Some rigidity results for the Hawking mass and a lower bound for the Bartnik capacity

We prove rigidity results involving the Hawking mass for surfaces immersed in a $3$-dimensional, complete Riemannian manifold $(M,g)$ with non-negative scalar curvature (resp. with scalar curvature bounded below by $-6$). Roughly, the main result states that if an open subset $\Omega\subset M$ satisfies that every point has a neighbourhood $U\subset \Omega$ such that the supremum of the Hawking mass of surfaces contained in $U$ is non-positive, then $\Omega$ is locally isometric to Euclidean ${\mathbb R}^3$ (resp. locally isometric to the Hyperbolic 3-space ${\mathbb H}^3$). Under mild asymptotic conditions on the manifold $(M,g)$ (which encompass as special cases the standard"asymptotically flat"or, respectively,"asymptotically hyperbolic"assumptions) the previous quasi-local rigidity statement implies a \emph{global rigidity}: if every point in $M$ has a neighbourhood $U$ such that the supremum of the Hawking mass of surfaces contained in $U$ is non-positive, then $(M,g)$ is globally isometric to Euclidean ${\mathbb R}^3$ (resp. globally isometric to the Hyperbolic 3-space ${\mathbb H}^3$). Also, if the space is not flat (resp. not of constant sectional curvature $-1$), the methods give a small yet explicit and strictly positive lower bound on the Hawking mass of suitable spherical surfaces. We infer a small yet explicit and strictly positive lower bound on the Bartnik mass of open sets (non-locally isometric to Euclidean ${\mathbb R}^{3}$) in terms of curvature tensors. Inspired by these results, in the appendix we propose a notion of"sup-Hawking mass"which satisfies some natural properties of a quasi-local mass.


Introduction
The goal of this paper is to prove some rigidity results involving the Hawking mass for surfaces immersed in an asymptotically flat (resp. asymptotically hyperbolic; and some suitable milder asymptotic assumptions), 3-dimensional, complete Riemannian manifold (M, g) with non-negative scalar curvature (resp. with scalar curvature bounded below by −6). Let us start by recalling some motivations for studying such a geometric setting, coming from Mathematical General Relativity.

Some background and motivation
In the framework of Mathematical General Relativity, a Riemannian manifold (M 3 , g) with non-negative scalar curvature Sc g ≥ 0 represents a "time-symmetric space-like slice" of a 4-dimensional space-time satisfying the so-called Dominant Energy Condition (DEC for short). The condition Sc g ≥ 0 is indeed a consequence of the DEC, coupled with the property of being a "time-symmetric space-like slice" (i.e. M has vanishing second fundamental form as a hypersurface in the ambient 4-dimensional space-time) satisfying the Einstein Constraint Equations.
The relationship between the notion of mass in General Relativity and the geometry of a manifold has been studied extensively in recent times. Let us recall some basic notation and fundamental results. First of all, let us mention that, unless otherwise specified, a manifold is allowed to have non-empty boundary. Definition 1.1 (Asymptotically flat Riemannian manifold with horizon boundary). A complete Riemannian manifold (M 3 , g) is said to be asymptotically flat (AF for short) if there exists a compact subset K ⋐ M such that M 3 \K is diffeomorphic to R 3 \B 1 (0) via a map Ψ which induces a system of coordinates at infinity, and in such coordinates the metric is written as for some τ > 1/2 and all multi-indices α, with |α| = 0, 1, 2, 3. We also require that the scalar curvature Sc of (M, g) is integrable. Moreover, if the boundary of M is non-empty, we assume that it is minimal and that there are no closed minimal surfaces intersecting the interior of M ; in this case we say that the manifold is asymptotically flat with horizon boundary.
Notice that in Definition 1.1 and throughout, we only consider one-ended AF manifolds with horizon boundary. For an AF Riemannian manifold (M, g) with horizon boundary as above there is natural notion of "total mass" of (M, g), well known under the name of ADM-mass (after Arnowitt-Deser-Misner [1]) and defined by (2) m ADM (M, g) = lim Bartnik [2] showed that m ADM (M, g) as defined in (2) is finite and independent of the chart at infinity.
The fundamental Positive Mass Theorem, proven first by Schoen-Yau [44] using minimal surfaces methods and then by Witten [52] via spinorial techniques, states that if (M 3 , g) is an AF Riemannian manifold with non-negative scalar curvature, then m ADM (M, g) ≥ 0 and equality holds if and only if (M 3 , g) is isometric to the Euclidean R 3 . Another landmark example of the interplay between mass and geometry is the Riemannian Penrose Inequality, proved independently by Huisken-Ilmanen [15] and Bray [4]. The Riemannian Penrose Inequality is a strengthening of the lower bound on the ADM mass in case of horizon boundary of an AF manifold with non-negative scalar curvature; namely, m ADM (M, g) ≥ |∂M| 16π .
In this paper, we further investigate the relationship between geometry and mass. More precisely between the geometric conditions of non-negative scalar curvature and asymptotic flatness (and some generalisations of it) on the one hand, and two notions of quasi-local mass: those of Hawking [11] and Bartnik [3]. Let us recall that, while in Newtonian gravity it is possible to define the mass contained in a region Ω simply by integration of a "mass density function" over Ω, defining a corresponding concept in the setting of General Relativity is much more subtle. Indeed, due to the Equivalence Principle, there is no pointwise notion of "gravitational energy density" (see for instance [48,Section 20.4] or [41]). Nevertheless there are several proposals for a notion of "quasi-local mass" contained in a region Ω (see for instance [22,Chap. 6] or [46]); we will focus on the Hawking [11] and Bartnik [3] quasi-local masses.
The Hawking mass Definition 1.2 (Hawking mass). Let (M 3 , g) be a Riemannian manifold with non-negative scalar curvature and let Σ ֒→ M be an immersed sphere. The Hawking mass of Σ is defined as: where |Σ| is the area, W (Σ) := Σ H 2 dV Σ is the Willmore functional and where we adopt the convention that the mean curvature H is the sum of the principal curvatures.
Evidently, if Σ is a minimal surface, then its Hawking mass is positive. It is also readily checked that the Hawking mass of a round sphere in R 3 is equal to zero, as its Willmore functional equals 16π. A classical inequality due to Willmore [51] asserts that the round sphere is the unique minimiser of the Willmore functional (up to scaling). Thus the Hawking mass of any surface in R 3 is less or equal to zero, with equality if and only if the surface is a round sphere. This fact is already suggesting that it could be appropriate to consider the supremum of the Hawking mass, over a suitable class of surfaces (see the Appendix 7.1 for an implementation of this idea for a notion of quasi-local mass). Nevertheless, Christodoulou and Yau [8] showed that the Hawking mass is non-negative for stable mean curvature spheres in 3-manifolds with non-negative scalar curvature (see also the more recent work [28] by Miao-Wang-Xie). The popularity of the Hawking mass is arguably due to the very powerful monotone property (Eardley, Geroch, Jang-Wald [16]) along the Inverse Mean Curvature Flow, which was key in Huisken-Ilmanen's [15] proof of the Riemannian Penrose inequality for a single black hole (see Bray [4] for the multiple black holes case and for a different proof).

The Bartnik mass
First of all let us mention that, after Bartnik [3] introduced the quasi-local mass named after him, several variants appeared in the literature. For convenience, here we adopt a version proposed in [4,15]. We refer to the recent [27,17] for reconciliation of some of the various versions and for a discussion of the topic.
Let (M 3 , g) be an AF Riemannian manifold (possibly with horizon boundary) with non-negative scalar curvature and let Ω ⊂ M be a bounded open set with smooth topological boundary ∂Ω. Recall that ∂Ω is said to be outer-minimising if P(Ω) ≤ P(Ω ′ ) for any set Ω ′ ⊂ M of finite perimeter (denoted with P(Ω ′ )) and finite volume such that Ω ⊂ Ω ′ . Define the Bartnik mass m B (Ω) of Ω (also known as Bartnik capacity of Ω) as where A is the set of AF manifolds (possibly with horizon boundary) with non-negative scalar curvature into which Ω isometrically embeds such that ∂Ω ⊂M is outer-minimising. Notice that the Positive Mass Theorem (or the Riemannian Penrose inequality, in case all elements in A have non-empty horizon boundary) immediately yields the non-negativity of the Bartnik mass. Another advantage of the Bartnik mass is that it satisfies a natural monotonicity property under inclusion (5) m B (Ω 1 ) ≤ m B (Ω 2 ) ∀Ω 1 ⋐ Ω 2 ⊂ M , with ∂Ω 1 outer-minimising in Ω 2 and ∂Ω 2 outer minimising in M .
Moreover, as a consequence of the proof of the Riemannian Penrose inequality via Inverse Mean Curvature Flow by Huisken-Ilmanen [15], it holds that for every ∂Ω ⊂ M outer-minimising in the AF manifold (M, g) with non-negative scalar curvature. Since every smooth extension (M ,g) ∈ A induces the same mean curvature on ∂Ω ⊂M , the inequality (6) combined with the very definition (4) implies (7) m B (Ω) ≥ m H (∂Ω).
Computing the Bartnik mass of a subset Ω is in general a non-trivial task. For some recent works in this direction, see for instance Mantoulidis-Schoen [25], CabreroPacheco-Cederbaum-McCormick-Miao [6] and Miao-Xie [29]: while it is clear from the definition that it is conceivable to expect upper bounds by direct comparison with somewhat explicit competitors, the issue of finding explicit lower bounds is more subtle. The latter is one of the goals of the present paper.

Statement of the main results
The first main result can be informally stated as follows: if the Hawking quasi-local mass is non-positive locally everywhere in an open set Ω having non-negative scalar curvature, then Ω is locally isometric to Euclidean R 3 . Below is the precise statement: Theorem 1.3 (Quasi-local rigidity Theorem 4.1). Let (M 3 , g) be a three-dimensional Riemannian manifold and let Ω ⊂ M be an open subset with non-negative scalar curvature. Assume that every p ∈ Ω \ ∂M admits a neighbourhood U ⊂ M \ ∂M such that Then Ω \ ∂M is locally isometric to Euclidean R 3 .
In order to obtain a global rigidity result out of the quasi-local rigidity Theorem 1.3, it is useful to add a suitable asymptotic condition at infinity. A fairly mild asymptotic assumption is the next one (that to the best of our knowledge has not appeared in the literature before). In order to state it, recall that a sequence {p n } ⊂ M is said to be diverging if, for some (and thus for any) fixedp ∈ M , we have d(p, p n ) → ∞ as n → ∞, where d is the Riemannian distance function induced by g. Definition 1.4. Let (M, g) be a complete Riemannian manifold. We say (M, g) is asymptotically locally simply connected (ALSC for short) if it is non-compact and for every R > 0, and any diverging sequence {p n } ⊂ M , there exists N = N (R) ≥ 1 such that the ball B g R (p n ) is simply connected, for every n > N (R). Remark 1.5. Note that the ALSC condition is satisfied by AF manifolds since, in this case, the balls B g R (p n ) are eventually diffeomorphic to Euclidean balls, which of course are simply connected. However, ALSC is a much milder condition than AF, as it merely requires a local (mild) topological control and no assumption on the metric tensor and on the global topology of M (outside of a compact set). Non AF examples of ALSC manifolds include for instance asymptotically conical manifolds, the Bryant soliton [5] and, more generally, C 0 -asymptotically locally Euclidean manifolds [33]. Theorem 1.6 (Global rigidity Theorem 4.3). Let (M 3 , g) be a connected, complete Riemannian manifold without boundary and with non-negative scalar curvature. If every p ∈ M admits a neighbourhood U satisfying the local non-positivity condition (8) for the Hawking mass, then (M 3 , g) is isometric to a flat space form. Furthermore, if (M 3 , g) is ALSC then it is globally isometric to Euclidean R 3 .
We mention again that, inspired by the above rigidity results, in the Appendix 7.1 we propose a notion of "sup-Hawking mass" which satisfies some natural properties of a quasi-local mass.
For the sake of the introduction, we confined the presentation to manifolds with non-negative scalar curvature. The reader interested in the extensions to the case of scalar curvature bounded below by a negative (or strictly positive constant) is referred to Section 6. Let us recall that such ambifent spaces are particularly relevant when the cosmological constant in Einstein's Equations is non-zero.
In order to prove the above rigidity results, we will compute accurate Taylor expansions of the Hawking mass on suitable spherical surfaces of small area (see Section 1.3 for some idea of the methods, and Proposition 3.2 for the precise statement). In Theorem 5.7, we will show that such spherical surfaces are outer-minimising, and thus provide a lower bound on the Bartnik mass (thanks to the monotonicity property (5) and the bound (7)). As a consequence, we obtain the next lower bound on the Bartnik mass in terms of curvature tensors. Theorem 1.7 (Lower Bound on the Bartnik mass). Let (M 3 , g) be an AF Riemannian manifold with non-negative scalar curvature and with (possibly empty) horizon boundary ∂M . Let Ω ⊂ M be a bounded open set with smooth boundary ∂Ω. Let p ∈ Ω \ ∂M and letρ =ρ(p) := inf q∈∂M∪∂Ω d(p, q). Then, for all ρ ∈ (0,ρ/2) the following lower bound holds: where S := Ric − 1 3 Sc g denotes the trace-free Ricci tensor and lim sup ρ→0 + ρ −6 |O p (ρ 6 )| < ∞. In particular, if m B (Ω) = 0 then Ω \ ∂M is locally isometric to Euclidean R 3 . Remark 1.8. Notice that (9) gives a strictly positive (yet small) lower bound on the Bartnik mass of Ω, provided that • either: Ω contains a point p with Sc p > 0 • or: Sc ≡ 0 on Ω and there exists p ∈ Ω with S p = 0.
As observed in the proof of Theorem 4.1, if Ω has non-negative scalar curvature and it does not have constant sectional curvature, then one of the two conditions above must be satisfied. Remark 1.9. Note that the lower bound (9) is interesting only when Ω does not contain some connected component of ∂M . Indeed if Ω contains some connected component Σ of ∂M , then the Riemannian Penrose Inequality [4,15] yields that m B (Ω) ≥ |Σ| 16π . This would give a definite lower bound on m B (Ω) in contrast with the infinitesimal lower bound (9). However, if ∂M ∩ Ω = ∅, the lower bound (9) seems to be new and interesting.
Let us also mention the recent work [53] by Wiygul, where the first order Taylor expansion of the Bartnik mass is computed for closed geodesic balls of small radius ρ > 0 and center p ∈ M , giving 1 12 Sc p ρ 3 . Under the additional condition that the Riemann curvature tensor vanishes at p, the first order Taylor expansion of the Bartnik mass for such geodesic balls is given by 1 120 ∆Sc(p)ρ 5 . Note that these results are in accordance with the lower bound given in (9), which holds without the assumption that the Riemann curvature tensor vanishes at p. Remark 1. 10. The fact that m B (Ω) = 0 forces Ω \ ∂M to be locally isometric to Euclidean R 3 was obtained with different methods by Huisken-Ilmanen (see [15,Positivity Property 9.2]). In the same paper, Huisken-Ilmanen write "careful examination of the proof (of the positivity property 9.2) should give a small, but explicit lower bound for m B (Ω) in terms of the scalar curvature in a small region". Our methods permit to implement this: indeed (9) gives a small, but quantitative lower bound for m B (Ω) in terms of the scalar curvature (at first order, and in terms of other curvature tensors for higher order) in a small region (in our arguments, such a region will be a perturbed geodesic sphere of small radius).

Some ideas of the proofs
In order to prove the quasi-rigidity Theorem 1.3, it is of course key to identify suitable competitors in order to test the condition (8). A first attempt would be to use geodesic spheres S p,ρ contained in the open set U . However such surfaces are not "optimal enough" for the Hawking mass. Indeed, using the expansions of the paper (which in turn build on top of [40,30,31]), one can check that Note that such expansions give no interesting information in case Sc ≡ 0. The idea is thus to "optimally perturb" the geodesic spheres by suitable normal graphs, motivated by the fact (not strictly necessary for the arguments, but useful as a motivation) that the maximisers of the Hawking mass under small area constraint are indeed perturbed geodesic spheres. This principle has been already observed by Lamm-Metzger [20], who proved W 2,2 -closeness to a geodesic sphere under a small energy assumption, and by Laurain-Mondino [32], who proved smooth convergence to a geodesic sphere under a milder energy assumption. For the reader's convenience, we will give a self-contained proof of this fact in the exact framework of the present paper in Proposition 7.10 in the appendix. In order to compute the Hawking mass for such optimal competitors, in Lemma 3.1 we prove that the graph function w p,ρ for an optimally perturbed geodesic sphere must satisfy a precise expansion in terms of ρ and of curvature tensors at p; namely it has the form (10) w p,ρ (Θ) = − 1 6 Ric(Θ, Θ) + 1 18 where Θ ∈ S 2 is the parametrising coordinate, and lim sup ρ→0 Plugging such an expansion of the optimal normal graph w p,ρ into the definition of the Hawking mass gives, after some computational efforts, the expansion of m H on such optimal competitors: This is proved in Proposition 3.2, which represents the key technical result of the paper. Indeed, once (11) is proved, it is not hard to obtain the quasi-local rigidity Theorem 1.3 (see Theorem 4.1 in the body of the paper): basically the assumption Sc ≥ 0 on Ω coupled with the condition that m H (S p,ρ (w p,ρ )) ≤ 0 for every p ∈ Ω forces Ω to be Ricci flat and then flat, thanks to the expansion (11). The global rigidity Theorem 1.6 (Theorem 4.3 in the body of the paper) follows from the quasi-local rigidity Theorem 1.3 and the classification of flat manifolds (see for instance [54]), plus a case by case analysis when applying the ALSC condition.
In order to obtain the lower bound on the Bartnik mass (Theorem 1.7) out of the expansion (11), we prove that the optimally perturbed geodesics spheres S p,ρ (w p,ρ ) are outer-minimising. This is achieved in Theorem 5.7 via a blow-up argument. In order to gain strong enough compactness in such an argument, we employ the regularity theory for quasi-minimisers of the perimeter by Tamanini [47], refining previous celebrated results by De Giorgi [10]. Once it is established that S p,ρ (w p,ρ ) are outer-minimising, Theorem 1.7 follows from the expansion (11) combined with the monotonicity property (5) and the bound (7).
We adopt the standard index notation R µν := Ric(E µ , E µ ). A key tensor for this paper is the traceless Ricci tensor (12) S := Ric − 1 3 Sc g.
It is easily seen that S 2 = Ric 2 − 1 3 Sc 2 . 3) Let (M ,g) ֒→ (M, g) be an isometrically immersed, closed, 2-sided, 2-dimensional surface with inward pointing normal unit vector N . The (scalar) second fundamental form h is defined by (13) h for X, Y ⊂ T Σ vector fields tangent to Σ. The two eigenvalues k 1 and k 2 of h at p ∈ Σ are called the principal curvatures. We set H := k 1 + k 2 and D : where H is called the mean curvature. Letg denote the restriction of g to T Σ (with matrixg ij with respect to a fixed set of coordinates) and letg ij denote the matrix ofg −1 . It holds 4) Large positive constants are always denoted by C. The value of C is allowed to vary from formula to formula and also within the same line. When we want to stress the dependence of the constants on some parameter (or parameters), we either add subscripts to C, e.g. C δ , or we mention the dependence with parenthesis, e.g. C(δ). Also constants with subscripts (or with parenthesis) are allowed to vary.
Call with Θ i the coordinate vector fields on S 2 : andθ i orΘ i the corresponding normalised vectors: We next define the perturbed geodesic spheres S p,ρ (w) in the 3-dimensional Riemannian manifold (M, g). Fix a point p ∈ M \ ∂M and consider the exponential map Exp p centred at p. For ρ > 0 small enough, the sphere ρS 2 ⊂ T p M is contained in the injectivity radius of the exponential map. We call S p,ρ the geodesic sphere of center p and radius ρ, parametrised by The perturbed geodesic spheres are normal graphs on geodesic spheres, by a function w belonging to the following suitable function spaces (chosen for technical reasons in order to apply Schauder estimates in Lemma 3.1).
Denote with C 4,α (S 2 ) (or simply C 4,α ) the set of the C 4 functions on S 2 whose fourth derivatives, with respect to the tangent vector fields, are α-Hölder continuous (0 < α < 1). The Laplace-Beltrami operator on S 2 is denoted by ∆ S 2 or, if there is no ambiguity, by ∆. The fourth order elliptic operator ∆(∆ + 2) induces the following orthogonal splitting of L 2 (S 2 ): note that the splitting makes sense since the kernel is finite (four) dimensional, thus a closed subspace.
If we consider C 4,α (S 2 ) as a subspace of L 2 (S 2 ), we can define Of course C 4,α (S 2 ) ⊥ is a Banach space with respect to the C 4,α norm; it is the space from which we will draw the perturbation w. If there is no confusion C 4,α (S 2 ) ⊥ will be called simply C 4,α ⊥ . We can now define the perturbed geodesic spheres that we will use as "test" surfaces for the Hawking mass. Fix p ∈ M , ρ > 0 and a small C 4,α (S 2 ) function w; the perturbed geodesic sphere S p,ρ (w) is the surface parametrised by The tangent vector fields on S p,ρ (w) induced by the canonical polar coordinates on S 2 are denoted by Z i .
Following the notation of [40], given a ∈ N, we denote with L (a) p (w) an arbitrary linear combination of the function w together with its derivatives with respect to the tangent vector fields Θ i up to order a. The coefficients of L (a) p may depend on ρ and p but, for all k ∈ N, there exists a constant C = C p > 0 independent on ρ ∈ (0, 1) such that Similarly, given a, b ∈ N, we denote with Q (b)(a) p (w) an arbitrary nonlinear combination, of order at least b, of the function w together with its derivatives with respect to the tangent vector fields Θ i up to order a such that Q (w) in powers of w and its partial derivatives may depend on ρ and p but, for all k ∈ N, there exists a constant C = C p > 0 independent on ρ ∈ (0, 1) such that provided w l C a (S 2 ) ≤ 1, l = 1, 2. We also agree that O p (ρ d ) denotes an arbitrary smooth function on S 2 that might depend on p but which is bounded by a constant (possibly dependent on p) times ρ d in C k (B 1 (p)) topology, for all k ∈ N.

Expansions of geometric quantities
In this subsection we recall the Taylor expansion of the geometric quantities associated to the a perturbed geodesic sphere S p,ρ (w), appearing in the Willmore functional and its first derivative (for the proofs see [40] and [31, Section 3.1]). These will be used in later sections and hold for any Riemannian 3-manifold. For the following expansions we will fix the (polar) coordinate vector fields Θ i on S 2 defined in (14), and the corresponding coordinate vector fields Z i on S p,ρ (w); i.e. we will use the notationg ij :=g(Z i , Z j ) (resp. (g S 2 ) ij := g(Θ i , Θ j )) and analogously h ij := h(Z i , Z j ). The derivatives of the function w : S 2 → R with respect to Θ i are denoted by w i . All the curvature terms, all the covariant derivatives and all the scalar products are meant to be evaluated at p (since we fixed normal coordinates centred at p, at p the metric is Euclidean).
The following expansions hold: For the details of the derivation of the next expansion, the interested reader can see the Appendix 7.3.
Recalling that H =g ij hij , the combination of (17) and (18) gives: and thus The determinant of the first fundamental form will be useful to compute integrals on Sp,ρ(w) and can be expanded as:

Hawking mass of an optimally perturbed geodesic sphere
Motivated by Proposition 7.10 in the appendix, it is natural to choose the class of perturbed geodesic spheres Sp,ρ(w) as surfaces to "test" the positivity of the Hawking mass. More precisely, by using the Euler-Lagrange equation of the Willmore functional (under area constraint), we will find an expansion of the perturbation w which is necessary for the perturbed geodesic sphere Sp,ρ(w) to be a critical point (under area constraint). Let us stress that in Lemma 3.1 we do not claim to construct area-constrained Willmore surfaces centred at any point (that would be false, as a necessary condition would be that such a point is a critical point for the scalar curvature; see [20,32]). The goal of Lemma 3.1 is thus merely to suggest an ansatz for an "optimal" perturbation (so the reader could take for granted that a convenient choice of perturbation w is given by the expansion (22)). Such "optimally" perturbed geodesic spheres will be the key geometric objects to prove our main theorems. Throughout the section, (M, g) will be an arbitrary Riemannian 3-manifold.

An optimal pertubation
In this subsection we compute the expansion (as ρ → 0) that a perturbation w has to satisfy if the perturbed geodesic sphere Sp,ρ(w) is a critical point (under area constraint) of the Willmore functional or, equivalently, of the Hawking mass.
Moreover, for every (p, ρ) ∈ K × (0, ρ0] the following expansion holds: where Θ ∈ S 2 is the parametrising coordinate, and lim sup ρ→0 Proof. Step 1: We show that, for every fixed compact subset K ⋐ M , there exists ρ0 > 0, r > 0 and a map w (·,·) :  [21] for a derivation of the formula): where λ ∈ R plays the role of Lagrange multiplier and N is the inward pointing unit normal vector. As proved in [ In particular, setting P : Since the operator ∆ S 2 (∆ S 2 + 2) is invertible on the space orthogonal to its Kernel and w ∈ C 4,α ( equation (25) is equivalent to the fixed point problem Using Schauder estimates one can check that, for every fixed compact set K ⋐ M , there exist ρ0 > 0 and r > 0 such that for all p ∈ K and ρ ∈ [0, ρ0] the map is a contraction (see [30,Lemma 4.4] for the details). Thus, for every p ∈ K and ρ ∈ (0, ρ0] there exists a unique wp,ρ ∈ B(0, r) ⊂ C 4,α (S 2 ) ⊥ such that the surface Sp,ρ(wp,ρ) is an area-constrained Willmore surface. By continuous dependence on parameters of fixed points in contractions, it also follows that limρ→0 wp,ρ C 4,α (S 2 ) = 0 (again, see [30,Lemma 4.4] for the details).
Step 2: we show that the expansion (22) holds. We set the ansatz (27) wp,ρ = ρ 2w In order to show that wp,ρ given in step 1 satisfies the ansatz (27) and the expansion (22), we need to improve the expansion of the Euler-Lagrange equation of the Willmore functional (23)- (24). To this aim, using the expansions of Section 2.1.2, one can check that (see the proof of [31, Proposition 3.9] for more details) which, plugged into (23), give Inserting the ansatz (27), that we write as w =wρ 2 + Op(ρ 3 ), in (28) yields We solve this PDE by Fourier methods, using the knowledge of the eigenfunctions of ∆ S 2 . To this aim, writing the radial unit vector as Θ = x µ Eµ ∈ TpM where {Eµ}µ=1,2,3 is an orthonormal basis of TpM , we have where µ Rµµ = Sc(p) is the scalar curvature at p and where we used that µ ( Recall that the eigenfunctions of ∆ S 2 relative to the second eigenvalue λ2 is an element of the eigenspace relative to λ2 = −6 (and analogously for the others (x µ ) 2 ). Therefore, is an eigenfunction of ∆ S 2 with eigenvalue −6. We can then rewrite equation (29) as Setting the value of the Lagrange multiplier as λ = 2 3 Sc(p) and noting that we conclude that Summarising, we showed that wp,ρ as in the ansatz (27) withwp =w given in (30) solves (28) or, equivalently, (23). The proof is complete once we recall that, by step 1, the solution w of (28) is unique provided w C 4,α (S 2 ) < r.

Computation of the Hawking mass
Proposition 3.2. Let Sp,ρ(wp,ρ) we an optimally perturbed geodesic sphere, i.e. wp,ρ is given in (22). Then the Hawking mass of Sp,ρ(wp,ρ) has the following expansion: where Sc is the scalar curvature and S := Ric − 1 3 Sc g is the traceless Ricci tensor.
Proof. In order to keep notation short, throughout the proof we will write w in place of wp,ρ.
Step 1: the Willmore functional integrand. To compute the Hawking mass, we find the expansion for the Willmore functional W (Sp,ρ(w)) : Using (21), the Taylor expansion ) and the fact that w = Op(ρ 2 ), we have: Multiplying (32) with (20) we obtain the integrand of the Willmore functional evaluated on a perturbed geodesic sphere: Inserting w =wρ 2 + Op(ρ 3 ) yields: Step 2: simplifying the Willmore functional integrand. We will use the following computations for the derivatives ofw = − 1 6 Ric(Θ, Θ) + 1 18 Sc(p): which, combined with the fact that Using (34) and recalling that ∆ S 2w = −6w, we can rewrite the first line of the terms multiplying ρ 4 in (33) as: The second line of the terms multiplying ρ 4 in (33) can be rewritten as: where the first equality follows because the only non-zero terms occur when i = j, k = n and m = l (note that ∂ig S 2 jk is only non-zero when j = k = 2 and i = 1), and for the second equality we used (34).
The third line of the terms multiplying ρ 4 in (33) can be rewritten as: 2g 11 1g 22 For the second equality above we have used that Θ11 = −Θ which means that all the i = k = 1 terms are zero thanks to the symmetries of the Riemann tensor. The last equality follows by applying (35) and (34). Combining (38) and (39) shows that the second and third lines of the fourth order term in (33) become: where we have used the definition of the Ricci tensor and the symmetries of the Riemann tensor to rewrite Ric(Θ,Θ2) = g(R(Θ, Θ1)Θ1,Θ2) and Ric(Θ, Θ1) = g(R(Θ,Θ2)Θ2, Θ1).
Then Ω \ ∂M is locally isometric to Euclidean R 3 .
Recall the Definition 1.4 of asymptotically locally simply connected (ALSC for short). The global rigidity result (Theorem 4.3, below) will follow by the combination of Theorem 4.1 with the following proposition of independent interest. Proof. By the classical Killing-Hopf Theorem, we have that (M 3 , g) is isometric to a space form R 3 /Γ, for some discrete sub-group Γ of the affine transformations E(3) of R 3 acting freely (see for instance [54, Theorem 3.3.2]). Denote with π : R 3 → R 3 /Γ ≃ M the covering map. Since E(3) is isomorphic to the semi-direct product O(3)⋉R 3 , we can write an arbitrary element γ ∈ Γ as γ = (r, a) where r ∈ O(3) and a ∈ R 3 . Denote with n ∈ {0, 1, 2, 3} the rank of the maximal abelian subgroup of Γ. After choosing an appropriate orthonormal basis of R 3 adapted to Γ, we can write R 3 = R n × R 3−n in such a way that each γ ∈ Γ can be expressed uniquely as where δ = (r| R n , a), a ∈ R n , and ψ = (r| R 3−n , 0). We next discuss separately the different cases for the value of n.
• n = 1. We claim that for any γ ∈ Γ, the corresponding translation a ∈ R must vanish. Assume by contradiction this is not the case and fix such a γ ∈ Γ with corresponding non-trivial translation 0 = a ∈ R. Fix an arbitrary point (u, v) ∈ R × R 2 and consider the curve α(s) = (u + sa, v) ⊂ R 3 for s ∈ [0, 1]. It is easily seen that, for any radius R > |a|, π • α is a loop contained in B M R (π(u, v)) ⊂ M and that π • α is not contractible in B M R (π(u, v)). Indeed, if π • α were homotopic to the constant loop c(·) ≡ π(u, v) then, by the uniqueness of path liftings in the covering space (see for instance [36, Lemma 54.1, Chapter 9]), the lifts of c(·) and π • α starting at (u, v) would end at the same point. But this is not true as the lift of c(·) is the constant loop in R × R 2 taking value (u, v), whereas the lift of π • α is α, which ends at (u + a, v) = (u, v). Therefore B M R (π(u, v)) ⊂ M is not simply connected for every R > a and every (u, v) ∈ R × R 2 , contradicting the ALSC assumption on M . This concludes the proof that for every γ ∈ Γ the corresponding translation a ∈ R must vanish. It follows that Γ ⊂ O(3) and thus, as in the case n = 0, we infer that Γ = {Id} must be trivial and (M, g) is isometric to R 3 .
• n = 2. We claim that for any γ ∈ Γ, the corresponding translation a ∈ R 2 must vanish. Assume by contradiction this is not the case and fix such a γ ∈ Γ with corresponding non-trivial translation 0 = a ∈ R 2 . Fix an arbitrary point (u, v) ∈ R 2 × R and consider the curve α(s) = (u + sa, v) ⊂ R 3 for s ∈ [0, 1]. The analogous argument as in the case n = 1 shows that, for any radius R > |a|, π • α is a loop contained in B M R (π(u, v)) ⊂ M and that π • α is not contractible in B M R (π(u, v)). This contradicts the ALSC assumption and thus, as in the n = 1 case, we conclude that Γ = {Id} must be trivial and (M, g) is isometric to R 3 .
• n = 3. In this case, the quotient M = R 3 /Γ is compact, contradicting the ALSC assumption. Therefore, the only possibility is Γ = {Id} and (M, g) isometric to R 3 . Theorem 4.3. Let (M 3 , g) be a connected, complete, ALSC Riemannian manifold without boundary and with nonnegative scalar curvature. Assume that every p ∈ M admits a neighbourhood satisfying (52) or, more generally, that the infinitesimal non-positivity of mH (53) holds for every p ∈ M .
Then (M 3 , g) is globally isometric to Euclidean R 3 .
Proof. The result follows by the combination of Theorem 4.1 and Proposition 4.2.

A lower bound on the Bartnik mass
In this section we are interested in asymptotically flat Riemannian manifolds as defined in the introduction, with non-negative scalar curvature. In order to give a lower bound on the Bartnik mass, in the next subsection we show that perturbed geodesic spheres Sp,ρ(w) are outer-minimising. We will then combine the monotonicity of the Bartnik mass (5), with the fact that Bartnik mass is bounded from below by the Hawking mass (7) and the geometric expansions of the Hawking mass on perturbed geodesic spheres obtained in the previous sections to get the result.

Some auxiliary lemmas about sets of finite perimeter
The goal of this section is to establish some auxiliary lemmas that will be useful in the proof of the main result in the next section, that is Theorem 5.7. Note that ∂M never contributes to the perimeter. Given a point p ∈ M \ ∂M , let Inj p > 0 be the injectivity radius at p and denote with φ p g : B g Inj p (p) → R 3 a normal coordinate chart centred at p with respect to the Riemannian metric g. It will be convenient to consider the rescaled metrics gρ := ρ −2 g, for ρ ∈ (0, 1]. It is a standard fact that, in gρ-normal coordinates centred at p, it holds: for some suitable C = C(p, k) > 0. From the very definition of perimeter (58), it is easily seen that We next establish some lemmas linking the relative perimeter Pg ρ (E, B gρ r (q)) of a finite perimeter set E ⊂ M , and the Euclidean relative perimeter P δ (φg ρ (E), B δ r (φg ρ (q))) of its image φg ρ (E) ⊂ R 3 . We remark that, in such relative perimeter calculations, we can ignore the fact that E may not be contained in the domain of φg ρ because it is enough that B gρ r (q) is contained there (which, for small ρ, it will be). To this aim, we will distinguish between the normal coordinate charts centred at either at a point p or q nearby, denoting them by φ p gρ and φ q gρ respectively. Proof. Let q ∈ B gρ 10 (p) and consider gρ-normal coordinates centred at q. Notice that, by smooth dependence of initial data in the geodesic equation, the constant appearing in (59) (when applied to normal coordinates centred at q ∈ B gρ 10 (p)) may be chosen independently on q, so it still only depends on p. For a C 1 -vector field X supported in B gρ r (q) with X ∞,δ ≤ 1, we thus have: where in the first inequality we used that V δ φ q gρ (E) ∩ B δ r (φ q gρ (q)) ≤ V δ B δ r (φ q gρ (q)) ≤ Cr 3 , and in the third inequality we have used that X is admissible in the definition of perimeter if and only if −X is too. Taking the supremum over all C 1 -vector fields X supported in B gρ r (q) with X ∞,δ ≤ 1 gives the claim.
Proof. From (59) and the smooth dependence on coefficients in the geodesic equation, it is standard to check that for any point q ∈ B gρ 10 (p) the gρ-exponential map satisfies: where we consider both maps (for exp gρ q , via the pullback metric) as diffeomorphisms on a ball in R 3 . Note that exp δ φ p gρ (q) = T φ p gρ (q) , where T φ p gρ (q) is the translation by the vector φ p gρ (q). Since the normal coordinate chart is the inverse of the exponential map, this means: It follows that Since the translation is an isometry of R 3 , it is not hard to see that the last estimate implies the claim.
We next recall a well known consequence of the monotonicity formula for a finite perimeter set which is stationary for the perimeter functional; actually this holds more generally for stationary rectifiable varifolds, see for instance [45,Chap. 4.3]. Pg(E, B g r (q)) ≤ Cr 2 , for all B g r (q) ⊂ U and all r ∈ (0, r0]. The following lemma can be proved along the same lines as [35, (6-9)]: We conclude this short section by recalling a regularity result of Tamanini [47], refining previous celebrated works by De Giorgi [10]. To this aim recall that, given a finite perimeter set E ⊂ R 3 and an open bounded subset V ⊂ R 3 , the excess of E relative to V is defined as Ψ(E, Br(q)) ≤ Cr 2+2α , for all q ∈ U and r ∈ (0, R).
Then, up to replacing E by the suitable a.e. representative, it holds that the reduced boundary ∂ * E coincides with the topological boundary ∂E, and ∂E is a C 1,α -hypersurface in U . Moreover, assuming that (64) holds uniformly for a sequence (E k ) k∈N convergent to E∞ in L 1 loc -topology, then E∞ satisfies (64) as well; moreover for any sequence of points q k ∈ ∂E k converging to q∞ ∈ ∂E∞, the outward-pointing unit normal to ∂E k at q k converges to the outward-pointing unit normal to ∂E∞ at q∞. If (M, g) is an AF, complete, three dimensional Riemannian manifold, the following holds:

Perturbed geodesic spheres are outer-minimising
This can be proved along the same lines as [38], noticing that the AF assumption guarantees that the pointed limit manifolds at infinity used in the proof of [38] coincide with the Euclidean space R 3 (thus one can relax the assumption of C 4,α bounded geometry, used in [38] to guarantee that the limit manifolds at infinity are C 3,α with C 2,α Riemannian metric). Proof. From [37] we know that I : (0, ∞) → (0, ∞) is a continuous function. Using (66), we conclude that the function [0, V0] ∋ v → v −2/3 I (M,g) (v) is continuous and never vanishes, giving the claim (67).
The goal of the present section is to prove the next theorem, which will allow us to use the expansion for the Hawking mass of perturbed spheres in order to get a lower bound on the Bartnik mass in Theorem 1.7.
Throughout this section, we will denote by Bp,ρ(w) the perturbed geodesic ball enclosed by Sp,ρ(w).
Proof. Using the standard L 1 loc -compactness and lower-semicontinuity of the perimeter, observe that, for any ρ > 0 and w ∈ C 1 (S 2 ), there exists a set of finite perimeter Ωp,ρ,w ⊂ M minimising the perimeter among all sets of finite perimeter and finite volume containing Bp,ρ(w). Such Ωp,ρ,w ⊂ M is called minimising hull of Bp,ρ(w). Clearly
We will show that, up to choosing the suitable a.e. representative, Ωp,ρ,w = Bp,ρ(w) for any ρ, w C 1 small enough. Note that, under such a smallness condition, it holds that Bp,ρ(w) ⋐ M \ ∂M ; thus, in the blow up arguments of steps 2-4, it is not restrictive to assume that ∂M = ∅.
The constant coming from Lemma 5.3 is independent of ρ because the Ωp,ρ,w have uniformly bounded perimeter and gρ is smoothy converging to the Euclidean metric δ on B δ 11 (0). Combining (74) and (75) with Lemma 5.1 gives the claim (73).
For every perturbed geodesic sphere Sp,ρ(w) ⊂ Ω, with ρ > 0 and w C 1 sufficiently small (only depending on p), we have: where the first inequality follows from (7) applied to Sp,ρ(w) and the second follows by the monotonicity property (5), which applies thanks to the outer-minimising property of Sp,ρ(w) proved in Theorem 5.7. For every p ∈ Ω\∂M and for ρ > 0 sufficiently small (depending only on p), let wp,ρ ∈ C 4,α (S 2 ) ⊥ ⊂ C 1 (S 2 ) be given by Lemma 3.1, i.e. wp,ρ is the optimal perturbation extremizing (in C 4,α (S 2 ) ⊥ ) the Hawking mass under area constraint. Combining the expansion (31) and the inequality (79), specialised to w = wp,ρ, gives the claimed lower bound on the Bartnik mass: If mB(Ω) = 0, from (79) we infer that mH (Sp,ρ(w)) is non-positive for every perturbed geodesic sphere of sufficiently small radius. We can then apply Theorem 4.1 to Ω to infer that Ω \ ∂M is locally isometric to Euclidean R 3 . ✷

Other rigidity results
In this section, we collect other rigidity results involving the Hawking mass in various different settings. As the reader will appreciate, the proofs will be quite straightforward thanks to the work done in the previous sections (in particular we will make repeated use of the expansion of the Hawking mass obtained in Proposition 3.2).

Rigidity for the generalised Hawking mass (i.e. for non-zero cosmological constant)
The standard Hawking mass (3) is relevant when the ambient space is a Riemannian 3-manifold with non-negative scalar curvature. Such metrics are natural when the cosmological constant is zero. Instead, when the cosmological constant Λ is negative (resp. positive), it is is more natural to consider metrics with scalar curvature bounded below by a negative (resp. positive) constant. Indeed, the Dominant Energy Condition coupled with the Einstein Constraint Equations imply that the scalar curvature of a totally geodesic space-like hypersurface (i.e. the socalled time-symmetric case) is bounded below by 2Λ. When Λ is negative (resp. positive) it is standard to choose the normalization Λ = −3 (resp. Λ = 3) and it is natural to compare the geometry of a totally geodesic space-like hypersurface with a space-form of constant sectional curvature K = −1 (resp. K = 1).
Arguing along the lines as the proof Theorem 4.3 we obtain the following rigidity result involving the generalised Hawking mass. Recalling (50), we obtain the following expansion for the generalized Hawking mass of Sp,ρ(wp,ρ): Scp ρ 4 + Op(ρ 5 ) .
Recalling that in dimension three the Riemann curvature tensor can be written as Rm = Ric ∧ g − 1 4 Sc g ∧ g where ∧ is the Kulkarni-Nomizu product (see for instance [23,Corollary 7.26]) we conclude that g has constant sectional curvature K. Even though the proof of Theorem 5.7 made use of the AF assumption, we expect that for ρ > 0 sufficiently small the surface Sp,ρ(wp,ρ) is outward minimising also in a locally asymptotically hyperbolic framework. We did not push in that direction since it does not seem to be useful in order to obtain a lower bound on the hyperbolic analogue of the Bartnik mass, in the spirit of Theorem 1.7. Indeed, if for ρ > 0 sufficiently small the surface Sp,ρ(wp,ρ) is outward-minimising, one can start the (weak) Inverse Mean Curvature Flow in the sense of Huisken-Ilmanen [15] and the generalised Hawking mass is monotone non-decreasing also in this setting (see for instance [39] for more details). However, as proved by Neves [39], it may happen that the asymptotic limit of the Hawking mass along the IMC flow exceeds the hyperbolic-ADM mass of the manifold, thus preventing us to repeat the proof of Theorem 1.7 in the K = −1 case.

R 3 and H 3 rigidity in the homogeneous setting
In this section we replace the ALSC assumption in the rigidity Theorem 4.3 with the homogeneity condition. Recall that a Riemannian manifold (M, g) is said to be homogeneous if its isometry group Isom(M, g) acts transitively on M . In other words, if for every p, q ∈ M , there exists γ ∈ Isom(M, g) such that γ(p) = q.
Whilst of course our spatial universe is not exactly homogenous, at cosmological scales the homogeneity provides a useful idealisation. Indeed, spatial homogeneity is a standard assumption in Cosmology. For instance, it leads to an exact solution of Einstein's field equations, known as the Robertson-Walker metric for space-time [50,12].
Theorem 6.4. Let K ∈ {−1, 0, 1} and let (M 3 , g) be a connected, homogeneous Riemannian manifold with scalar curvature Sc ≥ 6K. If for every p ∈ M there is a neighbourhood U of p such that the generalised Hawking mass (80) of every embedded sphere contained in U is non-positive, then (M 3 , g) is isometric to Proof. Since homogeneity implies completeness and that ∂M = ∅, Theorem 6.2 yields that M has constant sectional curvature K. The conclusion now follows from the classical classification of homogenous spaces of constant sectional curvature (see for instance [54,Theorem 2.7.1]). Notice that in the case K = 1 one can be more precise about the type of quotients appearing, at the price of a more technical statement.

R 3 and H 3 rigidity under global asymptotic volume growth assumptions
In this section we replace the ALSC assumption in the rigidity Theorem 4.3 with a global volume growth assumption (satisfied, for instance, by asymptotically flat spaces and asymptotically hyperbolic spaces). Such an assumption is obtained by comparing the volume growth of metric balls in the space under consideration with metric balls in an appropriate model space, as the radius goes to infinity. Definition 6.5. Let K ∈ {−1, 0} and let (M 3 , g) be a complete Riemannian manifold without boundary and with Sc ≥ 6K. We say that (M 3 , g) satisfies the K-Global Asymptotic Volume property (K-GAVP) if: where VolK(r) denotes the volume of a metric ball of radius r in the 3-dimensional simply connected space of constant sectional curvature K.
There are various ways to define an asymptotically hyperbolic manifold. For the conformal compactification approach, see for instance Wang [49]. In closer analogy to Definition 1.1, we take the asymptotic chart approach (see for instance [9,13,43] for discussions on the physical relevance of such metrics): Definition 6.6. A 3-dimensional Riemannian manifold (M, g) is said to be asymptotically hyperbolic (AH) if there is a compact subset C ⊂ M and a diffeomorphism φ : M \ C → R 3 \ B1(0) such that the metric satisfies: for some s > 0 in the chart φ. Here H 3 = (R 3 , g H 3 ) denotes the standard hyperbolic space with metric g H 3 = 1 1+r 2 dr 2 + r 2 g S 2 (in polar coordinates). Remark 6.7. AF (resp. AH) manifolds satisfy the 0-Global Asymptotic Volume property (resp. −1-GAVP). For the AF case, consider a straight line segment γ in R 3 , parameterised on the interval (0, √ r) by γ(t) = √ rtâ, for some unit vectorâ ∈ R 3 . It is easy to see that if (M, g) is AF with metric g| M \C =ḡ + h, with h = O(r −s ) for some s > 0, then there exists a constant A > 0 such that giving that B δ r (p) ⊂ B g Ar(1+Ar −s ) (p). Therefore Vol δ (B δ r (p)) ≤ Volg(B g A+r(1+Ar −s ) (p)). Sending r → ∞ yields the 0-GAVP: The hyperbolic case is analogous, replacing straight lines by minimising geodesics in H 3 and applying Definition 6.6.
Theorem 6.8. Let K ∈ {−1, 0}. Let (M 3 , g) be a connected, complete Riemannian manifold without boundary, with scalar curvature Sc ≥ 6K, and satisfying the K-Global Asymptotic Volume property. If for every p ∈ M there is a neighbourhood U of p such that the generalised Hawking mass of every embedded sphere contained in U is non-positive, then Proof. By recalling Theorem 6.2, the assumptions on the scalar curvature and Hawking mass imply that (M 3 , g) has constant sectional curvature K. Therefore the Ricci curvature of (M 3 , g) is identically equal to 2Kg. The Bishop-Gromov Theorem (see for instance [23,42]) gives that the ratio is non-increasing and is bounded above by 1.
Moreover, if equality holds for some r > 0, then the metric ball B g r (p) in (M, g) is isometric to a metric ball of radius r in the simply connected 3-dimensional space of constant sectional curvature K. It follows that with equality if and only if for every r > 0 the metric ball B g r (p) in (M, g) is isometric to a metric ball of radius r in the simply connected 3-dimensional space of constant sectional curvature K or, equivalently, if (M, g) is globally isometric to the simply connected 3-dimensional space of constant sectional curvature K. Since the equality in (85) is forced by the K-GAVP, the result follows.

A sup-Hawking mass for AF manifolds with non-negative scalar curvature
Inspired by the results of the present paper, it is natural to propose a slight variant of the Hawking mass. Indeed, the standard Hawking mass, while being very useful in applications (for instance in the proof of the Riemannian Penrose inequality via Inverse Mean Curvature Flow by Huisken-Ilmanen [15]), has some inconvenient features. For instance, it can be negative and it has no clear monotone property under inclusion. Even though the list of properties that are desirable for a quasi-local mass is open for debate, let us mention some natural ones.
Let (M 3 , g) be an asymptotically flat Riemannian manifold with non-negative scalar curvature and with (possibly empty) horizon boundary ∂M . According to Bartnik [2], a "good" notion of quasi-local mass m(Ω) for subsets Ω ⊂ M should satisfy: (iv) Asymptotic to ADM mass: If {Ω k } ∞ k=1 is an exhaustion of M , then m(Ω k ) → mADM(M, g) as k → ∞. Even if not explicitly requested by Bartnik, it is also natural to require: (v) Compatibility with Schwartzshild: Let m Sch > 0 and consider R 3 \ B δ 2m Sch (0) ≃ S 2 × [2m Sch , ∞) endowed with the Schwarzshild metric g m Sch of mass m Sch > 0: Then the quasi-local mass of every subset (satisfying suitable geometric conditions) containing the horizon {r = 2m Sch } is equal to m Sch .
We propose the next definition. It is clear that, if ∂Ω ≃ S 2 is smooth then A benefit of the proposed sup-Hawking mass (87) is that it satisfies (a suitable version) of all the requirements (i)-(v). Property (i) is clearly satisfied, so let us discuss the others. The proof of (ii) is a nice application of the present paper as it involves basically all of the main results. Proof.
If Ω ′ ⊂Ω1 satisfies that ∂Ω ′ ≃ S 2 is smooth and outer-minimising inΩ1 and ∂Ω1 is outer-minimising in Ω2, then ∂Ω ′ ≃ S 2 is smooth and outer-minimising inΩ2 as well. The monotonicity is then a direct consequence of the definition (87).
In order to prove (a suitable version of) property (iv), we first establish the next two results of independent interest. Proof. Since ∂Ω is outer-minimising in M , every Ω ′ ⊂Ω with ∂Ω ′ ≃ S 2 smooth and outer-minimising inΩ is also outer-minimising in M . Thus, one can run the (weak version of the) Inverse Mean Curvature Flow by Huisken-Ilmanen [15] starting from Ω ′ and obtain that mH(∂Ω ′ ) ≤ mADM(M, g). The claim follows now from the very definition (87) of mSH(Ω). Proof. It is well known that for ρ ≫ 1 sufficiently large, the coordinate sphere Σρ satisfies: • Σρ is outer-minimising in M (a careful reader has probably noticed that this fact can also be proven by a blow-down argument analogous to the proof by blow-up of Theorem 5.7); • limρ→∞ mH (Σρ) = mADM(M, g), see for instance [22,Exercise 4.25].
The combination of the last property and (88) gives on the one hand that On the other hand, using Proposition 7.4 and that Σρ is outer-minimising, we infer that mSH(Bρ) ≤ mADM(M, g) for every ρ ≫ 1. The conclusion follows.
We can now prove the following (suitable version of) property (iv).
Also, using that {Ω k } ∞ k=1 is an exhaustion of M , we have that for every ρ ≫ 1 there exists k0 > 0 such that where Bρ is as in Lemma 7.6. The monotonicity property of mSH (see Proposition 7.3) and the fact that ∂Bρ = Σρ is outer-minimising in M (see the proof of Lemma 7.6) yield that for every ρ ≫ 1 sufficiently large there exists k0 > 0 such that: mSH(Bρ) ≤ mSH(Ω k ), ∀k ≥ k0.
Recalling Lemma 7.6, we thus obtain The conclusion follows by the combination of the first and last displayed formulas in the proof.
We next establish property (v). To obtain the reversed inequality, observe that the horizon Σ2m Sch := {r = 2m Sch } is outer-minimising and satisfies that mH(Σ2m Sch ) = m Sch . The definition (87) of mSH yields: The conclusion follows by combining the two inequalities displayed in the proof.
We wish to conclude this appendix by suggesting a possible research direction, drawing a more precise connection between the sup-Hawking mass and the Bartnik mass. Notice indeed that the sup-Hawking mass corresponds to "defining a quasi-local mass by taking the supremum from inside" while the Bartnik mass corresponds to "defining a quasi-local mass by taking the infimum from outside". In Remark 7.5, we observed that for every Ω ⊂ M with ∂Ω outer-minimising it holds that mSH(Ω) ≤ mB(Ω), however it is natural to expect that the two objects coincide (under appropriate conditions on Ω and/or M ): Open Problem: find appropriate (necessary and/or sufficient) conditions on Ω and/or M so that mSH(Ω) = mB(Ω). (iii) is satisfied in the same form as in Proposition 7.3.
However, property (iv), the upper bound analogous to Proposition 7.4, and the analogue of Proposition 7.8 have all little chances to hold for K = 0 (see Remark 6.3 for more details). This is why the presentation here is focused on the AF case with non-negative scalar curvature.

Local area-constrained maximisers of m H are perturbed geodesic spheres
In this paper, we often estimated the sup of the Hawking mass using perturbed geodesic spheres Sp.ρ(w). Even if not strictly necessary for such arguments, in the next proposition we prove that such a choice of competitors is very natural. Indeed we show that optimal competitors for the supremum of the Hawking mass among surfaces contained in a small ball are given by perturbed geodesic spheres Sp,ρ(w) with w ∈ C 4,α (S 2 ) ⊥ , w C 4,α ≤ Cρ 2 . Thus, such optimal competitors satisfy the expansions given in Lemma 3.1 and Proposition 3.2. For related results in this direction, see Lamm-Metzger [20], who proved W 2,2 -closeness to a geodesic sphere under a small energy assumption, and Laurain-Mondino [32], who proved smooth convergence to a geodesic sphere under a milder energy assumption. Recall that, by definition C 4,α (S 2 ) ⊥ := C 4,α (S 2 ) ∩ Ker ∆ S 2 (∆ S 2 + 2) ⊥ , where Ker ∆ S 2 (∆ S 2 + 2) ⊥ ⊂ L 2 (S 2 ) denotes the L 2 -orthogonal space to the finite (actually four) dimensional space Ker ∆ S 2 (∆ S 2 + 2) .
Using normal coordinates centred atp and estimating the difference between the Riemannian and Euclidean geometric quantities (see for instance [34]), it is not hard to check that (90) and (91) yield the existence of C > 0 such that for any sequence Σj ⊂ M Hausdorff converging to a point p, with Σj either non-orientable or of genus at least one, it holds: (92) lim sup j→∞ mH(Σj ) |Σj | ≤ −C.
Thus, the assumptions that Σj ⊂ M are a sequence of maximisers of mH under area constraint and Hausdorff converging to a pointp ∈ M yield that Σj must be a topological sphere, for j large enough, with W (Σj) < 8π. Now we can apply [32, Corollary 1.3] (see also [19]) to infer that ∇Sc(p) = 0 and that, if we rescale (M, g) around p in such a way that the rescaled surfacesΣj have fixed area 1, thenΣj converge smoothly (up to a subsequence) to a round sphere in the three-dimensional Euclidean space. In particular Σj is a normal graph over a geodesic sphere, of small radius and graph function (in any C k norm). Now, by a contraction mapping argument one can find pj →p, ρj ↓ 0, wj ∈ C 4,α (S 2 ) ⊥ with lim sup j→∞ ρ −2 j wj C 4,α (S 2 ) < ∞ such that Σj = Sp j ,ρ j (wj).

Second fundamental form on a perturbed geodesic sphere
In this section we give a self-contained proof of the Taylor expansion (18) for the second fundamental form on a perturbed geodesic sphere. We follow the approach in [40], however we compute more terms, since they are needed for the main results in this paper. First of all let Zi, i = 1, 2, be the coordinate vector fields on Sp,ρ(w): Zi = exp * (∂ θ i ρ(1 − w)Θ) = ρ((1 − w)Θi − wiΘ).