Doubling of Asymptotically Flat Half-spaces and the Riemannian Penrose Inequality

Building on previous works of Bray, of Miao, and of Almaraz, Barbosa, and de Lima, we develop a doubling procedure for asymptotically flat half-spaces (M, g) with horizon boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma \subset M$$\end{document}Σ⊂M and mass \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\in {\mathbb {R}}$$\end{document}m∈R. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3\le \dim (M)\le 7$$\end{document}3≤dim(M)≤7, (M, g) has non-negative scalar curvature, and the boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial M$$\end{document}∂M is mean-convex, we obtain the Riemannian Penrose-type inequality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} m\ge \left( \frac{1}{2}\right) ^{\frac{n}{n-1}}\,\left( \frac{|\Sigma |}{\omega _{n-1}}\right) ^{\frac{n-2}{n-1}} \end{aligned}$$\end{document}m≥12nn-1|Σ|ωn-1n-2n-1as a corollary. Moreover, in the case where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial M$$\end{document}∂M is not totally geodesic, we show how to construct local perturbations of (M, g) that increase the scalar curvature. As a consequence, we show that equality holds in the above inequality if and only if the exterior region of (M, g) is isometric to a Schwarzschild half-space. Previously, these results were only known in the case where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dim (M)=3$$\end{document}dim(M)=3 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document}Σ is a connected free boundary hypersurface.


Introduction
Let (M, g) be a connected, complete Riemannian manifold of dimension 3 ≤ n ≤ 7 with integrable scalar curvature R(g) and non-compact boundary ∂ M with integrable mean curvature H (∂ M, g). Here, H (∂ M, g) is computed as the divergence along ∂ M of the normal −ν(∂ M, g) pointing out of M.
Here, R n + = {x ∈ R n : x n ≥ 0} is the upper half-space andḡ the Euclidean metric.
Escobar has studied asymptotically flat half-spaces in the context of the Yamabe problem for compact Riemannian manifolds with boundary; see [17] and also the related works of Brendle [9] and of Brendle and Chen [10]. Almaraz [1, pp. 2628-2629] and Almaraz, Barbosa, and de Lima [2, p. 674] have studied asymptotically flat half-space in detail and associated to them a global geometric invariant called the mass. This mass, whose definition is attributed to Marques on [2, p. 677], is given by x i (∂ j g)(e i , e j ) − (∂ i g)(e j , e j ) dμ(ḡ) x i g(e i , e n ) dl(ḡ) (2) where the integrals are computed in the asymptotically flat chart. Here, ω n−1 = |{x ∈ R n : |x|ḡ = 1}|ḡ denotes the Euclidean area of the (n − 1)-dimensional unit sphere and e 1 , . . . , e n is the standard basis of R n . In analogy with the work [34] of Schoen on closed manifolds, Escobar has established a connection between the magnitude of the Yamabe-invariant of a compact manifold with boundary and the sign of the mass (2) of an associated asymptotically flat half-space in [17]. Almaraz, Barbosa, and de Lima have showed that the mass (2) is a geometric invariant and, in fact, non-negative provided that (M, g) satisfies suitable energy conditions. As noted in [2, p. 675], previous results in this direction had been obtained by Escobar [17,Appendix] and by Raulot [33,Theorem 23]. is isometric to (R n + ,ḡ). Remark 2. As explained in [31, §2], the assumption that H (∂ M, g) ≥ 0 can and should be viewed as a non-negativity condition for the scalar curvature R(g) across ∂ M in a distributional sense; see also [7, p. 207]. We note that this condition also has a natural physical interpretation; [3,Remark 2.7].
Theorem 1 is fashioned after the positive mass theorem for asymptotically flat initial data for the Einstein field equations, which has been proved by Schoen and Yau [36] using minimal surface techniques and by Witten [40] using certain solutions of the Dirac equation. In the presence of a so-called outermost minimal surface in the initial data set, a heuristic argument due to Penrose [32] suggests a stronger, quantitative version of the positive mass theorem which has been termed the Riemannian Penrose inequality. This inequality has been verified by Huisken and Ilmanen in [21] in dimension n = 3 when the outermost minimal surface is connected, by Bray in [7] in the case of a possibly disconnected outermost minimal surfaces, and by Bray and Lee in [8] in the case where 3 ≤ n ≤ 7 and the outermost minimal surface may be disconnected. We provide more details on asymptotically flat manifolds without boundary, the positive mass theorem, and the Riemannian Penrose inequality in Appendix A.
Almaraz, de Lima, and Mari [3] have studied the mass (2) of initial data sets with non-compact boundary in a spacetime setting; see also the recent survey [15] of de Lima. Moreover, they argue that, in the presence of an outermost minimal surface, a Riemannian Penrose-type inequality should hold for asymptotically flat half-spaces as well; see [3,Remark 5.6].
To describe recent results in this direction, we recall the following definitions from [23]; see  We say that an asymptotically flat half-space (M, g) has horizon boundary ⊂ M if is a non-empty compact minimal hypersurface with the following properties.
• The connected components of are either free boundary hypersurfaces or closed hypersurfaces. • Every minimal free boundary hypersurfaces or minimal closed hypersurfaces in M( ) is a component of .
Theorem 4 ([23, Theorem 1.2]). Let (M, g) be an asymptotically flat half-space of dimension n = 3 with horizon boundary ⊂ M such that the following three conditions hold.
• is a connected free boundary hypersurface.
with equality if and only if (M( ), g) is isometric to a Schwarzschild half-space (3).

Remark 5.
Previous results in direction of Theorem 4 for asymptotically flat half-spaces arising as certain graphical hypersurfaces in Euclidean space had been obtained by Barbosa and Meira [5].
Remark 6. The method of weak free boundary inverse mean curvature flow employed in the proof of Theorem 4 in [23] had been studied previously by Marquardt in [27]. It appears to the authors of this paper that the scope of this method is essentially limited to the case where n = 3 and is a connected free boundary hypersurface; see [23, p. 16].
Remark 7. Theorem 4 is related to a Penrose-type inequality for so-called asymptotically flat support surfaces conjectured by Huisken and studied by Volkmann; see [39, p. 38] and [23, Lemma 2.1].
Outline of our results. Comparing Theorem 4 with the Riemannian Penrose inequality for asymptotically flat manifolds, stated here as Theorem 41, suggests that the assumptions that n = 3 and that be a connected free boundary hypersurface in Theorem 4 are not necessary. The goal of this paper is to address this conjecture using a strategy different from that in [23]. In fact, we demonstrate how the gluing method developed by Miao in [31], which in turn expands on an idea of Bray [7], can be used to develop a doubling procedure for asymptotically flat half-spaces that reduces the Riemannian Penrose inequality for asymptotically flat half-spaces to the Riemannian Penrose inequality for asymptotically flat manifolds. For the statement of Theorem 8, recall from Appendix A the definition of an asymptotically flat manifold (M,g), of its massm, of its horizon boundary˜ , and of the exterior regionM(˜ ). Theorem 8. Let (M, g) be an asymptotically flat half-space of dimension 3 ≤ n ≤ 7 with horizon boundary ⊂ M such that the following two conditions hold.
Let ε > 0. There exists an asymptotically flat manifold (M,g) with horizon boundarỹ ⊂M such that Remark 9. Gluing constructions related to the one used in the proof of Theorem 8 have also been studied by Miao and McCormick in [28] and by Lu and Miao in [26].
Combining Theorem 8 with Theorem 41, we are able to extend the Riemannian Penrose inequality for asymptotically flat half-spaces to dimensions less than 8 and to horizon boundaries that may be disconnected.

Corollary 10.
Let (M, g) be an asymptotically flat half-space of dimension 3 ≤ n ≤ 7 with horizon boundary ⊂ M such that the following two conditions hold. (4) Remark 11. Carlotto and Schoen have showed in [12,Theorem 2.3] that there is an abundance of asymptotically flat Riemannian manifolds with non-negative scalar curvature that contain a Euclidean half-space isometrically. Note that Corollary 10 shows that the Riemannian Penrose inequality, stated here as Theorem 41, can be localized to the geometrically non-trivial part of such initial data.
The approximation argument used to prove Corollary 10 cannot be applied to characterize the case of equality in (4) directly. Yet, we observe that (M, g) can be locally perturbed to increase the scalar curvature near non-umbilical points of the boundary ∂ M. Combining this insight with a variational argument used by Schoen and Yau [36] to characterize the case of equality in the positive mass theorem, we are able to prove the following rigidity result. Theorem 12. Let (M, g) be an asymptotically flat half-space of dimension 3 ≤ n ≤ 7 with horizon boundary ⊂ M such that the following two conditions hold.
Remark 13. Bray and Lee [8] have proved rigidity of the Riemannian Penrose inequality for asymptotically flat manifolds (M,g) of dimension 3 ≤ n ≤ 7 under the additional assumption that (M,g) be spin; see Theorem 41. Building on previous work [29] by McFeron and Székelyhidi, Lu and Miao [26,Theorem 1.1] have showed that the spin assumption can be dispensed with. Using the techniques developed in this paper, we are able to give a short alternative proof of this fact; see Theorem 39.

Remark 14.
We survey several important contributions to scalar curvature rigidity results preceding Theorem 12 in Appendix F.

Remark 15.
For the proofs of Theorem 8, Corollary 10, and Theorem 12, it is sufficient to require the metric g to be of class C 2,α . For the sake of readability, we will assume throughout that g is smooth.
Outline of the proof. Let (M, g) be an asymptotically flat half-space with horizon boundary ⊂ M and suppose that The basic idea to prove Theorem 8 is to consider the double (M,g) of (M, g) obtained by reflection across ∂ M. The metricg is only C 0 across ∂ M. The condition H (∂ M, g) ≥ 0 suggests that the scalar curvature ofg is non-negative in a distributional sense; see Remark 2. Moreover, since ⊂ M is an outermost minimal surface that intersects ∂ M orthogonally, its double˜ ⊂M is an outermost minimal surface without boundary.
The difficulty in rendering this heuristic argument rigorous is that (M,g) needs to be smoothed in a way that allows us to keep track of the mass, the horizon boundary, and the relevant energy conditions all at the same time. To this end, we first adapt an approximation procedure developed by Almaraz, Barbosa, and de Lima in [2, Proposition 4.1] to arrange that g is scalar flat and conformally flat at infinity and that ∂ M is totally geodesic at infinity; see Proposition 16. In particular, the reflected metricg is C 2 outside of a bounded open set W ⊂ M. Moreover, using a local conformal perturbation of the metric, we may arrange that is strictly mean convex; see Lemma 20.
Next, we smoothg near W ∩ ∂ M using a technique developed by Miao in [31]. In this step, the mean convexity of ∂ M ensures that the scalar curvature of the smoothed metric remains uniformly bounded from below near ∂ M; see Lemma 24. Moreover, we show that the strict mean convexity of˜ is not affected by this procedure; see Lemma 25.
By a conformal transformation similar to that developed by Miao in [31, §4], building in turn on [36,Lemma 3.3], we remove the small amount of negative scalar curvature that may have been created close to ∂ M in the approximation process. This conformal transformation only changes the mass of the smoothed manifold by a small amount; see Proposition 31. Finally, using˜ as a barrier, it follows that the smoothed metric has horizon boundary. Since ⊂ M is area-minimizing, it follows that the area of the new horizon boundary is at least as large as that of˜ ; see Lemma 32. This is how we obtain Theorem 8.
To prove Theorem 12, we first construct a global conformal perturbation of (M( ), g) that preserves the conditions R(g) ≥ 0 and H (∂ M, g) ≥ 0, strictly decreases m(g) unless R(g) = 0, and which changes the area of only marginally. Second, if the second fundamental form h(∂ M, g) of ∂ M does not vanish, we construct a local perturbation of (M( ), g) that increases R(g), preserves the condition H (∂ M, g) ≥ 0, and changes neither m(g) nor | | g . We note that a perturbation with these properties could not possibly be conformal; it has to be fine-tuned to the geometry of ∂ M. Consequently, if equality in (4) holds, then ∂ M is totally geodesic and the double (M,g) is C 2 -asymptotically flat. Theorem 12 now follows from Theorem 41.

Reduction to Conformally Flat Ends
In this section, we assume that (M, g) is an asymptotically flat half-space of dimension 3 ≤ n ≤ 7 and decay rate τ > (n − 2)/2. We also assume that (M, g) has horizon boundary ⊂ M and that The goal of this section is to approximate the Riemannian metric g by a sequence {g i } ∞ i=1 of Riemannian metrics g i on M that are scalar flat, conformally flat, and such that h(∂ M, g i ) = 0 outside of some compact set.
Here and below, and ∂ M are oriented by their unit normal vectors ν( , g) and ν(∂ M, g) pointing towards M( ). H ( , g) and H (∂ M, g) are computed as the divergence of −ν( , g) along and the divergence of −ν(∂ M, g) along ∂ M, respectively.

Proposition 16.
Let τ ∈ R be such that (n − 2)/2 < τ < τ. There exist sequences {g i } ∞ i=1 of Riemannian metrics g i on M and {K i } ∞ i=1 of compact sets K i ⊂ M such that (M, g i ) is an asymptotically flat half-space with horizon boundary i ⊂ M( ) and such that the following properties hold.
Proof. Arguing as in the proof of [ • g i is conformally flat in M\K i .
, and (5) holds. By (5), there is λ 0 > 1 such that the hemispheres R n + ∩ S λ (−1/2 λ e n ) have negative mean curvature with respect to g i and meet ∂ M at an acute angle with respect to g i provided that λ ≥ λ 0 and i is sufficiently large. We consider the class of all embedded hypersurfaces of M( ) that are homologous to in M( ) and whose boundary is contained in ∂ M and homotopy equivalent to ∂ in M( ) ∩ ∂ M. Since H ( , g i ) = 0, it follows from [30, Theorem 1] that there is an outermost minimal hypersurface i ⊂ M( ) that is homologous to in M( ), whose boundary is homotopy equivalent to ∂ in M( ) ∩ ∂ M, and whose components are either free boundary hypersurfaces or closed hypersurfaces. Moreover, Recall from [23, Lemma 2.3] that is area-minimizing with respect to g in M( ). Consequently, as i → ∞, Finally, using (6), the curvature estimate [23,Lemma 3.3], and standard elliptic theory, it follows that, passing to another subsequence if necessary, { i } ∞ i=1 converges to a minimal surface 0 ⊂ M( ) with respect to g in C 1,α (M) and smoothly away from ∂ M, possibly with finite multiplicity. Since M( ) is an exterior region, it follows that 0 = . Since i is area-minimizing in M( i ) with respect to g i , we obtain that, as i → ∞, The assertion follows.

Gluing of Asymptotically Flat Half-spaces
In this section, we assume that (M, g) is an asymptotically flat half-space of dimension 3 ≤ n ≤ 7 with horizon boundary ⊂ M such that the following properties are satisfied.
• (M, g) is conformally flat outside of a compact set.
The goal of this section is to double (M, g) by reflection across ∂ M and to appropriately smooth the metric of the double.

Lemma 17.
There is δ 0 > 0 with the following property. The map Proof. Clearly, is a local diffeomorphism and surjective. Moreover, by compactness and Lemma 45, using the fact that g is asymptotically flat (1), it follows that is injective. where We consider the map We obtain a smooth structure onM by requiring that the map˜ be smooth. Moreover, . It follows that the Riemannian metricg onM defined bỹ For the following lemma, recall from Appendix A the definitions (29) of an asymptotically flat metric and (30) of the mass of an asymptotically flat manifold without boundary.
Lemma 19.g is of class C 0 and C 2 -asymptotically flat. Moreover, the following properties hold.
Proof. The assertions follow from the above construction, using that (M, g) is conformally flat at infinity, that ∂ M is totally geodesic at infinity, that intersects ∂ M orthogonally, and that is area-minimizing in its homology class and boundary homotopy class in M( ); see Fig. 3.

of Riemannian metrics g i on M and a neighborhood W
M of such that g i and g are conformally equivalent, g i = g outside of W, and g i → g in C 2 (M). Moreover, Proof. This follows as in Lemma 17, using also that g i = g outside of W and that g i → g in C 2 (M); see Lemma 20.
As before, we consider the maps Since g and g i are conformally equivalent, the maps˜ i are of class C 2 . As before, given t ∈ (−δ 0 , 0], we define γ (g i ) t = γ (g i ) −t and obtain a continuous metricg i given bỹ To smooth the metricsg i , we recall some steps from the construction in [31, §3]. To Given an integer i ≥ 1 and t ∈ (−δ 0 , δ 0 ), we define the Riemannian metric on ∂ M.
We obtain a Riemannian metricg i δ onM of class C 2 given bỹ The following lemma is obtained by direct computation using Lemma 20; cp. [31, pp. 1168-1170]. For the statement, we choose a smooth reference metricǧ onM that agrees withg outside of a compact set.

Lemma 23. There holds
In the next lemma, the assumption that H (∂ M, g i ) ≥ 0 is used.

Lemma 24 ([31, Proposition 3.1]
). There holds, as δ 0 and uniformly for all i, Proof. Without loss of generality, we may assume that is a connected free-boundary hypersurface. It follows that˜ ⊂M is a connected, compact hypersurface without boundary of class C 1,1 that is smooth away from π −1 (∂ ). Let The following error estimates are independent of the choice of x 0 . Note that and in U δ . Using Lemma 23, we obtaiñ Using Lemma 23 again, we conclude that, on U δ ∩˜ \π −1 (∂ ), Here, denotes a Christoffel symbol. Using Lemma 23 once more, we have for all 1 ≤ ≤ n − 2. Using also (9) and (10), we see that Moreover, using (9) and (10), the same argument that led to Lemma 23 shows that This follows by approximation using mean curvature flow as in the proof of [ We then choose a function χ ∈ C ∞ (M) with We define the Riemannian metricĝ i δ onM bŷ There holds, as δ 0 and uniformly in i, Moreover, outside of a compact subset ofM,ĝ i δ =g i =g for all i. Proof. This follows from the construction using Lemmas 20 and 23.

Lemma 28.
There holds, as δ 0 and uniformly in i, On the other hand, note that, as δ 0, Moreover, recall that, in local coordinates, Using Lemmas 23, 24, and 27, we conclude that, as δ 0, The assertion follows from these estimates.
for any j ≥ j 0 . Using Lemma 26, it follows that˜ i δ ⊂M(˜ ).
By Lemma 29, (M,g i ) has horizon boundary˜ i =˜ i δ i ⊂M(˜ ). By comparison with a large coordinate hemisphere, we see that Moreover, we have π −1 (π(˜ i )) =˜ i . In fact, by area-minimization, there is a closed embedded minimal hypersurface that encloses π −1 (π(˜ i )). Since˜ i is outermost, this minimal surface coincides with˜ i .
Fix α ∈ (0, 1). By [21, Regularity Theorem 1.3 (ii)], Lemma 23, and compactness, it follows that, passing to another subsequence if necessary,˜ i converges to an embedded hypersurface˜ 0 ⊂M(˜ ) of class C 1,α in C 1,α possibly with multiplicity. By standard elliptic estimates, this convergence is smooth away from π −1 (∂ M) and there holds Since is area minimizing in M( ), we have, using also Lemma 19, Passing to a further subsequence if necessary, we have i → 0 ⊂ M( ) in C 1,α possibly with multiplicity, where 0 satisfies In particular, using also (11), we see that˜ i converges to˜ with multiplicity one. The assertion follows.

Conformal Transformation to Non-negative Scalar Curvature
In this section, we assume thatM is a smooth manifold of dimension 3 ≤ n ≤ 7 and that g is a Riemannian metric on M of class C 0 . We also assume thatg is C 2 -asymptotically flat and that there is a closed separating hypersurface˜ ⊂M of class C 1,1 . Moreover, we assume that {g i } ∞ i=1 is a sequence of Riemannian metricsg i onM of class C 2 with the following properties.
Finally, we assume that, as i → ∞, and that, for some τ > (n − 2)/2, In this section, we construct Riemannian metricsĝ i conformally related tog i which have non-negative scalar curvature.
Proof. This has been proved in [31, §4.1] in the special case where˜ i =˜ = ∅. Compared to [31, (35)], we takeũ i ∈ C ∞ (M(˜ )) to be the unique solution of As shown in [36,Lemma 3.2], the existence of such a solution follows from (12) and the fact that, for every α ∈ (0, 1),˜ i is of class C 1,α . We may now repeat the proofs of [31, Lemma 4.1, Proposition 4.1, and Lemma 4.2]; the only difference is that the elliptic estimates for the functionũ i now also depend on estimates on the C 1,α -regularity of˜ i , which, by assumption, are uniform in i. By Lemma 44, we have that H (˜ i ,ĝ i ) = 0.
The assertion now follows from Proposition 31, using that˜ i is area-minimizing iñ M(˜ i ) with respect tog i .

Lemma 33.
Suppose that there is a mapF :M →M with the following properties.
•F is an isometry with respect tog i for every i.
•g is C 2 in M ± and M ± ∩˜ ⊂ M ± is an outermost minimal hypersurface.
Proof. This follows as in the proof of Proposition 30.

Remark 34.
In the situation of Section 3, we may takeF :M →M to be the unique map with π •F = π andF = Id.

Proof of Theorem 8
Let Proof of Theorem 8. Using Proposition 16 to obtain a conformally flat approximation of (M, g), Proposition 30 to double the approximation, and Proposition 31, Lemmas 32, and 33 to conformally transform the double to non-negative scalar curvature, we see that there exists a smooth manifoldM of dimension n and a sequence {ĝ i } ∞ i=1 of Riemannian metricsĝ i onM with the following properties.
The assertion follows.

Mass-Decreasing Variations and Rigidity
Let (M, g) be an asymptotically flat half-space of dimension 3 ≤ n ≤ 7 with horizon boundary ⊂ M such that R(g) ≥ 0 in M( ) and H (∂ M, g) ≥ 0 on M( ) ∩ ∂ M. We also assume that equality holds in (4), i.e. , that The goal of this section is to show that (M( ), g) is isometric to the exterior region of a Schwarzschild half-space (3). The on M( ). Note that g t is asymptotically flat for every t ∈ [0, 1). Moreover, by Lemma 44, we have, for every t ∈ (0, 1), Arguing as in the proof of Proposition 16, we find that (M, g t ) has horizon boundary t ⊂ M( ) and that t → smoothly as t 0. Using that v = 0 on and that H ( , g) = 0, we conclude that Next, we compute m(g t ) in the asymptotically flat chart of (M, g). By (2), Using that g is asymptotically flat (1) and that v ∈ C 2,α τ (M( )), we have It follows that In conjunction with Lemma 51, we conclude that and, in particular, that As this is not compatible with Corollary 10, the assertion follows. • ψ has compact support in U ∩ ∂ M.
Let 0 < t 0 < sup x∈M |ψ|. We define the family {g t } t∈[0,t 0 ) of Riemannian metrics smoothly as t 0. Using that g t = g near and that M( ) is an exterior region, we conclude that t = for every t ∈ (0, t 0 ) sufficiently small. On the other hand, clearly, m(g t ) = m(g) for every t ∈ (0, t 0 ). It follows that m(g t ) = 1 2 n n−1 | t | g t ω n−1 n−2 n−1 for every t ∈ (0, t 0 ) sufficiently small. As this is not compatible with Lemma 35, the assertion follows.
The following lemma is the key technical step in the proof of Theorem 12.

Using that γ s = A t A and that σ ε,δ | R n−1 = A T E A, we obtain
Note that We conclude that The assertion follows. g t u t + R(g t ) u t = 0 i ni n t (M( )), such that (u t − 1) ∈ C 2,α τ (M( )). Moreover, the limiṫ every β ∈ (0, α); see [36, pp. 73-74]. By (26), Using that (u t − 1) ∈ C 2,α τ (M( )), we see thatĝ t is C 2 −asymptotically flat and, using also Lemma 44, that R(ĝ t ) = 0 in M( ) and H (∂ M,ĝ t ) = 0 on M( ) ∩ ∂ M. Using Lemma 37, we see thatu is non-constant. By the maximum principle,u < 0 in M( )\ and D(g) ν( ,g)u < 0 on . Now, Lemmas 44 and 37 imply that H ( ,ĝ t ) > 0 for all t > 0 sufficiently small. As in the proof of Proposition 16, it follows that (M,ĝ t ) has horizon boundaryˆ t ⊂ M( ) and thatˆ t → smoothly as t 0. On the one hand, using that g t = g on ,u = 0 on , and H ( , g) = 0, we conclude that Moreover, arguing as in the proof of Lemma 35, we have In conjunction with Lemma 51, we conclude that On the other hand, by Corollary 10, we have This is not compatible with (27) and (28).
The assertion follows.

Rigidity in the Riemannian Penrose Inequality
In this section, we give an argument alternative to that in [26] to show that the assumption that (M,g) be spin in the rigidity statement of [8,Theorem 1.4], stated here as Theorem 41, is not necessary.
For the statement of Theorem 39 below, recall from Appendix A the definition of an asymptotically flat manifold (M,g), of its horizon boundary˜ , and of the exterior region M(˜ ). Then (M(˜ ),g) is isometric to the exterior region of a Schwarzschild space (31). Proof. Following the argument given in [8, §6], we aim to show that the manifold (M,ĝ) obtained by reflection of (M,g) across˜ is smooth so that the characterization of equality in the positive mass theorem, stated here as Theorem 40, applies to (M,û 4 n−2ĝ ). Here,û ∈ C 2 (M) is the unique harmonic function that approaches 1 respectively 0 in the two ends of (M,ĝ). To this end, it suffices to show that˜ is totally geodesic.
The argument presented in Lemma 35 shows that R(g) = 0. If h(˜ ,g) = 0, the argument presented in Lemma 37 shows that there exists a family {g t } t∈[0,t 0 ) of Riemannian metrics on (M,g) such that •g t =g outside of a compact set, •g t →g smoothly as t 0, • lim t 0 t −1 (|˜ |g t − |˜ |g) = 0, • lim t 0 t −1 H (˜ ,g t ) = 0, and lim t 0 t −1 R(g t ) ≥ 0 with strict inequality at some point.
Adapting the argument in the proof of Lemma 38 to the case of an asymptotically flat manifold, we see that this leads to a contradiction with the inequality in Theorem 41.
Acknowledgments The authors acknowledge the support of the START-Project Y963 of the Austrian Science Fund. The second-named author acknowledges the support of the Lise-Meitner-Project M3184 of the Austrian Science Fund. The authors thank Pengzi Miao for helpful feedback on the statement of Theorem 39 and for bringing the results in [26] to their attention. The authors thank the anonymous referees for their feedback which has improved the exposition of this paper. This paper is dedicated to the memory of Robert Bartnik.

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A. Asymptotically Flat Manifolds
In this section, we recall some facts about asymptotically flat manifolds.
Let 3 ≤ n ≤ 7. A metricg on {x ∈ R n : |x|ḡ > 1/2} is called C 2 -asymptotically flat if its scalar curvature is integrable and if there is τ > (n −2)/2 such that, asx → ∞, A complete connected Riemannian manifold (M,g) of dimension n is said to be an asymptotically flat manifold if the following properties all hold.
•g is of class C 2 .
• There is a non-empty compact subset ofM whose complement is diffeomorphic to the set {x ∈ R n : |x|ḡ > 1/2}. • The pull-back ofg by this diffeomorphism is C 2 -asymptotically flat.
We usually fix such a diffeomorphism and refer to it as the asymptotically flat chart. The mass of an asymptotically flat manifold is the quantitỹ see [4, p. 999]. Here, e 1 , . . . , e n are the canonical basis vectors of R n and ω n−1 = |S n−1 1 (0)|ḡ denotes the area of the (n − 1)-dimensional unit sphere. Bartnik has showed that the mass (30) of a C 2 -asymptotically flat manifold converges and does not depend on the choice of asymptotically flat chart; see [6,Theorem 4.2].
Let˜ ⊂ M be a compact hypersurface without boundary. We call the components of such a hypersurface closed. If˜ is separating, we orient˜ by the unit normal ν(˜ ,g) pointing towards the closureM(˜ ) of the non-compact component ofM\˜ . The mean curvature H (˜ ,g) is then computed as the divergence of −ν(˜ ,g) along˜ .
We say that (M,g) has horizon boundary if there is a non-empty hypersurfacẽ ⊂M with the following two properties.
• Each component of˜ is a closed minimal hypersurface.
• Every closed minimal hypersurfaces inM(˜ ) is a component of˜ .
If (M,g) has horizon boundary˜ ⊂ M, we say thatM(˜ ) is the exterior region of M and we call the horizon˜ an outermost minimal surface. An example of an exterior region with horizon boundary is the Schwarzschild space of massm > 0 and dimension n ≥ 3 defined by where˜ = x ∈ R n : |x|ḡ = m 2 The positive mass theorem has been proved by Schoen and Yau in [36] using minimal surface techniques and subsequently by Witten in [40] using certain solutions of the Dirac equation.
The Riemannian Penrose inequality has been proved by Huisken and Ilmanen in the case where the horizon boundary is connected using inverse mean curvature flow in [21]. For general horizon boundary, it has been obtained by Bray [7] using his quasi-static flow. Bray's technique has been extended to higher dimensions in his joint work [8] with Lee.

B. Riemannian Geometry
In this section, we recall some facts from Riemannian geometry. and suppose that ⊂ M is a two-sided hypersurface with unit normal ν( , g) and mean curvature H ( , g) computed as the divergence of ν( , g) along .

C. Laplace Operator on Asymptotically Flat Half-spaces with Horizon Boundary
In [2, Proposition 3.3], Almaraz, Barbosa, and de Lima have proved an existence and uniqueness result for the Laplace equation on asymptotically flat half-spaces. In this section, we explain how their result can be adapted to an asymptotically flat half-space with horizon boundary. Let (M, g) be an asymptotically flat half-space of rate τ > (n − 2)/2 with horizon boundary.
If ⊂ M is a compact hypersurface whose components are closed hypersurfaces or free boundary hypersurfaces, we consider the differentiable manifoldM We define the Riemannian metricĝ onM byĝ(x) = g(π(x)) where π([(x, ±1)]) = x. Note that (M,ĝ) has two asymptotically flat ends and thatĝ is of class C 2 away from π −1 ( ). Moreover, note that, althoughĝ is only Lipschitz, the coefficients of ĝ are still Lipschitz since is minimal.

D. Local Perturbations of a Riemannian Metric
In this section, we construct local perturbations of a Riemannian metric that are used in this paper.
Lemma 48. Let (M, g) be a Riemannian manifold of dimension n ≥ 3 with boundary ∂ M oriented by the unit normal ν(∂ M, g) pointing towards M. Let ⊂ M be a compact hypersurface whose components are either closed or free boundary hypersurfaces. There exists a sequence {ψ i } ∞ i=1 of functions ψ i ∈ C ∞ (M) with the following properties: Moreover, if W ⊂ M is a neighborhood of , then spt(ψ i ) ⊂ W for all but finitely many i.
Let β ∈ C ∞ (R) be non-negative such that Let a, b : M → R be given by a(x) = dist(x, , g) and b(x) = dist(x, ∂ M, g), respectively. Here, dist( · , , g) and dist( · , ∂ M, g) are the signed distance functions that become positive in direction of the respective unit normals. Given an integer i ≥ 1, we define ψ i : M → R by Note that ψ i is smooth provided that i is sufficiently large and that, as i → ∞, Moreover, if W ⊂ M is a neighborhood of , then spt(ψ i ) ⊂ W for all but finitely many i.
On , there holds Using that ν( , g)(y) ∈ T y ∂ M for every y ∈ ∂ , we obtain D(g) ν ( ,g) provided that i is sufficiently large.
On ∂ M, we have If i a(x) ≥ 1, we have If i a(x) < 1, we have, as before, (D(g) ν(∂ M,g) a)(x) = O(a(x)) while β(i a(x)) = 1. Consequently, provided that i is sufficiently large. The assertion follows.
By a direct computation as in the proof of Lemma 49, the function ρ : R n−1 → R given by ρ(x) = η(|x|ḡ) satisfies the asserted properties.

E. Asymptotic Growth Estimate for Subharmonic Functions
In this section, we derive an asymptotic growth estimate for subharmonic functions on asymptotically flat half-spaces. The corresponding estimate for subharmonic functions on asymptotically flat manifolds has been stated by Corvino in [13, p. 164] and proved in detail by Czimek in [14, Proposition 2.6]. We note that the argument in [14,Proposition 2.6] can be adapted to the setting of an asymptotically flat half-space. Below, we give a different, self-contained proof.
Note that, in some sense, Lemma 51 is a quantitative version of the Hopf boundary point lemma as stated in, e.g. , [18,Lemma 3.4].
Lemma 51. Let n ≥ 3 and g be a C 2 -asymptotically flat metric on R n + . Suppose that there are a negative function u ∈ C 2,α τ (R n + ) and a number λ 0 > 1 such that Proof. We first assume that g u = 0.
Consequently, lim sup We have already showed that Moreover, since g u is integrable, the argument that led to (39) shows that lim λ→∞ λ −1 n i=1 R n + ∩S n−1 λ (0) x i ∂ i u dμ(ḡ) exists. The assertion follows.

F. Scalar Curvature Rigidity Results
In this section, we give an overview of several techniques that have been used to derive scalar curvature rigidity results in mathematical relativity. The proofs given by Schoen and Yau in [37, Theorem 1] and, independently, by Gromov and Lawson in [19, Corollary A] of the following result are in some sense a precursor to the positive mass theorem, stated here as Theorem 40.
Theorem 52 ( [19,37]). Let n ≥ 3 be an integer and T n = S 1 × · · · × S 1 be the torus of dimension n. Let g be a Riemannian metric on T n with R(g) ≥ 0. Then g is flat.
The proofs in [19,37] show that any Riemannian metric g on T n with R(g) ≥ 0 must actually satisfy R(g) = 0. Studying the variation of scalar curvature (24), Bourguignon had previously observed that, unless Ric(g) = 0, such a metric can be perturbed to a metric of positive scalar curvature; see [22,Lemma 5.2].
Let (M,g) be an asymptotically flat manifold with R(g) ≥ 0 and m(g) = 0; see Appendix A. The rigidity statement in the positive mass theorem, stated here as Theorem 40, can be viewed as a generalization of Theorem 52 to non-compact spaces. By constructing a global variation ofg, Schoen and Yau have showed that, unless Ric(g) = 0, g can be perturbed to a metric of non-negative scalar curvature and negative mass; see [36, §3]. If (M,g) is spin, the alternative argument of Witten implies the existence of certain parallel spinors. The existence of these spinors implies that (M,g) is flat; see [40, §3]. We note that Shi and Tam have adapted this argument to settings with lower regularity; see [38, §3]. Recently, Lu and Miao [26, Proposition 2.1] have extended the rigidity statement in Theorem 40 to metrics with a corner. Their proof uses an argument of McFeron and Székelyhidi [29] based on the observation that the mass is constant along Ricci flow.
If n = 3, the proof of the positive mass theorem in [36] suggests that there are no non-compact properly embedded area-minimizing surfaces inM unless (M,g) is flat R 3 . This conjecture of Schoen has been confirmed by Chodosh and the first-named author in [11,Theorem 1.6]. In the proof, they use local perturbations ofg to construct a local foliation of a neighborhood of such a minimal surface by non-compact properly embedded area-minimizing surfaces obtained as limits of solutions of the Plateau problem. We remark that related rigidity results that restrict the topology of a horizon boundary are known; see [16,Corollary 1.4] and the references therein.
Let (M,g) be asymptotically flat of dimension 3 ≤ n ≤ 7 with horizon boundarỹ ⊂M. If n = 3 and˜ is connected, Huisken and Ilmanen have proved the Riemannian Penrose inequality, stated here as Theorem 41, by evolving˜ by inverse mean curvature flow to a large coordinate sphere in the asymptotically flat chart. By an explicit calculation, they have showed that the Hawking mass of the evolving horizon is non-decreasing and, in fact, constant if (M(˜ ),g) is scalar flat and foliated by totally umbilic constant mean curvature spheres. To prove Theorem 41 in the general case where 3 ≤ n ≤ 7 and is possibly disconnected, Bray and Lee have used a conformal flow of the metricg along which the mass is non-increasing while the area of the horizon boundary remains constant. Their proof shows that the mass is, in fact, constant along this flow if and only if a suitable conformal transformation of the double ofM(˜ ) obtained by reflection across˜ has zero mass. If (M,g) is spin, the rigidity results in [38, §3] apply to the possibly non-smooth double. As a consequence, (M(˜ ),g) is isometric to the exterior region of a Schwarzschild space (31). The work of Lu and Miao [26] shows that the assumption that (M,g) be spin is not necessary.
Finally, in Theorem, we give a short variational proof of rigidity in the Riemannian Penrose inequality that does not require the spin assumption. To this end, we show that if equality holds in (32), then the double of (M(˜ ),g) obtained by reflection across˜ is smooth. In fact, we observe that if˜ has non-vanishing second fundamental form, we can locally perturb the metricg to increase R(g) without decreasing the area of˜ . By a global conformal transformation to zero scalar curvature, we may then decreasem(g) without decreasing the area of˜ by much.