BOUNDARY GRAPH DEFORMATIONS FOR THE SPACETIME PENROSE INEQUALITY
A COMPLETE WALKTHROUGH
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.