Intrinsic flat stability of the positive mass theorem for asymptotically hyperbolic graphical manifolds

The rigidity of the Riemannian positive mass theorem for asymptotically hyperbolic manifolds states that the total mass of such a manifold is zero if and only if the manifold is isometric to the hyperbolic space. This leads to study the stability of this statement, that is, if the total mass of an asymptotically hyperbolic manifold is almost zero, is this manifold close to the hyperbolic space in any way? Motivated by the work of Huang, Lee and Sormani for asymptotically flat graphical manifolds with respect to intrinsic flat distance, we show the intrinsic flat stability of the positive mass theorem for a class of asymptotically hyperbolic graphical manifolds by adapting the positive answer to this question provided by Huang, Lee and the third named author.


Introduction
In the context of mathematical relativity, asymptotically hyperbolic manifolds correspond to initial data sets for the Einstein equations with a negative cosmological constant Λ. An asymptotically hyperbolic manifold is, roughly speaking, a Riemannian manifold (M n , g) such that the metric g approaches the metric of the n-dimensional hyperbolic space, H n , at infinity sufficiently fast. Under appropriate decay conditions on g there is a well defined notion of total mass of (M n , g) given by Chruściel and Herzlich [CH03], and Wang [Wan01].
From the constraint equations for the Einstein equations, it follows that if the initial data set is a time-symmetric, asymptotically hyperbolic manifold (M n , g), then with a suitable normalization of the cosmological constant, the dominant energy condition reduces to a lower bound on the scalar curvature, R(g) ≥ −n(n − 1). The positive mass theorem then asserts that the mass of this type of manifolds is non-negative, and it is equal to 0 if and only if the manifold is isometric to H n . The history of the positive mass theorem and its different proofs is rich. We refer the reader to a recent proof by Sakovich [Sak21], which also contains a complete description of the history of this result and previous results. The rigidity part was established in general by Huang, Jang and Martin in [HJM20].
From the rigidity statement of the positive mass theorem it is natural to ask whether a stability statement holds. The answer to this question is subtle, as can be seen in examples for the asymptotically flat setting given by Lee and Sormani in [LS12], showing that the answer is negative with respect to some usual topologies. Nonetheless, the stability of the positive mass theorem has been established in some cases. In particular, the Sormani-Wenger intrinsic flat distance [SW11] has shown to be an adequate notion of distance for this problem. In [SS17] Sakovich and Sormani obtained a stability result for the positive mass theorem for complete rotationally symmetric asymptotically hyperbolic manifolds with respect to this distance. Huang and Lee [HL15] showed stability of the positive mass theorem with respect to the Federer-Fleming flat distance for a class of asymptotically flat graphical manifolds. Subsequently Huang, Lee and Sormani showed stability of the positive mass theorem for a smaller subclass with respect to the intrinsic flat distance [HLS17]. While there were two gaps in [HLS17], see [HLS22], these are now completely filled in. The gap in [HLS17,Theorem 1.3] was filled in by work of Del Nin and the third named author [DNP22], and the gap in [HLS17,Theorem 1.4] by the work of Huang, Lee and the third named author [HLP22]. There is also a proof of [HLS17, Theorem 1.3] using a different approach to [HLS17] in [HLP22] (c.f. [AP21, Theorem 7.2] for a proof in the entire case).
Following the work in [HL15], the first named author showed in [Cab19] the stability of the positive mass theorem for a class of asymptotically hyperbolic graphical manifolds with respect to the Federer-Fleming flat distance. In this work, starting from [Cab19] and following [HLP22], we establish a stability result for a subclass of asymptotically hyperbolic graphs with respect to the intrinsic flat distance. We note that we cannot use the results in [DNP22] since they only ensure existence of intrinsic flat limits of the form (B R n (R), d R n , [[B R n (R)]]).
For asymptotically hyperbolic manifolds stability has been established in some other cases (see [SS17,All18]). The techniques in [HL15] have been also successfully applied to obtain a stability result of the Brown-York mass by Alaee, McCormick and the first named author in [ACPM21], and the techniques in [HL15,HLS17,AP21,HLP22] have also been applied to obtain flat and intrinsic flat stability results for tori with almost non-negative scalar curvature by the first and third named authors [CPKP20].
We will follow the definition of asymptotically hyperbolic graphs and the adaptation of the total mass for asymptotically hyperbolic manifolds to asymptotically hyperbolic graphs by Dahl, Gicquaud and Sakovich [DGS13]. We write H n+1 as the warped product H n × V R with metricb = b + V 2 ds 2 , where b is the metric of H n which we write in coordinates defined on R n = [0, ∞) × S n−1 and V (r) = cosh(r) where r represents the radial coordinate. We denote the open ball in H n of radius ρ around the origin by B b (ρ). We start by defining the class of asymptotically hyperbolic graphs that where studied in [Cab19] and which will be the basis for the subclass we will consider here. Definition 1.1. For n ≥ 3, define G n to be the space of graphs of functions, graph(f ) ⊂ H n+1 , where f : H n \ U → R is a continuous function which is smooth on H n \ U and U ⊂ H n is an open and bounded subset whose complement is connected, such that graph(f ) is a balanced asymptotically hyperbolic graph when endowed with the metric induced by H n+1 , with scalar curvature greater than or equal to −n(n − 1), and it is either entire or with minimal boundary. In addition, we require: (1) The mean curvature vector of graph(f ) in H n+1 points upward (2) For almost every h ∈ R, the level set f −1 (h) is star-shaped and outer-minimizing in H n .
We denote by m(f ) the total mass of any graph(f ) ∈ G n . The first named author showed stability of the class G n [Cab19]. In particular, after vertically translating graph(f ), it was shown that in any ball Bb(ρ) ⊂ H n+1 of radius ρ centered at the origin, Convergence with respect to the flat distance does not necessarily imply convergence with respect to the intrinsic flat distance. Hence, it is interesting to study the stability of the positive mass theorem with respect to intrinsic flat distance for the class G n . Since by Wenger's compactness theorem, compact oriented Riemannian manifolds with a uniform upper diameter bound and a uniform upper volume bound for the manifolds and their boundaries is precompact with respect to intrinsic flat distance, we will consider (in a similar way to [HLS17,HLP22]) a subclass of G n . Definition 1.2. For constants ρ 0 , γ, D > 0, we define G n (ρ 0 , γ, D) to be the space of ndimensional manifolds (M, g) (possibly with boundary) that admit a smooth Riemannian isometric embedding into H n+1 , Ψ : M → H n+1 , such that Ψ(M) = graph(f ) for some graph(f ) ∈ G n and that in addition the following is satisfied: (3) U ⊂ B b (ρ 0 /2) (4) For r ≥ ρ 0 2 , a uniform decay condition holds, (1.1) The region Ω(ρ 0 ) = Ψ −1 (B b (ρ 0 ) × R) has bounded depth, depth(Ω(ρ 0 )) = sup{d M (p, Σ(ρ 0 )) : p ∈ Ω(ρ 0 )} ≤ D, where d M denotes the length distance in M induced by g and Σ(ρ 0 ) = ∂Ω(ρ 0 ) \ ∂M.
(6) If f is not entire, a stronger condition for the minimal boundary holds, ∇ b ν f → ∞ while ∇ b X f remains bounded as one approaches ∂U (where ν is the local vector field obtained by extending the outward unit normal of ∂U to a neighborhood of ∂U by parallel transport along the flow lines of the normal exponential map and X is any vector field with X ⊥ b ν). We will additionally demand that our minimal boundary is mean convex, i.e., H ≥ 0, where H denotes the mean curvature of ∂U as a submanifold of H n , and star-shaped.
Conditions (3) and (4) of Definition 1.2 provide a uniform control on the exterior region, i.e. the complement of Ω(ρ 0 /2), of the sequence of manifolds, while (5) prevents the formation of deep "gravity wells" (c.f. [LS14,HLS17]). Conditions (3)-(5) provide a uniform intrinsic diameter bound of regions Ω j (ρ) and a uniform bound for vol(∂Ω j (ρ)). Conditions (1) and (2) together with the uniform diameter bound ensure convergence of vol(Ω j (ρ)) to vol(B b (ρ)). Thus, Wenger's compactness theorem ensures convergence to an integral current space. To get the precise limit space one has to use condition (6) to ensure that the regions Ω j (ρ) embed into suitable Riemannian manifolds diffeormorphic to B b (ρ) via a capping procedure as in [HLP22], Theorem A.1, in order to apply a compactness result by Allen and the third named author [AP21] (see Theorem 4.1). We note that to apply Theorem 4.1 one has to endow Ω(ρ) with the structure (Ω(ρ), d intr . The same is true when applying the compactness result from [AP21] in the proof of [HLP22, Theorem 3.2]. However, by [DNP22] the result in [HLS17] is correct for (Ω(ρ), d M , [[Ω(ρ)]]) as well. We also remark that boundedness of ∇ X f for tangential directions X as one approaches ∂U as specified in (6) was not originally stated in [HLS17], nor in [HLP22], though in the latter it was implicitly assumed.
We also obtain a pointed version. This is the analogue of [HLS17, Theorem 1.4].
Theorem 1.4. Let M j ∈ G n (ρ 0 , γ, D) be a sequence of asymptotically hyperbolic graph manifolds with lim j→∞ m(M j ) = 0 and p j ∈ Σ j (ρ 0 ) be a sequence of points. Then for almost every R > 0 we have This manuscript is organized as follows. In Section 2 we provide background material. In Section 3 we prove volume estimates for the regions Ω(ρ), uniform diameter and area bounds and also show their volume converges to the volume of a ball in hyperbolic space provided m(f ) → 0. Additionally, we show Gromov-Hausdoff and intrinsic flat convergence of annular regions, Ω(ρ ′ )\Ω(ρ), and that the inner boundaries, ∂M, converge to the zero integral current space. The proofs of the main results are given in Section 4. The proof of Theorem 1.3, in the entire case, consists in applying the compactness theorem of [AP21], Theorem 4.1, to the regions Ω(ρ), and in the non-entire case, we apply Theorem A.1, proven in Appendix A, where we enlarge the Ω(ρ)'s and use condition (6) to carefully construct diffeormorphisms from the enlargements to B b (ρ), so that Theorem 4.1 can be applied. Theorem 1.4 follows from Theorem 1.3 and Lemma 4.3, we believe the latter result is interesting in its own. It Christina Sormani for helpful and interesting discussions during the preparation of this work. AJCP is grateful for the generous support of the Carl Zeiss Foundation and the financial support of the Deutsche Forschungsgemeinschaft through the SPP 2026 "Geometry at Infinity". MG is grateful for the support of the Deutsche Forschungsgemeinschaft through the SPP 2026 "Geometry at Infinity". RP acknowledges support from CONACyT Ciencia de Frontera 2019 CF217392 grant. This project started during the Simons Center Mass in General Relativity Workshop, March 26-30, 2018 organized by Christina Sormani, Shing-Tung Yau, Richard Schoen, Mu-Tao Wang and Piotr Chrusciel to whom we are grateful.

Background
In this section we first collect some results from [DGS13] about asymptotic hyperbolic graphs and review material from [Cab19] about the class G n . In the second part, we define integral current spaces, intrinsic flat distance and state some results that we will apply in subsequent sections.
2.1. Asymptotically hyperbolic graphs. Here we give some definitions related to asymptotically hyperbolic graphs and state a Riemannian Penrose like inequality, obtained in [DGS13]. For a more detailed discussion about asymptotically hyperbolic graphs the reader is referred to [DGS13].
Let H n denote the hyperbolic space of dimension n and let b be its Riemannian metric, which in spherical coordinates (r, θ) ∈ [0, ∞) × S n−1 takes the form b = dr 2 + sinh 2 (r)σ, where σ represents the standard Riemannian metric of S n−1 .
We will consider graphs of functions over subsets of H n inside the (n + 1)-dimensional hyperbolic space H n+1 with Riemannian metricb, written in coordinates (r, θ, s) ∈ H n × R asb Given an open set U ⊂ H n and a continuous function f : and the Riemannian metric g induced by H n+1 . To simplify notation, we will often denote geometric quantities associated to graph(f ) using f instead of its metric. For example, we denote its scalar curvature as R(f ). This should not cause any confusion as the meaning of the symbols will be clear from the context. While there is a more general definition of asymptotically hyperbolic manifolds and their mass (see [CH03], [Wan01]) we will only consider asymptotically hyperbolic graphs and so, in the interest of simplicity, will only present the graph case here (following [DGS13]). Definition 2.1. Let n ≥ 3 and U ⊂ H n be an open (possibly empty) bounded subset with connected complement. Let f : H n \ U −→ R be a continuous function which is smooth on H n \ U . We say that (graph(f ), g) is an asymptotically hyperbolic graph (or simply f is asymptotically hyperbolic) with respect to the chart Π as in where B is a closed ball in H n that properly contains U and dvol b denotes the volume form induced by b.
In general, the total mass of an asymptotically hyperbolic manifold can be defined as the minimization of a functional, called the mass functional, that depends on its coordinate chart at infinity, say Ψ. If the mass functional is positive over an appropriate subset of a vector space, then Ψ can be chosen so that the mass takes a simpler form; this suitable diffeomorphism Ψ is then referred to as a set of balanced coordinates [DGS13]. For an asymptotically hyperbolic graph, if Π as in (2.1) is a set of balanced coordinates we say that f is balanced. In this case the mass takes the form given below [DGS13].
2. If f is an asymptotically hyperbolic and balanced function, its mass is given by where e := V 2 df ⊗ df , V (r) = cosh(r), S r is the coordinate sphere of radius r in H n , ν r the outward normal vector to S r and ω n−1 denotes the volume of the round sphere S n−1 .
Definition 2.3. We say that an asymptotically hyperbolic function f : We now give an example of an asymptotically hyperbolic graph with minimal (and mean convex) boundary which represents an initial data set for the AdS-Schwarzschild spacetime.
Example 2.4 (AdS-Schwarzschild manifolds as graphs). Recall that the spatial AdS-Schwarzschild is defined for m ≥ 0 and n ≥ 3, as the manifold (ρ + , ∞) × S n−1 with the metric given by where ρ + is the largest root of ρ n + ρ n−2 − 2m = 0.
The mass of this manifold is equal to m, and clearly, g 0 = b. To see this manifold as a graph over H n it is convenient to make the change of variables ρ = sinh(r) on H n , so that the metric on H n+1 is given bȳ b = (1 + ρ 2 )ds 2 + 1 1+ρ 2 dρ 2 + ρ 2 σ. Then, the AdS-Schwarzschild graph of mass m is given by the function that is this function is constant in the theta parameter, and it can be checked by direct computations that this graph has a minimal boundary at ρ = ρ + .
We now state a Riemannian Penrose like inequality, which was very useful when proving flat convergence of sequences contained in the class G n . We will use it in the proofs of Theorem 3.8 and Lemma 3.11, to establish uniform volume bounds for the regions ∂Ω j (ρ), and U j and to show that the inner boundaries of Ω j (ρ), ∂M j , converge to the zero integral current space, respectively. . Suppose that f : H n \ U → R is a balanced asymptotically hyperbolic graph in H n+1 with minimal boundary and scalar curvature R(f ) ≥ −n(n−1). Suppose that ∂U is mean convex (i.e., H ≥ 0, where H denotes the mean curvature of ∂U in H n ) and that U contains an inner ball centered at the origin of radius r 0 . Then, vol(∂U) ≤ 2ω n−1 V (r 0 ) m(f ) where V (r) = cosh(r) and ω n−1 denotes the volume of the round sphere S n−1 . In particular, vol(∂U) ≤ 2ω n−1 m(f ).
Let us remark at this point that we will from now on, by slight abuse of notation, use vol to denote volumes of Riemannian (sub-)manifolds regardless of their dimension and we will suppress specifying the metric in the notation unless there would be inequivalent canonical choices.
2.2. The class G n and stability of the PMT with respect to flat distance. In [Cab19] the first named author, inspired by the work of Huang and Lee [HL15], defined the class G n and proved the stability of the hyperbolic positive mass theorem with respect to the flat distance (also known as Federer-Fleming distance). Here we review parts of this work needed in subsequent sections.
We now give more details about the conditions appearing in Definition 1.1. Given an asymptotically hyperbolic function, f : H n \ U → R, let H denote the mean curvature vector of graph(f ) inside H n+1 and let n 0 := ( 0, 1) ∈ T p H n+1 , for any p ∈ H n+1 . We say that H points upward ifb(n 0 , H) is a non-negative function that does not vanish everywhere. The mean curvature vector convention is that deformations in the direction given by this vector decrease volume. In particular, the standard sphere has positive mean curvature with respect to the inner pointing unit normal vector field. We say that f −1 (h) is star-shaped if in some radial coordinates (r, θ) on H n \ {0} the set can be written as a smooth graph Note that this in particular implies that f −1 (h) is a differentiable (n − 1)-dimensional submanifold of H n . For a bounded and finite perimeter set E ⊂ H n , we say that ∂ * E (where ∂ * E denotes the reduced boundary of E) is outer-minimizing if for any bounded set F containing E we have H n−1 b (∂ * E) ≤ P (F ) where P denotes the perimeter of F and H n−1 b denotes the (n − 1) dimensional Hausdorff measure of H n .
The key idea to prove the stability of the hyperbolic positive mass theorem for G n was to find a suitable "height", h 0 (f ), which divides any graph(f ) in two parts. In the lower part, one can show that all level sets of f have volume bounded above by some function that depends on m(f ) which goes to zero as m(f ) goes to zero. Meanwhile, in the upper part, the quantity sup(f ) − h 0 (f ) is bounded above by a function that also depends on m(f ) and goes to zero as m(f ) does.
In order to define this height, one studies the function wheref is the extension of f to H n which is defined to be constant on U. Using condition (1) of the definition of G n (i.e. that the mean curvature vector of graph(f ) points upward), it is shown that there exists h max ∈ R so that f < h max everywhere and that V is finite for any h < h max . Using condition (2) it is shown that V is non-decreasing. Therefore, V is differentiable almost everywhere and a height can be defined. If m(f ) < 1 then it follows that h 0 (f ) = sup{h : After an appropriate rescaling of f , using that the mean curvature of the level sets is nonnegative, a suitable expression for m(f ) and the Minkowski inequality, one shows that V ′ (h) ≥ F (V(h)) for almost every h ≥ h 0 (f ) and some function F . Then the upper bound for sup(f ) − h 0 (f ) is obtained by comparing V to the solution of the equation Y ′ (h) = F (Y (h)) with initial condition equal to V(h 0 (f )). That is, one finds that Y ≤ V and Y goes to infinity at a finite height, implying that V also does and hence giving an upper bound to the rescaling of f . Rescaling back gives the desired inequality: Lemma 4.8]). Let f ∈ G n , then there exists a constant C = C(n) such that The stability of the hyperbolic positive mass theorem with respect to the flat distance reads as follows.

Theorem 2.8 is proven by explicitly choosing integral currents
in Bb(ρ) and M(A j ) + M(B j ) → 0 as j → 0, which implies the conclusion. All this can be guaranteed by applying Theorem 2.7, the isoperimetric inequality, the definition of h 0 (f ) and Theorem 2.5.
2.3. Integral currents, intrinsic flat distance and convergence of balls. We now give a brief introduction to integral currents in metric spaces, integral current spaces and intrinsic flat distance. For further details about integral currents in metric spaces we refer the reader to Ambrosio and Kirchheim [AK00], Lang [Lan11], and Lang and Wenger [LW11]. For the definition of integral current spaces and intrinsic flat distance between them we refer to Sormani and Wenger [SW10,SW11]. For results about point convergence we refer to Sormani [Sor18] and, Huang, Lee and Perales [HLP22]. The n-dimensional current T endows Z with a finite Borel measure, ||T ||, called the mass measure of T and set(T ) is the set of points in Z where the n-dimensional lower density of ||T || is positive: The mass of T is defined as M(T ) = ||T ||(Z) and it is known that spt(T ) = spt(||T ||) = set(T ). For any Lipschitz function ϕ : Z → Y the push-forward of T is the current ϕ ♯ T : The boundary of T , ∂T : ∂T (f, π 1 , ..., π n−1 ) = T (1, f, π 1 , ..., π n−1 ), where 1 : Z → R denotes the constant function equal to 1. For any Borel set A ⊂ Z, the restriction of T to A, is the current where 1 A : Z → R denotes the indicator function of A. In this case ||T A|| = ||T || A and so spt(T A) ⊂ A.
Remark 2.9. By [Lan11, Proposition 3.3], the current T A can be identified with a current defined in (A, d) and that we will denote in the same way, that is, The main examples of currents are the zero n-dimensional currents, that is, T (f, π 1 , . . . , π n ) = 0 for all (f, π 1 , . . . , π n ), and the ones given by where ϕ : A → Z is a Lipschitz function, A ⊂ R n is a Borel set and, θ ∈ L 1 (A, R). We will work with n-dimensional integral currents T which are n-dimensional currents that can be written as a sum of currents of the form ϕ i♯ [[θ i ]] as given above, with θ ′ i s integer valued, and so that ∂T is also a current. The class consisting of these currents will be denoted as I n (Z) and to be more precise we will sometimes write I n (Z, d).

2.3.2.
Integral current spaces and intrinsic flat distance. An n-dimensional integral current space Q = (X, d X , T ) consists of a metric space (X, d X ) and an n-dimensional integral current, T ∈ I n (X, d X ), where (X, d X ) is the metric completion of (X, d X ), and such that set(T ) = X. There is the notion of zero n-dimensional integral current space denoted as 0 = (X, d, T ), here T = 0 and set(T ) = ∅. We define M(Q) = M(T ), set(Q) = set(T ).
The boundary of Q, ∂Q, is an (n − 1)-dimensional integral current space and is defined in the following way. By Remark 2.9, ∂T : Lip b (X) × [Lip(X)] n−1 → R can be identified with a current that we denote in the same way, ∂T : We remark that set(∂T ) ⊂ X, and that the second entry of ∂Q is the metric of X restricted to set(∂T ), which by abuse of notation we write as d X .
(ii) ∂M can be regarded as an .
so by Remark 2.9 and abusing notation, we write . We reiterate our notational convention of denoting the intrinsic length distance on Y , obtained from considering (Y, g| Y ) as a Riemannian manifold, and it will be important for us to keep track of which distance we are using. (iv) If we do not endow M with a Riemannian metric we can still define , which is given by By Remark 2.9 and abusing notation, we let , as in the previous example. In the latter, we still have set( . Given an n-dimensional integral current space Q = (X, d X , T ). Then for any p ∈ X and for almost every r > 0, We say that an integral current space (X, d, T ) is precompact if (X, d) is precompact. The definition of intrinsic flat distance is as follows.
Definition 2.13 ([SW11, Definition 1.1]). Given two n-dimensional precompact integral current spaces, (X 1 , d 1 , T 1 ) and (X 2 , d 2 , T 2 ), the intrinsic flat distance between them is defined as where the infimum is taken over all complete metric spaces Z and all metric isometric embeddings ϕ j . The flat distance between two n-dimensional integral currents T 1 , In this case, we identify both integral currents spaces and so d F is a distance in the space of equivalence classes with this relation. If M i are compact oriented Riemannian manifolds, if and only if there is a Riemannian isometry between M 1 and M 2 that preserves their orientation. Wenger proved the following compactness theorem.
We will sometimes use the notation ( in the intrinsic flat sense. Intrinsic flat converging sequences have intrinsic flat converging boundaries and the mass functional is lower semicontinuous with respect to this distance.
2.3.3. Convergence of points and balls. Convergence of points and "balls" under intrinsic flat distance is more subtle than when using Gromov-Hausdorff distance. With intrinsic flat distance we can have sequences of points disappearing in the limit or balls converging to the zero integral current space (see some examples of this in [SW11,Appendix] When keeping track of the embeddings and the space, we write: Definition 2.17 (Gromov). Let (Y j , d j ), j ∈ N, be a sequence of compact metric spaces that converges in Gromov-Hausdorff sense to the compact metric space (Y ∞ , d ∞ ). Let y j ∈ Y j , j ∈ N, and y ∈ Y ∞ . We say that {y j } converges to y, y j → y, if there exist a compact metric space (Z, d) and isometric embeddings ϕ j : Since it will be important to keep track of the embeddings and space, in the previous case, we wite: For intrinsic flat convergence there is a similar result.
if and only if there exist a complete and separable metric space (Z, d) and isometric embeddings ϕ j : Similarly, when keeping track of the embeddings and the metric space, we write be a sequence of precompact ndimensional integral current spaces, j ∈ N, that converges in intrinsic flat sense to the precompact n-dimensional integral current space (Y ∞ , d ∞ , T ∞ ). Let y j ∈ Y j , j ∈ N, and y ∈ Y ∞ . We say that {y j } converges to y, y j → y, if there exist a complete and separable metric space (Z, d) and isometric embeddings ϕ j : Y j → Z, j ∈ N ∪ {∞}, as in the previous theorem, such that d Z (ϕ j (y j ), ϕ ∞ (y)) → 0.
In this case, to avoid any confusion, we will sometimes write: Note that y is not necessarily contained in Y ∞ and if this is the case we say that the sequence of points disappears in the limit and that y disappeared ([Sor18, Definition 3.2]). When a sequence converges in both Gromov-Hausdorff and intrinsic flat sense to the same limit space the same embeddings and metric space can be taken.
if and only if there exist a complete and separable metric space (Z, d) and isometric embed- Under the assumption of the previous theorem, it is easy to see that for any sequence of points y j ∈ Y j , j ∈ N, there exists y ∈ Y ∞ such that y j → y. That is, using the Hausdorff convergence of the compact sets ϕ j (Y j ) to the compact set ϕ ∞ (Y ∞ ), one can find a convergent subsequence of ϕ j (y j ) to a point z = ϕ ∞ (y) for some y ∈ Y ∞ .
In our main theorems, Theorem 1.3 and Theorem 1.4, the sequences we consider do not necessarily converge in Gromov-Hausdorff sense, but we will be able to prove that the sequence of points p j in Theorem 1.4 does not disappear in the limit by showing convergence in both Gromov-Hausdorff and intrinsic flat sense of annular subregions to the same limit space.
Then there exist a subsequence, also denoted d j , and a metric d ∞ satisfying (2.6) such that d j converges uniformly to d ∞ , where T j = ι j♯ T and ι j : (Y, d) → (Y, d j ) are all identity functions.
We recall the following useful lemma about intrinsic flat convergence of balls.
then there exists a subsequence y j k ∈ X j k such that for almost every r > 0 and all k ∈ N∪{∞} the triples S(y j k , r) and S c (y j k , r) are integral currents spaces, and using the same isometric embeddings we have, Now, by Example 2.12 we know that ||T j k ||(d −1 y j k (r)) = 0 and thus the mass measures of S(y j k , r) and S c (y j k , r) are ||T j k || B(y j k , r) and ||T j k || X j k \ B(y j k , r), correspondingly. Assume that our assertion does not hold, then the first inequality above must be a strict inequality, and we get which is a contradiction.
We now introduce notation similar to Example 2.11 and Example 2.12. Given an integral current space Q = (X, d, T ) and a subset Y ⊂ X, we define the restriction of Q to Y by and note that T Y is a current but might not be an integral current, so in particular the triple is not necessarily an integral current space.
We now see when a non-disappearing sequence of points y j ∈ Y j , with Y j ⊂ X j , is a non-disappearing sequence if we consider each y j as an element in X j .
Theorem 2.24 ([HLP22, Theorem 2.9]). Let Q j = (X j , d j , T j ) be a sequence of n-dimensional integral current spaces, j ∈ N ∪ {∞}, such that and let Y j ⊂ X j such that Q j Y j are n-dimensional integral current spaces that converge to some integral current space N ∞ , where y j ∈ Y j and y ∈ set(N ∞ ). If there exists r > 0 such that the metric ball B X j (y j , r) is contained in Y j for all large j, then there exists a subsequence y j k and a point y ∞ ∈ X ∞ such that We give a lower bound of the distance between a point in a limit space to the set of the boundary.
Proof. By Theorem 2.18 there exist a complete and separable metric space (Z, d) and isometric embeddings ϕ j : , we obtain the desired convergence by applying Theorem 2.15.
To end up this section we state a result that allow us to pass from subconvergence of balls for almost all radii to convergence of all balls for all radii provided one deals with manifolds and has volume convergence.
Theorem 2.27 ([HLP22, Theorem 2.12]). Let (M j , g j ) be Riemannian manifolds with p j ∈ M j , for j ∈ N ∪ {∞}. Assume that for every subsequence of {p j k } k∈N of {p j } j∈N there is a subsequence {p j k ℓ } ℓ∈N such that for almost every r > 0, vol(B(p j k ℓ , r)) → vol(B(p ∞ , r)) and S(p j kℓ , r) F −→ S(p ∞ , r). Then for all almost every r > 0, we have vol(B(p j , r)) → vol(B(p ∞ , r)) and S(p j , r) F −→ S(p ∞ , r).
In this section, we first calculate some estimates that will be used in the proof of Theorem 1.3. We establish uniform intrinsic diameter bounds for regions of the form Ω(ρ) and their volumes, and uniform volume bounds of their boundaries. We also show that for any sequence M j ∈ G n (ρ 0 , γ, D) such that m(M j ) → 0, the volumes of the Ω j (ρ)'s converge to the volume of the ball B b (ρ).
In the last part we show that the annular regions The key ingredients to obtain the estimates are an isoperimetric inequality for the hyperbolic space, the coarea formula and the fact that the manifolds are graphs that satisfy the properties listed in Definition 1.2.
We recall the following isoperimetric inequality applicable to domains in the hyperbolic space. We also recall the following useful fact originally stated in the Euclidean case in the proof of Theorem 3.1 in [HLS17].
Proof. By standard computations and the coarea formula, Now we calculate the first term on the right hand side. By Definition 2.6, for all regular values h ≤ h 0 (f ) of f , If necessary, taking a non decreasing sequence of regular values h i ∈ R of f with lim i→∞ h i = h 0 (f ) and for which f −1 (h i ) is star-shaped, and applying the isoperimetric inequality, Proposition 3.2, we have For the second term, we use again the fact that for any regular value h ≤ h 0 (f ) of f we have H n−1 (f −1 (h)) ≤ 2βω n−1 m(f ) and, we note that min f ≤ h 0 (f ). Hence,
Corollary 3.6. Let M ∈ G n (ρ 0 , γ, D) be a manifold so that m(M) < 1. Then for any Remark 3.7. Note that there is an easier way to obtain a uniform upper bound estimate for vol(Ω(ρ)). Indeed, by the coarea formula and Proposition 3.3, it follows that vol(Ω(ρ)) ≤ vol(B b (ρ)) + | max Nonetheless, this bound does not immediately imply convergence of the vol(Ω(ρ)) to vol(B b (ρ)) provided m(M) → 0. Since the estimate in Corollary 3.6 involves m(M), this estimate implies the aforementioned volume convergence as we will see in Theorem 3.8. This is important to be able to apply Theorem 4.1 in the proof of Theorem 1.3.

3.2.
Diameter bounds, area bounds and volume convergence. Now we prove that for any sequence {M j } ⊂ G n (ρ 0 , γ, D) with mass m(M j ) converging to zero, the sequence Ω j (ρ) has uniform intrinsic diameter bounds, the volumes converge to the volume of the ball B b (ρ) and, the boundaries have uniform volume bounds. These estimates will be used in the proof of Theorem 1.3 and Lemma 3.11.
1 Note however that then the estimate clearly also holds for the diameter of Ω j (ρ) with respect to d Mj because d Mj ≤ d intr Ωj (ρ) .
Since lim j→∞ m(f j ) = 0 we can assume that m(M j ) < 1 and apply Corollary 3.6,
By applying Theorem 2.21 to the sequence A j (ρ, ρ ′ ) we get the following.
Corollary 3.10. Let M j ∈ G n (ρ 0 , γ, D) be a sequence. Then for any R ≥ ρ ′ ≥ ρ > ρ 0 /2 and with the same notation as in the previous lemma, we have and Now we show that the inner boundaries converge to the zero integral current space and that the outer boundaries converge to the boundary of the limit space.
Lemma 3.11. Let {M j } ⊂ G n (ρ 0 , γ, D) be a sequence such that lim j→∞ m(f j ) = 0 and ρ ≥ ρ 0 . Then there exist a subsequence {(Ω j k (ρ), d intr Ω j k (ρ) , [[Ω j k (ρ)]])} and an integral current . With no loss of generality assume that (this is true up to a sign) for all j ∈ N, Proof. By Wenger's compactness theorem, Theorem 2.14, and Theorem 3.8 there exist a subsequence . Then by Theorem 2.15 we get

Proofs of the main theorems
Here we prove Theorem 1.3 and Theorem 1.4. A key tool to prove the former is Theorem 4.1. Unlike Wenger's compactness theorem, Theorem 2.14, Theorem 4.1 tell us which space is the limit. Theorem 4.1 was applied in [AP21,HLP22] to fill in a gap in the proof of the stability of the positive mass theorem for asymptotically flat graphical manifolds under intrinsic flat distance [HLS17,HLS22]. . Let D, A > 0 and λ j ∈ (0, 1] be a sequence such that lim inf j→∞ λ j = 1. Let (Ω j , g j ) be a sequence of ndimensional compact oriented Riemannian manifolds with non-empty boundary, j ∈ N∪{∞}, with g j continuous for j ∈ N and g ∞ smooth, so that any g ∞ -minimizing geodesic between two points in the interior of Ω ∞ lies completely in the interior. Assume that for all j ∈ N, Let us recall our first main theorem, Theorem 1.3.
In the non-entire case we prove Theorem 1.3 by enlarging each Ω j (ρ), as described in Appendix A, to get manifolds diffeormorphic to B b (ρ) and then apply Theorem 4.1 to the new sequence. The uniform diameter, volume and area bounds needed follow by the corresponding uniform diameter and area bounds shown in Theorem 3.8 and the way the enlargements and diffeomorphisms are chosen.
Proof of Theorem 1.3. We first assume that all the manifolds M j have non-empty boundary. Let λ j ∈ (0, 1] be a sequence such that lim inf j→∞ λ j = 1 and fix some L > D + 1 2 sinh(ρ 0 ) π 1 + γ 2 . We replace each (Ω j (ρ), g j ) by a manifold ( Ω j ,g j ) diffeomorphic to B b (ρ) by applying Theorem A.1 for λ = λ j . To obtain the conclusion of the first part of the theorem, by the triangle inequality, it is enough to prove that We get the first limit as follows. By the definition of intrinsic flat distance and given that by Theorem A.1 (Ω j (ρ), d intr Ω j (ρ) ) embeds in a distance preserving way into ( Ω j , d Ω j ), we get ). By Theorem A.1, Theorem 2.5 and our hypothesis on the masses we get vol( Ω j \ Ω j (ρ)) ≤ V (L, ρ 0 , ∂U j ) → 0. Thus, we get the first limit.
Since lim inf j→∞ λ j = 1 and g j ≥ λ j b by Theorem A.1, and by Remark 3.1 the ball B b (ρ) is totally convex, we can apply Theorem 4.1 to conclude that the second limit holds. Putting both limits together we get the conclusion, If the M j are the graphs over entire functions, then there is no need to enlarge the manifolds. The result follows immediately from Theorem 3.8 and Theorem 4.1. If the sequence M j contains subsequences of both entire and non-entire manifolds the result follows from the previous cases.
Next we will rewrite Theorem 1.4 and give its proof. However, before we do this, let us first establish the following lemmas.
are n-dimensional integral current spaces that converge to the integral current space N = (Y, d, S), for some S ∈ I n (Y ) • there exists r > 0 such that B X j (y j , r) is contained in Y j for all large j. Then for any complete metric space Z and isometric embeddings ϕ j : X j → Z that satisfy we can ensure that there exist a subsequence y j k and y ∞ ∈ X ∞ such that Proof. By Theorem 2.20 we have where W is a complete and separable metric space and ψ j are isometric embeddings. Since ψ ∞ (Y ) is a compact subset of a complete space, W , and there exists a subsequence of ψ j (y j ) that converges to ψ ∞ (y) for some y ∈ Y . Therefore Then by Theorem 2.24 there exists y ∞ ∈ X ∞ and a subsequence such that The next lemma compares d M and d intr Ω(ρ) on relevant regions of M for large ρ. This is important as it will allow us to use Theorem 1.3 in the proof of Theorem 1.4.
Lemma 4.4. Let (M, g) ∈ G n (ρ 0 , γ, D) and fixR > 1 + (4 + π sinh(ρ 0 + 1)) 1 + γ 2 . Then To show d intr Ω(ρ 0 +R) = d M on the desired subsets we need to show that for two points x, y ∈ B M (p, R) (resp. x, y ∈ A(ρ 0 − 1, ρ 0 + 1)) any curve of length d M (x, y) from x to y in M will remain in Ω(ρ 0 +R). This is satisfied if any curve of length bounded by diam( (B M (p, R) We first consider the case x ∈ B M (p, R). Let c be a curve in M starting at x of g-length L ≤ 2R. We may extend c to a curvec in M starting at p ∈ Σ(ρ 0 ) of length L ≤ 3R. Since M is a graph over H n , the b-length of the projection ofc onto H n is similarly bounded by 3R and this projection starts at ∂B b (ρ 0 ). So by the triangle inequality the projection must remain in B b (ρ 0 + 3R) ⊂ B b (ρ 0 +R). Hencec itself remains in Ω(ρ 0 +R).
In the following we abuse notation and do not change indexes when passing to subsequences.
Proof of Theorem 1.4. LetR > 1 + (4 + π sinh(ρ 0 + 1)) 1 + γ 2 . Then by Theorem 1.3 and Theorem 2.18 we know that for some metric space Z and isometric embeddings ϕ j . Assume with no loss of generality that ρ 0 > 2, so ρ 0 − 1 > ρ 0 /2, and A j (ρ 0 − 1, ρ 0 + 1) ⊂ Ω j (ρ 0 +R) sinceR > 1. Thus we can apply Corollary 3.10 to get that ) converges in intrinsic flat sense to some integral current space By Lemma 3.11 and Theorem 2.25, we get that So B b (p ∞ , R) ⊂ B b (ρ 0 +R) holds for almost every R ∈ (0,R) and using the notation from Example 2.12 where in the last equality we use the fact that for manifolds N i with integral current spaces if and only if there exists an orientation preserving isometry between the N i 's. Furthermore, by applying Example 2.12 and Lemma 4.4 item (1), we get that for almost every R ∈ (0,R 3 ) Now by (4.1), Lemma 2.22 and Remark 2.23, we get that for almost every R > 0 and a subsequence of p j , and M(S(p j , R)) → M(S(p ∞ , R)). Thus, for a subsequence of p j and almost every R ∈ (0,R 3 ) it holds and vol(B M j (p j , R)) → vol(B b (R)).
To finalize the proof, take a sequence of positive real numbersR i → ∞,R 1 > 1 + (4 + π sinh(ρ 0 +1)) 1 + γ 2 , and by a diagonalization argument, proceeding as above, get a further subsequence of the p j such that for almost all R > 0 we have ). Thus, we can apply Theorem 2.27 to conclude the proof.

Appendix A. Capping and construction of suitable diffeomorphisms
Let M n be an asymptotically hyperbolic manifold and let Ψ : M → H n+1 be an isometric embedding such that Ψ(M) = graph(f ) for some f : H n \ U → R, with non-empty minimal boundary ∂U. Recall that this implies, among other things, that f is constant at the boundary -for simplicity assume that f | ∂U = 0. As before, we set Ω(ρ) = Ψ −1 (B b (ρ) × R) for ρ > 0. We also identify Ω(ρ) with Ψ(Ω(ρ)) ⊂ H n+1 and equip it with the induced metric from H n+1 . Our goal is to "cap-off" Ω(ρ) to obtain a differentiable manifold Ω with a C 0 Riemannian metric and to construct a C 1 diffeomorphism Φ : B b (ρ) → Ω so that the conditions on the Riemannian metrics of Theorem 4.1 (cf. [AP21,HLP22]) are satisfied. More precisely, we are interested in proving the following theorem. The proof consists in a direct adaptation of the argument in [HLP22] to our setting.
The remainder of the appendix will be dedicated to working in detail through all the steps necessary in the construction from [HLP22] in our hyperbolic setting.
A.1. Normal exponential map, defining U and f c . Consider the normal exponential map of ∂U ⊂ H n . By compactness of U, there exists an ǫ * > 0 and an open neighborhood ) is a diffeomorphism and by shrinking ǫ * we may assume that E is still a diffeomorphism on I × ∂U for some open subset I with [−ǫ * , ǫ * ] ⊂ I.
With the previous notation, we are able to define a map φ from (−L, Note that φ is not a C 1 map (but piecewise smooth), so we will replace φ by a C 1 map Φ λ in order to construct Φ λ appearing in Theorem A.1. This will be done by introducing an appropriate scaling. The scaling will crucially rely on x)) → ∞ as t → 0 + . This will show that the image of Φ λ is indeed a C 1 submanifold of H n+1 . Now we want to construct a cap U which will be given as the graph of a capping function f c : U → [−1, 0], smooth on U and satisfying f c | ∂U = 0 and ∇ b ν f c = d dt f c (E(t, x)) → ∞ as t → 0 − , i.e. as one approaches ∂U, so that we will be able to attach it to the cylinder and will be able to make the resulting manifold C 1 in the same way as we will be doing for f above. We construct this capping now: We start by defining f c on E((−ǫ * , 0] × ∂U) ⊂ U ⊂ H n by setting By shrinking ǫ * if necessary, we further assume that for all s ∈ (−1, 0] we have where {ω t } is the family of Riemannian metrics on ∂U induced by the normal exponential map (see (A.1) below). Recall that ∂U is star-shaped and this implies that ∂U has a single connected component which is C 1diffeomorphic to S n−1 via the graph θ → (ρ ∂U (θ), θ) ∈ (0, ∞) × S n−1 ∼ = H n \ {0}. Thus, we may further shrink ǫ * to additionally ensure that the level sets are star-shaped as well: Since, by assumption, ρ ∂U : S n−1 → (0, ∞) is a C 1 function whose graph coincides with ∂U, the spherical coordinate vector field ∂ ρ can never be tangential to ∂U and ∂ ρ , ν b > 0 near ∂U and one can use the monotonicity of t → ρ(E(t, θ)) and an implicit function theorem argument to obtain a unique differentiable family of differentiable maps ρ d : This construction is very similar to the process we will encounter in defining the scalingα in Lemma A.2 hence we will skip the details here.
A.2. Construction of Φ λ . We already introduced the normal exponential map of ∂U, which gives a diffeomorphism E : (−ǫ * , ǫ * ) × ∂U → N ǫ * onto an open neighborhood N ǫ * of ∂U in H n for ǫ * > 0 as before. Pulling back the metric and using "t" to denote the coordinate on (−ǫ * , ǫ * ), standard properties of the normal exponential map give that the metric splits as Fix λ ∈ (0, 1). Let ε ≡ ε(λ, f, ∂U, L, f c ) ∈ (0, 1), ε < ǫ * , satisfy ε < ε 0 for ε 0 from the Lemmas A.3 and A.4, 2L ε ≥ 1 and inf where · e denotes the Euclidean norm on R n−1 . Note that this is possible because the function in the infimum above is continuous and equal to one on the diagonal.
Once we have defined Φ λ we will set giving us the desired C 1 diffeomorphism from B b (ρ) → Ω satisfying g( Φ λ * (u), Φ λ * (u)) ≥ λ b(u, u) everywhere because it is either a graph (where g ≥ b automatically since V ≥ 1) or given by Φ λ • E −1 which will be constructed to do so.
In the coordinates we are using, and we have: . Since the metric g on Ω is induced byb, we have where we suppressed the arguments for the second and third formula (as a rule of thumb: f and its derivatives will have argument (α(t, x), x), α and its derivatives (t, x) and ω and V = cosh are evaluated at pr H n (Φ λ (t, x)) = (α(t, x), x) and r(α(t, x), x) respectively) and used the form ofb = b + V (r) 2 ds 2 = dt 2 + ω t + V (r) 2 ds 2 = dt 2 + ω ij (t, x)dx i dx j + V (r) 2 ds 2 in our chosen coordinates. We introduce the following notation With this the above expressions can be rewritten as x) ((0, ε) × ∂U) be arbitrary and compute (ignoring basepoints/arguments for now for readability) In the first case we estimate where we used the Cauchy-Schwarz inequality and that V 2 ≥ 1. Now by (A.2) we have ω (α(t,x),x) (ū,ū) ≥ √ λω (t,x) (ū,ū) (note that this is scaling invariant so it is sufficient to consider u with Euclidean norm equal one) and we obtain the desired In the second case we proceed similarly and estimate again using the Cauchy-Schwarz inequality and (A.2).