Area-controlled end replacement and the Bondi Penrose inequality in the CKS class

1 Replacing the end and retaining the area

We ask whether a Cha–Khuri–Sakovich hyperboloidal end can be replaced by an asymptotically flat end while leaving a prescribed compact interior unchanged and making the ADM mass approach the Lorentz norm of the hyperboloidal charge. For the Penrose inequality, a mass comparison alone is insufficient. A surface enclosing the inner boundary can travel through the modified end, so its area must remain controlled wherever the replacement occurs.

We construct replacements that preserve the dominant energy condition and give a lower metric comparison on the entire exterior. This controls every enclosing surface at once, including disconnected surfaces and surfaces touching the original boundary. The construction applies to the precise asymptotic class of Cha, Khuri and Sakovich [7]. The end replacement is the main geometric result; the sharp Bondi inequality follows by applying the numerical asymptotically flat inequality to each completed replacement.

1.1 The exterior, its charges and its enclosing cuts

Let \Omega be a connected orientable smooth three-manifold with nonempty compact smooth boundary S. Let g be a smooth Riemannian metric and K a smooth symmetric covariant two-tensor, both smooth up to S. We require completeness as a metric space with S included, and exactly one coordinate end with compact complement. With G=c=1 and zero cosmological constant, the constraint densities are

The dominant energy condition (DEC) is \mu\geq |J|_g. For a two-sided surface with normal \nu pointing into its end side, write

Thus a future marginally outer trapped surface has \theta_+=0. Our spacetime convention is K(Y,Z)=\overline g(\overline\nabla_Yn_{\mathrm{future}},Z); the future unit hyperboloid in Minkowski space has K=g.

An enclosing cut is \Gamma=\partial D, the intrinsic manifold boundary of a connected smooth codimension-zero submanifold-with-boundary D\subset\Omega that is closed in \Omega, contains the whole sufficiently distant end, and has manifold interior in \operatorname{int}\Omega. Its full boundary must be compact, smooth, embedded and two-sided. Every component and every portion coinciding with S contributes its area. A cut need not be connected, trapped or minimal. Set

Taking D=\Omega shows that the class includes S itself. We neither assume attainment of the infimum nor identify it with \operatorname{Area}_g(S). This distinction matters for the Penrose inequality: Ben-Dov’s spherical example violates the bound using the apparent horizon’s own area, while leaving the minimum-enclosing-area formulation intact [5, Sections 1 and 4].

Let \sigma be the unit round metric on \Sph. In coordinates (r,\omega) on the end, put

The CKS class in this paper consists exactly of the expansions

The leading angular fields are smooth. For a coordinate component, O_j(r^{-q}) bounds r^{q+i}\partial_r^i\partial_\omega^\alpha through i+|\alpha|\leq j in a finite sphere atlas. Equivalently, one may use angular derivatives and scaled radial derivatives r\partial_r. No smallness, pointwise sign or symmetry of the leading fields is required. These component rates and derivative orders are those of [7, Equations (2.1)–(2.3)].

Writing n=(n^1,n^2,n^3) for the inclusion \Sph\subset\R^3, define

We call (E_B,P_B) the Bondi charge of the initial data. All statements below concern the explicit CKS functional (1.4) and its Lorentz norm; no spacetime development is assumed. Identifying an initial-data charge with the Bondi–Sachs charge of a cut of null infinity requires an appropriate compatible development and asymptotics; see, for example, [10, Theorem 5.3]. We assume E_B>|P_B|. A chart is balanced if P_B=0; then E_B=m_B>0. The second tensor is essential to the aspect M. For comparison with Chen, Wang and Yau, set p=K-g. Its leading tangential coefficient is m^K-m^g, so their mass aspect is

Their charge factor 1/(8\pi) therefore gives exactly (1.4) [8, Definitions 1.3–1.4]. Chen, Wang and Yau obtain rest-frame foliations in their asymptotic setting [8, Corollary 4.3]. Lemma 3.1 proves the Lorentz covariance and preservation of the finite derivative class (1.3) needed here.

1.2 The geometric theorem and its numerical consequence

For an asymptotically flat end with Cartesian coordinates x=rn, our ADM conventions are

In Cartesian coordinates O_j(r^{-q}) has the usual differentiated decay: an order-i derivative is O(r^{-q-i}).

The theorem separates the geometry of the end from a numerical inequality. In particular it retains the boundary’s original null expansion, whatever its sign. For a weakly future trapped boundary, the numerical spacetime Penrose inequality gives the following result.

This gives the sharp Bondi–Penrose bound in the specified CKS class, using the numerical spacetime Penrose theorem of [15, Definitions 1.1–1.2 and Theorem 1.3] in dimension three. Section 6.3 states that input and checks its full hypotheses for each fixed replacement. The geometric theorem also gives a conditional implication from the corresponding asymptotically flat inequality, recorded in Section 6.3. The sharpness examples exclude additional closed horizons of either time orientation.

1.3 Predecessors and the construction

Penrose’s mass–area comparison arose from gravitational collapse, cosmic censorship and the area theorem [16]. For time-symmetric asymptotically flat data, Huisken and Ilmanen’s weak inverse mean curvature flow proves the bound for each connected component of an outermost minimal horizon [13]. Bray’s conformal flow gives the sharp bound for the total area of a possibly disconnected outer-minimizing minimal horizon [6]. The extension to a general second fundamental form must retain an area comparison while controlling both constraint densities.

In the hyperboloidal setting, Cha, Khuri and Sakovich reduce the sharp inequality to a generalized Jang equation coupled to weak inverse mean curvature flow, with the required boundary behavior and asymptotics and a connected outermost apparent horizon [7, Theorem 3.2]. Their prescribed-warping existence theorem solves the generalized Jang equation alone [7, Theorem 3.4]; it does not establish existence for the coupled system. Sakovich’s ordinary Jang argument proves positive energy in a weighted asymptotically hyperbolic class [17, Theorem 1.1].

Null-geometric approaches make the role of the limiting cut and observer explicit. Mars and Soria compare a trapped cross-section with the limiting Hawking energy along a distinguished null foliation; identifying that limit with Bondi energy requires their large-sphere condition [14, Theorems 2–3]. In a controlled perturbative vacuum regime near Schwarzschild, Alexakis constructs a nearby marginally trapped sphere and an asymptotically round null foliation giving a Bondi-energy inequality at its new limiting cut [1, Theorem 1.3 and Section 2.1].

For nonsymmetric initial data, Allen, Bryden, Kazaras and Khuri prove a universal suboptimal-constant inequality for invariant ADM mass in the asymptotically flat case and for the chosen end’s energy under Wang-type hyperboloidal asymptotics [2, Theorems 1.1 and 1.3]. Their area is the least enclosing area of a selected outermost apparent horizon. Ellithy’s future-development approach bounds actual horizon-section area under quasi-final-state and further Bondi–Sachs hypotheses [12, Theorem 4.12, Remark 4.14 and Theorem C.1]. Its final Bondi quantity is defined in the specified Bondi–Sachs foliation. These results distinguish the initial-data norm, a foliation’s energy, and the area being bounded.

Our construction is related to earlier geometric changes of an end. The radial-lapse ansatz belongs to Bartnik’s quasi-spherical framework [3][4, Section 5]. Chruściel and Delay bend an exactly hyperbolic end to a flat end through a Minkowski graph, preserving the compact side and DEC [9, proof of Theorem 4.2]. Their model has zero charge and uses the opposite time orientation. Corvino and Huang’s localized deformation theory addresses the additional difficulty that a metric change also changes the norm of the momentum density [11, Introduction and Equation (1.1)]. Ellithy’s late-time comparison likewise attaches a quasi-spherical Riemannian end before applying inverse mean curvature flow [12, Appendix C, proof of Theorem C.1]. The target here is a DEC replacement of the fixed CKS initial-data exterior, with positive charge approximated by ADM mass and a global lower metric bound protecting every enclosing cut. The collar and bending estimates establish these properties directly.

The proof begins with a radial vacuum model in Section 2. It shows how the tangential tensor trace can be reduced to zero while retaining a mass variable. An arbitrary angular mass aspect creates two additional terms: a Laplacian in energy and a gradient in momentum. Heat evolution cancels the first, and a trace-free sphere tensor cancels the second. Its divergence equation has precisely a first-harmonic obstruction; Lorentz balancing removes that obstruction.

Before bending, the original leaves must be made round. Because the input DEC may be saturated, this cannot be justified by smallness alone. Section 3 supplies the constraint identities, and Section 4 constructs a collar whose added constraint vector lies in the future causal cone. The estimates retain the activation cutoff as a factor, even at its flat initial endpoint. Section 5 then performs the angularly corrected bending. Heat starts while the tangential tensor trace still has its hyperboloidal value; the trace-free correction enters as bending begins. Section 6 computes the ADM charges at fixed R, compares every cut, and applies the numerical theorem before letting R\to\infty. Section 7 supplies actual equality examples. Vanishing comparison errors alone do not classify equality for general hyperboloidal data.

2 The radial model and the angular obstruction

We first isolate the operation that will change the end. On a round sphere, the mean curvature and the tangential trace of K are distinct quantities. Their difference of squares determines a local mass. This permits us to lower the tensor trace without prescribing that the future null expansion should vanish.

Fix m>0 and a smooth function v=v(r). On any interval where 1+v^2>2m/r, define

The positivity condition holds, for example, on r>2m. The leaves have mean curvature H=2u/r and tensor trace p=2v/r. Thus

Choosing v=r gives the hyperboloidal scale, whereas v=0 gives the spatial Schwarzschild metric. The quantity reduced to zero is p=2v/r, not \theta_+.

These data are vacuum for every such choice of v. Indeed the warped-product scalar-curvature calculation gives

Their sum is zero by differentiating u^2=1+v^2-2m/r. The radial momentum equals

and the angular momentum vanishes by spherical symmetry. This is the basic flexibility of the construction: changing v alone need not cost any mass.

For general CKS data the corresponding mass is a function of direction. If m in (2.1) is replaced by F(r,\omega), then angular derivatives of the lapse appear in both constraints. The energy contains a leading multiple of \Delta_\sigma F and the angular momentum a leading multiple of \mathrm d_{\Sph}F. We will evolve the angular function by heat flow and add a trace-free tangential tensor to cancel these terms. The elementary obstruction to the latter step explains why balancing belongs at the beginning of the proof. For a smooth sphere function f, write \langle f\rangle=\avg{f}.

In a balanced chart f_*=M/4 has mean m_B and vanishing first harmonic. We next prove the existence of that chart and bring the original geometry to round leaves. The small increasing mass correction used in that collar will continue through the bending. It supplies a positive DEC margin after the two leading angular terms have canceled. The round-leaf radial-lapse metric is a zero-shift instance of Bartnik’s quasi-spherical framework [3, 4]. Here both the metric and the second tensor vary, and both constraint inequalities are proved directly rather than imported from a prescribed-scalar-curvature theorem.

3 Balancing and the constraint identities

The round model suggests which quantities to preserve, but the given end need not have round leaves or a radial mass aspect. We begin by choosing coordinates in which the Bondi momentum vanishes. This change does not alter the data; it removes the first spherical harmonic of the mass aspect, which is the obstruction to the angular divergence equation used later. We then write the constraints for the original, generally nonround leaves. These identities will let us round them without losing the dominant energy condition.

All remainder estimates below have the componentwise, differentiated meaning in (1.3). Only three derivatives of the metric remainder and two of the tensor remainder are used.

3.1 A balanced end chart

We use this balanced chart from now on and again call its coordinates (r,\omega). The original data have not been deformed. We have only put their total charge in its rest frame; the angular function f_* may still change sign and need not be small.

3.2 Sphere geometry and the constraints

The collar will change both the leaf geometry and the second tensor. To see which changes can preserve DEC, let (G,k) be smooth data on a sphere-foliated portion of the end, with the physical constraint normalization of the introduction. Write

Here \gamma is the induced sphere metric, s is a sphere-tangential vector field, and N is the unit normal toward increasing r. Unless otherwise stated, norms, contractions, traces, and differential operators on a leaf are taken with \gamma; in particular, \Delta_\gamma=\diver_\gamma\nabla^\gamma. Denote the leaf second fundamental form, mean curvature, and normal acceleration by

Decompose k into its normal, mixed, and tangential parts:

The following scalar variables will be central:

Here u is the rescaled mean curvature and v is the rescaled tangential trace of k. The quantity F is their mass combination; it agrees with the constant mass in the radial model. We use these variables only where u,d>0. The scalar d records the effect of the nonround leaves on the relation u=Ud. It is distinct from the differential \mathrm d; write \mathrm d_{\mathbb S^2} for the angular differential.

The identities separate the two adjustments needed in the collar. Changing L changes the constraints without introducing a radial derivative of L. Changing F produces the first-derivative term 2NF/(ur^2) in (3.9), where L has canceled. We will use a radial addition to L for the large null component of the constraint increment and a slowly increasing addition to F for the small component. Their sizes will also control the angular momentum increment.

3.3 The original data in foliation variables

Before changing the data, we identify the sizes of these variables. Put a subscript 0 on the quantities associated with G_0=g and k_0=K. In particular, the limit of F_0 must contain the leading coefficients of both tensors, because it is the Bondi mass aspect that must survive the replacement.

4 A collar with round leaves

Our next task is to replace the large leaves by round spheres while retaining the original data on their inner side. The metric change is small, but smallness alone does not preserve a saturated dominant energy condition. Instead, we arrange that the change in the constraint vector is itself future causal.

The construction has two stages on adjacent annuli. On the first we start a positive addition to F and a radial addition to L. On the second, with these additions fully active, we interpolate the remaining fields to their round values. The first addition controls the small null component C-Q; the second controls C+Q. Their product will dominate the square of the angular momentum change. This is why no strictness assumption on the original DEC is needed.

Throughout this section a subscript 0 denotes the quantities of the original data in the balanced chart. Constants denoted C_* or C_j may depend on these data and on fixed cutoff profiles, but not on the large parameter R. They are distinct from the constraint density C and its original value C_0.

5 Bending the round end

The collar has produced the round leaves needed for the model of Section 2. We now lower their tangential tensor trace 2v/r from 2 to zero. The angular mass function is not discarded: its mean will become the ADM energy, while its other modes are smoothed by heat evolution. The trace-free correction from Lemma 2.1 removes the associated angular momentum. The increasing mass term already present in the collar then absorbs all remaining constraint errors.

There are two radial scales. We begin reducing v/r at a radius comparable to R, but postpone its final cutoff to the annulus R^2\leq r\leq2R^2. By then v and its scaled derivatives are small, so the final cutoff is smooth without a singular estimate involving 1/v. Heat evolution is activated before either change, while v=r.

We apply Lemma 5.1 with the balanced aspect f_* and m=m_B. All constants below may depend on this smooth aspect and on fixed cutoff profiles, but not on the large parameter R. We use C_* for a positive bound that may increase from line to line, to distinguish such constants from the constraint quantity C.

The two coefficients in the bending prescription can be read from the round constraints. To see this, consider first the region after the joining annulus, where the added normal term has disappeared. Take \gamma=r^2\sigma, L=v', \eta=0, and set A=1+v^2 and u^2=A-2F/r, temporarily leaving F and \tau undetermined. Replacing u^2 by A in the angular coefficients, the terms to cancel in the scalar and tangential momentum constraints are, respectively,

The first expression suggests taking F=f(t(r),\omega)+c_R(r) with t'=1/(2rA); the radial padding then contributes only its positive mass derivative. For the second, since \diver_\sigma\mathcal T=\mathrm d_{\Sph}f and \diver_\gamma=r^{-2}\diver_\sigma on covariant sphere tensors, the coefficient of \mathcal T must be r(v'-v/r)/A. This calculation chooses the formulas. The proof below retains the exact coefficients and all quadratic terms, and starts heat evolution before changing v/r to handle the joining annulus.

6 Charges, completeness and enclosing area

We now finish Theorem 1.1. All estimates in this section concern the data constructed in Sections 4 and 5; in particular their DEC has already been proved. Subscripts R are suppressed when they are not needed.

6.1 The asymptotically flat output

For each fixed sufficiently large R, the outer construction has v=\beta=0 beyond a finite radius. Therefore L=0, \tau=0 and k=0 there, and

Lemma 5.2 bounds every fixed-order scaled radial and angular derivative of F. In Cartesian coordinates x=rn the metric takes the form

Every Cartesian derivative can be expressed as \partial_{x_i}=r^{-1}(n_iD+X_i), where D=r\partial_r and X_i is a smooth tangential field on the unit sphere. Repeatedly applying this identity to b n_i n_j proves G_{ij}-\delta_{ij}=O_j(r^{-1}) for every fixed finite j. In particular the O_6 metric requirement of the numerical input below holds. The round constraint estimates give C=O(r^{-7/2}) and J=0 on this tail. Since its volume element is comparable to r^2\mathrm dr\,\mathrm dA_\sigma, the constraint densities are integrable. The portion inside a fixed tail sphere is compact and smooth, so integrability holds on the whole exterior.

For completeness, the ADM energy calculation allows angular dependence of b. Direct differentiation, using \partial_jn_i=(\delta_{ij} -n_i n_j)/r, gives

The derivatives of b cancel after contraction with n; there is no assumption that F is radial. Since rb=2F+O(r^{-1}), uniformly on the sphere, (1.5) gives

The mean of the heat solution is constant, which proves the second equality. The vanishing of k on the tail proves P_R=0 directly. The bound in (4.1) yields 0\leq c_R(\infty)\leq2R^{-1/2}. Because m_B>0, the constructed charges satisfy the strict inequality E_R>|P_R|. These are the charges of one completed replacement, with R fixed. The constant in the refined tail estimate (5.13) need not be uniform in R.

6.2 A comparison for every cut

In the balanced chart the original metric satisfies |g-h|_h\leq C_* r^{-3} for r\geq R. On R\leq r\leq3R, Lemma 4.1 gives the same estimate for G. Beyond 3R, the new angular metric is exactly r^2\sigma and has no shift, while

Write M_*\geq\|F\|_\infty for a bound independent of R. Since 0\leq v\leq r, comparison of the radial coefficients gives

The angular coefficients already agree with those of h. Thus G\geq(1-2M_*r^{-3})h on the whole outer region, while g\leq(1+C_*r^{-3})h. Together with the collar comparison and the identity G=g on the compact interior, this gives a constant C_{\mathrm{met}} independent of R such that, with \varepsilon_R=C_{\mathrm{met}}R^{-3},

We increase R once so that 0\leq\varepsilon_R<1.

This also proves completeness, including the boundary. A G_R-Cauchy sequence is g-Cauchy by (6.3), and hence has a limit in \Omega. The smooth positive metrics induce the same local topology, so it converges in G_R as well. The manifold, its boundary, and its single end have not changed. Compact truncations remain compact after the balancing chart change because old and new radii are comparable sufficiently far out.

On the tangent plane of any smooth cut, the two relative eigenvalues of G_R with respect to g are at least 1-\varepsilon_R. Its area element is therefore at least (1-\varepsilon_R) times the original area element. The admissible cuts are the same geometric surfaces for the two metrics, with the same filled-side convention. Taking the infimum over all such surfaces gives

This includes cuts coinciding with the obstacle S. It uses neither existence of a minimizing cut nor any identification of its area with the area of S. Equations (6.2)– (6.4), together with the smooth DEC construction, prove Theorem 1.1.

6.3 The numerical input and the Bondi bound

The replacements now have the geometric properties needed for the numerical step. We use the dimension-three case of the companion [15, Definitions 1.1–1.2 and Theorem 1.3], stated here with our physical constraint normalization.

To identify this statement with the cited theorem, its densities satisfy

Thus \mu_{\mathrm N}=8\pi\mu and J_{\mathrm N}=8\pi J; the common positive factor preserves DEC and integrability. At dimension three its ADM factors are [2(n-1)\omega_{n-1}]^{-1}=1/(16\pi) and [(n-1)\omega_{n-1}]^{-1}=1/(8\pi), so the charges are unchanged. Its Definition 1.2 takes the full intrinsic boundary of the connected closed end-containing outer domain, including every component and all contact with the original boundary. This is precisely the cut class in (1.2).

There are two successive infinities in this proof. Radial infinity defines the ADM charges of one completed replacement. The replacement radius tends to infinity only after its numerical inequality has been proved. No convergence of comparison metrics or minimizing cuts is required, and the constant in O_R(r^{-3}) need not be uniform in the replacement radius.

Dependence on the numerical input. The proof uses the companion only through the displayed asymptotically flat inequality. More generally, this inequality is the sole numerical input needed for the same argument to give the Bondi bound for every weakly future trapped, possibly disconnected boundary under the other hypotheses of Corollary 1.2. In particular, it applies when S is connected and marginal, and the open exterior contains no compact smooth embedded two-sided separating union of closed future or past apparent horizons, allowing either sign on each component. These additional horizon conditions are not needed for the transfer. The Schwarzschild examples below satisfy them and attain the bound at every positive mass.

The inequality alone does not classify equality for the original data: the construction need not preserve an equality case at finite R. A classification, or an extension to weaker null-infinity expansions, would require a further argument.

7 Schwarzschild equality examples

The radial model gives equality examples once its inner endpoint is placed on the future Schwarzschild horizon. We verify their geometry at that endpoint, their CKS charges at infinity, and the area of every enclosing cut. The examples also exclude every closed interior apparent horizon of either time orientation.

Fix m>0 and let \Omega=[2m,\infty)\times\Sph. Choose a smooth function v(r) equal to -1 near 2m and to r for all sufficiently large r. Define

The function u is strictly positive at the boundary and throughout the exterior, including any interior zero of v. Thus these are smooth data up to S=\{2m\}\times\Sph. At infinity u\sim r, so the radial length \int^\infty u^{-1}\mathrm dr diverges logarithmically. Together with compactness of finite truncations, this proves completeness. Substitution in (5.7), or directly in (3.8), gives C=Q=Z=0: here F=m, \eta=\tau=0, and L=v'. In particular the data are vacuum.

7.1 Spacetime realization and the future boundary

To verify their Schwarzschild interpretation and sign, write h_m=1-2m/r. On r>2m the Schwarzschild metric is

Take a graph t_s=t_s(r) with

Its induced radial metric coefficient is u^{-2}. The future unit normal and outward radial unit tangent have (t_s,r)-components

The tangential sphere component of its second fundamental form is (v/r)r^2\sigma. The radial component of \overline\nabla_{e_r}n_{\mathrm{future}} is uv': the connection terms from \overline\Gamma^r_{t_st_s}=h_mh_m'/2 and \overline\Gamma^r_{rr}=-h_m'/(2h_m) cancel. Orthogonality to the future normal then gives k(e_r,e_r)=v', with mixed component zero. This proves exactly the convention used in (7.1).

In advanced time w=t_s+\int h_m^{-1}\mathrm dr, the spacetime metric is regular at the future horizon. Along the graph,

The latter expression is smooth at r=2m because v=-1 there and u=1. Thus the slice extends smoothly to the future horizon, not to the opposite horizon with the wrong expansion sign. At S,

For large r we have v=r, v'=1, and hence k=G. Moreover

The CKS expansion holds with m^r=2m and m^g=m^K=0. Consequently M=4m, E_B=m, P_B=0, and m_B=m>0.

7.2 No further horizon and the exact area infimum

Every interior coordinate sphere satisfies

because u^2-v^2=1-2m/r>0. This also rules out nonround apparent horizons. Indeed, let T be any compact closed embedded two-sided surface in the interior, and let x be a maximum of r|_T. The tangent plane at x is tangent to the coordinate sphere. Orient T locally by the radial outward normal. The surface Hessian identity and \Delta_T r(x)\leq0 give

Thus H_T(x)\geq2u/r>|\tr_{T_xT}k|. Reversing the orientation changes H_T only by sign, so neither \theta_+=0 nor \theta_-=0 can hold on T. The argument applies to every component of a separating union of closed surfaces, even if different components are assigned different horizon signs.

Finally let \pi:\Omega\to S be radial projection. Since r\geq2m and the metric is u^{-2}\mathrm dr^2+r^2\sigma, this map is length nonincreasing. A radial ray from S to infinity starts on the filled inner side, or on the cut itself, and eventually enters the retained outer domain. It therefore meets every enclosing cut \Gamma; at a point of boundary coincidence the initial point is already an intersection. Thus \pi|_\Gamma is surjective. The area formula, with absolute two-dimensional Jacobian at most one, implies

As S itself is an allowed cut, A_{\min,G}(S)=16\pi m^2>0. Equality in (1.7) follows. These examples prove the sharpness assertion in Corollary 1.2, including its connected marginal specialization.