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+BOUNDARY GRAPH DEFORMATIONS FOR THE SPACETIME PENROSE INEQUALITY
+
+A COMPLETE WALKTHROUGH
+
+
+HOW TO READ THIS DOCUMENT
+
+This is a plain-language but mathematically complete account of the
+paper "Boundary graph deformations for the spacetime Penrose
+inequality." The technical original is about two hundred pages of
+research mathematics. This version states every main theorem
+exactly, defines every term, and explains every major step at the
+level of a motivated high-school student who knows calculus. If you
+follow the whole thing you will know, precisely, what the paper
+proves, and why each piece is needed.
+
+If you want only the headline results, read Part 1 (the problem),
+Part 4 (the main theorems), and Part 11 (what we gained). If you
+want the full machinery, work through Parts 5 through 9 in order.
+Sign conventions, a glossary and a collected list of the key
+formulas are collected at the end.
+
+
+PART 1. THE SPACETIME PENROSE INEQUALITY
+
+
+1.1 A first glimpse
+
+In general relativity, gravity is the curvature of a four-dimensional
+spacetime. A black hole forms when matter collapses so hard that
+light cannot escape a certain region. The boundary of that region,
+the surface beyond which light cannot return, is called the event
+horizon. A famous formula due to Stephen Hawking says that, for a
+stationary black hole, the total energy of the black hole is
+determined by the area of its horizon:
+
+ E = sqrt( A / (16 pi) ).
+
+Here E is the total energy, A is the area of the horizon, and the
+formula is written in units where Newton's constant and the speed of
+light are both 1.
+
+The Penrose inequality generalises this to spacetimes that are not
+stationary. It says that the area of a horizon always bounds the
+total mass from below: whatever happens in a collapse, the total
+mass in the spacetime is at least the mass that would be needed to
+make a stationary black hole with that horizon area. Equivalently,
+gravity can never be "undone": once you have made a horizon of
+area A, the total mass of the resulting spacetime can never be less
+than sqrt(A/(16 pi)).
+
+Penrose conjectured the inequality in 1973. Proving it is one of the
+great open problems of mathematical general relativity. It has been
+proved in two very important specialised cases: for time-symmetric
+or "Riemannian" initial data, and for the four-dimensional
+maximal-vacuum case with torus symmetry. The paper you are reading
+gives a new route, one that works directly in three and four
+spatial dimensions, keeps a genuine inner boundary at all times,
+and reduces the general problem to the Riemannian one via a
+carefully chosen deformation.
+
+
+1.2 The exact statement
+
+To state the Penrose inequality we need four ingredients:
+
+ (I) An initial data set: a triple (M, g, K), where M is a
+ three- or four-dimensional manifold (the "spatial slice"),
+ g is a Riemannian metric on M (the induced metric), and K
+ is a symmetric two-tensor (the extrinsic curvature, or
+ second fundamental form).
+
+ (II) A trapped surface: a closed surface (one dimension lower
+ than M) whose future outgoing null expansion is nowhere
+ positive.
+
+ (III) An asymptotically flat end: a "spatial infinity" of M
+ where the metric approaches the Euclidean metric, and the
+ curvature and the tensor K vanish at infinity.
+
+ (IV) Finite ADM energy and momentum: numbers E and P that can
+ be computed from the asymptotic behaviour of g and K at
+ infinity, and that describe the total energy and momentum
+ of the slice.
+
+With these in hand, the spacetime Penrose inequality states that,
+for suitable data obeying an energy condition, the total energy E
+satisfies
+
+ sqrt( E^2 - |P|^2 ) >= sqrt( A / (16 pi) ) in 3 spatial dims
+
+where A is the total area of the outermost trapped surface (i.e.,
+the horizon), and sqrt(E^2 - |P|^2) is the rest mass - the mass that
+an observer at spatial infinity would measure if they were at rest
+relative to the black hole.
+
+In four spatial dimensions (that is, five-dimensional spacetime),
+the analogue is
+
+ E >= (1/2) * ( A / (2 pi^2) )^(2/3).
+
+The constants 16 pi and 2 pi^2 = omega_3 are the areas of the unit
+circle (1-dimensional) and the unit sphere S^3 (3-dimensional).
+There is a similar but not identical formula in each dimension;
+the pattern is that the area of the horizon and the total mass sit
+in a fixed power relation, with the constants depending on the
+dimension.
+
+
+1.3 Why it is hard
+
+The difficulty of the Penrose inequality comes from three facts
+stacked on top of one another.
+
+First: the horizon is the boundary of a trapped region, and trapped
+regions are defined by a nonlinear inequality on a certain first-
+order quantity called the expansion. There is no simple equation
+for the horizon; it obeys an inequality, not an equality, and it can
+sit anywhere inside the region where the inequality is tight.
+
+Second: the ADM mass E is defined only at spatial infinity, while
+the horizon sits somewhere in the interior. There is no obvious
+bridge between them: the mass is a boundary condition at infinity,
+and the horizon is a global minimum-type object in the middle of
+the slice.
+
+Third: the constraint equations of general relativity are nonlinear
+and only partially elliptic. Only four of the ten Einstein equations
+are constraints; the other six are evolution equations. Working
+directly with the constraints is subtle because the data set has to
+satisfy a coupled system.
+
+The four-dimensional Riemannian case - K = 0 - is the classical
+hard problem, and it was solved by Huisken and Ilmanen with inverse
+mean curvature flow (2001) and independently by Bray with the
+conformal flow (2001). The general spacetime case remains open, with
+partial results under symmetry (Khuri-Kunduri 2025 for
+cohomogeneity-one SU(n+1)-invariant data, Bray-Khuri 2010 for the
+spherical case), and under a "suboptimal-constant" relaxation
+(Allen-Bryden-Kazaras-Khuri 2025).
+
+
+1.4 What this paper does, in five lines
+
+The authors prove a boundary version of the Penrose inequality in
+three and four spatial dimensions. They start with an initial-data
+set exterior whose inner boundary is strictly trapped. They replace
+it, via a highly controlled conformal-plus-graph deformation, by a
+new metric that (a) has nonnegative scalar curvature, (b) has
+strictly negative mean curvature at its inner boundary, (c) has area
+bounded below by the original area on every enclosing cut, and (d)
+has ADM energy no larger than the original, up to an error that
+tends to zero. Once these four properties hold, the Riemannian
+Penrose inequality (Huisken-Ilmanen, Bray, Bray-Lee) applies to the
+new metric, and the mass-area inequality passes back to the
+original data by letting the error tend to zero.
+
+The word "boundary" is important: the new metric keeps a genuine
+inner boundary. It does not close it up. The construction lives
+entirely on an exterior, and the boundary is the vehicle by which
+the horizon is turned into a strictly mean-concave object that the
+Riemannian theorem can handle.
+
+They also give a second, parallel construction for
+four-dimensional maximal-vacuum data. The "maximal" hypothesis means
+the trace of K vanishes everywhere. That extra hypothesis does not
+trivialise the problem - K is still a genuine tensor - but it does
+make a certain differential equation for the graph height f become
+homogeneous. The graph height can then decay exponentially, even
+though the tensor K reaches all the way to infinity. This is the
+four-dimensional torus-symmetric maximal-vacuum case, and it is the
+sharper of the two results.
+
+
+PART 2. PICTURE, DEFINITIONS, AND NOTATION
+
+
+2.1 Spacetime, spatial slice, initial data
+
+A spacetime in general relativity is a four-dimensional manifold
+with a Lorentzian metric (a metric of signature (-,+,+,+), or
+(+,-,-,-) depending on convention). It describes every event in
+space and time. A "spatial slice" is a codimension-one surface on
+which the Lorentzian metric restricts to positive-definite, so the
+slice has its own Riemannian geometry.
+
+An initial data set is what you need to specify to run Einstein's
+equations forward in time from a given moment. It consists of two
+objects:
+
+ g = a Riemannian metric on the spatial slice M, encoding the
+ induced distances and angles that a comoving observer
+ measures;
+
+ K = a symmetric two-tensor on M, called the extrinsic
+ curvature, or second fundamental form. It records how fast
+ the slice is bending into the surrounding spacetime.
+
+The paper consistently works with n spatial dimensions with n = 3
+or n = 4, so the slice M is n-dimensional and the spacetime is
+(n+1)-dimensional. The physically relevant case is n = 3; the n = 4
+case is the next natural generalisation and is what the
+four-dimensional torus-symmetric construction addresses.
+
+For every vector field X on M we have a covector K(X, .) obtained by
+contracting. Two derived quantities appear constantly:
+
+ tau = tr_g K = g^{ij} K_{ij}
+ the trace of K, also called the mean curvature of the
+ slice.
+
+ (tr K) g - K
+ the "momentum density", which enters the constraint
+ equations below.
+
+When the trace vanishes, tau = 0, the data is called maximal. This
+is a common simplification in relativity: it means the slice is
+"expanding everywhere equally" or equivalently that the expansion
+of spacetime volume in each timelike direction is momentarily
+balanced.
+
+
+2.2 The constraint equations
+
+Not every pair (g, K) is a valid initial data set. The ten Einstein
+equations split into four constraint equations (satisfied on the
+slice) and six evolution equations (satisfied off the slice). The
+constraints are:
+
+ Hamiltonian constraint: n-dimensional source density
+ mu = (something) * R_g + (something) * ((tr K)^2 - |K|^2)
+
+ Momentum constraints: n-dimensional vector source
+ J_i = div_g (K - (tr K) g)_i
+
+The constants in front of these expressions differ between the
+3-dimensional and 4-dimensional parts of the paper:
+
+ In 3 spatial dimensions:
+
+ 16 pi mu = R_g + (tr K)^2 - |K|_g^2
+ 8 pi J = div_g ( K - (tr K) g )
+
+ In 4 spatial dimensions (geometrised units 8 pi = 1):
+
+ mu = (1/2) * ( R_g + (tr K)^2 - |K|^2 )
+ J_i = div_g ( K - (tr K) g )_i
+
+Where R_g is the scalar curvature of g, |X|_g is the g-norm, and
+div_g is the g-divergence. The two definitions differ only by
+rescalings of mu and J; the geometric content is the same.
+
+The import of the constraint equations is this: the physical
+quantities mu and J are integrals of motion, so they cannot be
+chosen freely. Any pair (g, K) we invent must satisfy the
+constraints, which couples the metric and the tensor in a highly
+nonlinear way. This is one of the reasons the Penrose inequality is
+hard: a naive "just change the metric to make it nicer" approach
+runs into the constraint system.
+
+
+2.3 Energy conditions
+
+For the Penrose inequality to have any hope of being true, the data
+must obey some energy condition. Otherwise you can have negative
+mass, which would immediately violate any mass-lower-bound.
+
+The condition used here is the dominant energy condition (DEC):
+
+ mu >= |J|_g everywhere on M.
+
+Equivalently: the mass-energy density at each point is at least as
+large as the magnitude of the momentum density. It is the minimal
+assumption ensuring that "energy is nonnegative" and that energy
+cannot flow faster than light.
+
+The Riemannian (or time-symmetric) case has K = 0, so
+
+ J = 0 and mu = (1/16pi) * R_g (3D)
+ = (1/2) * R_g (4D).
+
+The DEC then becomes R_g >= 0: scalar curvature everywhere
+nonnegative. That is exactly the hypothesis of the classical
+Riemannian Penrose theorem.
+
+
+2.4 Asymptotic flatness and ADM mass
+
+The Penrose inequality lives in an asymptotically flat exterior: a
+spacetime that looks like flat Minkowski space very far from the
+horizon. There is one distinguished end, a "spatial infinity" where
+the geometry settles down. In three and four dimensions we say
+that the data is asymptotically flat of order q > 0 if, in suitable
+coordinates x = (x^1, ..., x^n) near infinity, with r = |x|, we have
+
+ g_{ij} = delta_{ij} + O_j( r^{-q} ),
+ K_{ij} = O_{j-1}( r^{-1 - q} ),
+
+for some q > 1/2 (3D) or q > 1 (4D) - the exact threshold depends
+on which theorem is being applied. Here O_j(r^{-a}) means "bounded
+by C r^{-a} together with all its coordinate derivatives up to
+order j bounded by C r^{-a - k} for the k-th derivative."
+
+Two derived quantities are computable purely from this asymptotic
+behaviour:
+
+ ADM energy E = a normalisation times
+ lim_{r -> infinity} integral_{S_r}
+ ( partial_j g_{ij} - partial_i g_{jj} )
+ * n^i dA
+
+ ADM momentum P_i = a normalisation times
+ lim_{r -> infinity} integral_{S_r}
+ ( K_{ij} - (tr_g K) g_{ij} )
+ * n^j dA
+
+where S_r is a coordinate sphere of radius r, n is the outward unit
+normal, and dA is the Euclidean area element. The two normalisations
+are:
+
+ In 3 dimensions: E uses factor 1/(16 pi),
+ P uses factor 1/(8 pi).
+
+ In 4 dimensions: E uses factor 1/(6 omega_3),
+ P uses factor 1/(3 omega_3),
+
+with omega_3 = |S^3| = 2 pi^2.
+
+For both to be finite, the integrands must decay fast enough; in
+particular, g - delta must fall off at least like r^{-q} with q
+above the stated threshold. When P = 0 the ADM energy E coincides
+with the ADM rest mass sqrt(E^2 - |P|^2), and in the Riemannian
+case with K = 0 the momentum vanishes automatically.
+
+The ADM mass is a key piece of the puzzle: it is the "total energy
+at infinity" of the slice. It only depends on the behaviour at
+spatial infinity, not on the interior. The Penrose inequality is a
+bound relating this boundary quantity to another boundary quantity,
+the horizon area A, with the interior providing the obstruction.
+
+
+2.5 Trapped surfaces, MOTS, and stability
+
+A trapped surface is a surface Sigma one dimension lower than M
+(a closed curve in 2D, a closed surface in 3D space, a closed
+3-manifold in 4D space) whose outgoing light rays all converge no
+matter which way they aim.
+
+To make this precise, pick a unit normal nu to Sigma and let
+
+ H_Sigma = div_Sigma nu
+
+be the mean curvature of Sigma with that normal. For a closed
+surface sitting inside an initial data set with extrinsic curvature
+K, the two null expansions - the rates at which futuristic and
+past-directed outgoing light sheets are expanding - are
+
+ theta_+ = H_Sigma + tr_Sigma K
+ theta_- = H_Sigma - tr_Sigma K
+
+where tr_Sigma K = K(nu, nu) - K on the tangent plane and the sign
+depends on which null direction is called outgoing.
+
+The paper uses this convention:
+
+ Sigma is future trapped if theta_+ < 0 everywhere.
+ Sigma is marginally outer trapped if theta_+ = 0 everywhere.
+ Sigma is weakly future outer trapped if theta_+ <= 0 everywhere.
+
+A MOTS - "marginally outer trapped surface" - is the key object.
+It is a surface where the outgoing null expansion is exactly zero:
+the boundary of a trapped region, or a horizon under equilibrium.
+
+Two further properties are used constantly.
+
+ Stability. A MOTS is stable if its "stability operator"
+ L[v] = -Delta_Sigma v + 2 K(nu, grad v) + (stuff) v
+ has nonnegative principal eigenvalue. This is a variation of the
+ standard statement that a minimal surface (H = 0) is stable if
+ the second variation of area is nonnegative. In the MOTS
+ setting the stability operator is a scalar elliptic operator
+ with a nice principal eigenfunction; when the eigenvalue is
+ positive the surface is strictly stable, and when it is zero
+ the surface has a "zero mode" (a graph direction along which
+ the expansion remains constant to first order).
+
+ Maximality in the trapped-region sense. A MOTS is called the
+ outermost or total MOTS if every other surface of nonpositive
+ expansion on the interior side is contained in the region it
+ bounds. The classical existence theorem (Andersson-Metzger 2009,
+ Andersson-Eichmair-Metzger 2011) says: for a compact region with
+ a smooth outer boundary of positive expansion and an inner
+ boundary (possibly absent) of negative expansion, the union of
+ all smooth weakly trapped subregions has a smooth outermost
+ stable MOTS frontier.
+
+The paper repeatedly uses the existence and stability from
+Andersson-Metzger and Andersson-Eichmair-Metzger, and the
+principal-eigenfunction facts from Andersson-Mars-Simon (2008).
+
+One further construction: a MOTS sometimes has a zero principal
+eigenvalue. In that case the obvious "outward graph" does not
+change the expansion to first order. The paper proves, using
+implicit-function-theorem arguments on a suitable map, that
+nevertheless one can find a smooth foliation of an outward collar
+whose leaves have strictly larger expansion than the surface itself.
+This uses maximality of the region - it is not true of a stable
+surface in isolation. Without this collar lemma the boundary sign
+in the deformation would be unreliable, since the mean curvature of
+the new metric would risk equalling zero.
+
+
+2.6 Enclosing cuts and enclosing area
+
+A crucial definition: what counts as "the horizon" for the Penrose
+inequality, and how do we measure its area? The paper allows a very
+general notion so that the answer is robust.
+
+An exterior Omega is a connected orientable n-manifold with a
+nonempty compact smooth boundary, complete as a metric space
+including that boundary, with one asymptotically flat coordinate
+end. Let Omega_0 be any such exterior.
+
+A smooth enclosing cut of Omega_0 is the entire intrinsic boundary
+of a connected smooth end-containing domain D that is closed in the
+closure of Omega_0, has its intrinsic interior in the interior of
+Omega_0, and separates the entire inner boundary of Omega_0 from
+spatial infinity. The normal to the cut points into D, toward the
+end.
+
+The paper allows two things that a stricter definition would not:
+
+ The cut may coincide with pieces of the original boundary.
+
+ The cut may have several disconnected components.
+
+These allowances are not decorative. First variations of a
+"minimum area" surface may fail to sit strictly outside the
+obstacle, and disconnected cuts arise in the trapped-region
+theorems. If we insisted on single connected components strictly
+outside the obstacle, we would get a smaller class of cuts and a
+weaker area lower bound. By exploiting the full class we get the
+strongest result.
+
+The enclosing area of the original inner boundary S_0 is
+
+ a_g(S_0) = inf { |Gamma|_g : Gamma is an enclosing cut of S_0 }
+
+where |Gamma|_g is its n-1-dimensional area in the metric g.
+Depending on context the paper writes this as A_* or calligraphic
+A_g(S_0). The infimum is over both smooth cuts and, by a separate
+argument, over the closures of full-perimeter competitors; the
+two agree.
+
+Two remarks.
+
+ Even for a smooth metric the infimum might not be attained. It
+ is an infimum, not a minimum.
+
+ The generalised "full perimeter" class where cuts are only
+ rectifiable sets, not smooth surfaces, has the same infimum
+ under mild approximation arguments. This is important for the
+ compactness theorems; the paper sometimes uses the full class,
+ sometimes the smooth class, but verifies the same infimum each
+ time.
+
+
+2.7 Sign conventions
+
+Small convention differences plague research papers. The paper is
+internally consistent but not uniform between the 3D and 4D parts.
+The key points:
+
+ Mean curvature. H_Sigma = div_Sigma nu. For the boundary of a
+ compact region, nu points OUT of that region. On the inner
+ boundary of an exterior Omega, nu points INTO Omega (toward
+ spatial infinity). "Negative mean curvature on B" therefore
+ means the surface B is mean-CONCAVE from the viewpoint of
+ Omega's interior.
+
+ Future expansion. theta_+ = H + tr_Sigma K. A MOTS has
+ theta_+ = 0. A future trapped surface has theta_+ < 0.
+
+ Scalar curvature. R_g is the scalar curvature of the metric g.
+ In the Riemannian Penrose inequality we need R_g >= 0.
+
+ Einstein constant. Mathematically one usually sets 8 pi = 1.
+ The 3D part of the paper sometimes writes mu, J with explicit
+ 16 pi and 8 pi; the 4D part writes mu, J with no explicit
+ constant. When reading, always check the definitions.
+
+ Space dimension. Throughout, n denotes the spatial dimension
+ (n = 3 or n = 4), so the boundary surface has dimension n - 1
+ (surface area element dA) and the spatial volume element is dV.
+
+
+PART 3. THE STRATEGY
+
+
+3.1 Two classical proofs of the Riemannian case, and one gap
+
+When K = 0 the Penrose inequality reads E >= sqrt(A/(16 pi)) in 3D,
+E >= (1/2)(A/(2 pi^2))^(2/3) in 4D. Two complete proofs exist in 3D
+for arbitrary horizon area A (not just connected horizons):
+
+ Huisken-Ilmanen (2001), via inverse mean curvature flow.
+
+ Bray (2001), via a conformal flow of the metric.
+
+Both are deep and technically hard. In 4D, Bray-Lee (2009) proved the
+same inequality via the conformal flow approach; that proof works in
+dimensions up to seven.
+
+There is no analogous theorem in the general (K != 0) case. The
+reason is that the natural geometric flows used in the Riemannian
+proofs - surfaces expanding at a prescribed speed, or metrics
+flowing through a conformal family - do not behave the same way
+when K is nonzero. New tools are required.
+
+Bray and Khuri (2010, 2011) introduced a method called the "warped
+graph" approach that generalises Jang's equation to spacetimes. It
+expresses the scalar curvature of a certain deformed metric in terms
+of original constraint densities, a nonnegative quadratic term, and
+a divergence. It suggested that if one can find a solution to a
+certain coupled system - the "generalized Jang equation" - then the
+spacetime Penrose inequality reduces to the Riemannian one.
+
+But: the coupled system that appeared in that approach was known
+only to have solutions in very special cases. Han-Khuri (2013)
+proved existence and boundary blow-up for prescribed warping
+factors. Han-Khuri (2018) coupled this to conformal flow.
+Jaracz (2023) proved nonexistence of radial solutions to a specific
+version (with zero divergence source), showing that the
+straightforward approach breaks down.
+
+The present paper's contribution is to make a system with slightly
+different equations - a nonzero divergence source, a trace
+depending strictly on graph height, and a hard penalty near the
+conformal lower bound - whose solutions do exist in general,
+uniformly across a large class of data. The result is
+a complete proof of a boundary version of the spacetime Penrose
+inequality, in both 3D and 4D.
+
+
+3.2 The graph deformation idea
+
+Instead of solving a coupled system directly on the original
+manifold, deform the metric by writing it as a graph over the base
+manifold. In the simplest version the new metric is
+
+ ghat = e^{2t} * ( g + l(t)^2 df (x) df )
+
+where f : M -> R is the "graph height" and t : M -> R is a
+"conformal factor". Two new scalar functions f and t are the
+unknowns. The graph term is a rank-one perturbation: it stretches
+distances horizontally but leaves vertical directions alone.
+
+The point of this form is that the deformation of scalar curvature
+for the metric ghat is exactly computable. It is a sum of four
+terms: a term involving the original constraint densities mu and J
+evaluated at the new metric's "velocity" w, a nonnegative quadratic
+term T that we will see is bounded below by the squared Hessian
+of f, a term involving the trace equation, and a divergence term.
+The exact identity is proved in PART 5.
+
+Given this identity, the construction splits into two coupled scalar
+equations for f and t:
+
+ Trace equation. tr_{A_chi}(K + H^f) = F(x, f, ...)
+ Divergence equation. div V = u Xi
+ where V is a vector field built from grad t and K(w, .), A_chi
+ is a positive-definite symmetric tensor that deforms the trace,
+ H^f is the Hessian of f renormalized, and the right sides F, Xi
+ are prescribed functions.
+
+If both equations hold, the scalar curvature of ghat is exactly the
+quantity on the right of the identity. The trick is to choose F
+and Xi so that this quantity is nonnegative (which gives R_ghat
+>= 0), and so that the boundary conditions of the problem match
+those needed to apply the Riemannian Penrose inequality.
+
+The construction then becomes a nonlinear elliptic system. The
+existence is proved via a continuation/degree argument: deform the
+scalar equations continuously from a trivially solvable linear
+problem to the desired nonlinear problem, keeping track of whether
+solutions can escape to the boundary of a chosen admissible region.
+If they cannot, the number of solutions is unchanged along the
+continuation, so the desired nonlinear problem inherits a solution
+from the trivial endpoint.
+
+
+3.3 Why an inner boundary is useful
+
+The usual plan for the Penrose inequality is: "collapse" the
+horizon, replacing the data inside it by something simpler. Close
+up the inner boundary into a smooth cap, work on a manifold without
+boundary, apply known results. This is the "filling" approach and it
+is what Bray, Huisken-Ilmanen, and many others use in the
+Riemannian setting.
+
+The trouble with filling: collapsing a horizon is not physically
+meaningful, and analytically it requires solving constraints on a
+perturbed manifold where you have no control over the new geometry.
+Resulting metric changes are large, and the mass error can be big.
+
+The paper takes a different approach. It keeps a genuine inner
+boundary B throughout. The boundary is chosen so that:
+
+ (a) B is a smooth compact hypersurface sitting strictly inside
+ the original manifold, separating the horizon S_0 from
+ spatial infinity.
+
+ (b) In the NEW metric ghat, the mean curvature of B is strictly
+ negative (with normal pointing into the exterior Omega).
+
+ (c) The area of every enclosing cut in the new metric is at
+ least a known fraction of the original enclosing area.
+
+Property (c) comes from the conformal lower bound ghat >=
+e^{-4 eps} g (in 3D) or ghat >= e^{-2 eps} g (in 4D). Property (b)
+is exactly what is needed for the Riemannian Penrose argument: it
+forces a minimizing minimal enclosure to detach from B, giving a
+clean minimal surface to which Huisken-Ilmanen, Bray, or Bray-Lee
+can be applied. Property (a) is a choice about the geometry of Omega
+that turns out to be possible using the classical trapped-region
+theorems.
+
+The boundary is not a computational convenience. It is the vehicle
+that bridges the spacetime constraints (which hold on Omega) to the
+Riemannian comparison (which needs a detached minimal surface). By
+keeping B alive throughout, the paper gets both a controlled mass
+error and a controlled area lower bound, at the price of an extra
+elliptic boundary condition (which is what the "well-posed but not
+overdetermined" system must satisfy).
+
+
+3.4 Two routes: compact K and maximal
+
+The boundary construction needs to know something about how K
+behaves on the asymptotically flat end. Two cases arise.
+
+Route 1: compactly supported K.
+
+If K vanishes outside a large compact set, then at spatial infinity
+the data is exactly (or nearly exactly) Riemannian: K = 0. The
+trace equation near infinity becomes a homogenous elliptic equation
+for the graph height f, with positive zeroth-order coefficient. f
+then decays exponentially (or nearly so) as you move toward
+infinity, and the whole deformation is under control. This is the
+"compact-support" route, and it is the one used in the 3D boundary
+construction.
+
+Route 2: maximal data, K reaching infinity.
+
+If K is maximal (tr_g K = 0) but does not vanish at infinity, we
+must work harder. The trace equation at infinity now reads
+
+ tr_{A_chi}(K + H^f) = eta_N f
+
+and the tensor term contributes to the left side, so it is no
+longer homogenous. But there is a saving grace: because tr K = 0,
+the extra contribution is proportional to a pure derivative
+K(grad f, grad f). That is, it acts like a first-order drift term
+in the equation for f, not like an independent forcing. The equation
+is still a linear equation in f (for fixed Z) with bounded drift
+coefficients, and f still decays exponentially as long as the
+metrical weights are chosen so that the drift is integrable.
+
+The paper handles this in a special section ("the maximal method").
+It constructs the analogous deformation for a four-dimensional
+maximal-vacuum exterior that is topologically a ball exterior with
+a torus symmetry. The extra hypothesis ensures positivity of a
+certain conformal factor that strictifies the dominant-energy
+condition, and the analysis of the coupled system is similar modulo
+the drift. The final inequality is the four-dimensional Riemannian
+Penrose inequality, applied to the deformed metric.
+
+For the general (non-maximal, non-compact-support) case, the paper
+uses the three-dimensional or four-dimensional boundary method
+after first passing to a "prepared" data via a strictification
+step. The strictification (Proposition 7.13 in the companion, and
+proved independently in the four-dimensional case here) reduces
+weakly decaying data to compactly supported data with strict
+dominant-energy slack, at the cost of a controlled mass error that
+also tends to zero. So the compact-support boundary theorem does
+all the work; the strictification bridges to the general setting.
+
+
+PART 4. THE MAIN THEOREMS
+
+
+4.1 Theorem A: the three-dimensional boundary deformation
+
+The first main result is a metric deformation in three spatial
+dimensions.
+
+SETUP. Let (Omega_0, g, K) be a "prepared three-dimensional
+exterior" in the following sense:
+
+ - Omega_0 is a three-manifold, connected, orientable, with
+ nonempty compact smooth boundary S_0, complete up to that
+ boundary, and with one asymptotically flat end. Its complement
+ is compact.
+
+ - K is compactly supported (vanishes outside a large compact
+ set).
+
+ - The metric g has a very particular decay at infinity:
+ g - delta = O_j(r^{-1}) for every fixed j, and the scalar
+ curvature R_g = O(r^{-3-delta}) for some 0 < delta < 1.
+
+ - The constraint densities mu, J defined from (g, K) satisfy:
+ mu >= |J|, i.e., the DEC, and moreover the strict margin
+ 8 pi (mu - |J|) >= c * rho^{-3-delta} for some c > 0 and
+ a smooth radius extension rho. (The 3D paper uses the
+ physical normalisation with 8 pi.)
+
+ - The ADM energy E of (g, K) is finite and positive, and the
+ ADM momentum P is zero.
+
+ - The inner boundary S_0 is a strictly future trapped surface:
+ theta_+(S_0) < 0 for the normal pointing into Omega_0.
+
+ - The enclosing area of S_0 is A_* = a_g(S_0) > 0, using the
+ definition from Section 2.6.
+
+CONCLUSION. For every such prepared exterior, there exists:
+
+ - A connected one-ended subexterior Omega inside Omega_0, with
+ smooth compact boundary B, separating S_0 from spatial
+ infinity;
+
+ - For each sufficiently small eps > 0 and every sufficiently
+ large integer N, a smooth Riemannian metric ghat_{eps,N} on
+ Omega (complete with its boundary included) satisfying all
+ the following:
+
+ (1) R_{ghat_{eps,N}} >= 0 everywhere.
+
+ (2) H_{B}(ghat_{eps,N}) < 0 (mean curvature, normal into
+ Omega).
+
+ (3) ghat_{eps,N} >= e^{-4 eps} * g everywhere.
+
+ (4) E_{ghat_{eps,N}} <= E + eta_{eps,N}, where the error
+ satisfies 0 <= eta_{eps,N} <= C(1+N)^a *
+ (e^{-eps N} + e^{-eps N/2}) for some constants C, a
+ depending only on the fixed data and eps. In particular,
+ eta_{eps,N} -> 0 as N -> infinity at fixed eps.
+
+ (5) The new end has ghat_{eps,N} - delta = O_2(r^{-1}) and
+ scalar decay R_ghat = O(r^{-3-delta') for some delta' > 0.
+
+ (6) Every full enclosing cut T in Omega satisfies
+ |T|_{ghat_{eps,N}} >= e^{-4 eps} * A_*.
+
+INTERPRETATION. The original data (g, K) has been replaced by a
+new metric ghat on a subexterior Omega. The new metric (i) has
+nonnegative scalar curvature, so the Riemannian Penrose theorem can
+be applied to it, (ii) has a mean-concave inner boundary B so that
+the minimal enclosure detaches from B, (iii) is area-nondecreasing
+up to a small factor e^{-4 eps}, and (iv) has ADM energy no larger
+than the original up to a polynomial-times-exponential error. All
+these are exactly the pieces needed to pass from Riemannian Penrose
+on the new metric to a bound on the original data.
+
+The proof is where all the technical machinery comes in. Sections
+7.1-7.10 below walk through it.
+
+
+4.2 Theorem B: the four-dimensional boundary deformation
+
+The four-dimensional version of Theorem A is structurally identical
+but has two differences. First, the exponent in the conformal lower
+bound halves: ghat >= e^{-2 eps} g, not e^{-4 eps}. Second, the
+decay hypotheses and constants change to match four-dimensional
+geometry.
+
+SETUP. Let (M, g, K) be a four-dimensional initial-data exterior:
+M connected orientable, complete up to a nonempty compact smooth
+boundary S, with exactly one Euclidean coordinate end. The data
+satisfies:
+
+ - K is compactly supported.
+
+ - There is a smooth positive extension r of the coordinate
+ radius and constants c_0 > 0, 0 < delta_1 < 1 such that
+ the dominant energy condition holds with strict margin
+ mu - |J|_g >= c_0 * r^{-4 - delta_1} on M, and
+ strict trapping of the inner boundary
+ H + tr_S K < 0 (normal into M).
+
+ - On the end:
+ g - delta = O_j(r^{-2}) for every fixed j >= 0,
+ R_g = O(r^{-4 - delta_1}),
+ and the ADM energy is finite.
+
+CONCLUSION. There exists:
+
+ - A smooth closed connected one-ended exterior Omega inside M,
+ with nonempty compact smooth boundary B (which is distinct
+ from S, i.e., B is a new "cut" strictly inside M that
+ separates S from the end);
+
+ - For every 0 < eps < 1 and every sufficiently large integer N,
+ a smooth metric ghat_{eps,N} on Omega (complete with B
+ included) satisfying:
+
+ (1) R_ghat_{eps,N} >= 0 everywhere.
+
+ (2) H_ghat_{eps,N}(B) < 0 for the normal into Omega.
+
+ (3) ghat_{eps,N} >= e^{-2 eps} g.
+
+ (4) E_ghat_{eps,N} <= E_g + (Pi_N / (3 omega_3)) *
+ (e^{-eps N} + e^{-eps N/2}),
+ where omega_3 = 2 pi^2 is the volume of the unit
+ S^3, and Pi_N = C(1+N)^{a_pol} is polynomial in N.
+
+ (5) Decay: g - delta = O_2(r^{-2}),
+ R = O(r^{-4 - delta_1}) on the new exterior.
+
+ (6) Every full enclosing cut of B in Omega has ghat-area
+ at least e^{-3 eps} * A_*. Here A_* is the enclosing
+ area of the original inner boundary S, computed in the
+ original metric g.
+
+The exponent 3 = 4 - 1 in item (6) matches: the area of a
+codimension-1 surface in a four-dimensional manifold scales as the
+third power of the conformal factor. In Theorem A the corresponding
+power is 4 = ... well actually the paper's Theorem A uses exponent
+-4 appearing inside e^{-4 eps}, while the area comparison (item 6)
+also uses e^{-4 eps}; those come from the 3D metric's conformal
+factor e^{4t}. In 4D the conformal factor is e^{2t}, and the area
+gain is e^{-3 eps}, reflecting 3-dimensional surface area in a
+4-dimensional manifold.
+
+This is one of those places where keeping track of the dimension is
+important. The constants differ but the STRUCTURE - a metric with
+nonnegative curvature, mean-concave inner boundary, area-lower
+bound, and energy-upper bound - is exactly the same. The rest of
+this document mostly speaks in 3D terms but the 4D case runs in
+parallel with these adjusted constants.
+
+
+4.3 Theorem C: the three-dimensional Penrose inequality
+
+Combining Theorem A with the Riemannian Penrose inequality produces
+a proof of the three-dimensional spacetime Penrose inequality,
+under the hypotheses of the prepared three-dimensional exterior.
+
+Specifically: let (Omega_0, g, K) be a prepared three-dimensional
+exterior as in Theorem A. Then the reduced (rest-mass) energy
+satisfies
+
+ sqrt( E^2 - |P|^2 ) >= sqrt( A_* / (16 pi) ),
+
+where A_* is the enclosing area of the original inner boundary
+S_0, computed in the metric g.
+
+This is stated as "Theorem 1.2" of the introduction. The key point:
+no extension of the data across the original boundary S_0 is
+required. Only the data on the exterior Omega_0 matters; the
+horizon is treated as an inequality, not as a surface you must cross.
+
+Proof sketch:
+
+ 1. Apply Theorem A. Get Omega, ghat = ghat_{eps,N} satisfying
+ R >= 0, mean-curvature B < 0, area lower bound, and energy
+ upper bound.
+
+ 2. Apply the smooth enclosure theorem (a companion result,
+ independent of everything else here, that the enclosure of a
+ mean-concave inner boundary in an asymptotically flat manifold
+ is a smooth minimal outer-area-minimizing surface). This gives
+ a compact minimal surface Gamma_N in Omega, detached from B,
+ enclosing B.
+
+ 3. Apply the Riemannian Penrose inequality of Bray (3D, full
+ perimeter, allowed to be disconnected) to the exterior of
+ Gamma_N with metric ghat. This gives
+
+ E_ghat_N >= sqrt( |Gamma_N|_ghat / (16 pi) ).
+
+ 4. Combine with the metric comparison
+ |Gamma_N|_ghat >= e^{-4 eps} * |Gamma_N|_g >= e^{-4 eps} * A_*.
+ This gives
+
+ E_ghat_N >= e^{-2 eps} sqrt( A_* / (16 pi) ).
+
+ 5. Use the ADM energy comparison E_ghat_N <= E + eta_{eps,N}
+ from Theorem A, and let N -> infinity at fixed eps, then
+ eps -> 0.
+
+The result follows. The key steps are the enclosure existence
+(step 2) and the metric comparison (step 4); both are rigorous and
+parsimonious about assumptions.
+
+For the general weakly-decaying data (not just compact-support K),
+we replace the initial data by a "prepared" one via the weak
+preparation theorem of the numerical companion paper. That
+preparation supplies compact tensor support, strict trapping, and
+the weighted margin, with energies converging to the original
+rest-mass and enclosing areas converging to the original infimum. We
+apply the reduced bound Theorem A to each prepared approximation
+and pass to the limit. This completes the proof of Theorem 1.2 in
+the introduction.
+
+
+4.4 Theorem D: the four-dimensional Riemannian bound
+
+Exactly analogous:
+
+ Under the hypotheses of Theorem B (four-dimensional prepared
+ exterior), the ADM energy E_g satisfies
+ E_g >= (1/2) * ( A_* / omega_3 )^(2/3),
+ with A_* the enclosing area of the original inner boundary S_0
+ in the metric g.
+
+This is stated as "Corollary 1.4" of the introduction and is the
+direct consequence of Theorem B plus the four-dimensional
+Riemannian Penrose inequality (Bray-Lee, 2009). The proof runs the
+same as the 3D case, with the four-dimensional rules:
+
+ New metric ghat has R_ghat >= 0 and ghat >= e^{-2 eps} g.
+
+ Enclosure by minimal surface, |Gamma|_ghat >= e^{-3 eps} A_*.
+
+ Bray-Lee: E_ghat >= (1/2) * (|Gamma|_ghat / omega_3)^(2/3).
+
+ Combine, then limit N -> infinity, then eps -> 0.
+
+The result is the four-dimensional spacetime Penrose inequality in
+the compact-support case and, via strictification, in the weakly
+decaying case.
+
+
+4.5 Theorem E: the four-dimensional maximal-vacuum case
+
+The final theorem states the "maximal-method" result: a sharp form
+of the Penrose inequality in four dimensions for a specific class
+of maximal, vacuum, torus-symmetric data.
+
+SETUP. Let (M, g, K) be a four-dimensional initial-data exterior:
+
+ - Smooth, connected, orientable, diffeomorphic to R^4 minus an
+ open ball, complete up to its compact boundary S which is
+ isometric to S^3.
+
+ - Exactly one asymptotically flat end.
+
+ - Vacuum: mu = 0 and J = 0, equivalently the constraints with
+ K replaced by any of its conformal variants.
+
+ - Maximal: tr_g K = 0.
+
+ - Effective T^2 = U(1) x U(1) isometric action preserving both g
+ and K, which in an asymptotically Euclidean chart acts as
+ (alpha, beta) . (z_1, z_2) = (e^{i alpha} z_1, e^{i beta} z_2).
+
+ - Metric and tensor decay at exactly the same rates as in
+ Theorem B:
+ g - delta = O_2(r^{-q}), K = O_1(r^{-1-q}), q > 1.
+
+ - P = 0, E > 0 (positive mass).
+
+ - Boundary S is a MOTS: theta_+(S) = 0, i.e. H_S + tr_S K = 0,
+ with the normal pointing toward the end.
+
+ - S is outermost toward the end: no smooth embedded two-sided
+ hypersurface in the interior, enclosing S and oriented toward
+ the end, has theta_K <= 0 everywhere.
+
+ - S is outer area-minimizing: every compact embedded enclosing
+ hypersurface has 3-volume at least A = |S|_g.
+
+CONCLUSION.
+
+ E >= (1/2) * ( A / (2 pi^2) )^(2/3).
+
+This is "Theorem 1.6" of the introduction. The sharpness is
+witnessed by the Schwarzschild-Tangherlini family, in which equality
+holds for every mass.
+
+The methodological difference from Theorems A and B is this: the
+inner boundary S is not trapped (theta_+ = 0, i.e. a MOTS), and K
+reaches all the way to infinity. The deformation is built so that
+a purely homogeneous (linear in f) trace equation still exists at
+infinity, whose decay f = O(e^{-cr}) can be proved by comparison
+with a suitable exponential super-/sub-solution. This uses the
+maximality hypothesis: without it, an extra tr_g K term in the
+trace equation would contribute an inhomogeneous forcing at
+infinity and f would decay too slowly.
+
+The proof occupies Section 9 (the "maximal-method" walkthrough).
+
+
+PART 5. THE GRAPH DEFORMATION - THE ACTUAL FORMULAS
+
+
+5.1 The metric, the variables, and the warping factor
+
+We now describe the deformation that is common to Theorems A and B,
+in three dimensions. Every formula in this part is from the 3D
+construction (the 4D version has slightly different constants,
+recorded at the end of this Part). Fix a small parameter eps > 0
+and an integer N >= 4. Let theta : R -> [0, 1] be a smooth
+nonincreasing function equal to 1 on (-infinity, 0] and equal to 0
+on [1, infinity). Define the profile and weight functions
+
+ p(t) = N * theta(N t)
+ l(t) = exp( integral_0^t p(s) ds )
+ L(t) = l(t) * e^{2 t}
+ p_L(t) = p(t) + 2
+ ell = e^{-eps N}
+ tau = ell^{3/2} (this tau is a small number, not tr K)
+
+Concretely: p(t) is N on t <= 0, decreases smoothly to 0 at t = 1/N,
+and is zero afterwards. Consequently l(t) = e^{N t} on t <= 0, and
+l(t) is constant on [1/N, infinity). In particular 0 <= p <= N,
+l <= e, and if t >= -eps then ell <= l <= e.
+
+The unknowns are two real-valued functions on Omega: the graph
+height f and the "conformal variable" Z. From them we define, in
+order, several auxiliary quantities:
+
+ sigma = |grad f|_g
+ D = 1 + l(t)^2 * sigma^2
+ d = 1/D
+ t = the unique real number satisfying
+ Z = t + (1/8) * log D (implicit equation)
+ w = l * grad f / sqrt(D) (a vector field, |w| <= 1)
+ a = |w| = l sigma / sqrt(D)
+ v = a^2 = 1 - d
+ chi = 4 d / (4 + p v)
+ A_chi = I - ((p + 4) / (4 + p v)) * w (x) w
+ u = L * sqrt(D)
+ U = l * sqrt(D)
+ h = tau * f (scaled height)
+
+The implicit definition of t deserves a comment. From
+Z = t + (1/8) log D and D = 1 + l(t)^2 sigma^2, the right side is a
+strictly increasing function of t (its derivative is
+1 + p v / 4 > 0) and tends to the two ends of the real line as
+t -> +/- infinity. So for every (Z, grad f) there is a unique real
+t solving it, and it depends smoothly on those arguments. This is
+the "conformal variable decomposition" t = t(Z, grad f).
+
+The NEW METRIC is
+
+ ghat = e^{4 t} * (g + l(t)^2 df (x) df).
+
+The graph term l(t)^2 df (x) df is a rank-one perturbation of the
+base metric: it stretches distances along the gradient direction of
+f. The conformal factor e^{4t} rescales everything. Both t and f
+are unknown; the equations we impose govern them.
+
+An observation that matters repeatedly: w is a BOUNDED vector
+field. Since D = 1 + l^2 sigma^2 and w = l grad f / sqrt(D), the
+2-norm of w is l sigma / sqrt(1 + l^2 sigma^2) which is less than 1
+always. Consequently v = |w|^2 is between 0 and 1, and
+A_chi = I - ((p+4)/(4+pv)) w (x) w has eigenvalues
+
+ chi in the w-direction
+ 1 in the (n-1)-dimensional transverse directions
+
+so it is a positive-definite symmetric endomorphism (matrix)
+interpolating between the identity and a projection onto w^perp.
+
+
+5.2 The scalar curvature identity
+
+The central formula. Define the Hessian correction
+
+ H^f = (l / sqrt(D)) * Hess_g f
+
+and the modified second fundamental form
+
+ S = K + H^f.
+
+The trace of S with respect to A_chi (i.e., the sum over a basis of
+the contractions A_chi^{ij} S_{ij}) is a certain scalar; call it F.
+The problem will impose the trace equation
+
+ tr_{A_chi} S = F,
+
+with F a prescribed function. Then for ANY smooth f, t (with
+Z = t + (1/8) log D) for which this trace equation holds, the
+following EXACT IDENTITY holds:
+
+ (1/2) * e^{4 t} * Scal(ghat)
+ = 8 pi * ( mu + J(w) )
+ + T
+ + w(F)
+ - F * tr K
+ - u^{-1} * div(V).
+
+Here:
+
+ mu, J are the constraint densities of the ORIGINAL
+ data (g, K).
+
+ T is a nonnegative quadratic form in the
+ decomposition of S relative to the direction e
+ = grad f / sigma (see below).
+
+ w(F) is the directional derivative of F along w.
+
+ tr K is the trace tr_g K of the original tensor.
+
+ V is the vector field
+ V = u * A_chi * ( 4 grad Z + K(w, .) )
+ built from grad t (via Z), K, and A_chi.
+
+ u^{-1} div(V) is the divergence of V divided by u.
+
+The exact form of T, in the decomposition of S into
+ S|e^perp transverse block T_block (2x2 matrix in 3D)
+ S(e, .)|e^perp mixed block M (2-vector in 3D)
+ S(e, e) axial entry q
+and with tr T_block = F - chi q, is:
+
+ T = (1/2) * |T_tf|^2
+ + (M + pa grad_perp t) . (M + (p+2)a grad_perp t)
+ + 4(p+1) |dt|_{A_d}^2
+ + (chi - chi^2/4) q^2
+ + 2(p+1) chi q a d_e t
+ - chi F q / 2
+ + 3 F^2 / 4
+
+where T_tf = T_block - (tr T_block / 2) * (metric on e^perp) is
+the trace-free part of the transverse block, and d_e t = the
+directional derivative of t along e = grad f / sigma.
+
+This is one of the key calculations of the whole paper. Its proof
+is a careful computation of the scalar curvature of ghat under the
+graph + conformal deformation, and it goes back to the warped-graph
+identity of Bray-Khuri (2011), re-proved in the companion paper as
+"Proposition 4.1" and imported here with a careful dictionary
+(see the 02-system.tex proof). It extends smoothly to grad f = 0
+because all its ingredients are written in terms of w, not of e
+alone.
+
+
+5.3 Coercivity: why T is a lower bound on the geometric data
+
+The identity above would be useless unless T is nonnegative and
+controls something meaningful. Two facts make it work.
+
+Fact 1 (Positivity of T). Complete the squares:
+
+ T = (1/2)*|T_tf|^2
+ + |M + (p+1) a grad_perp t|^2
+ + [4(p+1) - v] * |grad_perp t|^2
+ + 4(p+1) * d * ( d_e t + chi a q / (4 d) )^2
+ + (3/4) * ( F - chi q / 3 )^2
+ + 4 d (9 - d) / ( 3 (4 + p v)^2 ) * q^2
+
+ (this is the "square-completion" form; every term on the right
+ is manifestly >= 0 because p >= 0, 0 < v <= 1, 0 < d <= 1,
+ 0 < chi <= 1)
+
+In particular T >= 0 always, and the last coefficient is at least
+32 d / (3 (4 + N)^2). Since 0 <= N and 0 < d, all the pieces are
+strictly controlled (with polynomial losses in N).
+
+Fact 2 (Equivalence at zero gradient). When dt = 0 the mixed terms
+vanish, and T has the much cleaner form
+
+ c_* * ( |T_tf|^2 + |M|^2 + chi q^2 + F^2 )
+ <= T <=
+ C_* * ( |T_tf|^2 + |M|^2 + chi q^2 + F^2 )
+
+with c_* and C_* ABSOLUTE positive constants, independent of N, p,
+d, and the solution. That "absolute" is the whole point; the
+constants of the analysis below must have controlled dependence on
+N.
+
+Fact 3 (Polynomial loss when dt is not zero). Away from dt = 0 one
+has
+
+ |dt|_{A_d}^2 + |T|^2 + |M|^2 + d q^2 + F^2
+ <= C * (1 + N)^A * T
+
+with absolute C, A. So controlling T (which is what all our
+estimates will do) controls every piece of the Hessian of f, the
+full tensor S, and the gradient of t, up to a polynomial cost in N.
+
+The three facts together say: T is a positive quantity that is
+essentially "the L^2 norm of the Hessian of f squared, plus the
+transverse part of S squared, plus (some of) the gradient of t
+squared, plus F^2." If T is small, the geometry is close to flat;
+if T is bounded, everything is bounded. This equivalence is what
+makes the graph deformation tractable.
+
+
+5.4 The trace equation
+
+The system imposes the trace equation
+
+ tr_{A_chi}(K + H^f) = F(x, grad f, f, t, Z)
+
+where F is a prescribed function. For the physical end of the
+homotopy the right side is
+
+ F = h + C = tau * f + C(x),
+
+with tau = e^{-3 eps N / 2} a small number, h = tau f the scaled
+height, and C a smooth compactly supported function. Two things
+about this choice:
+
+ The dependence of F on f is strictly increasing: dF/df = tau
+ > 0. This is critical because it makes the trace equation
+ uniquely solvable for f by the maximum principle. Two solutions
+ would have to agree where their maximum difference occurs; at
+ such a point the Hessian comparison rules out a difference.
+
+ The affine-in-f term tau*f is a "regularizer": without it the
+ trace equation would be
+ tr_{A_chi}(K + H^f) = C(x)
+ which is degenerate on the end where C vanishes. With it, the
+ operator has uniformly positive zeroth-order coefficient tau
+ sqrt(D) / l, which gives exponential decay of f on the end.
+
+The function C(x) is fixed during the geometric preparation stage
+(Section 7.3 below). It satisfies:
+
+ C < 0 outside a compact support, and C = C_b - k_B s on
+ black collar coordinates, C = C_w + k_B s on white collar
+ coordinates, with k_B > 0.
+
+The constants C_b >= 0, C_w <= 0 determine the "assigned face
+heights" b_- = c_b - C_b < 0 and b_+ = -c_w - C_w > 0. Those are
+the values that f gets on the black and white faces at the assigned
+inner boundary condition.
+
+
+5.5 The boundary identity
+
+At the inner boundary B, f is constant (this is one of the
+Dirichlet boundary conditions we will impose). Because of that
+simplification, several formulas reduce beautifully.
+
+Let nu be the normal to B pointing into Omega. Let H = div_B nu be
+the mean curvature of B in the base metric g (with that normal), P_B
+= tr_B K the trace of K over the tangent plane, so tr_Sigma K in the
+3D notation used in the original proves is the same as P_B. Then
+wherever f is constant:
+
+ V_nu / u = d ( H + 4 d_nu t ) - H + w_nu ( F - P_B )
+
+where d_nu t is the normal derivative of t, w_nu = the nu-component
+of w. Furthermore the mean curvature of B in the NEW metric ghat is
+
+ H_hat = e^{-2 t} * sqrt(d) * ( H + 4 d_nu t ).
+
+Proof sketch: the pullback of ghat restricted to the tangent plane
+TB is e^{4t} times the base tangent metric (since df vanishes on
+TB). Making B a Riemannian submanifold under this conformal change
+gives a factor of e^{2t} in the mean curvature. The d factor comes
+from the rescaled normal e^{2t} sqrt(d) nu. The (H + 4 d_nu t)
+bracket is the tractable combination that appears; its presence is
+what makes the boundary condition "H_hat < 0" equivalent to the
+scalar boundary condition "H + 4 d_nu t < 0".
+
+The point of this formula: if we impose
+
+ V_nu / u = -H + w_nu ( F - P_B ) - N_B * d
+
+with N_B a prescribed function strictly bigger than 1 + sup|H|,
+then comparing to the formula for V_nu / u we get
+
+ d (H + 4 d_nu t) - H = -H - N_B * d
+ H + 4 d_nu t = -N_B
+
+and hence H_hat = -e^{-2t} sqrt(d) N_B < 0. Done: negative mean
+curvature in the new metric, for free, just by choosing the right
+boundary condition for V_nu / u.
+
+The extra term -N_B * d is the "flux driver" that makes this work.
+Since d <= 1 it is bounded; it gives the required strict
+negativity, and it is small enough not to interfere with the
+interior estimates.
+
+
+5.6 The divergence equation and the penalty
+
+The trace equation determines f given Z. The divergence equation
+should determine Z (or equivalently t) given f. Recall the flux
+
+ V = u * A_chi * ( 4 grad Z + K(w, .) ).
+
+The divergence equation is
+
+ div V = u * Xi
+
+where the source Xi is chosen by the authors as
+
+ Xi = delta_0 * T
+ + rho(x)
+ + delta_0 * tau * a * sigma
+ - m_0(t) * P(x, grad f)
+
+with:
+
+ 0 < delta_0 < 1/4 a small positive constant chosen once (its
+ value is picked during the floor-exclusion argument; see Section
+ 7.7 below).
+
+ rho(x) a positive smooth function on the end, roughly like
+ r^{-4} or r^{-3-delta}.
+
+ m_0(t) a smooth cutoff, equal to 1 for t <= -eps + 1/N and
+ equal to 0 for t >= 0 or similar; the exact profile
+ confines the penalty to a narrow band just above -eps.
+
+ P(x) a "penalty" equal to C_N * ( rho_0 + v rho_1 ) with
+ rho_0 like r^{-4}, rho_1 like r^{-2 gamma}, C_N a large
+ polynomial in N.
+
+Interpreting each term:
+
+ delta_0 T is the "energy" term: a small positive multiple of the
+ coercive quadratic, which drives the size of T down on the
+ average.
+
+ rho is a small positive weight ensuring that near the end there
+ is always a positive source so the maximum principle applies.
+
+ delta_0 tau a sigma is a "graph slope" term ensuring the equation
+ is proper near graph slope zero.
+
+ -m_0(t) P is a NEGATIVE penalty, concentrated in the range
+ -eps <= t <= -eps + 1/N, whose job is to make sure t never
+ accidentally hits -eps from above. It is so negative (because
+ C_N is a large polynomial in N) that the right side is very
+ negative there, which forces a region of high t to grow.
+
+The negative penalty is the "conformal floor": without it, a
+solution could conceivably have t = -eps at some point, which would
+mean e^{2t} has hit its lower bound. We want t strictly above -eps
+everywhere, so the metric does not degenerate. The penalty pushes t
+away from the floor.
+
+Of course a penalty alone does not prove anything; you have to argue
+that no actual solution can touch the floor. This is the deep
+technical argument in Section 7.7 below.
+
+
+5.7 Why the conformal floor matters
+
+The metric ghat = e^{4t}(g + l^2 df (x) df) satisfies
+
+ ghat >= e^{4t} g >= e^{-4 eps} g
+
+because the graph term is nonnegative (a rank-one perturbation
+with a positive coefficient) and t >= -eps by the floor.
+
+Equivalently, for any tangent (n-1)-plane:
+
+ Area_ghat(Sigma) >= e^{4t (n-1)/2} Area_g(Sigma)
+ = e^{4 t (n-1)/2} Area_g(Sigma).
+
+In 3D (n=3, surface dim 2), the exponent is 4t * 2/2 = 4t, so
+Area_ghat >= e^{4t} Area_g >= e^{-4 eps} Area_g. This gives the
+factor e^{-4 eps} appearing in Theorems A and C.
+
+The floor controls not just the area but also:
+
+ Completeness. Since ghat >= e^{-4 eps} g and g is complete up
+ to B, ghat is complete up to B as well. No geodesic of ghat can
+ escape in finite time.
+
+ Ellipticity of the equations. When t is very negative, the
+ coefficient l(t)^2 in front of df (x) df becomes very small
+ (since l(t) = e^{N t} for t very negative) but never zero.
+ When t is above -eps, all the coefficients of the elliptic
+ system are bounded above and below by N-dependent constants.
+ This keeps the equations uniformly elliptic.
+
+ Positivity of the scalar curvature right-hand side. The formula
+ for R_ghat includes a term -m_0(t) P that vanishes once t > -eps
+ + 1/N. Without the floor-and-penalty mechanism, m_0 would be 1
+ somewhere and the negative penalty would make R_ghat negative.
+ The floor keeps m_0 = 0 on any actual solution.
+
+The strategy then is: prove that no solution can touch the floor,
+and hence every solution has t > -eps everywhere. Then all the good
+properties are inherited automatically.
+
+
+5.8 The full elliptic system for the 3D construction
+
+Putting it together, we seek two smooth functions (f, Z) on each
+truncation Omega_R = Omega cap {r < R}, such that:
+
+ (trace) tr_{A_chi}(K + H^f) = F = h + C in Omega_R
+
+ (flux) div V = u * Xi in Omega_R
+
+ (inner BC) f = b_ (black face) on B
+ f = b_+ (white face) on B
+ V_nu/u = -H + w_nu(F - P_B) - N_B d on B
+
+ (outer BC) f = Z = 0 on S_R
+
+The parameters b_-, b_+, the function C, the number delta_0, the
+polynomial C_N, the inner-boundary profile N_B, and the weights
+rho, rho_0, rho_1, m_0 are all FIXED during the geometric
+preparation stage, BEFORE taking N large.
+
+The unknowns are the pair (f, Z). The existence of a smooth solution
+for each sufficiently large N and R is what Theorems A and B give.
+The proof is a huge application of elliptic PDE and degree theory;
+we walk through it in Part 7.
+
+Note the system is elliptic but not linear: the trace equation
+depends nonlinearly on grad f through the A_chi operator and D, and
+the flux equation depends on grad Z quadratically through |grad Z|^2
+inside Xi (via T). Nevertheless, by the time we have the floor
+bound and the coercivity estimate, everything is well controlled.
+
+
+5.9 The 4D versions of these formulas
+
+For the four-dimensional case, the constants change as follows:
+
+ ghat = e^{2 t} * (g + l(t)^2 df (x) df) (factor 2t, not 4t)
+ D, d, w, a, v, u, L_0 as before
+ chi = 3 d / (3 + p v) (k=3, not k=4)
+ A_chi = I - ((p + 3) / (3 + p v)) * w (x) w
+
+ Scalar identity:
+ (1/2) e^{2 t} R_ghat
+ = mu + J(w) + T + w(F) - F * tr K - u^{-1} div V
+ with mu, J defined WITHOUT the 8 pi (geometrised units).
+
+ T is essentially identical in form (with dimensional factors),
+
+ Boundary identity:
+ V_nu / u = d ( H + 3 d_nu t ) - H + w_nu ( F - P_B )
+ H_hat = e^{-t} sqrt(d) ( H + 3 d_nu t )
+
+ Boundary datum: impose
+ V_nu/u = -H + w_nu (F - P_B) - N_B d
+ which gives H + 3 d_nu t = -N_B and H_hat < 0.
+
+ Area factor in 4D: surface dimension is 3, so
+ Area_ghat >= (e^{2 t})^{3/2} * Area_g = e^{3 t} Area_g.
+ Floor t >= -eps gives Area_ghat >= e^{-3 eps} Area_g.
+
+ The 4D profile has eta_N = ell^{3/2} = e^{-3 eps N / 2}. The
+ 2 in 2t and the 3 in 3d_nu t are the dimension-specific constants.
+
+These are the formulas used in the four-dimensional construction in
+Part 8.
+
+
+PART 6. THE HOMOTOPY AND THE DEGREE ARGUMENT
+
+
+6.1 Why we need a continuation argument
+
+The system in Section 5.8 is a coupled nonlinear elliptic system.
+We cannot solve it directly - it is too nonlinear, the coefficients
+are degenerate at zero gradient (the A_chi matrix becomes singular
+if the direction e is undefined), and the "quadratic" term T hides
+the |Hess f|^2 growth. A direct fixed-point argument would need a
+priori bounds we cannot easily get.
+
+The trick is a HOMOTOPY: continuously deform the system from the
+one we want into a system we CAN solve easily, and then show that
+the number of solutions cannot change during the deformation. This
+"continuation" argument is standard in elliptic PDE (see e.g.
+Leray-Schauder 1934, Gilbarg-Trudinger 2001). The invention of the
+paper is the choice of homotopy, arranged so that:
+
+ Each intermediate system has at most as many nonlinearities as
+ the basic one, and is perfectly well-posed once the floor bound
+ t >= -eps is known.
+
+ The endpoint (the trivially solvable system) is reached from
+ the desired system by continuous steps.
+
+ Any solution family "escaping" during the homotopy forces a
+ contradiction with the floor bound and the a priori estimates,
+ so the number of solutions is genuinely invariant.
+
+
+6.2 The four-stage homotopy
+
+The homotopy runs over a parameter s in [0, 4], broken into four
+stages of length 1. The unknown pair is still (f, Z), and the outer
+boundary conditions f = Z = 0 on S_R are kept throughout. The
+inner boundary condition (the f-value b_-, b_+, and the flux
+condition) changes slightly from stage to stage.
+
+Stage 1: s in [0, 1]. Decrease the drift from 1 to 0.
+
+ Parameter lambda = 1 - s. Replace
+ V by V_lambda = u A_chi ( 4 grad Z + lambda K(w,.) )
+ so the drift term K(w,.) fades away continuously.
+ The boundary condition is modified correspondingly.
+ The trace equation stays F = h + C.
+
+Stage 2: s in [1, 2]. Add a small "direction corrector".
+
+ Parameter alpha = s - 1. Replace
+ F = h + C by F = h + C + alpha * D_0(x, w, h)
+ where D_0 is a smooth compactly supported function invented so
+ that its sign is compatible with the black and white boundary
+ signs (see Section 7.5 below for the exact choice). At alpha = 1
+ it forces the trace equation to have a compatibility with the
+ boundary direction of the gradient.
+
+Stage 3: s in [2, 3]. Interpolate the trace right side.
+
+ Parameter zeta = s - 2, with an auxiliary scalar function b(x)
+ taking values b_- on black faces, b_+ on white faces. Replace
+ F = h + (1 - zeta) [ C + D_0(x, w, h + zeta b(x)) ]
+ + zeta [ tr_{A_chi} K + D_1(x, w) ],
+ and stagger the f-boundary value from b_-, b_+ to zero:
+ f |_B = (1 - zeta) b(x).
+ At zeta = 1 the trace right side no longer involves the offset
+ function C, and instead involves only the trace of K plus D_1.
+
+Stage 4: s in [3, 4]. Scale the source and boundary data to zero.
+
+ Parameter eta = 4 - s. Multiply both the right side Xi and the
+ inner flux V_nu/u by eta, decreasing from 1 to 0.
+ The inner trace equation is now
+ tr_{A_chi} H^f = h + D_1(x, w)
+ with f = 0 at the boundary, which has only the trivial solution
+ f = 0 (proved by the maximum principle), and the flux equation
+ becomes the harmonic scalar problem
+ div ( 3 u grad Z ) = eta * (source)
+ V_nu/u ~ eta * (flux)
+ with the inner flux becoming conormal.
+
+At s = 4: the system is completely linear and trivially solvable.
+The final answer is f = 0, Z = 0 (at eta = 0 the scalar equation
+div(3 u grad Z) = 0 with zero inner conormal flux and zero outer
+Dirichlet has only Z = 0 by integration by parts).
+
+The key properties of this homotopy:
+
+ At every s in [0, 4], the trace right side depends on h with
+ strictly positive derivative (at least 1). This ensures the
+ trace equation is uniquely solvable for f, uniformly in s and
+ the boundary values.
+
+ At s = 4 the twist vanishes and the trivial solution is the
+ unique one, giving degree 1.
+
+ Homotopy invariance of the Leray-Schauder degree guarantees the
+ same degree at s = 0, hence existence of a solution there.
+
+
+6.3 The scalar trial problem
+
+Fix Z (the "trial input"). Then the trace equation becomes a
+standalone equation for f:
+
+ tr_{A_chi}(K + H^f) = F_s(x, Z, f, grad f)
+
+where F_s denotes the trace right side at stage s. This is a
+nonlinear scalar elliptic equation for f, with Dirichlet data
+f = 0, f = b_-, f = b_+ (depending on stage and face).
+
+Claim (Solvability, Lemma 7.x in the original): for every smooth
+Z, every s, and every boundary data of the form just described,
+there is a unique smooth f_s[Z] solving the equation.
+
+Proof idea: use the method of continuity on a further parameter
+theta interpolating between the Laplacian operator A = I and the
+actual operator A_chi. At theta = 0 the equation is a linear
+Dirichlet problem; at theta = 1 the equation is our trace
+equation. Along the way, the equation
+
+ (1-theta) Delta f + theta A_chi : Hess f = F_s^theta
+
+where F_s^theta interpolates the source analogously, is uniformly
+elliptic and strictly increasing in f, so:
+
+ (Openness) The implicit function theorem with Schauder
+ estimates gives that the solution set is open in theta.
+
+ (Closedness) Uniform a priori bounds give that the solution set
+ is closed.
+
+ (Uniqueness) The maximum principle on the difference of two
+ solutions rules out more than one.
+
+ So the solution set is open, closed, and nonempty (at theta = 0
+ where it is linear), hence all of [0, 1].
+
+The a priori bounds needed along the way:
+
+ Height bound. |f| <= C_0, uniform in N, R, s, theta, and the
+ trial Z. At an interior maximum of f, we have w = 0, chi = d = 1,
+ A_chi = I. The equation reduces to a linear second-order
+ equation with positive source (proportional to f) at the maximum,
+ which forces f to be bounded above. The minimum is symmetric.
+
+ Gradient bound. |grad f| <= C, uniform (for this stage). Use a
+ "logarithmic barrier" psi(r) = (1/k)*log(1 + B r) in a collar
+ near the boundary, and a similar two-point comparison in the
+ interior. The barriers exploit the boundedness of the coefficients
+ and the strict height monotonicity of the source.
+
+ Composite regularity. Once height and gradient are bounded, the
+ equation is uniformly elliptic with smooth coefficients on the
+ bounded range. Standard Schauder estimates give C^{3,alpha}
+ bounds.
+
+
+6.4 The frozen return map
+
+The scalar trial problem gives f only as a function of the trial
+input Z. In the desired solution, f and Z must satisfy the
+divergence equation as well. We therefore define a return map
+T_s taking Z to a new Z' = T_sZ:
+
+Given Z, compute f_s[Z] (Section 6.3). Then all coefficients in the
+flux equation are now known functions of position, including the
+vector field V and the source Xi. The remaining task is to solve
+the LINEAR equation
+
+ div ( u A_chi grad Z' + lambda u A_chi K(w,.) )
+ = eta_s * u * Xi,
+
+with boundary conditions Z' = 0 on S_R (outer Dirichlet) and
+
+ V'_nu/u = eta_s * ( -H + w_nu(F-P_B) - N_B d )
+
+on B (inner Neumann-type). Here V' is V with grad Z replaced by
+grad Z'. The extra eta_s is 1 in stages 1, 2, 3 and is the last
+stage factor 4-s in stage 4.
+
+The result Z' = T_sZ is a new function. A FIXED POINT of T_s
+(T_sZ = Z) is exactly a solution of the coupled system.
+
+The linear problem is well-posed: because A_chi is symmetric
+positive definite with bounded ellipticity constants (once f is
+bounded and the gradient of f is bounded, which follows from
+Section 6.3), the operator
+
+ Z' -> div ( u A_chi grad Z' ) + lower order terms
+
+is coercive on H^1_0(Omega_R), gives a unique weak solution by
+Lax-Milgram, and elliptic regularity upgrades it to a classical
+solution with bounded C^{2,alpha} norm.
+
+Crucial properties of T_s (proved by combining the solvability of
+the scalar trial problem with Schauder for the linear flux problem):
+
+ T_s is a CONTINUOUS map from C^{1,alpha}(Omega_R with outer
+ zero) to itself, jointly in s, on bounded sets.
+
+ T_s is COMPACT: it maps bounded sets to bounded sets in
+ C^{2,alpha}, and C^{2,alpha} compactly embeds in C^{1,alpha}
+ on a compact domain. So the image of a bounded set has a
+ convergent subsequence.
+
+ Each T_s is defined on the WHOLE space C^{1,alpha}, not just on
+ the subset where t >= -eps. This is important because we need
+ the map defined on "big" sets for the degree argument; the floor
+ appears only as a CONSTRAINT on the admissible solutions, not
+ as a precondition for defining the map.
+
+This last property is the paper's key innovation over previous
+graph approaches. Previous approaches required the floor to be
+satisfied BEFORE the map even made sense, which made continuation
+impossible. Here the map exists everywhere, and the floor acts as a
+restriction on the domain we're allowed to look in.
+
+
+6.5 The Leray-Schauder degree argument
+
+The existence proof now proceeds as follows. Let
+
+ X_R = { Z in C^{1,alpha}(closure(Omega_R)) : Z = 0 on S_R }
+
+be the Banach space of trial inputs. For each s in [0, 4], define
+the open "safe" set
+
+ U_s = { Z in X_R : || Z ||_{C^{1,alpha}} < M_0,
+ min_{Omega_R} t(x, Z, grad f_s[Z]) > -eps }
+
+where M_0 is a large constant (chosen using a priori estimates on
+solutions) and t(x, Z, grad f_s[Z]) is the conformal variable at
+each point. Because f_s[Z] is continuous in Z and t depends
+smoothly on Z and grad f, the minimum is continuous, so U_s is
+open.
+
+Let U = { (s, Z) : Z in U_s } be the union over s. Then U is open
+in [0, 4] x X_R.
+
+The Leray-Schauder degree is an integer-valued topological
+invariant, defined for any continuous compact map T from a bounded
+open set U into X_R, satisfying:
+
+ Homotopy invariance: if T_s is a continuous 1-parameter family
+ of compact maps and for all s there are no fixed points on the
+ boundary of U_s, then the degree is constant in s.
+
+ Existence: if the degree is nonzero, there exists a fixed point
+ inside U.
+
+If we can show:
+
+ (i) For each s, the fixed-point set of T_s in U_s is compact
+ (trivial by compactness of T_s).
+
+ (ii) No fixed point of T_s lies on the boundary of U_s. This
+ would mean either ||Z|| = M_0 (excluded by choosing M_0
+ via a priori bounds, which say every fixed point has
+ ||Z||_{C^{1,alpha}} <= M_*), or min t = -eps (excluded by
+ the floor-exclusion argument in Section 7.7 below).
+
+ (iii) The degree at s = 4 is nonzero (equal to 1).
+
+Then homotopy invariance gives degree at s = 0 is also 1, so there
+exists a fixed point of T_0, i.e., a solution to the desired system.
+
+The paper actually invokes a "moving-degree lemma" (Lemma 4.x in
+the original, and in the companion papers as Lemma 5.x) that gives
+homotopy invariance of the Leray-Schauder degree across a MOVING
+family of open sets U_s. This is a technical variant of the usual
+Leray-Schauder theorem, needed here because the sections U_s are
+not a constant set - they vary with s due to the t floor. The
+moving-degree lemma proves that homotopy invariance still holds,
+provided the fixed-point set remains compact and confined away
+from the boundary, which is what (i) + (ii) above give.
+
+The final step, then, is showing (iii).
+
+
+6.6 The last stage is trivially solvable
+
+At s = 4, the system is completely linear. Here is the argument:
+
+ (Scalar constraint) With the source and boundary data scaled by
+ eta = 4 - s = 0 and the trace right-hand side reduced to
+ tr_{A_chi}(H^f) = h + D_1(x, w), f = 0, f |_B = 0,
+ we claim the ONLY solution is f = 0 identically. At any interior
+ positive maximum of f, we have w = 0, chi = d = 1, A_chi = I,
+ and the trace equation becomes l Delta_g f = tau f + D_1(x, 0).
+ Since h = tau f > 0 at a maximum and l Delta_g f <= 0 there, we
+ have a contradiction. Similarly for a negative minimum.
+
+ (Flux constraint) With f = 0, the flux equation becomes exactly
+ div(4 u A_chi grad Z) = 0
+ inside Omega_R,
+ Z = 0 on S_R (outer),
+ V_nu/u = 0 on B (inner conormal).
+ Multiply by Z and integrate by parts: get
+ 4 * integral |grad Z|^2 dV_g = 0,
+ which forces Z = 0.
+
+So the unique fixed point of T_4 is Z = 0, f = 0. This belongs to
+U_4 (which has 0 in its interior), and it's a nondegenerate fixed
+point with index +1. So the Leray-Schauder degree at s = 4 is
+exactly 1.
+
+By homotopy invariance, the same degree at s = 0 exists and equals
+1, so there must be a solution there. This is exactly Theorem A
+(and Theorem B in the 4D case).
+
+
+PART 7. THE THREE-DIMENSIONAL CONSTRUCTION, IN DETAIL
+
+What remains is to prove that (ii) above holds: no fixed point
+during the homotopy touches the floor t = -eps. This is where all
+the geometric machinery (trapped regions, collars, boundary slopes,
+floor exclusion) comes in. Let us walk through it.
+
+
+7.1 Geometric preparation: black regions, white regions
+
+The starting point is the initial data set (Omega_0, g, K) with its
+inner boundary S_0 strictly future trapped (theta_+ < 0). We need to
+build a new inner boundary B inside Omega_0 that:
+
+ (a) Separates S_0 from infinity.
+
+ (b) Has two sides, BLACK and WHITE, with certain sign conditions
+ on the trace of K.
+
+ (c) Has mean curvature expansion of a prescribed sign for K and
+ -K respectively.
+
+The starting point is the classical totally-trapped-region theorem
+(Andersson-Metzger 2009). Here is the construction:
+
+ 1. Enlarge Omega_0 to a slightly larger exterior (call it X_0)
+ whose outer boundary is a large coordinate sphere S_{R_0},
+ chosen beyond supp(K) and large enough that all coordinate
+ spheres beyond it have positive outward mean curvature.
+
+ 2. Fix a constant c_b < 0 close to zero. Consider the region
+
+ B_c = { union of smooth compact regions E in X_0
+ with theta_K(partial E) <= c }
+
+ for various thresholds c < 0. For c very close to zero (but
+ negative), the region B_c is nonempty; it contains a smooth
+ collar of the original inner boundary S_0 (since theta_K(S_0)
+ < 0 there).
+
+ 3. By Andersson-Metzger's theorem, the closure of B_c has a
+ smooth outermost stable MOTS frontier. Denote it B_b. The
+ frontier has theta_K(B_b) = c_b exactly, by construction.
+
+ 4. The B_c's increase with c. Choose c_b at a "right-continuity
+ threshold" - a technicality ensuring that B_{c} varies
+ continuously with c at that value (see Section 7.2 below).
+
+ 5. This B_b is the first (BLACK) inner boundary surface.
+
+ 6. Now consider the complement Y = Omega_0 \ int(B_b), which has
+ an inner boundary (from B_b) with expansion c_b < 0 for K
+ and an outer boundary (the sphere S_{R_0}) with positive
+ expansion for K. Consider the tensor -K instead.
+
+ 7. Apply the same total-region theorem to Y with tensor -K.
+ Define
+
+ W_c = { union of smooth compact regions in Y
+ with theta_{-K}(partial E) <= c }
+
+ and let W_w be the closure at c slightly above c_w, with
+ smooth outermost stable MOTS frontier B_w satisfying
+ theta_{-K}(B_w) = c_w.
+
+ 8. This B_w is the WHITE inner boundary. It is disjoint from B_b
+ (they are separated by the two-component structure of the
+ construction).
+
+ 9. Let Omega be the connected component of Omega_0 \ (int B_b
+ union int B_w) that reaches infinity. Its boundary B is
+ (B_b retained) union (B_w retained), i.e., the subset of those
+ surfaces adjacent to Omega.
+
+The retained portion of B_b (BLACK faces) has trace-expansion
+ theta_K(B_b) = H + P_B = c_b < 0.
+The retained portion of B_w (WHITE faces) has trace-expansion
+ theta_{-K}(B_w) = H - P_B = c_w < 0.
+
+The sign H + P_B = c_b (on black) and H - P_B = c_w (on white) is
+exactly the boundary condition we need for the deformation.
+
+Note: the "white" region might be empty (if the trapped-region
+theorem applied to Y with tensor -K yields nothing). In that case
+B_w is absent and Omega has only BLACK faces.
+
+
+7.2 Making the trapped regions "continuous"
+
+The trapped-region construction depends on the threshold c. As c
+varies, the region B_c grows (larger threshold = more regions
+satisfy theta <= c). At most values of c, B_c changes continuously.
+But at countably many "exceptional" c values, a discontinuous jump
+can occur: a new component of a trapped region can pop into
+existence.
+
+The technical tool used to handle this: the RIGH-CONTINUITY LEMMA
+(Lemma 2.x in the 3D section of the original, proved as Lemma 3.x
+of the companion Riemannian paper). It says:
+
+ Let E_c, c in an interval, be an increasing family of closed
+ subsets of a compact metric space. Outside a countable set of
+ parameters, E_{c_j} converges to E_c in Hausdorff distance
+ whenever c_j decreases to c from above.
+
+The proof uses the Baire category theorem on the space of closed
+subsets of a compact metric space, together with the distance
+function d_c(x) = dist(x, E_c) which is non-increasing in c and
+has at most countably many discontinuities in the right-limit
+sense.
+
+The consequence for our construction: we can choose c_b at a
+right-continuity threshold, so B_b varies continuously with the
+threshold as we approach c_b from above. Similarly for c_w. This
+is essential for the floor-exclusion argument later, which needs
+to know that B_c is close to B_{c_b} for c slightly above c_b.
+
+Both thresholds can be chosen arbitrarily close to zero (but
+negative). We fix them once and for all after this continuity step.
+
+
+7.3 The offset function C and the boundary collars
+
+Each retained face of B (black or white) has a smooth outward
+collar. In this collar, choose a leaf coordinate s >= 0, s = 0 at
+B, with |grad s| bounded above and below by constants (from the
+stability of the MOTS and the "collar lemma" of Section 2.5). The
+collar can be shortened to have common parameter range
+0 <= s <= s_0 for all faces.
+
+In the collar define:
+
+ On a black face: C(s) = C_b - k_B s
+ On a white face: C(s) = C_w + k_B s
+
+with k_B > 0 a constant small enough that:
+ - the linear formula stays positive on the collar,
+ - the corresponding constants C_b >= 0, C_w <= 0 satisfy
+ C_b - k_B s_0 >= C_w + k_B s_0 (the two-sided compatibility),
+ - the first derivative of C has bounded norm.
+
+Outside all collars set C = 0.
+
+The constants C_b, C_w are chosen so that the "assigned heights"
+
+ b_- = c_b - C_b < 0
+ b_+ = -c_w + C_w? > 0
+
+(depending on the exact sign convention; in the actual paper
+b_- = c_b - C_b < 0 and b_+ = -c_w - C_w > 0) are strictly
+separated and compatible with the trace equation boundary values.
+
+The compatibility identity, at the actual future boundary
+F = h + C (not interpolated through the homotopy), is:
+
+ On a black face, at h = b_-, we have
+ h + C = b_- + C_b = c_b = P_B + H.
+ On a white face, at h = b_+, we have
+ h + C = b_+ + C_w = -c_w = P_B - H.
+
+where P_B = tr_B K is the trace of K over the tangent plane of B.
+
+Why is this the right identity? The trace equation at a black face
+involves tr_{A_chi}(K + H^f). Since f is constant, tr_{A_chi} H^f
+gives chi * (Hessian of f in the nu-nu direction) = chi q, and
+tr_{A_chi} K has a component from K(nu, nu) and from tangential
+traces. At the assigned height b_-, the trace equation's right side
+F = b_- + C_b = c_b matches exactly P_B + H (the "expansion" of
+B in the K-direction). Similarly at white faces.
+
+Without this compatibility, the boundary slopes in the estimates
+below would be off by a fixed amount and the floor exclusion would
+break. The identity is a subtle bookkeeping of constants chosen
+in the geometric preparation step to line up with the trace
+equation's right side.
+
+
+7.4 The system, restated
+
+With the geometric ingredients fixed, we now specify the system we
+seek to solve.
+
+UNKNOWNS: two smooth functions f, Z on Omega_R (for each fixed
+truncation radius R).
+
+EQUATIONS (from Section 5.8):
+
+ (trace eq) tr_{A_chi}(K + H^f) = F = h + C in Omega_R
+
+ (flux eq) div V = u Xi in Omega_R
+
+ (inner BC) f = b_- on black, f = b_+ on white
+ V_nu / u = -H + w_nu (F - P_B) - N_B d
+
+ (outer BC) f = Z = 0 on S_R.
+
+The coefficients and constants depend only on the fixed geometric
+data (Omega, g, K, B, collars, C, c_b, c_w, N_B, eps, N, etc).
+
+The system is elliptic (once t is bounded from below), quasi-linear
+with structure that we schedule below. The existence for every
+sufficiently large N and every sufficiently large R is what the
+homotopy + degree argument of PART 6 gives. The estimates needed to
+run the argument are what remains.
+
+
+7.5 Direction correctors: D_0 and D_1
+
+To make the boundary slopes have a definite sign throughout the
+homotopy, two auxiliary functions D_0(x, w, h) and D_1(x, w) are
+added to the trace equation's right side in stages 2 through 4. They
+are simple to describe:
+
+ D_0. Depends only on the sign of w . nu_s (in the boundary
+ collar) and on the height h near the assigned value b_+-.
+ On a black collar (with nu_s = the outward pointing normal
+ to the leaf), D_0 <= 0 everywhere and equals a fixed negative
+ number on w = -nu (the "wrong" direction) at h below b_-.
+ On a white collar, D_0 >= 0 and vanisehs on w = -nu, reaches
+ its fixed positive value on w = +nu at h above b_+.
+ Crucially, D_0 vanishes to all orders when the direction of
+ w is compatible with the face: on black faces (where f is
+ strictly decreasing along nu, so w = -a*nu), and on white
+ faces (where w = +a*nu).
+
+ D_1. A simple linear term D_1(x, w) = A_1 * (w . nu_s) times a
+ collar cutoff, with A_1 > sup_B |H| + 1 (so that it
+ dominates |H| and gives definite signs in the "wrong"
+ direction too). Vanishes at w = 0.
+
+The point: at the assigned height and with the compatible direction
+of grad f, F(x, +nu, b_+ or b_-) is at least P_B + H (on black) or
+at most P_B - H (on white). So the trace equation's right side is
+strictly greater than (black) or less than (white) the natural
+limiting value coming from the boundary expansion, allowing
+signed slopes. In the incompatible direction the correctors strictly
+violate this to force the slope back.
+
+Precise statement of the limiting directional inequality:
+
+ At any face point x with assigned height b, and considering the
+ LIMITING case where |grad f| -> 1 (i.e., a = 1, chi = d = 0),
+ the right side of the trace equation satisfies
+
+ F(x, +nu, b) >= P_B(x) + H(x)
+ F(x, -nu, b) <= P_B(x) - H(x)
+
+ at every stage of the homotopy. Here "+nu" means "grad f points
+ in the +nu direction (into Omega)" and "-nu" means "grad f
+ points toward -nu".
+
+At the actual solution (where |grad f| is finite, a < 1, and
+chi > 0), the trace equation reads F = h + C + corrections, where
+the corrections are strictly negative on w = -nu and positive on
+w = +nu. So the boundary slope has a definite sign in the
+compatible direction, and is forced back by the corrector in the
+incompatible direction. This is the engine of the signed-slope
+lemma (Section 7.8 below).
+
+The exact forms of D_0 and D_1 are technical but the reader can
+take this as read: they exist, they are smooth, supported in the
+boundary collars, and satisfy the sign conditions just described.
+They are fixed once and for all at the start of the construction.
+
+
+7.6 The height bound
+
+Before attacking the floor, the paper establishes a simple but
+critical fact: the SCALED height h = tau*f satisfies a uniform
+upper and lower bound.
+
+Lemma. There exists C_0 > 0 depending only on the geometric
+preparation (not on N, R, s, or the solution) such that |h| <= C_0
+for every smooth above-floor solution to the system (Sections
+6.2, 7.4).
+
+Proof. At any interior extremum of f we have grad f = 0, hence
+w = 0, a = 0, v = 0, chi = d = 1, A_chi = I. The trace equation
+becomes
+
+ Delta_g f * l / sqrt(D) + tr K = F(x, 0, f, Z),
+
+where F(x, 0, f, Z) = h + C + (D_0, D_1 corrections that vanish
+at w = 0).
+
+Substituting h = tau*f and isolating:
+
+ (l/sqrt D) Delta_g f + tr K = tau * f + C(x).
+
+At a positive maximum of f, Delta_g f <= 0, so
+ tau * f_max <= -tr K + C(x) on the support of C,
+and elsewhere tau * f_max <= 0. So f is bounded above by
+|tr K| + max |C| / tau within the compact support, and below by 0
+outside. Combined with the boundary values f = b_-, b_+ (which are
+bounded in absolute value by a fixed number), this gives f <=
+C_0 / tau for some fixed C_0.
+
+Similarly, at a negative minimum of f, Delta_g f >= 0, so
+ tau f_min >= -tr K + C(x)
+and f is bounded below by -( |tr K| + |C| ) / tau. Combined with
+the boundary, f >= -C_0 / tau.
+
+Thus |f| <= C_0 / tau, hence |h| = tau |f| <= C_0.
+
+Two crucial features of this argument:
+
+ It uses the strict height monotonicity of F: dF/df = tau > 0.
+ Without it, no bound is available.
+
+ The bound C_0 is INDEPENDENT of the deformation parameters
+ N, R, s, and of the solution. It comes purely from tr K, C,
+ and the boundary values, all of which are fixed geometric data.
+
+The height bound is the first of many a priori bounds. It is the
+one that does NOT need the floor; all subsequent bounds (on
+gradient of f, on Z, on t) DO need the floor, as we will see.
+That is why the floor-exclusion argument comes next.
+
+
+7.7 The floor exclusion - the deepest part of the paper
+
+The heart of the whole construction. We must prove:
+
+ For every above-floor smooth solution to any of the four homotopy
+ stages, and for all sufficiently large R (depending on N, eps),
+ the minimum of t on Omega_R is strictly greater than -eps. That
+ is, t never touches the floor.
+
+The proof proceeds by contradiction: suppose t attains its minimum
+-eps at some point x_0. Then the paper shows a contradiction
+between the equations and the choice of small delta_0 and large C_N.
+
+Step 1: A Gauss-type comparison at the minimum.
+
+At any interior minimum of t, we have grad t = 0 and Hess t >= 0.
+Under these constraints, the exact Gauss equation for the graph
+metric ghat reduces to a clean inequality:
+
+ (1/2) e^{4 t} R_ghat <= (1/2) R_g - v Ric_g(e, e)
+ + S(H^f),
+
+where S is the "second fundamental form contribution":
+
+ S(A) = (1/4) (tr_perp A)^2 - (1/2) |A_perp_perp^tf|^2
+ + d A_ee tr_perp A - d |A_e_perp|^2.
+
+for any symmetric tensor A. This is the standard graph submanifold
+Gauss equation applied to the graph of f in the warped product
+Omega x R, at a minimum of t (so the warping factor and its
+derivatives have the "stationary" behaviour).
+
+Step 2: The algebraic identity connecting T and S(H^f).
+
+This is the KEY calculation (Lemma 4.3 of the companion paper, or
+Lemma 8.2 of the 3D paper). At a minimum of t (dt = 0), define the
+"T-minus-S" difference:
+
+ T - S(S) = |T_tf|^2 + (1+d) |M|^2
+ + chi (1 + d - chi/2) q^2
+ - d F q
+ + F^2 / 2
+
+where S = K + H^f is the modified second fundamental form. This
+identity shows that T dominates S(S) up to a term that is
+controllable in terms of F^2 and |K|^2, with a UNIVERSAL constant
+c_* > 0 and a POLYNOMIAL (in N) multiplicative constant.
+
+ T - S(H^f) >= c_* T - C_N ( F^2 + |K|^2 ) [general]
+ T - S(H^f) >= c_* T - C_N v F^2 [K = 0]
+
+The "v = 1 - d" in the second line is important: when v is small
+(near-graph-zero slope), the correction is small.
+
+The proof uses the Cauchy-Schwarz inequality |Fa q| <= chi q^2/2 +
+d^2 F^2/(2 chi), together with the denominator bound
+chi <= (n-1) d / (n-1 + pv) and the fact that p <= N.
+
+Step 3: The equation at the minimum.
+
+Combine the scalar identity (Section 5.2) with the Gauss comparison
+(Section 7.7.1) and the floor-algebra identity (Section 7.7.2).
+At the interior minimum of t:
+
+ u^{-1} div V
+ = mu + J(w) + T + w(F) - F tr K - (1/2) e^{4t} R_ghat (scalar identity)
+ >= mu + J(w) + T + w(F) - F tr K - (1/2) R_g + v Ric - S(H^f) (Gauss)
+ >= (1/2) [ (tr K)^2 - |K|^2 ] + J(w) + w(F) - F tr K
+ + c_* T - C_N ( F^2 + |K|^2 ) - v Ric_g(e, e) (algebra)
+
+After collecting terms using the identities for mu (containing
+tr K, |K|, R_g) and the constraint equation for J, this becomes
+
+ u^{-1} div V >= c_3 * T + tau a sigma - C_N ( rho_0 + v rho_1 )
+
+with c_3 > 0 an absolute constant, rho_0, rho_1 the end weights
+from the penalty, and C_N a polynomial. This is Equation
+5.6-with-lower-bound, extended via the flux-drift correction for the
+homotopy's replacement of V by V_lambda = V - lambda * A_chi*K(w,.).
+
+Step 4: Contradiction.
+
+At the proposed floor minimum, m_0 = 1 (the penalty is active).
+The right side of the divergence equation is
+
+ Xi = delta_0 * T + rho + delta_0 * tau a sigma - C_N(rho_0 + v rho_1).
+
+The left side (from Step 3) is at least
+
+ c_3 * T + tau a sigma - C_N(rho_0 + v rho_1).
+
+So we must have
+
+ c_3 T + tau a sigma - C_N(...) >= delta_0 T + rho + delta_0 tau a sigma - C_N(...)
+
+Rearranging,
+
+ (c_3 - delta_0) T + (1 - delta_0) tau a sigma + (C_N - C_N) (...) - rho >= 0.
+
+Choose delta_0 < min(c_3/2, 1/4). Choose the penalty coefficient
+C_N so large (as a polynomial in N) that the (C_N - C_N) term
+absorbs the -rho term. Then the inequality is IMPOSSIBLE: LHS is
+strictly negative, but RHS (the difference of the two sides of
+the divergence equation) is required to be zero.
+
+Contradiction. No interior minimum can equal -eps.
+
+Step 5: Boundary minimum.
+
+If the minimum of t is attained on the inner boundary B (or outer
+S_R), a similar argument works using the boundary identity
+(Section 5.5). At the boundary, t is decreasing away from B (since
+it's a minimum), so d_nu t < 0. But the boundary flux condition
+gives H + 4 d_nu t = -N_B, so 4 d_nu t = -N_B - H < 0. This is
+consistent, but then the boundary flux V_nu at the minimum has a
+specific value:
+
+ V_nu / u = -H + w_nu(F - P_B) - N_B d = d(-N_B - H) / d + ... = -N_B - H + ...
+
+which does not have the sign the equation demands at the minimum.
+This line of attack is spelled out in the original: comparing the
+boundary identity with the prescribed flux condition yields a
+contradiction.
+
+Step 6: The end.
+
+The floor never touches the outer boundary S_R because the
+explicit end-height estimate (which we'll discuss in the next
+section) shows t > -eps on S_R for large enough R. This is a
+consequence of the fact that at the outer sphere f = Z = 0 and
+the equation for t gives t bounded below by a large constant
+(not -eps).
+
+Conclusion. For delta_0 sufficiently small and C_N sufficiently
+large (both choices POLYNOMIAL in N and independent of R, s, and
+the solution), no smooth solution along the homotopy has t = -eps
+anywhere. In particular, every above-floor smooth solution has
+t > -eps strictly.
+
+This is the deepest and most delicate part of the whole paper.
+Everything else builds on it. The key idea is that the penalty
+C_N (rho_0 + v rho_1) is a large negative contribution that forces
+t away from the floor, and the coercive term delta_0 T is a large
+positive contribution that must be compensated by the penalty if
+the floor is approached. The scalar identity and Gauss equation
+combine to say: approaching the floor requires T to be large, but
+the trace equation only allows T to be large when the penalty is
+also large; and these are contradictory.