Spacetime Penrose Inequality for Asymptotically Hyperbolic Spherical Symmetric Initial Data

Home , Apparent horizon, Richard Arnowitt

U.U.D.M. Project Report 2020:27

Spacetime Penrose Inequality For Asymptotically Hyperbolic Spherical Symmetric Initial Data

Mingyi Hou

Examensarbete i matematik, 30 hp Handledare: Anna Sakovich Examinator: Julian Külshammer Augusti 2020

Department of Mathematics Uppsala University

Spacetime Penrose Inequality For Asymptotically Hyperbolic Spherically Symmetric Initial Data

Mingyi Hou

Abstract

In 1973, R. Penrose conjectured that the total mass of a space-time containing black holes cannot be less than a certain function of the sum of the areas of the event horizons. In the language of differential geometry, this is a statement about an initial data set for the Einstein equations that contains apparent horizons. Roughly speaking, an initial data set for the Einstein equations is a mathematical object modelling a slice of a space-time and an apparent horizon is a certain generalization of a minimal surface. Two major breakthroughs concerning this conjecture were made in 2001 by Huisken and Ilmanen respectively Bray who proved the conjecture in the so-called asymptotically flat Riemannian case, that is when the slice of a space-time has no extrinsic curvature and its intrinsic geometry resembles that of Euclidean space. Ten years later, Bray and Khuri proposed an approach using the so-called generalized Jang equation which could potentially be employed to deal with the general asymptotically flat case by reducing it to the Riemannian case. Bray and Khuri have successfully implemented this strategy under the assumption that the initial data is spherically symmetric. In this thesis, we study a suitable modification of Bray and Khuri’s approach in the case when the initial data is asymptotically hyperbolic (i.e. modelled on a hyperboloidal slice of Minkowski spacetime) and spherically symmetric. In particular, we show that the Penrose conjecture for such initial data holds provided that the generalized Jang equation has a solution with certain asymptotic behaviour near infinity. Furthermore, we prove a result concerning the existence of such a solution. Contents

1 Introduction1

2 Preliminaries4 2.1 Apparent Horizons...... 4 2.2 Asymptotically Flat Initial Data...... 5 2.3 Asymptotically Hyperbolic Initial Data...... 5 2.4 Spherically Symmetric Asymptotically Hyperbolic Initial Data...... 6

3 The Generalized Jang Equation8 3.1 The Generalized Jang Equation Approach...... 8 3.2 Spherical Symmetry...... 9

4 Motivation for Asymptotics and the Warping Factor 12 4.1 Behavior at Inﬁnity...... 12 4.2 Proof of the Inequality...... 15

5 Existence and Asymptotics 18 5.1 Existence...... 18 5.2 Asymptotics...... 19 Chapter 1

Introduction

In the 1970s, physicist R. Penrose conjectured that the total mass of a spacetime containing black holes with event horizons of total area A should be at least pA/16π. In his argument, cosmic censorship, a famous conjecture saying that singularities arising in the solutions of the Einstein equations are typically hidden within event horizons, is one of the fundamental ingredients. In fact, a proof of a suitable version of the Penrose inequality would give indirect support to cosmic censorship and a counter example of the inequality would very likely involve a spacetime for which cosmic censorship fails to hold [Mar09]. Thus, finding a proof for the inequality has become an active area of research. An important case of the Penrose inequality is a statement about asymptotically flat initial data. Recall that a spacetime is a Lorentzian manifold (N , gN ) satisfying the Einstein equations, 1 Ric − R g = 8πT (1.1) N 2 N N where T is the so-called stress-energy tensor, RicN is the Ricci curvature tensor and RN is the scalar curvature. We call a triple (M, g, K) an initial data set (or Cauchy data) if (M, g) is a Riemannian manifold and K is a symmetric 2-tensor, such that the so-called constraint equations ( 1 µ = (R − |K|2 + tr (K)2) 2 g g g (1.2) J = divg(K − trg(K)g) are satisfied. Here Rg is the scalar curvature and µ and J are physically energy and momentum density respectively. In the case when (M, g) is a space-like slice of a spacetime (N , gN ), with unit normal n, we have µ = T (n, n) and J = T (n, ·). In what follows, we will assume that any given initial data satisfies the so-called dominant energy condition

µ > |J| (1.3) which is known to be satisﬁed in all physically reasonable spacetimes. Note that if (1.2) is satisﬁed then (M, g, K) is indeed the initial data required to pose an initial value problem for the Einstein equations in the sense that the time evolution of the initial data is a spacetime (N , gN ) satisfying (1.1) and (M, g) is a space-like hypersurface in the spacetime with second fundamental form K [BI04].

1 Roughly speaking, an initial data (M, g, K) is asymptotically flat if near infinity g approaches the Euclidean metric δ and K approaches 0. For an asymptotically flat initial data set there is a well known notion of mass [ADM08] that in a certain sense measures how fast the initial data set becomes “flat” at infinity. The argument which can be used to motivate the Penrose inequality roughly goes as follows [Mar09]. Assume a spacetime (N , gN ) which is asymptotically flat, in the sense that it looks like the Minkowski spacetime [ONe83] at infinity. Intuitively, the event horizon H, is the boundary of the region around a black hole which not even the light can escape. Note that it is a null hypersurface which is at least Lipschitz continuous [Wal84]. Assume now, that the spacetime admits an asymptotically flat slice with total ADM mass MADM and which intersects H on a cut S. If H is a smooth hypersurface, then this cut is a smooth embedded surface which has a well-defined area |S|. Consider any other cut S1 to the future of S along the event horizon. The black hole area law [Haw71; Haw72] states |S1| > |S|, provided that the null energy condition, which is a certain strengthening of (1.3), holds. From physical considerations, the spacetime is expected to settle down to some equilibrium configuration. Assuming that all matter fields are swallowed by the black hole in the process, uniqueness theorems for stationary black holes (see e.g. [Heu96]) imply that the spacetime must approach the Kerr black hole spacetime, where the area of the section S of the event horizon is independent of the cut and takes the value p 2 2 2 2 AKerr = 8πM(M + M − L /M ) 6 16πM , where M is the total mass and L is the total angular momentum of the spacetime. In particular, M should be the asymptotic value of the Bondi mass along the future null infinity. Since gravitational waves carry positive energy, the Bondi mass cannot increase to the future [BVM62; SB62]. The Penrose inequality p MADM > |S|/16π follows provided that the Bondi mass approaches the ADM mass MADM which is known to be true under certain assumptions (see e.g. [Hay03; Kro03; AM79]). After a period of heuristic proofs and partial results, important breakthroughs have been made in the past decades. First, Huisken and Ilmanen [HI01] were able to prove the Penrose inequality in the so called time symmetric case, i.e. when the initial data has K = 0, using a geometric evolution equation known as the inverse mean curvature flow. Shortly afterwards, another proof using the so-called conformal flow was found by Bray. Roughly speaking, the result of [HI01] is for a single black hole, while that of [Bra01] allows for multiple ones. In summary, we have the following result in the time symmetric case:

Theorem 1 (Riemann Penrose Inequality). Let (M, g) be a complete, smooth, asymptotically ﬂat 3-manifold with nonnegative scalar curvature, total mass M, and with the boundary (possibly disconnected) which is an outermost minimal surface of total area A. Then r A M (1.4) > 16π with equality if and only if (M, g) is isometric to a Schwarzschild manifold 3 (R \{0}, gSchw).

2 The Penrose inequality is a strengthening of the famous positive mass theorem, saying that an asymptotically flat initial data (M, g, K) satisfying dominant energy condition has MADM > 0, with equality if and only if it is a slice of the Minkowski spacetime. This result was proven in [SY82] by reducing it to the time symmetric case K = 0 that was earlier settled in [SY79]. The proof uses a certain quasi-linear elliptic partial differential equation, the so-called Jang equation. It was suggested by Bray and Khuri (see [BK11; BK10]) that a generalization of the Jang equation could be a way to reduce the general Penrose inequality to the case of time symmetry. This idea has been successfully implemented in the asymptotically flat spherically symmetric case [BK10]. In this thesis, we provide some further support to the generalized Jang equation approach of Bray and Khuri by implementing it in the case when the initial data set is asymptotically hyperbolic spherically symmetric. This setting is of great interest for both geometrical and physical reasons, in particular, it is important for the study of gravitational radiation [Fra04]. The thesis is organized as follows: In Chapter 2, we introduce terminology required in this thesis. In Chapter 3, we briefly review the generalized Jang equation and rewrite it as an ordinary differential equation under the assumption of spherical symmetry. In Chapter 4, we prove a theorem stating that the Penrose inequality for asymptotically hyperbolic spherically symmetric initial data holds provided that a solution to the generalized Jang equation exists and has certain asymptotic behavior at infinity. Finally, in Chapter 5, we prove the existence of a solution to the generalized Jang equation and obtain a (non-optimal) result regarding its asymptotics.

3 Chapter 2

Preliminaries

In this chapter, we introduce a few deﬁnitions that will be used in this thesis.

2.1 Apparent Horizons

An important concept involved in the Penrose conjecture is that of event horizon. However, in the setting of the initial data set formulation of the Penrose inequality, it is problematic to compute the area of the event horizons of black holes. Indeed, the event horizons are not determined by the local geometry of the initial data, but one is required to solve the Einstein equation for detecting them. For this reason one usually replaces the notion of event horizon by the concept of apparent horizon. Deﬁnition 2 (Apparent Horizons). Let (M, g, K) be an initial data set. A compact oriented hypersurface S ⊂ M is called an apparent horizon (or a marginally outer respectively inner trapped surface) if S satisﬁes

HS ± trS(K) = 0 (2.1) where HS is the mean curvature of S. In fact, we may deﬁne the so-called null expansions

θ± = HS ± trS(K) (2.2) which measure the rate of change of area for a shell of light emitted by the surface in the outward future (respectively past) direction. Thus the gravitational ﬁeld is interpreted as being strong near S if θ+ < 0 or θ− < 0, in which case S is referred to as a future (past) trapped surface. Future (past) apparent horizons arise as boundaries of future (past) trapped regions and satisfy the equation θ+ = 0 (θ− = 0). Here, an outermost future (respectively past) apparent horizon refers to a future (respectively past) apparent horizon outside of which there is no other apparent horizon. Such a horizon may have several components, each having spherical topology [Gal08]. 3 For example, consider the initial data (R \{0}, gsc,K = 0) representing t = 0 slice of the Schwarzschild spacetime [ONe83], the simplest nontrivial solution to the 4 Einstein equations, where (gsc)ij = (1 + m/2r) δij and m is a positive constant which can be associated with the mass (see below). This manifold has zero scalar curvature everywhere, is spherically symmetric, and it has an outermost apparent horizon at r = m/2.

4 2.2 Asymptotically Flat Initial Data

We now introduce the rigorous notion of asymptotically flat initial data that has been used in the heuristic argument given in the Introduction. Definition 3 (Asymptotically Flat Initial Data). An initial data set (M, g, K) is called asymptotically flat if outside a compact set M is diffeomorphic to the comple- ment of a ball in (R3, δ) and if in the Cartesian coordinates xi, i = 1, 2, 3, induced by this diffeomorphism g and K have the following asymptotic behavior: 2 2 −1 −2 |gij − δij| + r|∂kgij| + r |∂klgij| = O(r ),Kij + r|∂kKij| = O(r ) (2.3)

p i j as r → ∞ where r = |x| = δijx x . The total mass of an initial data set is essentially a geometrical invariant that depends on the geometry “at infinity” of a non-compact Riemannian manifold. For example in the Schwarzschild spacetime, time-like geodesics (which represent test particles) curve in the coordinate chart as if they were accelerating towards the center of the spacetime at a rate asymptotic to m/r2 in the limit as r goes to infinity. Hence, to be compatible with Newtonian physics we must define m to be the total mass of the Schwarzschild spacetime. For asymptotically flat manifolds the total ADM energy-momentum vector is defined through the coordinate expressions Z 1 i δ EADM = lim (∂jgij − ∂igjj)ν dµ (2.4) r→∞ 2(n − 1)ωn Sr Z 1 j δ (PADM )i = lim (Kij − trg(K)gij)ν dµ (2.5) r→∞ (n − 1)ωn Sr where Sr is the coordinate sphere of radius r, ν is the unit outward normal to Sr, n δ ωn is the volume of the n-dimensional unit sphere in R , dµ is the area form on Sr induced by δ. Under some additional decay conditions on g and K these quantities are independent of the choice of coordinates xi [Bar86]. The total ADM mass is defined as q 2 2 MADM = EADM − |PADM | (2.6)

2 ij where |PADM | = δ (PADM )i(PADM )j.

2.3 Asymptotically Hyperbolic Initial Data

The terminology of asymptotically hyperbolic setting is rather sophisticated. For this paper, it will suffice to use the following definition [Sak20]. Let H3 denote the 3-dimensional hyperbolic space and let gH denote its metric. In polar coordinates it dr2 2 2 can be expressed by gH = 1+r2 + r σ, where σ is the standard round metric on S . Definition 4 (Asymptotically Hyperbolic Initial Data). An initial data set (M, g, K) is said to be asymptotically hyperbolic if there exists a compact subset Ω and a dif- n ¯ 3 feomorphism Φ: M \ Ω → H \ BR, where BR is a ball in H , such that dr2 Φ g = + r2(σ + mr−3 + O (r−4)) ∗ 1 + r2 2 (2.7) −1 −2 Φ∗(K − gH )|T S2×T S2 = pr + O1(r )

5 where m ∈ C2(S2) and p ∈ C1(S2) are symmetric 2-tensors on S2 and the notation −n n+i i α Ol(r ) asserts that r |∂r∂ p| ≤ C for all i + |α| ≤ l. For the convenience of the reader, we briefly review the definition of total mass for asymptotically hyperbolic initial data sets. First we define a vector space V := ∞ n {V ∈ C (H )|HessgH V = V gH }. It has the basis of functions √ 2 i V(0) = 1 + r ,V(1) = x r, i = 1, 2, 3 where x1, x2, x3 are the coordinate functions on R3 restricted to S2. If we consider 3 3,1 H as the upper unit hyperboloid in Minkowski space R , then the functions V(i) are the restrictions to H3 of the coordinate functions of R3,1. The mass functional of an asymptotically hyperbolic initial data set with respect to the chart Φ is then a linear functional HΦ on V defined by Z gH gH HΦ(V ) = lim V (divg e − dtrg (e)) + trg (e)dV − (e + 2η)(∇ V, ·) (νr)dµ r→∞ H H H Sr √ 2 where e = Φ∗g − gH , η = Φ∗(K − g) and νr = 1 + r ∂r is the unit outer normal to Sr. This functional transforms appropriately under isometries of the hyperbolic space. The components of the energy-momentum vector (E,P ) are given by 1 1 E = H (V ),P = H (V ), i = 1, 2, 3. (2.8) 16π Φ (0) i 16π Φ (i) A computation shows that in the case when the initial data has asymptotics (2.7) the energy is given by Z 1 σ E = (trσm + 2trσp)dµ . (2.9) 16π S2 In what follows we abuse the terminology slightly by calling E the mass, while physically it is the total energy. For more details, we refer the reader to [Mic11].

2.4 Spherically Symmetric Asymptotically Hyperbolic Initial Data

The main object of interest in this thesis is the spherically symmetric asymptotically hyperbolic initial data. From now on we make the assumption that all initial data sets in this thesis are spherically symmetric which means that the manifold M 3 2 is diﬀeomorphic to R \ B(r0) for some ball with radius r0 and g = g11dr + ρσ, K = 2 κdr + γσ for some smooth functions g11, ρ, κ, γ depending only on r. Note that for such an initial data the outermost apparent horizon has only one connected component so we can assume it is the boundary of the manifold. A straightforward computation shows that in this case we can write the null expansions as follows

p ρ0 2γ θ = g11 ± . (2.10) ± ρ ρ

6 Recalling the deﬁnition of asymptotically hyperbolic initial data, the following assumption will be imposed on g11, ρ, κ, γ for the rest of the thesis, for r → ∞ 1 mg g = , ρ = r2(1 + + O (r−4)), 11 1 + r2 r3 3 (2.11) 1 mk κ = (1 + O (r−3)), γ = r2(1 + + O (r−4)) 1 + r2 2 r3 2 where mg, mk ∈ R. Since we are in dimension 3, the quantities mg, mk encode mass through the formula Z g k 1 g k m + 2m m = trσ m σ + 2m σ = . (2.12) 16π S2 2

7 Chapter 3

The Generalized Jang Equation

In this chapter, we will restrict our attention to spherically symmetric initial data. The ﬁrst part contains a discussion of the Jang equation approach to the Penrose conjecture suggested by Bray and Khuri. In the second part, we transform the generalized Jang equation into an ODE under the assumption of spherical symmetry.

3.1 The Generalized Jang Equation Approach

The classical Jang equation was introduced by physicist P. S. Jang in 1978 and was successfully employed to reduce the positive mass theorem for general initial data to the case of time symmetry [SY81]. However, the Jang equation has been shown to be unsuitable for addressing the Penrose inequality [MM04]. For this reason, a generalization of this equation has been proposed in [BK11]. In this section, we introduce the so-called generalized Jang equation. In the proof of the time symmetric Penrose inequality, nonnegative scalar curvature plays a central role. Thus, in the general case, one is motivated to deform the initial data in an appropriate way so that we can get the positivity of the scalar curvature for the deformed metric. The deformation used to prove the positive mass theorem is given byg ¯ = g + df 2, for some function f deﬁned on M. Note thatg ¯ is the induced metric on the graph Σ = {t = f(x)} in the product 4-manifold (M×R, g+dt2). It turns out that the scalar curvature will be “almost” nonnegative when Σ satisﬁes Jang’s equation

HΣ − trΣ(K) = 0 where HΣ is mean curvature of Σ and K is trivially extended to M×R. Motivated by the case of equality for Penrose inequality [BK10], we will consider the Jang surface Σ in a warped product manifold (M × R, g + u2dt2), where u is a nonnegative function deﬁned on M. It is also shown in [BK10] that in order to achieve “almost” positivity for scalar curvature we would like the Jang surface Σ to satisfy an equation with the same structure but with K extended non-trivially, namely ¯ HΣ − trΣ(K) = 0 (3.1)

8 where ¯ K(∂xi , ∂xj ) = K(∂xi , ∂xj ), for 1 ≤ i, j ≤ 3, ¯ K(∂xi , ∂t) = 0, for 1 ≤ i ≤ 3, 2 (3.2) ¯ u g(∇f, ∇u) K(∂t, ∂t) = . p1 + u2|∇f|2 Here xi, i = 1, 2, 3, are local coordinates on M. This particular extension yields an optimal positivity property for the scalar curvature of the solutions to (3.1). A long calculation [BK10; BK11] gives the following result for the scalar curvature of Σ satisfying equation (3.1) with extension (3.2): 2 R¯ = 16π(µ − J(w)) + |h − K¯ | |2 + 2|q| − div (uq) (3.3) Σ g¯ g¯ u g¯ whereg ¯ = g + u2df 2 and h are the induced metric and second fundamental form ¯ ¯ of Σ respectively, K|Σ is the restriction to Σ of the extended tensor K, and q is a 1-form and w is a vector with |w|g ≤ 1 given by ,i ,j uf ∂xi uf ¯ w = , qi = (hij − (K|Σ)ij) p 2 2 p 2 2 1 + u |∇gf| 1 + u |∇gf|

,i ik where f = g f,k and f,k = ∂kf. If the dominant energy condition is satisﬁed, then all terms appearing on the right-hand side of (3.3) are nonnegative, except for possibly the last term. When the tensor K is extended according to (3.2), we will refer to equation (3.1) as the generalized Jang equation, and the graphical submanifold Σ = {t = f(r)} will be called the Jang surface throughout the paper. In local coordinates the generalized Jang equation takes the following form: u2f if j u∇ f + u f + f u gij − ij i j i j − K = 0 (3.4) 2 2 q ij 1 + u |df|g 2 2 1 + u |df|g

2 where ∇ijf is the {i, j} component of covariant hessian ∇ f with respect to g, i.e. k k ∇ijf = ∂j∂if − Γij∂kf where Γij are Christoﬀel symbols of g.

3.2 Spherical Symmetry

In the general case the generalized Jang equation is an elliptic PDE. However, under spherical symmetry, we can rewrite it as an ODE. First we observe that equation (3.4) consists of two parts, namely the radial part and the spherical part u2(g11)2(f 0)2 u∇ f + 2u0f 0 u∇ f g11 − rr − K + gab ab − K = 0 2 2 q 11 q ab 1 + u |df|g 2 2 2 2 1 + u |df|g 1 + u |df|g where a, b = φ1, φ2 (here and in what follows a and b refer to spherical coordinates). Then we make the following substitution [BK10]: upg11f 0 v := (3.5) p1 + u2g11(f 0)2

9 and compute its ﬁrst derivative with respect to r coordinate √ pg11u0f 0 + g (g11)0uf 0/2 + pg11uf 00 v0 = 11 p1 + g11u2(f 0)2 u2u0(g11)3/2(f 0)3 + u3pg11(g11)0(f 0)3/2 + u3(g11)3/2(f 0)2f 00 − . (1 + u2g11(f 0)2)3/2

We can multiply this by pg11 to obtain

0 0 0 p u∇rrf + 2u f p u g11v0 = g11(1 − v2) − g11(1 − v2) v. p1 + u2g11(f 0)2 u Hence the radial part becomes u2(g11)2(f 0)2 u∇ f + 2u0f 0 g11 − rr − k 2 2 q 11 1 + u |df|g 2 2 1 + u |df|g 0 0 11 2 u∇rrf + 2u f = g (1 − v )( − k11) q 2 2 1 + u |df|g p p u0 = g11v0 + g11(1 − v2) v − g11(1 − v2)κ. u On the other hand, the spherical part is

0 ab u∇abf p 11 ρ 2γ g ( − kab) = g v − . q 2 2 ρ ρ 1 + u |df|g

Therefore adding the two parts we get the following equation

p √ u0 p ρ0 2γ g11v0 + g11(1 − v2)( g v − κ) + g11 v − = 0. (3.6) 11 u ρ ρ As will be explained in next chapter, for the proof of Penrose inequality it is reasonable to take √ 1 − v2 ρ0 u = √ √ . (3.7) g11 2 ρ As mentioned before we expect our warping factor to be nonnegative and this is indeed the case for (3.7) since

p ρ0 1 H = g11 = (θ + θ ) > 0 ∀r > r (3.8) Sr,g ρ 2 + − 0 and |v(r)| 6 1 for all r > r0 by (3.5). With the above choice of u we obtain an ordinary diﬀerential equation

p p ρ00 g0 ρ0 g11(1 − v2)v0 + g11(1 − v2) − 11 − v ρ0 2g 2ρ 11 (3.9) p ρ0 2γ −g11(1 − v2)κ + g11 v − = 0. ρ ρ

10 Note that for v 6= ±1 we can write this as

p 11 2 0 2 g (1 − v )v + (1 − v )F±(r, v) ∓ θ± = 0 (3.10) where 00 0 0 p 11 ρ p 11 g11 11 p 11 ρ v 1 F±(r, v) = g 0 v − g v − g κ − g − . ρ 2g11 ρ 2 −v ± 1 With our application in mind, we would like a blow-up (or down) solution at the horizon which translates to v(r0) = ∓1 for the outermost future (past) horizon. Hence we are interested in establishing the existence and regularity results for the following initial value problem: (p g11(1 − v2)v0 + (1 − v2)F (r, v) ∓ θ = 0 ± ± (3.11) v(r0) = ∓1

11 Chapter 4

Motivation for Asymptotics and the Warping Factor

Recall that in Chapter 3 we made a specific choice of the warping factor u (see equation (3.7)). In this chapter, we motivate it and discuss the desired asymptotic behavior at infinity for a solution of (3.11). For the proof of the Penrose inequality, we will need the mass of the Jang surface to coincide with the mass of our initial data. Hence in the first section, we investigate asymptotics that can preserve mass.

4.1 Behavior at Inﬁnity

The generalized Jang equation for asymptotically flat initial data set is studied in detail in [HK13]. In that setting the warping factor is assumed to have the following asymptotics u u(r) = 1 + 0 + O (r−2+ε) (4.1) r 2 where u0 is a constant to be determined. Also, f and hence v are expected to vanish at infinity. Similar asymptotics for u are expected in the asymptotically hyperbolic setting. However, the asymptotic behavior of the solution of the generalized Jang equation is more complicated. In analogy with the approach to the positive mass theorem [SY82; Sak20] we impose the following asymptotics as r → ∞ √ 2 −1+ε f(r) = 1 + r + 2m log r + O3(r ) (4.2) for any ε > 0. Furthermore, we will assume that at the horizon f blows up (or down) and u vanishes just like in the asymptotically flat setting. Motivation for imposing (4.1) and (4.2) partly comes from the fact in this case the induced metric on the Jang surface will be asymptotically flat so that we can compute the ADM mass of (Σ, g¯) where

2 2 2 0 2 2 g¯ = g + u df = g11 + u (f ) dr + ρσ.

Lemma 5. Consider the Jang surface Σ = {t = f(r)} in the warped product manifold (M × R, g + u2dt2). If f and u satisfy (4.2) and (4.1) respectively, then the Jang metric g¯ is asymptotically ﬂat, and the ADM mass of the Jang metric is given by m¯ ADM = 2m + u0.

12 3 Proof. It is clear that the manifold (Σ, g¯) has an end diﬀeomorphic to R \Br0 , with coordinates y = (r, φ1, φ2) as in (2.11). Let x denote the associated Cartesian coordinates, related to y through the usual spherical transformation. In what follows, i, j, . . . are indices for x-coordinates and α, β, . . . are indices for y-coordinate. It follows that ∂yα ∂yβ g¯ = g + u2f f = g + u2f f ij ij ,i ,j αβ ,α ,β ∂xi ∂xj ∂r ∂r ∂ya ∂yb = g + u2(f 0)2 + g . 11 ∂xi ∂xj ab ∂xi ∂xj From (2.11), (4.1) and (4.2) we get

2 0 2 −1 2 −1 g11 + u (f ) = 1 + O(r ), gab = ρσab = r σab + O(r ), which implies ∂yα ∂yβ (g + u2f f ) = δ + O(r−1) αβ ,α ,β ∂xi ∂xj ij since the leading terms are exactly the standard coordinate transformation of Eu- clidean metric from spherical to Cartesian. Next we compute ∂g¯ ∂(g + u2f f ) ∂yα ∂yβ ∂yα ∂yβ ∂g¯ = ij = αβ ,α ,β + (g + u2f f ) + ij ∂xk ∂xk ∂xi ∂xj αβ ,α ,β ∂xk∂xi ∂xk∂xj ∂(g + u2f f ) ∂yγ ∂yα ∂yβ ∂yα ∂yβ = αβ ,α ,β + (g + u2f f ) + ∂yγ ∂xk ∂xi ∂xj αβ ,α ,β ∂xk∂xi ∂xk∂xj ∂g ∂yγ ∂yα ∂yβ ∂r ∂r ∂r = αβ + 2u0u(f 0)2 + 2u2f 00f 0 ∂yγ ∂xk ∂xi ∂xj ∂xk ∂xi ∂xj ∂yα ∂yβ + (g + u2f f ) + . αβ ,α ,β ∂xk∂xi ∂xk∂xj Since ∂r/∂xi = O(1), ∂ya/∂xi = O(r−1) and ∂ya/∂xi∂xj = O(r−2), together with −2 2 assumption (4.2) and (4.2) we get |∂g¯ij| = O(r ). Similarly we can get |∂ g¯ij| = O(r−3). Details of the computation for mass can be found in [CKS16]. Remark. From the computation for mass it follows that we can actually relax the assumptions (4.1) and (4.2) a little bit by assuming

u0 −2+ε 00 −3 u(r) = 1 + + O1(r ), u (r) = O(r ), √ r 2 −1+ε 000 −3 f(r) = 1 + r + 2m log r + O2(r ), f (r) = O(r ). To provide further motivation for the above assumptions, we examine them for our substitution (3.5). Suppose v is a smooth function that has the following series expansion at inﬁnity

2 −3 v(r) = 1 + B/r + C/r + O2(r ). It follows that 2B 2C + B2 α α 1 − v2(r) = − − + O (r−3) = − 1 − 2 + O (r−3). r r2 2 r r2 2

13 We then substitute the expansions into (3.10). Since

p −α1B g11(1 − v2)v0(r) = − + O(r−3), r2 00 0 0 p 11 2 ρ (r) g11(r) ρ (r) −α1 −α1B − α2 −3 g (1 − v )v(r)( 0 − − ) = + + 2 + O(r ), ρ (r) 2g11 2ρ(r) r r α α g11(1 − v2)κ(r) = − 1 − 2 + O(r−3), r r2 p ρ0(r) 2B 1 + 2C g11 v(r) = 2 + + + O(r−3), ρ(r) r r2 2γ(r) = 2 + O(r−5), ρ(r) we conclude from (3.10) that 2B −2α B − α + α + 1 + 2C + 1 2 2 + O(r−3) = 0. r r2 Consequently, if v is a solution we get the following relations

2B = 0

−2α1B + 1 + 2C = 0

2 −3 It follows that v(r) = 1 − 1/2r + O2(r ) for r large enough. 2 3 −4+ε Now we know that v(r) = 1−1/2r +D/r +O2(r ) so that we can obtain more terms in the expansion for v . After a long computation we get D = (2mk +mg)/2 = m. So the mass shows up in the solution of the generalized Jang equation. Next we show that we can indeed recover (4.2) and (4.1) from the asymptotics of v. By deﬁnition

u(r)pg11f 0(r) 4ρv2 v(r) = ⇒ (f 0(r))2 = . p1 + u2(r)g11(f 0(r))2 (1 − v2)2(ρ0(r))2(g11)2 It follows that √ 2 g 3 −4 0 2 ρv 1 − 1/2r + (m /2 + D)/r + O2(r ) f (r) = 2 0 11 = −2 (1 − v )ρ g 1 − 2D/r + O2(r )

r 2m −2+ε =√ + + O2(r ). 1 + r2 r Similarly, we obtain √ 1 − v2 ρ0(r) √ √ (2r − mg/r2 + O (r−3)) u(r) = √ = 1 + r2 1 − v2 2 p p 2 g −2 g11 2 ρ(r) 2 r + m /r + O3(r ) 1 D =(1 + + O(r−4))(1 − + O (r−2))(1 + O (r−3)) 2r2 r 2 2 −m =1 + + O (r−2+ε). r 2

This gives us u0 = −m andm ¯ ADM = m. All in all, the computation indicates that our assumptions (4.2) and (4.1) are convincing and provide the direction for proving the asymptotics.

14 4.2 Proof of the Inequality

In this section, we prove the Penrose inequality in the case of spherical symmetry by assuming that a solution to the generalized Jang equation exists and satisfies (4.2) and (4.1). Recall that in this case the scalar curvature of the Jang surface is given by (3.3). The important point to note here is that the Jang surface is also spherically symmetric. √ 2 2 For convenience we write η = ρ so thatg ¯ =g ¯11dr + η σ. Recall the definition of the Hawking mass in a 3-dimensional manifold: r Z Ag¯(Sr) 1 2 m¯ H (r) = 1 − HSr,g¯dσg¯ (4.3) 16π 16π Sr where √ 1 − v2 ρ0(r) HSr,g¯ = √ . g11 ρ(r) Hence in the case of spherical symmetry we have 1 m¯ = η(1 − η2 ) (4.4) H 2 ,s where s is the geodesic coordinate on Σ: √ Z r Z r g p 2 0 2 11 s = g11 + u (f ) = √ . 2 0 0 1 − v Furthermore, its derivate involves the scalar curvature of Σ η2η m¯ 0 (r) = ,s R,¯ (4.5) H 4 ¯ 2 2 where R = η2 (1−η,s −2ηη,ss) is the scalar curvature of Σ with respect to the induced metricg ¯ [LS14]. Integrating (4.5) with respect to r coordinate we get Z 1 ¯ g¯ m¯ H (∞) − m¯ H (r0) = η,sR dµ (4.6) 4π Σ where dµg¯ is the volume form on the Jang surface Σ. Now we may apply Schoen-Yau identity (3.3) and choose u in a way that it cancels with the term in front of the divergence, i.e. √ √ 2 2 1 − v 1 − v ρ,r u = η,s = √ η,r = √ √ . (4.7) g11 g11 2 ρ Note that u is nonnegative due to null expansions assumption (see (3.7) and (3.8)). It follows from divergence theorem that Z 1 2 2 g¯ m¯ H (∞) − m¯ H (r0) = η,s 2(µ − J(w)) + |h − K|Σ|g¯ + 2|q|g¯ dµ 4π Σ 1 Z − divg¯(uq)dωg¯ (4.8) 2π Σ Z 1 g¯ > − uhq, ng¯ig¯dµ 2π ∂Σ∪∂∞ where ng¯ is the unit outer normal (viewed as a 1-form). Here in the last inequality, we use the fact that dominant energy condition (1.3) is satisfied.

15 Lemma 6. If a blow up (or down) solution of IVP (3.11) exists and satisﬁes (4.2) p (4.1) then m¯ H (∞) = m, m¯ H (r0) = A(S0)/16π and Z uhq, ng¯ig¯dσg¯ = 0. (4.9) ∂Σ∪∂∞ where (Σ, g¯) is the Jang surface as described above. Proof. First we look at the Hawking mass of Σ. Since the solution blows up at the horizon we conclude H = 0 and consequentlym ¯ (r ) = pA(S )/16π. Sr0 ,g¯ H 0 0 The Hawking mass at inﬁnity is the limit r Z Ag¯(Sr) 1 2 m¯ H (∞) = lim 1 − HS ,g¯dσg¯ r→∞ r 16π 16π Sr rρ = lim 1 − u2(r) r→∞ 4

= − u0. It is a well-known fact that for asymptotically flat spherically symmetric manifolds we havem ¯ ADM =m ¯ H (∞). Futhermore, by Lemma (5) we know thatm ¯ ADM = 2m + u0. It follows that 2m + u0 = −u0 som ¯ ADM = −u0 = m. Now we prove (4.9). Consider Z uhq, ng¯ig¯dσg¯ (4.10) Sr where Sr are coordinate spheres. Since v(r0) = ±1, u vanishes at the horizon. Consequently, the integral vanishes at the horizon. For the spatial infinity, we consider the following limit Z lim uhq, ng¯ig¯dσg¯. r→∞ Sr Recall the definition of q in (3.3): ug11f 0 q1 = (π11 − K11), q2 = q3 = 0 p1 + u2g11(f 0)2 where 0 0 u∇rrf + 2u f 1 2u0 + 2m −4+ε π11 = = 2 − 3 + O(r ). q 2 2 r r 1 + u |df|g

√ g Note that n = ds = √ 11 dr, dσ = ρ(r)dσ, hence g¯ 1−v2 g¯ √ √ 2 11 g11 1 − v uhq, ng¯ig¯dσg¯ = ug¯ q1 √ ρdσ = uq1ρ √ dσ. 1 − v2 g11

We may now plug the above in (4.2), (4.1) and the expression for q1 to get √ √ 2 2 1 − v p 11 1 − v lim uq1 √ ρ = lim u g v(π11 − K11) √ ρ = −(2u0 + 2m). r→∞ g11 r→∞ g11

16 Hence Z ug¯(q, ng¯)dσg¯ = −4π(2u0 + 2m). S¯r

This vanishes as u0 = −m. All in all, we obtain the following result

Theorem 7. Let (M, g, K) be a spherically symmetric asymptotically hyperbolic initial data set satisfying (2.11) and ∂M is an outermost future (past) horizon. If the IVP (3.11) admits a solution and if for r → ∞

r m −4+ε v(r) = √ + + O2(r ), 1 + r2 r3 then r A m ≥ 16π where m and A are the mass of the initial data and the area required to enclose the outermost apparent horizon, respectively. If the equality holds then the initial data set arises from an embedding into the Schwarzschild spacetime.

Proof. The inequality follows from the above lemma directly. For the case of equality, inspecting the proof of Lemma (6), we see that

r Z A 1 2 2 g¯ 0 = m − ≥ η,s 2(µ − |J|g) + |h − K|Σ|g¯ + 2|q|g¯ dµ . 16π 4π Σ Hence ¯ µ − |J|g = 0, h − K|Σ = 0, q = 0.

It follows from (3.3) that R¯ = 0. We may now apply Theorem1 to (Σ , g¯) to obtain ∼ g¯ = gsc, that isg ¯ is isometric to the Schwarzschild manifold 2m g = 1 − −1dr2 + r2σ. sc r Hence 2m g¯ = 1 − −1, ρ = r2 ⇔ η = r. 11 r Therefore

p 2m −1/2 u = η = g¯11η = 1 − . ,s ,r r

2 2 2 2 4 2 2 Since g =g ¯ − u df = gsc − u df , the map G :(M, g) → (SC , gsc − u dt ), where x 7→ (x, f(x)), provides an isometric embedding. Finally, a calculation [BK10] ¯ 4 shows that h − K|Σ = 0 implies that the second fundamental form of G(M) ⊂ SC is precisely given by the initial data K.

17 Chapter 5

Existence and Asymptotics

In this chapter, we study the initial value problem (3.11) as it is the ingredient left to complete the proof of the Penrose inequality for asymptotically hyperbolic spherically symmetric initial data. In the ﬁrst part, we establish the existence of a solution. Then we study the asymptotic behavior of the solution in detail. As a preliminary, we recall the following comparison result in [Sak20] for ordinary diﬀerential equations which will help us to prove the existence and establish the asymptotics.

Lemma 8. Let F :[r0, ∞) × [−1, 1] → R be continuous in both variables. If 0 0 functions l = l(r) and k = k(r) satisfy l + F (r, l) < k + F (r, k) and l(r0) 6 k(r0) then l(r) 6 k(r) for r > r0. The proof is rather simple and uses contradiction. This lemma allows us to bound the solution by suitable functions which may be called barriers (or super and sub- solutions).

5.1 Existence

The main diﬃculty in proving the existence of a solution is that the IVP (3.11) degenerates for v = ±1, that is v0 is “killed” by 1 − v2 when v = ±1, which prevents us from using classical theorems for ODEs. In order to handle the degeneracy, we will perform another substitution to “hide it”. Moreover, it will be convenient to use the geodesic coordinate near horizon, i.e. Z r √ τ = g11, (5.1) r0 in which (3.11) becomes

(1 − v2(τ))v0(τ) + (1 − v2(τ))F (τ, v) ∓ θ (τ) = 0 ± ± (5.2) v(0) = ∓1 where ρ00(τ) ρ0(τ)v(τ) 1 F (τ, v) = v(τ) − κ(τ) − − . ± ρ0(τ) ρ(τ) 2 −v(τ) ± 1 We will work with this coordinate system in this section.

18 Proposition 9. Consider the initial value problem (5.2). If θ±(τ) > 0 for τ > 0 and θ+(0) = 0, θ−(0) = 0 or θ±(0) = 0 depending on the type of horizon and if 0 0 furthermore θ+ > 0 (θ− > 0 for past horizon case), then it admits at least one solution. Moreover, the solution satisfies |v(τ)| < 1 for all τ > 0. Proof. We first observe that the degeneracy, the term 1 − v2 in front of v0, can be “hidden” by the following substitution v3 z = v − (5.3) 3 2 2 It follows that for v ∈ [−1, 1] we have z ∈ [− 3 , 3 ] and it is increasing. The inverse function v = v(z) is also strictly increasing. Moreover, we can extend z(v) outside [−1, 1] continuously so that its inverse remains continuous and increasing, for example using linear functions. We first consider the outermost past horizon case. It is straightforward to check that (5.2) takes the form

0 2 z (τ) + (1 − v (τ))F−(τ, v) + θ−(τ) = 0 2 (5.4) z(0) = 3

Note that F− is continuous in both arguments due to our assumptions on the initial data. Similarly θ− is smooth for all τ > 0. Then it follows by Peano’s existence theorem [Har64] that there exists at least one solution in a rectangle τ ∈ [0, a] and |z − 2/3| 6 b. Next, we prove an a priori estimate for z on the whole interval [0, ∞) and then extend the solution. It is obvious that ±2/3 are two barriers for all solutions in between when τ > 0. Hence it remains to show that the solution z enters that region, i.e. z(ε) < 2/3 for arbitrarily small ε > 0. Since we already know that z(0) = 2/3 and z0(0) = 0 from (5.4), we need to look at the second derivative of z at τ = 0. From (5.3) it is obvious that z00(0) = −2(v0(0))2. We can then get a quadratic equation for v0(0) by diﬀerentiating (5.2) at τ = 0

0 2 0 0 2(v (0)) + 2F−(0, 1)v (0) − θ−(0) = 0. (5.5) 0 0 Now we use the assumption that θ−(0) > 0 to obtain v (0) is real and bounded. This gives z00(0) is indeed negative and consequently z is decreasing at 0. Finally, we can extend the solution to the whole interval by the extension theorem in [Har64]. The same argument applies to the outermost future horizon case, i.e. θ+(0) = 0. We conclude that the initial value problem admits a solution v ∈ C∞((0, ∞)) ∩ C1([0, ∞]) and |v| < 1 for τ > 0.

5.2 Asymptotics

In this section, we investigate the asymptotics of solutions of IVP (3.11) whose existence was proven in Section 5.1. We prove a rough estimate in Proposition 10, and subsequently reﬁne it in Proposition 11. In both cases we use suitable barrier functions to achieve our goal. Proposition 10. Let v = v(r) be the solution of IVP (3.11). Then for any suﬃ- ciently small ε > 0 there exists r1 > r0 such that 1 v(r) = 1 − + O(r−3+ε). (5.6) 2r2

19 Proof. Consider the generalized Jang equation p ρ0 2γ J(v) := (1 − v2)v0 + (1 − v2) a(r)v − g11κ + v − = 0 (5.7) ρ ρpg11 where 00 0 0 ρ g11 ρ a(r) = 0 − − . ρ 2g11 2ρ The asymptotics of the terms in the Jang equation when r is big enough are as follows: 1 1 3mg a(r) = − + + O(r−5), r r3 r4 p 1 1 g11κ = − + O(r−4), r 2r3 ρ0 2 3mg = − + O(r−5), ρ r r4 2γ 2 1 mk − mg = − + + O(r−5). ρpg11 r r3 r4

3−ε 1 αr− 1 Suppose v− = 1 − 2 − 3−ε , where α = 2 − 2 . Without loss of generality we can 2r r 2r− 2 assume 1 < α < 2 for r− big. Hence 0 6 1 − v− 6 1 and 1 α(3 − ε)r3−ε v0 = + − . − r3 r4−ε Then

2 0 p 11 (1 − v−)(v− + a(r)v− − g κ) α(2 − ε)r3−ε αr3−ε 3mgαr3−ε αr3−ε =(1 − v2 ) − + O(r−4) + − − − + O(r−5) − − r4−ε r6−ε r7−ε r3−ε α(2 − ε)r3−ε C αr3−ε − + 1 + − 6 r4−ε r4 r6−ε and ρ0 2γ v− − ρ ρpg11 2αr3−ε 2m 3mgαr3−ε αr3−ε = − − − + O(r−5) + − + O(r−5) − r4−ε r4 r7−ε r3−ε 2αr3−ε C − − + 2 . 6 r4−ε r4

r− Note that here we use the fact 0 < r 6 1, so C1,C2 depend only on the initial data and are independent of r−. It follows that

0 2 0 p 11 ρ 2γ J(v−) =(1 − v )(v + a(r)v− − g κ) + v− − − − ρ ρpg11 αεr3−ε C αr3−ε εr3−ε C 2r3−ε − − + − + − − − + − + − 6 r4−ε r4 r6−ε 6 r4−ε r4 r6−ε

20 where C− = C1 + C2. Thus J(v−) < 0 provided that r− > 0 is chosen so that 3 −εr− + 2r− + C− < 0. 1−ε 1 r+ Similarly, we can construct a super-solution by considering v+(r) = 1− 2r2 + 2r3−ε . 2 Hence 0 6 1 − v+ 6 1 and 1 (3 − ε)r1−ε v0 = − + + r3 2r4−ε Then similarly

2 0 p 11 (1 − v+)(v+ + a(r)v+ − g κ) (2 − ε)r1−ε r1−ε 3mgr1−ε r1−ε =(1 − v2 ) − + + O(r−4) − + + + + O(r−5) + + 2r4−ε 2r6−ε 2r7−ε 2r3−ε (2 − ε)r1−ε C − + − 3 > 2r4−ε r4 and ρ0 2γ v+ − ρ ρpg11 2r1−ε 2m 3mgr1−ε r1−ε = + − + O(r−5) − + + O(r−5) + 2r4−ε r4 2r7−ε 2r3−ε 2r1−ε C + − 4 . >2r4−ε r4 It follows that 0 2 0 p 11 ρ 2γ J(v+) =(1 − v )(v + a(r)v+ − g κ) + v+ − + + ρ ρpg11 εr1−ε C + − + >2r4−ε r4 where C− = C3 + C4. Thus J(v+) > 0 provided r+ > 2C+/ε. 1 −3+ε Finally, we can conclude v = 1 − 2r2 + O(r ) for r > r1 = max{r−, r+}. Proposition 11. Let v = v(r) be the solution of IVP (3.11). Then for any suffi- ciently small ε there exists r2 > r0 such that 1 m v(r) = 1 − + + O(r−4+ε) (5.8) 2r2 r3 1 Proof. We improve the result from Proposition 10. Consider v = 1 − 2r2 + ψ, then −3+ε ψ = O(r ) by Proposition 10. We fix ε and r1 as in Proposition 10. It follows 3−ε that |ψ| 6 C(ε, r1)/r where C(ε, r1) is fixed and depends only on ε and r1. We can now estimate the term 1 − v2 more accurately: 1 C(ε, r ) 1 − v2 1 − 1 − − 1 6 2r2 r3−ε 1 2C 1 C C2 1 + − − − 6 r2 r1−ε 4r2 r3−ε r4−2ε 1 (1 + C ) 6 r2 0

21 where C0 is a constant. 3 1−ε 4−ε First, we construct a lower barrier. Suppose ψ− = m/r − (Cr2 + mr2 )/r , 3−ε 1 then ψ−(r2) = −C/r2 . Therefore v(r2) v−(r2) := 1 − 2 + ψ−(r2). Recall the > 2r2 left hand side of the generalized Jang equation J(v) = 0 is given by (5.7). We plug in v = v−: ρ0 2m J(v ) =(2 − v2 )(ψ0 + a(r)ψ + O(r−4)) + ( − + O(r−5)) − − − − ρ r4 3m (4 − ε)ϕ r =(1 − v2 )(− + − 2 + O(r−4) − r4 r5−ε r φ ϕ r 3mgϕ r ϕ r − 2 − + − 2 − − 2 + O(r−5) − 2 ) r5−ε r7−ε r8−ε r4−ε 2ϕ r 3mgϕ r ϕ r + (− − 2 + O(r−5) + − 2 + O(r−5) − 2 ). r5−ε r8−ε r4−ε

ε Here we denote ϕ− := C + m/r2. For a ﬁxed ε > 0, ϕ− is bounded provided r2 is big enough. Hence 3ϕ C 2ϕ r C J(v ) (1 − v2 )( − + 1 ) + (− − 2 + 2 ) − 6 − r4−ε r4 r5−ε r5 1 3ϕ C 2ϕ r C (1 + C )( − + 1 ) − − 2 + 2 6r2 0 r4−ε r4 r5−ε r5 ϕ r C − − 2 + − . 6 r5−ε r5

1+ε Thus J(v−) < 0 provided −2ϕ−r2 + C− < 0. 1 Similarly we can construct an upper barrier v+ = 1 − 2r2 + ψ+, where ψ+ = m 1−ε 4−ε r3 + (Cr2 − mr2 )/r . Finally, we conclude that 1 m v(r) = 1 − + + O(r−4+ε). 2r2 r3

Remark. Note that in order to complete the proof of the spacetime Penrose inequality for asymptotically hyperbolic spherically symmetric initial data we need to show 1 m −4+ε 0 3 4 −5+ε that v(r) = 1 − 2r2 + r3 + O2(r ), in particular, v = 1/r − 3m/r + O(r ) and v00 = O(r−4). At the same time, plugging v into (3.10) and solving for v0 only gives v0 = O(r−3+ε), which is related to the fact the equation degenerates as v → 1. Note that this diﬃculty is not present in the asymptotically ﬂat setting of [BK10] as v → 0 for r → ∞. The question whether the optimal decay for v0 and v00 can be established will be addressed elsewhere.

22 Bibliography

[ADM08] Richard Arnowitt, Stanley Deser, and Charles W. Misner. “Republi- cation of: The dynamics of general relativity”. In: General Relativity and Gravitation 40.9 (Sept. 2008), pp. 1997–2027. issn: 1572-9532. doi: 10.1007/s10714-008-0661-1. [AM79] Abhay Ashtekar and Anne Magnon-Ashtekar. “Energy-Momentum in General Relativity”. In: Phys. Rev. Lett. 43 (3 July 1979), pp. 181–184. doi: 10.1103/PhysRevLett.43.181. [Bar86] Robert Bartnik. “The mass of an asymptotically flat manifold”. In: Com- munications on Pure and Applied Mathematics 39.5 (1986), pp. 661– 693. doi: 10.1002/cpa.3160390505. eprint: https://onlinelibrary. wiley.com/doi/pdf/10.1002/cpa.3160390505. [BI04] Robert Bartnik and Jim Isenberg. “The Constraint Equations”. In: (June 2004). doi: 10.1007/978-3-0348-7953-8_1. [BK10] Hubert L. Bray and Marcus A. Khuri. “A Jang equation approach to the Penrose inequality”. In: Discrete & Continuous Dynamical Systems - A 27 (2010), p. 741. issn: 1078-0947. doi: 10.3934/dcds.2010.27.741. [BK11] Hubert L. Bray and Marcus A. Khuri. “P. D. E.’s Which Imply the Penrose Conjecture”. In: Asian J. Math. 15.4 (Dec. 2011), pp. 557–610. [Bra01] Hubert L. Bray. “Proof of the Riemannian Penrose Inequality Using the Positive Mass Theorem”. In: J. Differential Geom. 59.2 (Oct. 2001), pp. 177–267. doi: 10.4310/jdg/1090349428. [BVM62] Hermann Bondi, M. G. J. Van der Burg, and A. W. K. Metzner. “Grav- itational waves in general relativity, VII. Waves from axi-symmetric iso- lated system”. In: Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 269.1336 (1962), pp. 21–52. doi: 10. 1098/rspa.1962.0161. eprint: https://royalsocietypublishing. org/doi/pdf/10.1098/rspa.1962.0161. [CKS16] Ye Sle Cha, Marcus Khuri, and Anna Sakovich. “Reduction arguments for geometric inequalities associated with asymptotically hyperboloidal slices”. In: Classical and Quantum Gravity 33.3 (Jan. 2016), p. 035009. doi: 10.1088/0264-9381/33/3/035009. [Fra04] JörgFrauendiener. “Conformal Infinity”. English. In: Living reviews in relativity 7.1 (2004), p. 1. [Gal08] Gregory J. Galloway. “Rigidity of marginally trapped surfaces and the topology of black holes”. English. In: Communications in analysis and geometry 16.1 (2008), pp. 217–229.

23 [Har64] Philip Hartman. Ordinary differential equations. English. New York: Wi- ley, 1964. [Haw71] S. W. Hawking. “Gravitational Radiation from Colliding Black Holes”. In: Phys. Rev. Lett. 26 (21 May 1971), pp. 1344–1346. doi: 10.1103/ PhysRevLett.26.1344. [Haw72] S. W. Hawking. “Black holes in general relativity”. In: Communications in Mathematical Physics 25.2 (June 1972), pp. 152–166. issn: 1432-0916. doi: 10.1007/BF01877517. [Hay03] Sean A. Hayward. “Spatial and null infinity via advanced and retarded conformal factors”. In: Phys. Rev. D 68 (10 Nov. 2003), p. 104015. doi: 10.1103/PhysRevD.68.104015. [Heu96] Markus Heusler. Black Hole Uniqueness Theorems. Cambridge Lecture Notes in Physics. Cambridge University Press, 1996. doi: 10 . 1017 / CBO9780511661396. [HI01] Gerhard Huisken and Tom Ilmanen. “The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality”. In: J. Differential Geom. 59.3 (Nov. 2001), pp. 353–437. doi: 10.4310/jdg/1090349447. [HK13] Qing Han and Marcus Khuri. “Existence and Blow-Up Behavior for Solutions of the Generalized Jang Equation”. In: Communications in Partial Differential Equations 38.12 (2013), pp. 2199–2237. doi: 10 . 1080/03605302.2013.837919. eprint: https://doi.org/10.1080/ 03605302.2013.837919. [Kro03] Juan Antonio Valiente Kroon. “Early radiative properties of the develop- ments of time-symmetric conformally flat initial data”. In: Classical and Quantum Gravity 20.5 (Feb. 2003), pp. L53–L59. doi: 10.1088/0264- 9381/20/5/102. [LS14] Dan A. Lee and Christina Sormani. “Stability of the positive mass theorem for rotationally symmetric Riemannian manifolds”. In: Journal für die reine und angewandte Mathematik (Crelles Journal) 2014.686 (2014), pp. 187–220. [Mar09] Marc Mars. “Present status of the Penrose inequality”. In: Classical and Quantum Gravity 26.19 (Sept. 2009), p. 193001. doi: 10.1088/0264- 9381/26/19/193001. [Mic11] B. Michel. “Geometric invariance of mass-like asymptotic invariants”. In: Journal of Mathematical Physics 52.5 (2011), p. 052504. doi: 10.1063/ 1.3579137. eprint: https://doi.org/10.1063/1.3579137. [MM04] Edward Malec and Niall O´ Murchadha. “The Jang equation, apparent horizons and the Penrose inequality”. In: Classical and Quantum Gravity 21.24 (Nov. 2004), pp. 5777–5787. doi: 10.1088/0264-9381/21/24/007. [ONe83] Barrett O’Neill. Semi-Riemannian geometry with applications to relativity. English. Vol. 103. New York: Academic Press, 1983. isbn: 0125267401. [Sak20] Anna Sakovich. The Jang equation and the positive mass theorem in the asymptotically hyperbolic setting. 2020. arXiv: 2003.07762 [math.DG].

24 [SB62] R. K. : Sachs and Hermann Bondi. “Gravitational waves in general relativity VIII. Waves in asymptotically ﬂat space-time”. In: Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 270.1340 (1962), pp. 103–126. doi: 10.1098/rspa.1962.0206. eprint: https://royalsocietypublishing.org/doi/pdf/10.1098/ rspa.1962.0206. [SY79] Richard Schoen and Shing Tung Yau. “On the proof of the positive mass conjecture in general relativity”. In: Comm. Math. Phys. 65.1 (1979), pp. 45–76. [SY81] Richard Schoen and Shing Tung Yau. “Proof of the positive mass theorem. II”. In: Comm. Math. Phys. 79.2 (1981), pp. 231–260. [SY82] Richard Schoen and Shing Tung Yau. “Proof That the Bondi Mass is Positive”. In: Phys. Rev. Lett. 48 (6 Feb. 1982), pp. 369–371. doi: 10. 1103/PhysRevLett.48.369. [Wal84] Robert M. Wald. General relativity. English. Chicago: University of Chicago Press, 1984. isbn: 0226870324.