Effective versions of the positive mass theorem
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Abstract
The study of stable minimal surfaces in Riemannian 3manifolds (M, g) with nonnegative scalar curvature has a rich history. In this paper, we prove rigidity of such surfaces when (M, g) is asymptotically flat and has horizon boundary. As a consequence, we obtain an effective version of the positive mass theorem in terms of isoperimetric or, more generally, closed volumepreserving stable CMC surfaces that is appealing from both a physical and a purely geometric point of view. We also include a proof of the following conjecture of Schoen: An asymptotically flat Riemannian 3manifold with nonnegative scalar curvature that contains an unbounded areaminimizing surface is isometric to flat \(\mathbb {R}^3\).
1 Introduction
The geometry of stable minimal and volumepreserving stable constant mean curvature surfaces in asymptotically flat 3manifolds (M, g) with nonnegative scalar curvature is witness to the physical properties of the spacetimes containing such (M, g) as maximal Cauchy hypersurfaces; see e.g. [9, 10, 20, 37, 38, 55, 64]. The transition from positive to nonnegative scalar curvature of (M, g), which is so crucial for physical applications, is a particularly delicate aspect in the analysis of such surfaces. Here we establish optimal rigidity results in this context that apply very generally. We apply them to obtain a precise understanding of the behavior of large isoperimetric or, more generally, closed volumepreserving stable constant mean curvature surfaces in (M, g) that extends the results of Brendle and Eichmair [13] and Eichmair and Metzger [27, 28, 29]. In combination with existing literature, this yields a rather complete analogy between the picture in (M, g) and classical results in Euclidean space.
We review the standard terminology and conventions that we use here in Appendix A. In particular, we follow the convention that stable minimal surfaces are by definition twosided.
Observe that this line of reasoning cannot establish the rigidity part (only Euclidean space has vanishing mass) of the positive mass theorem. Conversely, a beautiful idea of Lohkamp [41, Sect. 6] shows that the rigidity assertion of the positive mass theorem implies the nonnegativity of ADMmass in general. Indeed, he shows that it suffices to prove that an asymptotically flat Riemannian 3manifold with horizon boundary and nonnegative scalar curvature is flat if it is flat outside of a compact set.
The ideas of Schoen and Yau described above are instrumental to our results here. We record the following technical variation on their work as a precursor of Theorems 1.2 and 1.3 below.
Proposition 1.1
(Sect. 6 in [28]) Let (M, g) be an asymptotically flat Riemannian 3manifold. Assume that \(\Sigma \subset M\) is the unbounded component of an areaminimizing boundary in (M, g), and that the scalar curvature of (M, g) is nonnegative along \(\Sigma \). Then \(\Sigma \subset M\) is totally geodesic and the scalar curvature of (M, g) vanishes along this surface. Moreover, for all \(\rho > 1\) sufficiently large, \(\Sigma \) intersects \(S_\rho \) transversely in a nearly equatorial circle. The Gauss curvature is integrable and \(\int _\Sigma K = 0\).
We also mention that other proofs of the positive mass theorem (including that of Witten [70] based on the Bochner formula for harmonic spinors and that of Huisken and Ilmanen [37] based on inverse mean curvature flow) have been given.
The discoveries of Schoen and Yau have incited a remarkable surge of activity investigating the relationship between scalar curvature, locally areaminimizing (or stable minimal) surfaces, and the physical properties of spacetimes evolving from asymptotically flat Riemannian 3manifolds according to the Einstein equations. This has lead to spectacular developments in geometry and physics. We refer the reader to [2, 10, 22, 31, 32, 37, 59] to gain an impression of the wealth and breadth of the repercussions.
The following rigidity result for scalar curvature was first proven by the firstnamed author under the additional assumption of quadratic area growth for the surface \(\Sigma \). Subsequently, the quadratic area growth assumption was removed independently (in the form of Theorem 1.2 below) by the firstnamed author [16] and (in the form of Theorem 1.3 below) in a joint project of the second and thirdnamed authors. The proof of Theorem 1.3 is included in this paper.
Theorem 1.2
[16] Let (M, g) be an asymptotically flat Riemannian 3manifold with nonnegative scalar curvature. Let \(\Sigma \subset M\) be a noncompact properly embedded stable minimal surface. Then \(\Sigma \) is a totally geodesic flat plane and the ambient scalar curvature vanishes along \(\Sigma \). Such a surface cannot exist under the additional assumption that (M, g) is asymptotic to Schwarzschild with mass \(m >0\).
Theorem 1.3
Let (M, g) be a Riemannian 3manifold with nonnegative scalar curvature that is asymptotic to Schwarzschild with mass \(m > 0\) and which has horizon boundary. Every complete stable minimal immersion \(\varphi : \Sigma \rightarrow M\) that is proper is an embedding of a component of the horizon.
To obtain these results, it is necessary to understand how nonnegative scalar curvature keeps in check the a priori wild behavior at infinity of the minimal surface. This difficulty does not arise in the original argument by Schoen and Yau. The proofs of Theorems 1.2 and 1.3 use properness in a crucial way. Moreover, the embeddedness assumption is essential in the derivation of Theorem 1.2 in [16].
In spite of their geometric appeal, we cannot apply Theorems 1.2 and 1.3 to prove effective versions of the positive mass theorem such as Theorem 1.10 below. This is intimately related to the fact that properness is not preserved by convergence of immersions. Our first main contribution here is the following technical result that rectifies this:
Theorem 1.4
Let (M, g) be an asymptotically flat Riemannian 3manifold with nonnegative scalar curvature. Assume that there is an unbounded complete stable minimal immersion \(\varphi : \Sigma \rightarrow M\) that does not cross itself. Then (M, g) admits a complete noncompact properly embedded stable minimal surface.
Using this, we obtain the following substantial improvement of Theorems 1.2 and 1.3:
Theorem 1.5
Let (M, g) be a Riemannian 3manifold with nonnegative scalar curvature that is asymptotic to Schwarzschild with mass \(m > 0\) and which has horizon boundary. The only nontrivial complete stable minimal immersions \(\varphi : \Sigma \rightarrow M\) that do not cross themselves are embeddings of components of the horizon.
For the proof of Theorem 1.4, we develop in Sect. 4 a general procedure of extracting properly embedded top sheets from unbounded complete stable minimal immersions that do not cross themselves. The method depends on a purely analytic stability result—Corollary C.2—that restricts the topological type of complete stable minimal immersions into (M, g).
The proof of the positive mass theorem suggests the following conjecture [60, p. 48] of Schoen: An asymptotically flat Riemannian manifold with nonnegative scalar curvature that contains an unbounded areaminimizing surface is isometric to Euclidean space. We include here a proof of this conjecture and that of a related rigidity result for slabs, both due to the second and thirdnamed authors:
Theorem 1.6
The only asymptotically flat Riemannian 3manifold with nonnegative scalar curvature that admits a noncompact areaminimizing boundary is flat \(\mathbb {R}^3\).
We recall the precise meaning of areaminimizing boundaries in Appendix I.
Theorem 1.7
Let (M, g) be an asymptotically flat Riemannian 3manifold with nonnegative scalar curvature and with horizon boundary. Any two disjoint connected unbounded properly embedded complete minimal surfaces in (M, g) bound a region that is isometric to a standard Euclidean slab \(\mathbb {R}^2 \times [a, b]\).
The proofs of Theorems 1.6 and 1.7 are inspired by the recent refinement due to Liu [40] of a strategy of Anderson and Rodríguez [2] to prove rigidity results for complete manifolds with nonnegative Ricci curvature.
We point that that we may excise the slab in the conclusion of Theorem 1.7 from (M, g) to produce a new smooth asymptotically flat Riemannian 3manifold with nonnegative scalar curvature that contains a properly embedded totally geodesic flat plane along which the ambient scalar curvature vanishes.
For comparison, we recall the following consequence of a gluing result due to the firstnamed author and Schoen:
Theorem 1.8
[17] There exists an asymptotically flat Riemannian metric \(g = g_{ij} \, dx^i \otimes dx^j\) with nonnegative scalar curvature and positive mass on \(\mathbb {R}^3\) such that \(g_{ij} = \delta _{ij}\) on \(\mathbb {R}^2 \times (0, \infty )\).
The coordinate planes \(\mathbb {R}^2 \times \{z\}\) with \(z > 0\) in Theorem 1.8 are stable minimal surfaces. In particular, the areaminimizing condition in Theorem 1.6 cannot be relaxed. We also see that the condition that (M, g) be asymptotic to Schwarzschild in Theorem 1.5 is necessary.
There is a rich theory of rigidity results for (compact) minimal surfaces in Riemannian 3manifolds with a lower scalarcurvature bound. We refer the reader to the papers [1, 7, 8, 15, 43, 44, 49, 54] for several recent results in this direction, and to the introductions of these papers for a complete overview.
Corollary 1.9
We are grateful to Lucas Ambrozio and Pengzi Miao for valuable discussions concerning this point.
We now turn our attention to the role played by closed volumepreserving CMC surfaces in asymptotically flat manifolds.
In the next two main results, we investigate the question of global uniqueness results for large volumepreserving stable CMC surfaces in asymptotically flat manifolds.
Theorem 1.10
In conjunction with the uniqueness results from [38, 57], we obtain the following consequence:
Corollary 1.11
Let (M, g) be a Riemannian 3manifold with nonnegative scalar curvature that is asymptotic to Schwarzschild with mass \(m > 0\) and which has horizon boundary. Let \(p \in M\). Every connected closed volumepreserving stable CMC surface \(\Sigma \subset M\) that contains p and which has sufficiently large area is part of the canonical foliation.
Theorem 1.10 was proven by the thirdnamed author and Metzger in [27] under the (much) stronger assumption that (M, g) has positive scalar curvature. As we have already mentioned, our improvement here is closely tied to the generality of Theorem 1.4.
In [13], Brendle and the thirdnamed author have constructed examples of Riemannian 3manifolds asymptotic to Schwarzschild with positive mass that contain a sequence of larger and larger volumepreserving stable CMC surfaces that diverge to infinity together with the regions they bound. Thus, in the uniqueness results of [38, 57], a proviso that the surfaces enclose some given set is certainly necessary. When the assumption of Schwarzschild asymptotics is dropped, the examples in Theorem 1.8 show even more dramatically that some such a condition is necessary to obtain uniqueness results. Theorem 1.10 extends the results of [38, 57] optimally in this sense.
We remark that much progress has been made recently in developing analogues of the results of [38, 57] in general asymptotically flat Riemannian 3manifolds, see e.g. [36, 42, 53].
Christodoulou and Yau [20] have noted that the Hawking mass of volumepreserving stable CMC spheres in asymptotically flat Riemannian 3manifolds with nonnegative scalar curvature is nonnegative. This observation makes these surfaces particularly appealing from a physical standpoint. Geometrically, they arise in the variational analysis of the fundamental question of isoperimetry. The results described above beg the question whether each leaf of the canonical foliation \(\{\Sigma _H\}_{H \in (0, H_0]}\) has least area for the volume it encloses, and whether it is uniquely characterized by this property. This global uniqueness result was established by Metzger and the thirdnamed author in [28]. (In exact Schwarzschild, this was proven by Bray in his dissertation [9].) Unlike the results based on stability that we have described above, the existence and global uniqueness of isoperimetric regions of large volume has been verified in higher dimensions as well [29].
Theorem 1.12
Assume now that \(n = 3\) and that the scalar curvature of (M, g) is nonnegative. Remarkably, isoperimetric regions \(\Omega _V\) exist in (M, g) for all volumes \(V > 0\) in this case. This follows from a beautiful observation due to Shi [66], as we explain in Appendix K. It is natural to wonder about the behavior of \(\Omega _V\) for large volumes \(V>0\). For simplicity of exposition, we assume for a moment that M has empty boundary. Let \(\Sigma _i = \partial \Omega _{V_i}\) where \(V_i \rightarrow \infty \). It has been shown in [28] that these surfaces either diverge to infinity as \(i \rightarrow \infty \), or that alternatively a subsequence of these surfaces converges geometrically to a noncompact areaminimizing boundary \(\Sigma \subset M\). In view of Theorem 1.6, the latter is impossible unless (M, g) is flat \(\mathbb {R}^3\). We arrive at the dichotomy that large isoperimetric regions in (M, g) are either drawn far into the asymptotically flat end, or they contain the center of the manifold.
Corollary 1.13
Let (M, g) be an asymptotically flat Riemannian 3manifold with nonnegative scalar curvature and positive mass. Let \(U \subset M\) be a bounded open subset that contains the boundary of M. There is \(V_0 > 0\) so that for every isoperimetric region \(\Omega \subset M\) of volume \(V \ge V_0\), either \(U \subset \Omega _V\) or \(U \cap \Omega _V\) is a thin smooth region that is bounded by the components of \(\partial M\) and nearby stable constant mean curvature surfaces.
Note that the conclusion of the corollary is wrong for flat \(\mathbb {R}^3\). When the scalar curvature of (M, g) is everywhere positive, this result was observed as Corollary 6.2 of [28]. The role of Theorem 1.6 here is that of Theorem 1.5 in the proof of Corollary 1.11.
2 Sheeting of volumepreserving stable CMC surfaces
Proposition 2.1
Proof
Let \(\pi : \Sigma \rightarrow \tilde{\Sigma }\) be the universal cover of \(\tilde{\Sigma }\). Let \(x^{*} \in \Sigma \) be a point such that \(\pi (x^{*}) = \tilde{x}^{*}\). Consider the immersion \(\varphi = \tilde{\varphi }\circ \pi : \Sigma \rightarrow M\).
In the argument below, we denote the second fundamental forms of the submanifolds \(\Sigma _k\) and the immersion \(\varphi : \Sigma \rightarrow M\) by \(h_k\) and by h respectively. Let \(U \subset \Sigma \) be open, bounded, connected, and simply connected with \(x^{*} \in U\). Fix \(r > 0\) sufficiently small.
Assume that there is a point in \(\Sigma \) where \(h^{2} + R\circ \varphi > 2 \delta \) for some \(\delta > 0\). Let \(U \subset \Sigma \) be a subset as above that contains this point. Fix \(k \ge 1\) sufficiently large. Then, for each \(j \in \{1,\dots ,n(k)\}\), this implies that the surface \(\Sigma _k^j\) contains a subset where \(h_k^2 + R > \delta \) whose area is bounded below independently of k. Because n(k) can be taken arbitrarily large, this contradicts (20). It follows that \(\varphi : \Sigma \rightarrow M\) is totally geodesic and \(R \circ \varphi = 0\).
To see that \(\varphi : \Sigma \rightarrow M\) is stable, it is enough to show that every bounded open subset \(U \subset \Sigma \) admits a positive Jacobi function. The argument below is similar to [67, p. 333], [45, p. 732], or [46, p. 493]. We may assume that U is simply connected and that \(x^{*} \in U\). By the argument above, \(\Sigma _k\) contains two disjoint pieces that appear as small graphs above U whose unit normals approximately point in the same direction. The defining functions of these graphs are ordered. They tend to zero in \(C^2_{loc}(U)\) as \(k \rightarrow \infty \). These functions satisfy the same graphical prescribed constant mean curvature equation on U. Hence, their difference is a positive solution of a linear uniformly elliptic partial differential equation. By the Harnack principle, the supremum and the infimum of this solution are comparable on small balls. As in [67, p. 333], we may rescale and pass to a subsequence that converges to a positive solution of the Jacobi equation on U.
It follows from [32, Theorem 3(ii)] that \(\Sigma \) with the induced metric is conformal to the plane. \(\square \)
3 Bounded complete stable minimal immersions
Proposition 3.1
Let (M, g) be an asymptotically flat Riemannian 3manifold with horizon boundary. Every complete minimal immersion \(\varphi : \Sigma \rightarrow M\) with uniformly bounded second fundamental form is either unbounded or an embedding of a component of the horizon.
Proof
Assume that the trace \(\varphi (\Sigma )\) of the immersion \(\varphi : \Sigma \rightarrow M\) is contained in a compact set. Let S be the union of the horizon and the closure of \(\varphi (\Sigma )\). There is a closed minimal surface in M that contains S. To see this, let \(r > 1\) large be such that \(S \subset B_r\) and such that the mean curvature of the coordinate sphere \(S_r\) with respect to the outward pointing unit normal is bounded below by \(H_0 > 0\).
4 Top sheets
Lemma 4.1
Proof
All rescalings take place in the chart at infinity.
Proposition 4.2
Let (M, g) be an asymptotically flat Riemannian 3manifold with nonnegative scalar curvature. Assume that there is an unbounded complete stable minimal injective immersion \( \varphi :\Sigma \rightarrow M. \) Then there is a proper such embedding.
Proof
All rescalings take place in the chart at infinity.
5 Proofs of main theorems
Proof of Theorem 1.3
Remark 5.1
The argument from [32] applied as in the last step of the preceding proof shows that the surface \(\Sigma \subset M\) in Proposition 1.1 is intrinsically flat.
Proof of Theorem 1.4
The domain \(\Sigma \) with the induced metric is conformal to the plane by Corollary C.2. If the immersion is injective, the result follows from Proposition 4.2. If not, it follows from Remark F.3 and Lemma F.5 that the immersion \(\varphi : \Sigma \rightarrow M\) factors to an unbounded complete stable minimal immersion \(\tilde{\varphi }: \tilde{\Sigma }\rightarrow M\) through a sidepreserving covering \(\pi : \tilde{\Sigma }\rightarrow \Sigma \). Note that \(\tilde{\Sigma }\) is cylindrical by topological reasons. This is impossible by Corollary C.2. \(\square \)
Proof of Theorem 1.5
This is immediate from Theorems 1.4 and 1.3, Lemma D.2, and Proposition 3.1. \(\square \)
Proof of Theorem 1.6
We first deal with the case where the boundary of M is empty.
Let \(r_0 > 0\) be as in Appendix J. Let \(\rho _0 > 1\) be such that \(S_\rho \) is convex for all \(\rho \ge \rho _0\). Every closed minimal surface of (M, g) is contained in \(B_{\rho _0}\).

\(B_{\rho _0}\) is disjoint from \(\{x \in M : \mathrm {dist}_g(x, p) < 4 r\}\);

\(r = \mathrm {dist}_g(\Sigma , p)/2 < r_0\);

\(\Sigma \) intersects \(\{x \in M : \mathrm {dist}_g(x, p) < 4 r\}\) in a single component, and the function \(\mathrm {dist}_g( \, \cdot \, , p)\) is decreasing in the direction of the unit normal of this component that is pointing into \(M_+\).
 (i)
\(g(t) \rightarrow g\) smoothly as \(t \rightarrow 0\);
 (ii)
\(g(t) = g\) on \(\{x \in M : \mathrm {dist}_g (x, p) \ge 3 r \}\);
 (iii)
\(g(t) \le g\) as quadratic forms on M, with strict inequality on \(\{x \in M : r< \mathrm {dist}_g (x, p) < 3 r\}\);
 (iv)
The scalar curvature of g(t) is positive on \(\{x \in M : r< \mathrm {dist}_g(x, p) < 3 r\}\);
 (v)
The region \(M_+\) is weakly meanconvex with respect to g(t).
By taking \(\epsilon >0\) smaller if necessary, we may assume that all closed minimal surfaces of (M, g(t)) are contained in \(B_{\rho _0}\).
Using standard convergence results from geometric measure theory, we now find a connected unbounded properly embedded separating surface \(\Sigma (t) \subset M\) as a subsequential geometric limit of \(\Sigma _\rho (t)\) as \(\rho \rightarrow \infty \). By construction, \(\Sigma (t)\) is contained in \(M_+ \cup \Sigma \) where it is areaminimizing with respect to g(t). Moreover, \(\Sigma (t)\) intersects \(\{x \in M : \mathrm {dist}_g (x, p) \le 3 r\}\). If \(\Sigma (t)\) intersects \(\{x \in M : \mathrm {dist}_g(x, p) < 3 r\}\), then it also intersects \(\{x \in M : \mathrm {dist}_g(x, p) \le r\}\) because of (iv) and Proposition 1.1. Passing to a subsequential geometric limit as \(t \rightarrow 0\), we obtain a connected unbounded properly embedded separating surface \(\Sigma _+ \subset M\) that is contained in \(M_+ \cup \Sigma \) where it minimizes area with respect to g. Using now the areaminimizing property of \(\Sigma \), we see that \(\Sigma _+\) is in fact areaminimizing in all of M. Note that \(\Sigma \) intersects \(\{x \in M : \mathrm {dist}_g(x, p) < 3 r\}\) while it is disjoint from \(\{x \in M : \mathrm {dist}_g(x, p) \le r\}\). It follows from the maximum principle that \(\Sigma \) and \(\Sigma _+\) are disjoint.
We may repeat this argument, beginning with any surface \(\Sigma _{+}\) constructed as above. It follows that an open neighbourhood of \(\Sigma \) in (M, g) is flat and in fact isometric to standard \(\mathbb {R}^2 \times ( \epsilon , \epsilon )\) for some \(\epsilon > 0\). Moreover, the surfaces in M that correspond to \(\mathbb {R}^2 \times \{z\}\) where \(z \in ( \epsilon , \epsilon )\) are all areaminimising. Using standard compactness properties of such surfaces and a continuity argument, we conclude that (M, g) is isometric to flat \(\mathbb {R}^3\).
We now turn to the general case where M has boundary. Consider \(\Omega \in {\mathcal {F}}\) with noncompact areaminimizing boundary \(\Sigma \subset M\). The unique noncompact component \(\Sigma _0 \subset M\) of \(\Sigma \) is a separating surface. Let \(M_\) and \(M_+\) denote the two components of its complement in M. Note that the interior of \(\Omega \cap M\) agrees with either \(M_\) (Case 1) or \(M_+\) (Case 2) outside of \(B_{\rho _0}\). The proof that g is flat in \(M_+\) proceeds exactly as above, except for the following change. In Case 1, we let \(\Sigma _\rho (t)\) have least area among properly embedded surfaces with boundary \(\Gamma _\rho \) that bound together with \(\Sigma _0 \cap B_\rho \) in \(M_+ \cap B_\rho \) and relative to \(M_+ \cap \partial M\). In Case 2, we let \(\Sigma _\rho (t)\) have least area among properly embedded surfaces with boundary \(\Gamma _\rho \) that bound together with \(M_+ \cap S_\rho \) in \(M_+ \cap B_\rho \) and relative to \(M_+ \cap \partial M\). Theorem 1.6 follows upon switching the roles of \(M_\) and \(M_+\). \(\square \)
Remark 5.2
The use of the conformal change of metric in this proof is inspired by an idea of Liu in his classification of complete noncompact Riemannian 3manifolds with nonnegative Ricci curvature [40]. The observation (12) is crucial in the proof of Theorem 1.6, as we use it to be sure that the surfaces \(\Sigma _\rho (t)\) do not run off as \(\rho \rightarrow \infty \). This observation is not needed for Theorem 1.7 below, since the solutions of Plateau problems considered in the proof cannot escape the slab as we pass to the limit. We point out that at a related point in the work of Anderson and Rodríguez [2], their assumption of nonnegative Ricci curvature is used tacitly in their delicate estimation of comparison surfaces [2, (1.5)].
Proof of Theorem 1.7
Since (M, g) has horizon boundary, M is diffeomorphic to the complement of a finite union of open balls with disjoint closures in \(\mathbb {R}^3\). Let \(\Omega \subset M\) be the connected region bounded by two disjoint unbounded properly embedded complete minimal surfaces \(\Sigma _0, \Sigma _1 \subset M\). By solving a sequence of Plateau problems in \(\Omega \cap B_r\) with boundary on \(\Omega \cap S_r\) and passing to a subsequential geometric limit as \(r \rightarrow \infty \), we obtain an unbounded properly embedded boundary \(\Sigma \subset M\) that is contained in \(\bar{\Omega }\) where it is areaminimizing with respect to g. In particular, every component of \(\Sigma \) is a stable minimal surface. By the maximum principle, if such a component intersects with \(\Sigma _0\) or \(\Sigma _1\), then it coincides with the respective surface. By Theorem 1.2, every unbounded component is a totally geodesic flat plane along which the ambient scalar curvature vanishes. We may now proceed as in the proof of Theorem 1.6. \(\square \)
Proof of Theorem 1.10
Remark 5.3
Proof of Theorem 1.12
Footnotes
 1.
 2.
In fact, either \(1 = \limsup _{r \rightarrow \infty }{\text {length} (\partial \Sigma _r)}/{2 \pi r}\) or \(2 \le \limsup _{r \rightarrow \infty }{\text {length} (\partial \Sigma _r)}/{2 \pi r} \).
 3.
The proof of Theorem 1.2 simplifies considerably for surfaces with quadratic area growth. Indeed, the arguments in [27, Sections 3 and 4] show that \(\int _{\Sigma }K = 0\). It follows from [32, p. 209] that \(\Sigma \) is flat with its induced metric. Lemma E.5 is quite elementary for surfaces with quadratic area growth, see the argument in [27, Lemma 3.5]. Finally, the Gauss equation rearrangement argument applied in the manner of Schoen and Yau leads to a contradiction.
 4.
It seems to us that the proof given in [65] “only” shows that there are no stable minimal immersions of the cylinder into (M, g) if the ambient scalar curvature is positive; see the argument given in [65, top of p. 216] and also the sentence after the statement of their Theorem 2. In the proof given in [50], consider the integral over the ball \(B_r\) at the bottom of page 292. In the evaluation of this integral using conformal invariance as suggested on the next page, we do not see how the geometry of the “conformally changed” domain is controlled so that the “order” of the test functions on the cylinder carries over.
 5.
In fact, the relevant twodimensional case of the splitting theorem is due to CohnVossen.
 6.
In other words, every sequence \(\{x_i\}_{i=1}^\infty \subset \Sigma \) that is Cauchy with respect to the induced Riemannian distance either has a limit in \(\Sigma \) or is such that \(\varphi (x_i) \rightarrow 0\).
 7.
Recall that in this paper a stable minimal immersion is by definition twosided.
 8.
We could also work with the larger class of all 3dimensional submanifolds with locally finite boundary area. However, by standard geometric measure theory, every such submanifold with areaminimizing boundary is properly embedded.
Notes
Acknowledgments
The firstnamed author wishes to express his gratitude to Richard Schoen for introducing him, with great professionality and unparalleled enthusiasm, to the mathematical challenges of general relativity. He also thankfully acknowledges the support of André Neves through his ERC Starting Grant. The secondnamed author would like to convey his deepest thanks to his advisor Simon Brendle for his invaluable support and continued encouragement. His research was supported in part by an NSF fellowship DGE1147470 as well as the EPSRC grant EP/K00865X/1. The thirdnamed author expresses his gratitude to Hubert Bray, Simon Brendle, Gregory Galloway, Gerhard Huisken, Jan Metzger, and Richard Schoen. A part of this paper was written up during his invigorating stay of two months at Stanford University, which was supported by their Mathematical Sciences Research Center. The second and thirdnamed authors would also like to thank the ErwinSchrödingerInstitute of the University of Vienna for its hospitality during the special program “Dynamics of General Relativity: Numerical and Analytic Approaches” in the summer of 2011. Open access funding for this article was provided by University of Vienna. It is a pleasure to sincerely congratulate Richard Schoen on the occasion of his 65th birthday.
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