A}|^4_{L^\infty (\Sigma _t)}\le -cR_e^{2}\partial _t{\mathcal {W}}(\Sigma _t)+cR_e^{-6}. \end{aligned}$$
(58)

We conclude

$$\begin{aligned} \int \nolimits _{\Sigma _t} \nu \Delta H\mathrm{{d}}\mu =\int \nolimits _{\Sigma _t} \nu _S \Delta _S H_S\mathrm{{d}}\mu _s +{\mathcal {O}}(\eta R_e^{-3}-\eta R_e^{3}\partial _t{{\mathcal {W}}}(\Sigma _t)). \end{aligned}$$
(59)

The claim then follows from (54), (55), (56), (57) and (59). \(\square \)

At this point, the central observation lies in the fact that the first three terms in (47) are a multiple of the variation of the Schwarzschild Willmore energy along a translation. Through approximation by a sphere, we can therefore explicitly compute \(\partial _t \tau _g\) up to an error. We denote the geometric quantities of \(\Sigma \) as usual, the geometric quantities of \(S:=S_{a_e}(R_e)\) are denoted using a tilde.

Lemma 5.2

Let \(\Sigma _t\) be an admissible area preserving Willmore flow. If \(\delta \) is chosen sufficiently small (depending on m), then there holds

$$\begin{aligned} \partial _t \tau _g \le -160 m^2R_g^{-6}\tau _g+c(\tau _g^2+\epsilon +\eta )R_g^{-6}-c\partial _t{\mathcal {W}}. \end{aligned}$$

Proof

Using (10), it is easy to see that the first three terms of (47) are four times the variation of the Willmore energy in the Schwarzschild space under a translation in direction \(-b_g\), where \(b_g:=a_g/|a_g|\). We therefore have

$$\begin{aligned} b_g\cdot \int \nolimits _{\Sigma _t} \nu _S(\Delta _S H_S +H_S{ \text {Rc}_{\mathrm{S}}}( \nu _S,\nu _S)+H_S|\mathring{A}_S|_S^2)\mathrm{{d}}\mu _S=-\frac{\mathrm{{d}}}{\mathrm{{d}}s}\bigg |_{s=0}\int \nolimits _{\Sigma _t} H_S^2\mathrm{{d}}\mu _S. \end{aligned}$$
(60)

Here the variation is given by \(s\mapsto F+sb_g\), where F is the embedding of \(\Sigma _t\rightarrow M\). With the help of Lemma 2.3, we compute

$$\begin{aligned} \int \nolimits _{\Sigma _t} H_S^2\mathrm{{d}}\mu _S= & {} \int \nolimits _{\Sigma _t} H_e^2 \mathrm{{d}}\mu _e-4m\int \nolimits _{\Sigma _t}H_er^{-2}\phi ^{-1}\partial _r\cdot \nu _e\mathrm{{d}}\mu _e\\&+\,4m^2\int r^{-4}\phi ^{-2}(\partial _r\cdot \nu _e)^2\mathrm{{d}}\mu _e. \end{aligned}$$

The first term on the right-hand side is four times the Euclidean Willmore energy and therefore invariant under translations. In fact, as translations are isometries of \({\mathbb {R}}^3\), the quantities \(H_e,\mathrm{{d}}\mu _e,\nu _e\) are all invariant under translations. From this, it is easy to see that

$$\begin{aligned} -\frac{\mathrm{{d}}}{\mathrm{{d}}s}\bigg |_{s=0}\int \nolimits _{\Sigma _t} H_S^2 \mathrm{{d}}\mu _S = -4m\int \nolimits _{\Sigma _t} H_e \partial _s(r^{-2}\partial _r)|_{s=0}\cdot \nu _e \mathrm{{d}}\mu _e+{\mathcal {O}}(R_e^{-3}). \end{aligned}$$
(61)

The terms summarized in \({\mathcal {O}}(R_e^{-3})\) can be expressed in terms of the approximating sphere using Lemma 2.4 and the fact that \(|H_e|_{L^2(\Sigma _t)}^2\) is bounded which follows from Lemma 2.3 and (16).Footnote 2 Using Lemmas 2.3 and  4.3, the resulting error can then be estimated by

$$\begin{aligned} cR_e^{-3}|\mathring{A}_e|_{L^2(\Sigma _t)} \le cR_e^{-4}-cR_e^{2}\partial _t{\mathcal {W}}(\Sigma _t). \end{aligned}$$
(62)

Consequently, we summarize such terms by the letter \(\Gamma \) and focus on the only term which is of higher order and compute

$$\begin{aligned}&m\int \nolimits _{\Sigma _t} H_e \partial _s(r^{-2}\partial _r)|_{s=0}\cdot \nu _e \mathrm{{d}}\mu _e\nonumber \\&\quad =m\int \nolimits _{\Sigma _t} H_e r^{-3}(b_g-3\partial _r\cdot b_g\partial _r )\cdot \nu _e \mathrm{{d}}\mu _e \nonumber \\&\quad =\int \nolimits _{\Sigma _t} H_e \text {Rc}_S(\nu _S,b_g)\mathrm{{d}}\mu _S+\Gamma \nonumber \\&\quad =\int \nolimits _{\Sigma _t} (H_e-{{\tilde{H}}}_e) \text {Rc}_S(\nu _S,b_g)\mathrm{{d}}\mu _S+\Gamma \nonumber \\&\quad =m\int \nolimits _{\Sigma _t} (H_e-{{\tilde{H}}}_e) r^{-3}(b_g\cdot \nu _e-3b_g\cdot \partial _r \partial _r\cdot \nu _e) \mathrm{{d}}\mu _S+\Gamma \nonumber \\&\quad =(-2mR_e^{-3})\int \nolimits _{\Sigma _t} \nu _e\cdot b_g(H_e-{{\tilde{H}}}_e) \mathrm{{d}}\mu _e+{\mathcal {O}}((\tau _e^2+R_e^{-1}+\eta )R_e^{-3}-R_e^{3}\partial _t{\mathcal {W}}(\Sigma _t))+\Gamma \nonumber \\&\quad ={\mathcal {O}}((\tau _e^2+R_e^{-1}+\eta )R_e^{-3}-R_e^{3}\partial _t{\mathcal {W}}(\Sigma _t))+\Gamma . \end{aligned}$$
(63)

In the second equation, we used (5). In the third equation, we used the fact that

$$\begin{aligned}\int \nolimits _{\Sigma _t} \text {Rc}_{\mathrm{S}}(\nu _S,b_g)\mathrm{{d}}\mu _S=0.\end{aligned}$$

This follows from the so-called Pohozaev identity, see (5.12) in [24]: In Schwarzschild, it states that for any vector field X and any compact, null-homologous and smooth domain \(\Omega \), the following relation holds

$$\begin{aligned} \frac{1}{2}\int \nolimits _{\Omega } g_S({\mathcal {D}}X, \text {Rc}_S)\mathrm{{d}}\mu _S=\int \nolimits _{\partial \Omega }\text {Rc}_S(X,\nu _S)\mathrm{{d}}\mu _S, \end{aligned}$$
(64)

where \({\mathcal {D}}\) denotes the conformal killing operator.Footnote 3. One then picks \(X=b_g\) which is a conformal killing vector field and \(\Omega \) to be the region bounded by \(\Sigma _t\) and a coordinate sphere with radius tending to infinity. In the fourth equation, we again used (5) again. Then, in the fifth equation, we used Lemma 2.4 to replace r by \(R_e\), \(\partial _r\) by \({{\tilde{\partial }}}_r\) and then \(\tilde{\partial }_r=r^{-1}({{\tilde{\nu }}}_e R_e+a_e)\) by \(\nu _e\), yielding errors that can, with the help of the estimate \(|H_e-\tilde{H}_e|_{L^2(\Sigma _t)}\le c|\mathring{A}|_{L^2(\Sigma _t)}\), be estimated by

$$\begin{aligned} R_e^{-2}|\mathring{A}_e|_{L^2(\Sigma _t)}(|\mathring{A}_e|_{L^2(\Sigma _t)}+\tau _e) \le R_e^{-1}|\mathring{A}|^2_{L^2(\Sigma _t)}+\eta R_e^{-3}+\tau _e^2 R_e^{-3}. \end{aligned}$$

The final equality follows from the translation invariance of the Euclidean area and the translation invariance of the Euclidean volume, which imply, respectively, that

$$\begin{aligned} \int \nolimits _{\Sigma _t} H_e b_g\cdot \nu _e \mathrm{{d}}\mu _e=\int \nolimits _{\Sigma _t}b_g\cdot \nu _e \mathrm{{d}}\mu _e=0. \end{aligned}$$

The term \(\Gamma \) can then also be expressed in terms of the approximating sphere yielding as before error terms than can be estimated as in (62). On the other hand, we can also use the translation invariance of the Euclidean area to conclude

$$\begin{aligned} \lambda \int \nolimits _{\Sigma _t} b_g\cdot \nu _SH_S\mathrm{{d}}\mu _S&=\int \nolimits _{\Sigma _t}b_g\cdot \nu _e H_e\mathrm{{d}}\mu _e-2m\lambda \int \nolimits _{\Sigma _t}\nu _e r^{-2}\phi ^{-1}\partial _r\cdot \nu _e\mathrm{{d}}\mu _e \nonumber \\&=-2m\lambda \int \nolimits _{S_t} {{\tilde{\nu }}}_e {{\tilde{r}}}^{-2}(1-\frac{m}{2\tilde{r}}) {{\tilde{\partial }}}_r\cdot {{\tilde{\nu }}}_e d{{\tilde{\mu }}}_e +{\mathcal {O}}(\lambda (|\mathring{A}|_{L^2(\Sigma _t)}+ R_e^{-2})). \end{aligned}$$
(65)

In the second line, we expanded \(\phi ^{-1}\) up to order \(r^{-1}\) and used Lemma 2.4. We then estimate using Lemma 4.3

$$\begin{aligned} \lambda (|\mathring{A}|_{L^2(\Sigma _t)}+ R_e^{-2}) \le \lambda ^2 R_e^2 +|\mathring{A}|_{L^2(\Sigma _t)}^2R_e^{-2}+ R_e^{-4} \le cR_e^{-4}-cR_e^2\partial _t{\mathcal {W}}. \end{aligned}$$
(66)

Combining (60)–(66), we have shown that

$$\begin{aligned} \partial _t \tau _g =&\frac{(3-2mR_g^{-1})}{|\Sigma _t|R_g}\bigg (-\frac{\mathrm{{d}}}{\mathrm{{d}}s}\bigg |_{s=0}\int \nolimits _{S_{R_e}(a_e+sb_g)} {{\tilde{H}}}_S ^2 d{{\tilde{\mu }}}_S\nonumber \\&-2m\lambda \int \nolimits _{S_{R_e}(a_e)} {{\tilde{\nu }}}_e\cdot b_g {{\tilde{r}}}^{-2}(1-\frac{m}{2{{\tilde{r}}}}) \tilde{\partial }_r\cdot {{\tilde{\nu }}}_e d{{\tilde{\mu }}}_e \nonumber \\&+{\mathcal {O}}((\tau _g^2+R_e^{-1}+\eta )R_e^{-3}-R_e^{3}\partial _t{\mathcal {W}}(\Sigma _t)) \bigg ). \end{aligned}$$
(67)

Due to the rotational symmetry of \(g_S\), it is easy to see that the function \(u(a):=\int \nolimits _{S_{R_e}{(a)}} H_S^2 \mathrm{{d}}\mu _S\) only depends on \(\tau _e\) and is in fact analytic in \(\tau _e\). Moreover, there holds \(d/ds|_{s=0} \tau _e =b_g\cdot b_e R_e^{-1}\) where \(b_e=a_e/|a_e|\). It therefore suffices to compute \(u(a_e)\) up to terms of order \(R_e^{-2}\). Dropping the tilde notation and writing \(S=S_{R_e}(a_e)\), we have

$$\begin{aligned} u(a_e)&=\int \nolimits _S H_e^2 -4m \int \nolimits _S 2R_e^{-1} (r^{-2}-\frac{m}{2} r^{-3})\partial _r\cdot \nu _e \mathrm{{d}}\mu _e \\&\quad +4m^2 \int \nolimits _S r^{-4} (\partial _r\cdot \nu _e)^2\mathrm{{d}}\mu _e+{\mathcal {O}}(R_e^{-3}). \end{aligned}$$

After a rotation, we may assume that \(a_e=(0,0,|a_e|)\). We choose the parametrization

$$\begin{aligned} (\theta ,\varphi ) \mapsto a_e+R_e(\sin \theta \sin \varphi ,\sin \theta \cos \varphi ,\cos \theta ) \end{aligned}$$

and compute the quantities

$$\begin{aligned} \nu _e=(\sin \theta \sin \varphi ,\sin \theta \cos \varphi ,\cos \theta ), \qquad r^2=R_e^2(1+\tau _e^2)+2R_e^2\tau _e\cos \theta . \end{aligned}$$

Furthermore, one can check that

$$\begin{aligned} \cos (\theta )=(r^2-R_e^2-|a_e|^2)/(2R_e|a_e|)=(r^2-R_e^2(1+\tau _e^2))/(2R_e^2\tau _e) \end{aligned}$$

and consequently

$$\begin{aligned} \partial _r\cdot \nu _e =r^{-1}(R_e+|a_e|\cos \theta )= (r^2+R_e^2(1-\tau _e^2))/(2rR_e). \end{aligned}$$

Next, we have \(\mathrm{{d}}\mu _e=R_e^2d\varphi d\theta \) and since there is no \(\varphi \) dependence, integration of \(\varphi \) solely adds a factor of \(2\pi \). Finally, we can substitute \(\theta \rightarrow r\) where the area element transforms via

$$\begin{aligned} \mathrm{{d}}\theta /\mathrm{{d}}r=-r/(R_e|a_e|\sin \theta )=-r/(R_e^2\tau _e\sin \theta ) \end{aligned}$$

and the boundary data are mapped to \((R_e-|a_e|,R_e+|a_e|)=(R_e(1-\tau _e),R_e(1+\tau _e))\) in an orientation reversing way. This gives

$$\begin{aligned} u(a_e)=&16\pi -\frac{8\pi m}{R_e^2 \tau _e} \int \nolimits _{R_e(1-\tau _e)}^{R_e(1+\tau _e)}(r^{-2}-\frac{m}{2}r^{-3})(r^2+R_e^2(1-\tau _e^2))\mathrm{{d}}r \\&+\frac{2\pi m^2}{R_e^2 \tau _e} \int \nolimits _{R_e(1-\tau _e)}^{R_e(1+\tau _e)}r^{-5}(r^2+R_e^2(1-\tau _e^2))^2\mathrm{{d}}r+{\mathcal {O}}(R_e^{-3}) \\&=16\pi -32\frac{\pi m}{R_e}+\frac{6\pi m^2}{R_e^2\tau _e}\log {\frac{1+\tau _e}{1-\tau _e}}+\frac{4\pi m^2}{R_e^2}\frac{5-3\tau _e^2}{(1-\tau _e^2)^2}+{\mathcal {O}}(R_e^{-3}) \\&=16\pi -32\frac{\pi m}{R_e}+32\frac{\pi m^2}{R_e^2}+\frac{32\pi m^2 \tau _e^2}{R_e^{2}}+{\mathcal {O}}(\tau _e^4R_e^{-2}+R_e^{-3}), \end{aligned}$$

where the last equality follows from Taylor’s theorem, provided \(\delta \) is chosen sufficiently small. Hence from the analyticity, it follows that

$$\begin{aligned} \frac{\mathrm{{d}}}{\mathrm{{d}}s} u(a_e+sb_g)\bigg |_{s=0}&= 64 \pi m^2 R_e^{-3}\tau _e b_g\cdot b_e+{\mathcal {O}}(\tau _e^2R_e^{-3}+R_e^{-4}) \nonumber \\&= 64 \pi m^2 R_e^{-3}\tau _e +{\mathcal {O}}(\tau _e^2R_e^{-3}+R_e^{-4}), \end{aligned}$$
(68)

where we used that \(b_g\cdot b_e=1+{\mathcal {O}}(R_e^{-1})\). On the other hand, it is easy to see that for any vector \(b_e^\perp \) perpendicular to \(b_e\), one has

$$\begin{aligned} \int \nolimits _S \nu _e\cdot b_e^\perp r^{-2}(1-\frac{m}{2r})\partial _r\cdot \nu _e \mathrm{{d}}\mu _e =0. \end{aligned}$$

In a similar fashion as above, we thus find

$$\begin{aligned}&-2m\lambda \int \nolimits _S \nu _e\cdot b_g r^{-2}(1-m/2r^{-1})\partial _r\cdot \nu _e \mathrm{{d}}\mu _e \nonumber \\&\quad =-\frac{\pi m\lambda }{R^3\tau _e^3} b_e\cdot b_g \int \nolimits _{R_e(1-\tau _e)}^{R_e(1+\tau _e)} (r^2-R_e^2(1-\tau _e^2)) r^{-2}(1-m/2 r^{-1})(r^2+R_e^2(1-\tau _e^2))\mathrm{{d}}r \nonumber \\&\quad = \frac{16\pi m\lambda }{3}\tau _e +{\mathcal {O}}((\tau _e^2+R_e^{-1}) \lambda ) \nonumber \\&\quad = \frac{32\pi m^2\tau _e}{3R_e^3}+{\mathcal {O}}(\tau _e^2 R_e^{-3}+cR_e^{-4}-R_e^{3}\partial _t{\mathcal {W}}(\Sigma _t)), \end{aligned}$$
(69)

where we used Lemma 4.3 in the last inequality. Combining (67)–(69) and replacing \(\tau _e\) by \(\tau _g\), the claim follows as \(3(64-32/3)= 160\). \(\square \)

We are now ready to prove the main result:

Theorem 5.3

Let (Mg) be \(C^3\)-close to Schwarzschild with decay coefficient \(\eta >0\) and mass \(m>0\) and let \(\Sigma \) be an embedded sphere. There exist constants \(\eta _0(m), \epsilon (m,\eta _0), \delta (m,\eta _0)>0\) and \(R_0(m,\eta _0)>0\) such that if \(\eta \le \eta _0\), \(r_{\min }\ge R_0\), \(\tau _e <\delta /2\) and \(|\mathring{A}|^2_{L^2(\Sigma )}<\epsilon \), then the area preserving Willmore flow starting at \(\Sigma \) exists for all times and converges smoothly to one of the leaves in the foliation \(\{\Sigma _\lambda \}\).

Proof

First, we choose \(\delta , \epsilon , R_0^{-1}\) small enough such that every admissible surface satisfies the constraints of the previous lemmas. According to Lemma 2.8, we can choose suitable initial conditions such that the area-preserving Willmore flow can only cease to exist if \(\tau _e\) reaches \(\delta \). We assume that \(\tau _e(\Sigma )=\delta /2\). We would then like to show that \(\tau _e\le \delta \) for all times. For this to hold, it is enough to show that \(\tau _g \le 9\delta /10 \) for all times if \(R_0\) is chosen sufficiently large. We suppose for contradiction that there is a first time \(T^*>0\) such that \(\tau _g(T^*)=9\delta /10\). We may assume that \(\tau _g(0) = 6\delta /10 \) and that \(\tau _g(t)>6\delta /10\) for all \(t>0\). Additionally, we require \(c(\delta ^2+\epsilon +\eta ) \le 80m^2\delta \) as well as \(c\epsilon <\delta /10\). The previous lemma implies that

$$\begin{aligned} \partial _t \tau _g \le -160 m^2R_g^{-6}\tau _g+80m^2\delta R_g^{-6}-c\partial _t{\mathcal {W}} \end{aligned}$$

for any time \(t\in [0,T^*]\). Hence, by integration and the excess estimate Lemma 2.8, we infer that

$$\begin{aligned} 3\delta /10 = \tau _g(T^*)-\tau _g(0)\le \delta /10 +80m^2R_g^{-6}\int \nolimits _0^{T^*}(-2 \tau _g+\delta )<\delta /10, \end{aligned}$$
(70)

which is of course a contradiction. Hence, no such time \(T^*\) exists and Theorem 2.2 gives long time existence and convergence of a subsequence to a surface \(\Sigma ^*\) of Willmore type satisfying (14). Applying Lemma 4.3 to the stationary surface \(\Sigma ^*\), it follows that \(\Sigma ^*\) must be strictly mean convex. We may then further decrease \(\delta \) such that the uniqueness statement of Theorem 2.1 can be applied and it follows that \(\Sigma ^*\) is part of the foliation constructed in [24]. From this, full convergence follows. \(\square \)

Applying the previous result to a small \(W^{2,2}\)-perturbation of one of the leaves \(\Sigma _\lambda \), we obtain the following:

Corollary 5.4

The leaves \(\Sigma _\lambda \) are strict local area-preserving maxima of the Hawking mass with respect to the \(W^{2,2}\)-topology. More precisely, there exists \(\tau _0,\Lambda >0\) depending only on \(m,\eta \) such that for any \(\lambda <\Lambda \) and any surface \(\Sigma \) enclosing \(\Sigma _\Lambda \) and satisfying \(\tau _e\le \delta \) as well as \(|\Sigma |=|\Sigma _\lambda |\) there holds \(m_H(\Sigma )\le m_H(\Sigma _\lambda )\). Equality holds if and only if \(\Sigma =\Sigma _\lambda \). Moreover, if \(\text {Sc}\ge 0\), then there holds \(m_H(\Sigma )\le m\) with equality if and only if \(\Sigma =\Sigma _\lambda \) and the non-compact component of \(M\setminus \Sigma _\lambda \) is isometric to the Schwarzschild manifold with mass m and a solid, centred ball removed. In particular, \(\Sigma _\lambda \) must be a centred sphere.

Proof

Fix \(\epsilon >0\) and \(\Lambda >0\) sufficiently small such that the previous theorem can be applied with \(R_0:=r_{\min }(\Sigma _\Lambda )\). Let \(\delta =\delta (\epsilon ,R_0,\eta ,m)>0\) be the constant from the previous lemma and \(\Sigma \) be a surface enclosing \(\Sigma _\Lambda \). As \(\Sigma _\Lambda \) is strictly mean convex and star-shaped (see [24]) it follows that \(|\Sigma |=|\Sigma _\lambda |\) for some \(\lambda <\Lambda \). If \(|\mathring{A}|^2_{L^2(\Sigma )}\ge \epsilon \) it follows from the integrated Gauss equation (15) that after possibly reducing \(\Lambda \) (and thus increasing \(R_0\)) there holds \(m_H(\Sigma )<0\). In the other case, we can apply the previous lemma and note that the area preserving Willmore flow increases the Hawking mass unless \(\Sigma \) is a surface of Willmore-type to deduce that \(m_H(\Sigma )\le m_H(\Sigma _\lambda )\) with equality if and only if \(\Sigma =\Sigma _\lambda \). If the scalar curvature is non-negative, we can use the inverse mean curvature flow (see [16]) starting at \(\Sigma _\lambda \) to show that \(m_H(\Sigma _\lambda )\le m\). If equality holds, the rigidity statement in [16] readily implies that \(\Sigma _\lambda \) must be a centred sphere in the Schwarzschild manifold. \(\square \)

Remark

It would of course be desirable to know if, and if so, in what sense, the leaves \(\Sigma _\lambda \) are global maximizers of the Hawking mass. Even in the exact Schwarzschild space, this seems to be a difficult problem: Connecting several spheres close to the horizon by small catenoidal handles, one can construct centred spherical surfaces with arbitrarily large Hawking mass which are additionally homologous to the horizon. Hence, one cannot expect any maximizing property without excluding a large compact set. On the other hand, a straight forward computation reveals that the Hawking mass of the sphere \(S_R(R^\beta e_3)\) tends to m as \(R\rightarrow \infty \) for any \(1/2<\beta <1\). Such surfaces eventually avoid any compact set and become totally off-centred.