The Area Preserving Willmore Flow and Local Maximizers of the Hawking Mass in Asymptotically Schwarzschild Manifolds

We study the area preserving Willmore flow in an asymptotic region of an asymptotically flat manifold which is C3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^3$$\end{document}-close to Schwarzschild. It was shown by Lamm, Metzger and Schulze that such a region is foliated by spheres of Willmore type, see (Lamm et al. in Math Ann 350(1):1–78, 2011). In this paper, we prove that the leaves of this foliation are stable under small area preserving W2,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{2,2}$$\end{document}-perturbations with respect to the area preserving Willmore flow. This implies, in particular, that the leaves are strict local area preserving maximizers of the Hawking mass with respect to the W2,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{2,2}$$\end{document}-topology.


Introduction
Let (M, g) 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 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] where is a compact surface bounding a region 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 η and ADM mass m if in the chart at infinity, there holds g = g S + h, where h is a symmetric two tensor satisfying for any 0 ≤ j ≤ k. Here, g S is the Schwarzschild metric with mass m, ∂ 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 or the area of its boundary . 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 . 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 W which is defined to be 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). Here, a surface of Willmore type is a critical point of the area prescribed Willmore energy. More precisely, every λ satisfies the equation 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 λ → 0. Given the results obtained for the isoperimetric problem, one is tempted to believe that in an asymptotic region, the leaves λ 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 ADMmass 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 λ in the foliation constructed in [24] in asymptotically Schwarzschild spaces regarding the maximization of the Hawking mass. In fact, we show the following: 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 (M, g) be C 3 -close to Schwarzschild with decay coefficient 0 < η ≤ η 0 (m) and mass m > 0 and let be an embedded sphere which is obtained as a small area-preserving W 2,2 -perturbation of a leave of the foliation { λ } from Theorem 1.1. Then the area preserving Willmore flow starting at exists for all times and converges smoothly to one of the leaves in the foliation { λ }.
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 2projection 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 (M, g) 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 η > 0. More precisely, we assume that there is a compact set K such that M \ K is diffeomorphic to R 3 \ B σ (0) for some σ > m/2 and that the following estimate holds on M \ K : Here, r denotes the radial function of the asymptotic chart R 3 \ B σ (0), ∇ g the gradient of the ambient space and Rm g and 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 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 (M, g). Finally, we assume that the scalar curvature of (M, g), denoted by Sc g , satisfies Such manifolds were called asymptotically Schwarzschild in [24] and we will adopt this terminology from now on. The Schwarzschild manifold (M S , g S ) = (R 3 \{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: where f = (2 − φ)/φ 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 We consider an immersed, closed and orientable surface ⊂ M and denote its first fundamental form by γ , its connection by ∇, its outward normal by ν, its second fundamental form by A, the traceless part byÅ, the mean curvature by H and the area element by dμ. Moreover, we denote the induced curvature by Rc and Rm , respectively. can also be regarded as an embedded surface in (R 3 \ B σ (0), g e ) or 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, η in a nondecreasing 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 , respectively. As all of our results concern surfaces which are contained in the asymptotic region, we will always assume that r min ≥ R 0 for some positive constant R 0 (η, m) which is to be determined. We let x be the position vector in the asymptotic region. If is contained in the asymptotic region, we define the Euclidean barycentre, approximate radius and centring parameter by respectively, and reiterate these definitions for the metrics g and g S . 1 The Hawking mass m H of a surface is defined by where W denotes the Willmore energy, that is, 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 for some scalar parameter λ. 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. 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 { t |0 ≤ t ≤ T }, 0 = flows by the area preserving Willmore flow with initial data if it satisfies the following evolution equation: where the Lagrange parameter λ is given by 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 ζ , there holds On the other hand, integration by parts reveals that Plugging in ζ = W + λH , integrating by parts and using the identity where we used Hölder's inequality in the last step. Similarly, where we used (11). In [ 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 is admissible if it is an embedded sphere satisfying the following conditions: for some constants , δ > 0. Moreover, we say that an area preserving Willmore flow is admissible up to time T > 0 if every t with 0 ≤ t ≤ 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 be an embedded sphere satisfying r min ≥ R 0 for some R 0 (m, η) sufficiently large. There is a universal constant c such that as well as

By conformal invariance, there holds
Finally, there holds Admissible surfaces enjoy various properties: Müller and de Lellis showed (see [13,14]) that in the Euclidean space, a surface 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 . Using the conformal parametrization, geometric quantities on S and 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.
Then is a sphere and there exists a universal constant c independent of and a conformal parametrization ψ : where id : S → R 3 is the identity, E the conformal factor of ψ and ν e ,ν e the outward normals of ψ and id, respectively. Moreover, the Sobolev embedding theorem implies

and one easily obtains
Using the previous two lemmas, we can relate r min , r max , R e provided , δ are chosen sufficiently small in (14): There are constants υ, τ 0 > 0 such that provided |Å e | L 2 < υ and τ e ≤ τ 0 . In particular, , δ in (14) can be chosen such that the previous estimate is true for any admissible surface.
We also need to compare the quantities | |, R, τ, a with respect to the different background metrics.

Lemma 2.6 Let be an admissible surface. Then we have the following estimates
Another useful tool is an integrated version of the Gauss equation. More precisely, any sphere satisfies where G = Rc − 1 2 gSc is the Einstein tensor. It follows from (15), Lemma 2.5 and (5) that any admissible surface enjoys uniform bounds on |H | L 2 ( ) and |A| L 2 ( ) , that is, 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 ≥ R 0 for some R 0 (η, m) sufficiently large, then any smooth function ω satisfies 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.
, be an admissible Willmore flow starting at a surface 0 = which satisfies |Å| L 2 ( ) < /2, τ e < δ/2 and r min ≥ R 0 for some constant > 1. If R 0 and are sufficiently large, then the following estimates hold: Proof Lemma 2.5 and Lemma 2.6 imply the crude estimate G(ν, ν)dμ ≤ cR −1 e . Using (15) and the fact that the flow decreases the Willmore energy we obtain Hence, the claim follows if R 0 is sufficiently large such that cR −1 e ≤ cR −1 0 < /4. For the second claim, we trivially estimate For the last claim, we note that according to Lemmas 2.5 and 2.6, there holds r , the claim now follows if is chosen sufficiently large.
Remark According to a classical inequality by Li and Yau ( [25]), embeddedness is automatically implied by the smallness of |Å| 2 L 2 ( t ) . It follows from the previous lemma that a flow can only cease to be admissible if τ e reaches δ. In order to study the evolution of τ 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 is spherical, that the conclusion of Lemma 2.5 holds, that r min ≥ 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 by ∇ and the connection of the ambient space by ∇. 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: Let ω ∈ k ( ) for some k ∈ N, p ∈ and e 1 , e 2 be and orthonormal frame of at p. Let X i ∈ {e 1 , e 2 }, 1 ≤ i ≤ k. There holds We would now like to use the previous lemma to express certain geometric quantities in terms of the Willmore operator W defined by W = H + H Rc(ν, ν) + H |Å| 2 . To this end, we first need the following lemma.

Lemma 3.2
The following identities hold: Proof With the convention the orthonormal frame satisfies Chosing ω = A in Lemma 3.1 we obtain as ∇ν is tangential. We clearly have ∇ * Q(X 1 , A straight forward calculation gives Evaluated at (X 1 , X 2 ), this yields (with slight abuse of notation) If we furthermore assume that the e i are principal directions, we obtain 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 the right-hand side becomes (again with abuse of notation) where we used thatÅ is trace free. Together with (22) this clearly implies As ∇γ = 0, there holds (γ H ) = γ H and we obtain the first claim. Choosing ω = ∇ H , we find Q ∇ H = 0 by the symmetry of second derivatives and evaluating at X 1 = e 1 , we obtain (again, with abuse of notation) as ∇ * Å = − 1 2 ∇ H + Rm * 1. This implies the second and third claim. Finally, if ω = ∇Å, then at (X 1 , X 2 , X 3 , X 4 ), we have This gives the very rough identity Remark In fact, one can actually show the more precise identitẙ see (1.7) in [24]. This can be seen by using the identity ∇ * Å = − 1 2 ∇ H + Rc T (·, ν). Integrating by parts and using the definition of W , we obtain The claimed inequality now follows from combining these estimates, absorbing the |∇Å| 2 term and using |∇ A| 2 ≤ c|∇Å| 2 + cR −6 e . Finally, we clearly have |∇ H | ≤ c|∇ A|.

Lemma 3.4 Under the assumptions of the previous lemma, there holds
Proof Multiplying (18) From the estimate |∇ A| 2 ≤ c|∇Å| 2 + cR −6 e and (17) it follows that The second and third row can be estimated by Using partial integration and the identity −∇ * Å = 1 2 ∇ H + Rm * 1, the first term in the first row can be computed to be In the last equation, the third term can be estimated as before and the second term can be estimated by Finally, we note that Combining all these inequalities, choosing κ > 0 sufficiently small, absorbing all terms when possible and noting that We now need the following Sobolev-type interpolation inequality.

Lemma 3.6
Let be a compact surface satisfying r min ≥ R 0 . If R 0 is chosen sufficiently large, then any smooth k-form ω satisfies Proof The proof of Theorem 5.6. in [22] caries over to our setting as 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].
We also need the following multiplicative Sobolev inequality.

Lemma 3.7 Under the assumptions of the previous lemma we have
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.
At this point, a small curvature assumption allows us to absorb the term on the lefthand side of the previous equation in Lemma 3.5. This finally yields an L ∞ -estimate forÅ.
In particular, we have Proof This is a direct consequence of the previous three lemmas.
It turns out that the integral curvature estimates also imply an improved estimate for |Å| L 2 ( ) .

Lemma 3.9 Under the assumptions of Lemma 3.3 there exists a constant
Proof Integrating (23) and using integration by parts as well as the identity ∇ * Å = − 1 2 ∇ H + Rc(ν, ·) T yields Next, using Lemma 2.7 and Hölder's inequality, we obtain Hence, this term can be absorbed. Again with the Michael-Simon-Sobolev inequality and Hölder's inequality, we get From this the claim follows.
Finally, we prove two useful W 2,2 -and L ∞ -estimates for the mean curvature.

Lemma 3.10 Under the assumptions of Lemma 3.3 there holds
Proof We multiply (19) by ∇ H to obtain the Bochner-type identity In order to state the L ∞ -estimate, we introduce the quantity 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
c . Now, Lemma 3.6 and Lemma 3.10 as well as |H S | ≤ cR −1 e imply that The third term can be absorbed using (26) and estimating |Å| 2 L 2 ( t ) ≤ . Another application of (26) yields Using this to estimate the second term, the claim follows.

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 t be an admissible area preserving Willmore flow. Then every t satisfies
Here, Proof Using Lemma 2.3, we may assume that η = 0. Let e i be an orthonormal frame of T p t at a point p. There holds 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 By Young's inequality and Lemma 2.3, this error can be further estimated via Now, for a round sphere, there holds∂ r =r −1 (a e + R eνe ). Hence, (∂ r −∂ r ·ν eνe ) · ∂r =r −2 (a e + R eνe − a e ·ν eνe − R eνe ) · (a e + R eνe ) =r −2 (a e · a e − (a e ·ν e ) 2 ).
As this term can be estimated by cτ 2 e , the first claim follows. The second claim is a straight-forward application of Lemmas 2.4 and 2.6.
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.
Proof Lemma 3.8 implies that Next, using (26) and R −1 0 ≤ we obtain Integrating by parts and using Young's inequality, we conclude that Now we use these estimates to obtain an estimate for | H | L 2 ( t ) . Recall that (c.f. Integrating by parts, using (29), (16), |Rc| ≤ cR −3 e and (28), we obtain t W Hdμ Again by (28), |Rc| ≤ cR −3 e and the definition of W we find Combining (30), (31) and (32), we find provided is sufficiently small. Returning to (30) and (31) we conclude 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), |Rc| ≤ cR −3 e and the first and third estimate to conclude that Next, in the situation of Lemma 3.9 we apply (25) and the estimate valid for any κ > 0, together with (26). Absorbing the |Å| 2 L 2 ( t ) 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.
As promised, we now prove the refined a-priori estimates.
Proof First, let us recall that the asymptotic behaviour of the metric implies that |Rc| ≤ cR −3 e as well as |Rc − Rc S | ≤ cR −4 e . Moreover, according to Lemma 2.3, there holds |ν − ν S | ≤ cR −2 e as well as |dμ − dμ S | ≤ cR −4 e while (16) states that |H | L 2 ( t ) ≤ c. We will use these estimates at various points without explicitly stating them. There holds We denote the 12 terms in the last equation by the Latin numbers I − X I I . There holds where we used Lemma 4.2 in the last inequality. We now focus on I V and X . We replace Rc by Rc S , ν by ν S and dμ by dμ S . This results in error terms that can be estimated by In the last inequality, we used the crude estimate |∇ H | 2 L 2 ( t ) ≤ c| H | L 2 ( 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 → t by ψ and indicate the respective geometric quantities of S by a tilde. According to Lemma 2.4 we have |r −3 as well as |∂ r −∂ r | ≤ |Å e | L 2 ( t ) while Lemma 2.5 and Taylor's theorem imply that |φ −6 −1| ≤ cR −1 e . Since |ν e − ν e | ≤ c|∇ e ψ −∇ e Id| we find Consequently, it follows from Hölder's inequality and Lemma 2.4 that replacing Rc S (ν S , ν S )dμ S by m R −3 e (1 − 3(∂ r ·ν e ) 2 )dμ e in I V and X results in error terms that can be estimated by In the first inequality, we used Young's inequality, Hölder's inequality and the esti- In the second inequality we used the estimate as well as Lemma 2.3 to replace |Å e | L 2 ( t ) by |Å| L 2 ( t ) . Performing the same two procedures with V I yields an error term that can be estimated by In the first inequality we used Young's inequality as well as the estimate |H | 2 In the second inequality we used Young's inequality again and the fact that |Å| L 2 ( t ) + τ e + R −1 e is bounded. In the third inequality we used Young's inequality one more time and Lemma 4.
. Performing this procedure on X I I yields a similar error term. On the other hand, there holds∂ r ·ν e = r −1 (R eνe + a e ) ·ν e . Again,r −1 can be replaced by R −1 e and since τ e = |a e |/R e , we find that Integrating by parts, we find that Combining this with (36), (37) and (39) we obtain In a similar way we can use (38) to find Here, we estimated τ e R −3 e |HÅ| 2 L 2 ( t ) ≤ cτ 2 e R −6 e + c|Å| 4 L ∞ ( t ) and then used Lemma 4.2. In I I and V I I I we first replace every H by H S . In light of Lemma 4.2, the error can be estimated by From this we find that In the second inequality, we used the Poincare inequality with zero mean. In the third inequality, we expressed ∇ in terms of ∇ and A, see (21). In the fourth inequality, we used (25) and the estimate |(∇Rc)(ν, ν)| 2 L 2 ( t ) ≤ c|Å| 2 L 2 ( t ) R −5 e + cτ 2 e R −6 e + cR −7 e , which can be shown in the same fashion as (25). Finally, we used |A| 2 and estimated these terms in the usual way. Combining (34),(35),(40),(41) and (42), we finally obtain In light of the inequality which follows from (26) and the divergence theorem, we can eventually absorb the second term on the right-hand side in (43) to obtain Next, using Lemma 3.9, (25) and Young's inequality we find for any κ > 0 In the last step, we have used (26), and (45). Absorbing yields the claimed estimate for |Å| 2 L 2 ( t ) . Reinserting into (45) gives the claimed estimate for | H | 2 L 2 ( t ) . This then implies as claimed. Finally, we recall the definition of λ, see (9). We have Using the same methods as before, the last term can be computed explicitly to give The claim follows.

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 τ g . In the next Lemma, we show that the evolution of τ g is linked to the translation sensitivity of W in terms of the Schwarzschild background. We note that the symbol "· indicates the Euclidean inner product.

Lemma 5.1 Let t be an admissible area preserving Willmore flow. Then the following holds
where b g = a g /|a g |. Moreover, we have Proof The first identity is a straight forward computation using the flow equation (8) and the fact that H W + λH 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 ))ν. To this end, we first replace H by H S in the second line of (46) which according to (26) yields an error term that can be estimated by Similarly, Lemmas 2.4 and 2.3 imply that replacing (x − a e ) by (x − a e ) = R eνe and thenν e by ν e results in error terms that can be estimated by and then ν S = φ −2 ν e by ν leads to an error that can be estimated by Here, we also used Lemma 2.6. We observe that Therefore, we obtain Moreover, using Lemma 4.3 and Young's inequality, we estimate Combining (48)-(53) shows that 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 |dμ−dμ S | ≤ cη R −2 e dμ, |ν − ν S | ≤ cη R −2 e as well as |H − H S | ≤ cη R −3 e we find Using Lemma 4.3, this error can be estimated via Next, we have According to Lemma 2.3 there holds . From this, it follows that replacing |Å| 2 by |Å S | 2 in the previous equation yields an error that can be estimated by In the first inequality, we used the crude estimate |H S | ≤ cR −1 e +|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, Using the asymptotic behaviour of Rc, ν, dμ, it is easy to see that Regarding the last term, we note It is then straightforward to see that any smooth function u satisfies the estimate | S u − u| ≤ cη(|∇ 2 u|R −2 e + |∇u|R −3 e ). Hence replacing by S results in an error that can be estimated by where we used Lemmas 4.2 and 4.3. Integrating by parts, we obtain There holds ∇ S ν S ≤ cR −2 e +|A S | ≤ cR −2 e +c|A|. Hence, using Lemma 2.3 to replace ∇ S H by ∇ S H S gives an error that can be estimated by According to Lemmas 3.3 and 4.3 there holds We conclude The claim then follows from (54), (55), (56), (57) and (59).
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 ∂ t τ g up to an error. We denote the geometric quantities of as usual, the geometric quantities of S := S a e (R e ) are denoted using a tilde.
In the second equation, we used (5). In the third equation, we used the fact that 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 , the following relation holds where D denotes the conformal killing operator. 3 . One then picks X = b g which is a conformal killing vector field and to be the region bounded by 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 , ∂ r by∂ r and theñ ∂ r = r −1 (ν e R e + a e ) by ν e , yielding errors that can, with the help of the estimate The final equality follows from the translation invariance of the Euclidean area and the translation invariance of the Euclidean volume, which imply, respectively, that t H e b g · ν e dμ e = t b g · ν e dμ e = 0.
The term 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 3 There is an easy way to see this using the potential function f = (2 − φ)/φ and the fact that g S is static, compare (4). In fact, there holds div S Rc S (X , Commuting derivatives and using S f = 0 we find −∇ * S ∇ 2 S f (X ) = Rc S (x, ∇ S f ). Using the static equation one more time it follows that div S Ric S (X , ·) = 1 2 g S (D X , f −1 ∇ 2 S f ) and (64) follows from the divergence theorem.
In the second line, we expanded φ −1 up to order r −1 and used Lemma 2.4. We then estimate using Lemma 4.3 Combining (60)-(66), we have shown that Due to the rotational symmetry of g S , it is easy to see that the function u(a) := S Re (a) H 2 S dμ S only depends on τ e and is in fact analytic in τ e . Moreover, there holds d/ds| s=0 τ e = b g · b e R −1 e where b e = a e /|a e |. It therefore suffices to compute u(a e ) up to terms of order R −2 e . Dropping the tilde notation and writing S = S R e (a e ), we have After a rotation, we may assume that a e = (0, 0, |a e |). We choose the parametrization (θ, ϕ) → a e + R e (sin θ sin ϕ, sin θ cos ϕ, cos θ) and compute the quantities ν e = (sin θ sin ϕ, sin θ cos ϕ, cos θ), Furthermore, one can check that and consequently ∂ r · ν e = r −1 (R e + |a e | cos θ) = (r 2 + R 2 e (1 − τ 2 e ))/(2r R e ).
Next, we have dμ e = R 2 e dϕdθ and since there is no ϕ dependence, integration of ϕ solely adds a factor of 2π . Finally, we can substitute θ → r where the area element transforms via dθ/dr = −r /(R e |a e | sin θ) = −r /(R 2 e τ e sin θ) and the boundary data are mapped to (R e −|a e |, R e +|a e |) = (R e (1−τ e ), R e (1+τ e )) in an orientation reversing way. This gives where the last equality follows from Taylor's theorem, provided δ is chosen sufficiently small. Hence from the analyticity, it follows that where we used that b g · b e = 1 + O(R −1 e ). On the other hand, it is easy to see that for any vector b ⊥ e perpendicular to b e , one has In a similar fashion as above, we thus find where we used Lemma 4.3 in the last inequality. Combining (67)-(69) and replacing τ e by τ g , the claim follows as 3(64 − 32/3) = 160.
Proof First, we choose δ, , R −1 0 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 τ e reaches δ. We assume that τ e ( ) = δ/2. We would then like to show that τ e ≤ δ for all times. For this to hold, it is enough to show that τ g ≤ 9δ/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 τ g (T * ) = 9δ/10. We may assume that τ g (0) = 6δ/10 and that τ g (t) > 6δ/10 for all t > 0. Additionally, we require c(δ 2 + + η) ≤ 80m 2 δ as well as c < δ/10. The previous lemma implies that for any time t ∈ [0, T * ]. Hence, by integration and the excess estimate Lemma 2.8, we infer that 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 * of Willmore type satisfying (14). Applying Lemma 4.3 to the stationary surface * , it follows that * must be strictly mean convex. We may then further decrease δ such that the uniqueness statement of Theorem 2.1 can be applied and it follows that * is part of the foliation constructed in [24]. From this, full convergence follows.
Applying the previous result to a small W 2,2 -perturbation of one of the leaves λ , we obtain the following: Proof Fix > 0 and > 0 sufficiently small such that the previous theorem can be applied with R 0 := r min ( ). Let δ = δ( , R 0 , η, m) > 0 be the constant from the previous lemma and be a surface enclosing . As is strictly mean convex and star-shaped (see [24]) it follows that | | = | λ | for some λ < . If |Å| 2 L 2 ( ) ≥ it follows from the integrated Gauss equation (15) that after possibly reducing (and thus increasing R 0 ) there holds m H ( ) < 0. In the other case, we can apply the previous lemma and note that the area preserving Willmore flow increases the Hawking mass unless is a surface of Willmore-type to deduce that m H ( ) ≤ m H ( λ ) with equality if and only if = λ . If the scalar curvature is non-negative, we can use the inverse mean curvature flow (see [16]) starting at λ to show that m H ( λ ) ≤ m. If equality holds, the rigidity statement in [16] readily implies that λ must be a centred sphere in the Schwarzschild manifold.
Remark It would of course be desirable to know if, and if so, in what sense, the leaves λ 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 β e 3 ) tends to m as R → ∞ for any 1/2 < β < 1. Such surfaces eventually avoid any compact set and become totally off-centred. and by Simon's identity A = ∇ H + P 1 This should be compared to Proposition 3.5 in [22]. The next step is then to rewrite the terms ∇ k Rc (ν) in terms of ambient derivatives of Rc and derivatives of the second fundamental form. The resulting terms can then be estimated and sometimes absorbed using Young's inequality. Moreover, lower order terms are estimated by their L ∞norm. This process is quite lengthy but rather straight-forward and results in the following evolution inequality.
This should be compared to Proposition 4.5 in [22]. A similar evolution inequality can also be derived for m = 0; however in this case, a delicacy appears: The ultimate goal is to interpolate between the highest order term and |A| 2 L 2 . Hence, we cannot make use of L ∞ -estimates in the evolution equation for m = 0. To circumvent this problem, one uses the Michael-Simon-Sobolev inequality and absorbs the higher order terms created by this procedure. This is very similar to the proof of Lemma 4.3 in [22]. It is only possible to absorb the higher order term if one assumes a small energy condition for some constant 0 > 0. In fact, one obtains the following lemma.
As can be seen in Proposition 4.4. in [22], the same phenomenon appears in the Euclidean case and is consequently not caused by the effects of the ambient curvature.
A-priori estimates Under the small curvature assumption, the integral curvature estimates can be turned into L ∞ -estimates by Gronwall's inequality and by Sobolev-type interpolation estimates in the spirit of Lemma 2.8 in [21]. Arguing essentially as in Proposition 4.6 in [22] while keeping track of the additional terms arising from the ambient curvature and the Lagrange parameter λ one obtains Lemma A.5 If |A| 2 L 2 ( t ∩{η>0}) < 0 holds, then for any k ∈ N there exist constants C,C depending on k, T , i , sup t∈(0,T ) |η > 0|, the estimates for the ambient curvature, |λ| L 2 ((0,T )) and |∇ i A| L 2 ( 0 ∩{η>0}) for i ≤ k and i ≤ k + 2, respectively, such that Similarly, interior estimates without the dependence on the initial data can be proven, c.f. Theorem 3.5 in [21].
Lemma A. 6 If |A| 2 L 2 ( t ∩{η>0}) < 0 then for any k ∈ N and any t 0 ∈ (0, T ) there exists a constant C depending on k, t −1 0 , T , i , sup t∈(0,T ) |η > 0|, the estimates on the ambient curvature and |λ| L 2 ((0,T )) such that Curvature concentration at a singularity In the next step, it is shown that a singularity can only occur, if the curvature concentrates around one point, a condition that will be specified later. To this end, one notes that a singularity can only occur if |λ| L 2 ((0,T )) or |A| C k ( t ) blows up for some k ∈ N, because otherwise the flow converges smoothly as time approaches T and can then be restarted. Assuming for now that |λ| L 2 ((0,T )) is bounded, by Lemma A.5 one obtains uniform curvature estimates if the small curvature condition is satisfied on every ball of a small radius ρ > 0. For a small ρ, this is certainly the case at t = 0, one can in fact assume that t ∩B ρ |A| 2 dμ < 1 2 0 for each ball of radius ρ. Integrating the inequality in Lemma A.4, one sees that the curvature can only concentrate after an amount of timeT depending on ρ, the ambient curvature bounds and |λ| L 2 ((0,T )) . The last condition seems somewhat restrictive; however, the point is to apply the above to a time close to the singular time. Namely, if |λ| L 2 ((0,T )) is bounded, then |λ| L 2 ((t,T )) approaches zero as t → T . From this, one can then deduce that a singularity can only occur if the curvature concentrates in a ball of arbitrarily small radius, as otherwise the flow could be continued past the singular time. More precisely, similarly to Theorem 1.2 in [22] we have Lemma A.7 Let λ ∈ L 2 ((0, T )) and T < ∞. Then there exists δ > 0, radii ρ i → 0 and times t i → T such that essentially uses again that the original flow stays in a compact region). In the same way, the L 2 -estimate for λ from the previous subsection implies that the rescaled Lagrange parameter satisfies |λ i | L 2 ([−1,ξ ]) → 0. The rest of the argument is then essentially the same as in [21]: one obtains uniform gradient estimates and shows that the rescaled flow converges to a stationary Willmore immersion in R 3 . While the L 2 -estimates for λ required some additional work, a lower bound on the area of the flow is automatic (contrary to the normal Willmore flow, see Theorem 5.2 in [21]) which implies that the limiting surface is non-compact. Moreover, the curvature around the point of concentration is not lost in the limit which implies that the limiting surface is not a plane.
We therefore obtain Lemma A.8 If a curvature concentration occurs, the surface ρ −1 i F( , t j ) converges locally uniformly to a non-compact Willmore immersionˆ in R 3 which is not a plane.
The same reasoning can also be applied if the curvature concentration occurs at T = ∞. Now, if one additionally requires that W( ) ≤ 8π − ρ holds for some ρ > 0, then for R 0 sufficiently large, this inequality also holds for the Euclidean Willmore energy. Moreover, the inequality is also true for the Euclidean Willmore energy ofˆ . Then, by a spherical inversion, one obtains an embedded compact Euclidean Willmore surface with a point singularity and Euclidean Willmore energy less than 8π . According to Lemma 5.1 in [23], the point singularity can be removed. But then by the classification of genus 0 Willmore surfaces by Bryant, see [7], the inversion ofˆ has to be a round sphere. This, however, implies thatˆ is a plane, a contradiction. From this we deduce the following result. Lemma A.9 Let W( ) < 16π − ρ and R 0 be sufficiently large. Then an area preserving Willmore flow enclosing the Ball B R 0 (0) for all times cannot develop a singularity at any finite or infinite time.
Finally, the previous lemma gives uniform curvature estimates for all times and then a standard compactness argument implies the subsequential convergence to a surface of Willmore type.