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For a compact marginally trapped surface N in M,

analogous results can be proved by studying the

stability operator defined with respect to the direc-

tion .LetJ be the operator defined in terms of a

variation of 

by J = 





. Then

J ¼

 þ2s



Scal

s

þD



j

G

þ





Here, s

= (1=2)hL



, D

iand G

þ

is the Einstein

tensor evaluated on L

, L



. We may call N stable if

the real part of the spectrum of J is non-negative. A

sufficient condition for N to be stable is that N is

locally outermost. This can be formulated, for

example, by requiring that a neighborhood of N in M

contains no trapped surfaces exterior to N.Inthiscase,

assuming that the DEC holds, N is a sphere or a torus,

and if the real part of the spectrum of J is positive then

N is a sphere. If N is a torus, then the ambient

curvature and shear vanishes at N, s

is a gradient, and

N is flat. One expects that in addition, global rigidity

should hold, in analogy with the minimal surface case.

This is an open problem. If N satisfies the stronger

condition of strict stability, which corresponds to the

spectrum of J having positive real part, then N is in the

interior of a hypersurface H of the ambient spacetime,

with the property that it is foliated by marginally

trapped surfaces (Andersson et al. 2005). If the NEC

holds and N has nonvanishing shear, then H is

spacelike at N. A hypersurface H with these proper-

ties is known as a dynamical horizon.

Jang’s Equation

Consider a Cauchy data set (M, g

, K

). Extend K

to a tensor field on M  R, constant in the vertical

direction. Then the equation for a graph

N ¼fðx; tÞ2M  R; t ¼ f ðxÞg

such that N has mean curvature equal to the trace of

the projection of K

to N with respect to the induced

metric on N, is given by

i;j



ð1 þjrf j

1=2



f r

1 þjrf j

¼ 0 ½8

an equation closely related to the equation



= 0. Equation [8] was introduced by P S Jang

(Jang 1978) as part of an attempt to generalize the

inverse mean curvature flow method of Geroch from

time-symmetric to general Cauchy data.

Existence and regularity for Jang’s equation were

proved by Schoen and Yau (1981) and used to

generalize their proof of the positive-mass theorem

from the case of maximal slices to the general case.

The solution to Jang’s equation is constructed as the

limit of the solution to a sequence of regularized

problems. The limit consists of a collection N of

submanifolds of M  R. In particular, component

near infinity is a graph and has the same mass as M.

N may contain vertical components which project

onto marginally trapped surfaces in M, and in fact

these constitute the only possibilities for blow-up of

the sequence of graphs used to construct N. If the

DEC is valid, the metric on N has non-negative

scalar curvature in the weak sense that

Scal



þ 2jr j

> 0

for smooth compactly supported functions . If the

DEC holds strictly, the strict inequality holds and in

this case the metric on N is conformal to a metric

with vanishing scalar curvature.

Jang’s equation can be applied to prove existence

of marginally trapped surfaces, given barriers. Let

(M, g

, K

) be a Cauchy data set containing two

compact surfaces N

, N

which together bound a

compact region M

in M. Suppose the surfaces N

and N

have 

< 0onN

and 

> 0onN

Schoen recently proved the following result.

Theorem 3 (Existence of marginally trapped sur-

faces). Let M

, N

be as above. Then there is a

finite collection of compact, marginally trapped

surfaces {

} contained in the interi or of M

, such

that [

is homologous to N

. If the DEC holds,

then 

is a collection of spheres and tori.

The proof proceeds by solving a sequence of

Dirichlet boundary-value problems for Jang’s equation

with boundary value on N

, N

tending to 1and 1,

respectively. The assumption on 

is used to show the

existence of barriers for Jang’s equation. Let f

be the

sequence of solutions to the Dirichlet problems. Jang’s

equation is invariant under renormalization f

! f

for some sequence c

of real numbers. A Harnack

inequality for the gradient of the solutions to Jang’s

equation is used to show that the sequence of solutions

, possibly after a renormalization, has a subsequence

converging to a vertical submanifold of M

 R, which

projects to a collection 

of marginally trapped

surfaces. By construction, the zero sets of the f

are homologous to N

and N

. The estimates on

the sequence {f

} show that this holds also in the limit

k !1. The statement about the topology of the 

follows by showing, using the above-mentioned

inequality for Scal

, that if DEC holds, the total

Gauss curvature of each surface 

is non-negative.

506 Geometric Analysis and General Relativity

Center of mass

Since by the positive-mass theorem m > 0 unless the

ambient spacetime is flat, it makes sense to consider

the problem of finding an appropriate notion of

center of mass. This problem was solved by Huisken

and Yau who showed that under the asymptotic

conditions [6] the isoperimetric problem has a unique

solution if one considers sufficiently large spheres.

Theorem 4 (Huisken and Yau 1996). There is an

> 0 and a compact region B

such that for each

H 2 (0, H

) there is a unique constant mean curva-

ture sphere S

with mean curvature H contained in

MnB

. The spheres form a foliation.

The proof involves a study of the evolution

equation

¼ðH 



HÞ ½9

where



H is the average mean curvature. This is the

gradient flow for the isoperimetric problem of

minimizing area keeping the enclosed volume con-

stant. The solutions in Euclidean space are standard

spheres. Equation [9] defines a parabolic system, in

particular we have

H ¼ H þðRicð; ÞþjAj

ÞðH 



HÞ

It follows from the fall-off conditions [6] that the

foliation of spheres constructed in Theorem 4 are

untrapped surfaces. They can therefore be used as

outer barriers in the existence result for marginally

trapped surfaces, (Theorem 3).

The mean curvature flow for a spatial hypersur-

face in a Lorentz manifold is also parabolic. This

flow has been applied to construct constant mean

curvature Cauchy hypersurfaces in spacetimes.

Maximal and Related Surfaces

Let N be the hypersurface x

= u(x

, ..., x

)in

Minkowski space R

1þn

with line element  dx

þþdx

. Assume jruj < 1 so that N is

spacelike. Then N is stationary with respect to

variations of area if u solves the equation

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 jruj

¼ 0 ½10

N maximizes area with respect to compactly

supported variations, and hence is called a maximal

surface. As in the case of the minimal surface

equation, eqn [10] and more generally the

Lorentzian prescribed mean curvature equation, is

quasilinear elliptic, but it is not uniformly elliptic,

which makes the regularity theory more subtle.

A Bernstein principle analogous to the one for the

minimal surface equation holds for the maximal

surface equation [10]. Suppose that u is a solution to

[10] which is defined on all of R

. Then u is an

affine function (Cheng and Yau 1976). An impor-

tant tool used in the proof is a Bochner type identity,

originally due to Calabi, for the norm of the second

fundamental form. For a hypersurface in a flat

ambient space, the Codazzi equation states r



= 0. This gives the identity

A

¼r

H þ A

þ A

Ric

½11

The curvature terms can be rewritten in terms of A

if the ambient space is flat. Using [11] to compute

jAj

gives an expression which is quadratic in rA,

and fourth order in jAj, and which allows one to

perform maximum principle estimates on jAj. Gen-

eralizations of this technique for hypersurfaces in

general ambient spaces play an important role in the

proof of regularity of minimal surfaces, and in the

proof of existence for Jang’s equation as well as in

the analysis of the mean curvature flow used to

prove existence of round spheres. The generalization

of eqn [11] is known as a Simons identity.

For the case of maximal hypersurfaces of

Minkowski space, it follows from further maximum

principle estimates that a maximal hypersurface of

Minkowski space is convex, in particular, it has

nonpositive Ricci curvature. Generalizations of this

technique allow one to analyze entire constant mean

curvature hypersurfaces of Minkowski space.

Consider a globally hyperbolic Lorentzian mani-

fold (V, ). A C

hypersurface is said to be weakly

spacelike if timelike curves intersect it in at most one

point. Call a codimension-2 submanifold   V a

weakly spacelike boundary if it bounds a weakly

spacelike hypersurface N

Theorem 5 (Existence for Plateau problem for

maximal surfaces (Bartnik 1988)). Let V be a

globally hyperbolic spacetime and assume that the

causal structure of V is such that the domain of

dependence of an y compact domain in V is compact.

Given a weakly spacelike boundary  in V, there is a

weakly spacelike maximal hypersurface N with  as

its boundary. N is smooth except possibly on null

geodesics connecting points of .

Here, maximal hypersurface is understood in a

weak sense, referring to stationarity with respect to

variations. Due to the nonuniform ellipticity for the

maximal surface equation, the interior regularity

Geometric Analysis and General Relativity 507

which holds for minimal surfaces fails to hold in

general for the maximal surface equation.

A time-oriented spacetime is said to have a crushing

singularity to the past (future) if there is a sequence 

of Cauchy surfaces so that the mean curvature

function H

of 

diverges uniformly to 1(1).

Theorem 6 (Gerhardt 1983). Suppose that (V, ) is

globally hyperbolic with compact Cauchy surfaces

and satisfies the SEC. Then if (V, ) has crushing

singularities to the past and future it is globally

foliated by constant mean curvature hypersurfaces.

The mean curvature  of these Cauchy surfaces is a

global time function.

The proof involves an application of results from

geometric measure theory to an action E of the form

discussed earlier. A barrier argument is used to control

the maximizers. Bartnik (1984, theorem 4.1) gave a

direct proof of existence of a constant mean curvature

(CMC) hypersurface, given barriers. If the spacetime

(V, ) is symmetric, so that a compact Lie group acts

on V by isometries, then CMC hypersurfaces in V

inherit the symmetry. Theorem 6 gives a condition

under which a spacetime is globally foliated by CMC

hypersurfaces. In general, if the SEC holds in a

spatially compact spacetime, then for each  6¼ 0,

there is at most one constant mean curvature Cauchy

surface with mean curvature .IncaseV is vacuum,

Ric

= 0, and 3 þ 1 dimensional, then each point x 2

V is on at most one hypersurface of constant mean

curvature unless V is flat and splits as a metric product.

There are vacuum spacetimes with compact Cauchy

surface which contain no CMC hypersurface

(Chrusciel et al. 2004). The proof is carried out by

constructing Cauchy data, using a gluing argument, on

the connected sum of two tori, such that the resulting

Cauchy data set (M, g

, K

) has an involution which

reverses the sign of K

. The involution extends to the

maximal vacuum development V of the Cauchy data

set. Existence of a CMC surface in V gives, in view of

the involution, barriers which allow one to construct a

maximal Cauchy surface homeomorphic to M.This

leads to a contradiction, since the connected sum of

two tori does not carry a metric of positive scalar

curvature, and therefore, in view of the constraint

equations, cannot be imbedded as a maximal Cauchy

surface in a vacuum spacetime. The maximal vacuum

development V is causally geodesically incomplete.

However, in view of the existence proof for CMC

Cauchy surfaces (cf. Theorem 6), these spacetimes

cannot have a crushing singularity. It would be

interesting to settle the open question whether there

are stable examples of this type.

In the case of a spacetime V which has an

expanding end, one does not expect in general that

the spacetime is globally foliated by CMC hyper-

surfaces even if V is vacuum and contains a CMC

Cauchy surface. This expectation is based on the

phenomenon known as the collapse of the lapse; for

example, the Schwarzschild spacetime does not

contain a global foliation by maximal Cauchy

surfaces (Beig and Murchadha 1998). However, no

counterexample is known in the spatially compact

case. In spite of these caveats, many examples of

spacetimes with global CMC foliations are known,

and the CMC condition, or more generally pre-

scribed mean curvature, is an important gauge

condition for general relativity.

Some examples of situations where global

constant or prescribed mean curvature foliations

are known to exist in vacuum or with some types of

matter are spatially homogeneous spacetimes,

and spacetimes with two commuting Killing fields.

Small data global existence for the Einstein equa-

tions with CMC time gauge have been proved for

spacetimes with one Killing field, with Cauchy

surface a circle bundle over a surface of genus > 1,

by Choquet-Bruhat and Moncrief. Further, for

(3 þ 1)-dimensional spacetimes with Cauchy

surface admitting a hyperbolic metric, small data

global existence in the expanding direction has been

proved by Andersson and Moncrief. See Andersson

(2004) and Rendall (2002) for surveys on the

Cauchy problem in general relativity.

Null Hypersurfaces

Consider an asymptotically flat spacetime contain-

ing a black hole, that is, a region B such that future

causal curves starting in B cannot reach observers at

infinity. The boundary of the trapped region is

called the event horizon H. This is a null hypersur-

face, which under reasonable conditions on causality

has null generators which are complete to the future.

Due to the completeness, assuming that H is

smooth, one can use the Raychaudhuri equation

[7] to show that the null expansion 

of a spatial

cross section of H must satisfy 

 0, and hence

that the area of cross sections of H grows mono-

tonously to the future. A related statement is that

null generators can enter H but may not leave it.

This was first proved by Hawking for the case of

smooth horizons, using essentially the Raychaudhuri

equation. In general H can fail to be smooth.

However, from the definition of H as the boundary

of the trapped region it follows that it has support

hypersurfaces, which are past light cones. This

property allows one to prove that H is Lipschitz

and hence smooth almost everywhere. At smooth

points of H, the calculations in the proof of

508 Geometric Analysis and General Relativity

Hawking apply, and the monotonicity of the area of

cross sections follows.

Theorem 7 (Area theorem (Chrusciel et al. 2001)).

Let H be a black hole event horizon in a smooth

spacetime (M, g). Suppose that the generators are

future complete and the NEC holds on H. Let

, a = 1, 2, be two spacelike cross sections of H and

suppose that S

is to the future of S

. Then

A(S

) A(S

The eikonal equation r



u = 0 plays a central

role in geometric optics. Level sets of a solution u are

null hypersurfaces which correspond to wave fronts.

Much of the recent progress on rough solutions to the

Cauchy problem for quasilinear wave equations is

based on understanding the influence of the geometry

of these wave fronts on the evolution of high-

frequency modes ‘in the background spacetime. In

this analysis many objects familiar from general

relativity, such as the structure equations for null

hypersurfaces, the Raychaudhuri equation, and the

Bianchi identities play an important role, together

with novel techniques of geometric analysis used to

control the geometry of cross sections of the wave

fronts and to estimate the connection coefficients in a

rough spacetime geometry. These techniques show

great promise and can be expected to have a

significant impact on our understanding of the

Einstein equations and general relativity.

Acknowledgments

The author is grateful to, among others, Greg

Galloway, Gerhard Huisken, Jim Isenberg, Piotr

Chrusciel, and Dan Pollack for helpful discussions

concerning the topics covered in this article. The

work of the author has been supported in part by

the NSF under contract number DMS 0407732 with

the University of Miami.

See also: Computational Methods in General Relativity:

The Theory; Einstein Equations: Initial Value

Formulation; Einstein’s Equations with Matter; Geometric

Flows and the Penrose Inequality; Hamiltonian Reduction

of Einstein’s Equations; Holomorphic Dynamics;

Lorentzian Geometry; Minimal Submanifolds; Mirror

Symmetry: A Geometric Survey; Spacetime Topology,

Causal Structure and Singularities; Stability of Minkowski

Space.

Further Reading

Andersson L (2004) The global existence problem in general

relativity. In: Chrusciel P and Friedrich H (eds.) The Einstein

Equations and the Large Scale Behavior of Gravitational Fields,

pp. 71–120. Basel: Birkhauser.

Andersson L, Galloway GJ, and Howard R (1998) A strong maximum

principle for weak solutions of quasi-linear elliptic equations with

applications to Lorentzian and Riemannian geometry. Commu-

nications on Pure and Applied Mathematics 51(6): 581–624.

Andersson L, Mars M, and Simon W (2005) Local existence of

dynamical and trapping horizons. Physical Review Letters 95:

111102.

Bartnik R (1984) Existence of maximal surfaces in asymptotically

flat spacetimes. Communications in Mathematical Physics

94(2): 155–175.

Bartnik R (1988) Regularity of variational maximal surfaces. Acta

Mathematica 161(3–4): 145–181.

Beig R and Murchadha NO

(1998) Late time behavior of the

maximal slicing of the Schwarzschild black hole. Physical

Review D 57(8): 4728–4737.

Cai M and Galloway GJ (2000) Rigidity of area minimizing tori

in 3-manifolds of nonnegative scalar curvature. Communica-

tions in Analysis and Geometry 8(3): 565–573.

Shiu Yuch Cheng and Shing Tung Yau (1976) Maximal space-like

hypersurfaces in the Lorentz–Minkowski spaces. Annals of

Mathematics (2) 104(3): 407–419.

Chrusciel PT, Delay E, Galloway GJ, and Howard R (2001)

Regularity of horizons and the area theorem. Ann. Henri

Poincare´ 2(1): 109–178.

Chrusciel PT, Isenberg J, and Pollack D (2004) Gluing initial data

sets for general relativity. Physical Review Letters 93: 081101.

Federer H (1969) Geometric measure theory. Die Grundlehren der

mathematischen Wissenschaften, Band 153. New York: Springer.

Galloway GJ (2000) Maximum principles for null hypersurfaces and

null splitting theorems. Annales Henri Poincare´ 1(3): 543–567.

Gerhardt C (1983) H-surfaces in Lorentzian manifolds. Commu-

nications in Mathematical Physics 89(4): 523–553.

Gilbarg D and Trudinger NS (1983) Elliptic partial differential

equations of second order. In: Grundlehren der Mathema-

tischen Wissenschaften (Fundamental Principles of Mathema-

tical Sciences), 2nd edn., vol. 224. Berlin: Springer.

Huisken G and Yau ST (1996) Definition of center of mass for

isolated physical systems and unique foliations by stable spheres

with constant mean curvature. Inventiones Mathematicae

124(1–3): 281–311.

Jang PS (1978) On the positivity of energy in general relativity.

Journal of Mathematical Physics 19(5): 1152–1155.

Lawson HB Jr. (1980) Lectures on Minimal Submanifolds, vol. 1.

Mathematics Lecture Series 9 (2nd edn). Wilmington, DE:

Publish or Perish Inc.

Rendall AD (2002) Theorems on existence and global dynamics

for the Einstein equations. Living Reviews in Relativity 5: 62

pp. (electronic).

Schoen R and Yau ST (1981) Proof of the positive mass theorem II.

Communications in Mathematical Physics 79(2): 231–260.

Simon L (1997) The minimal surface equation. In: Geometry, V,

Encyclopaedia of Mathematical Science vol. 90, pp. 239–272.

Berlin: Springer.

Geometric Analysis and General Relativity 509

Geometric Flows and the Penrose Inequality

H Bray, Duke University, Durham, NC, USA

This article was originally published as ‘‘Black holes,

geometric flows, and the Penrose inequality in general

relativity.’’ Notices of the American Mathematical Society,

49(2002), 1372–1381.

Introduction

In a paper, R Penrose (1973) made a physical

argument that the total mass of a spacetime which

contains black holes with event horizons of total area

A should be at least

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

A=16

. An important special

case of this physical statement translates into a very

beautiful mathematical inequality in Riemannian

geometry known as the Riemannian Penrose inequal-

ity. The Riemannian Penrose inequality was first

proved by Huisken and Ilmanen (1997) for a single

black hole and then by the author in 1999 for any

number of black holes. The two approaches use two

different geometric flow techniques. The most general

version of the Penrose inequality is still open.

A natural interpretation of the Penrose inequality

is that the mass contributed by a collection of

black holes is (at least)

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

A=16

. More generally,

the question ‘‘How much matter is in a given region

of a spacetime?’’ is still very much an open problem.

(Christodoulou and Yau 1988). In this paper, we

will discuss some of the qualitative aspects of mass

in general relativity, look at examples which are

informative, and describe the two very geometric

proofs of the Riemannian Penrose inequality.

Total Mass in General Relativity

Two notions of mass which are well understood in

general relativity are local energy density at a point

and the total mass of an asymptotical ly flat space-

time. However, defining the mass of a region larger

than a point but smaller than the entire universe is

not very well understood at all.

Suppose (M

, g) is a Riemannian 3-manifold

isometrically embedded in a (3 þ 1)-dimensional

Lorentzian spacetime N

. Suppose that M

has zero-

second fundamental form in the spacetime. This is a

simplifying assumption which allows us to think of

, g)asa‘‘t = 0’’ slice of the spacetime. (Recall that

the second fundamental form is a measure of how

much M

curves inside N

. M

is also sometimes

called ‘‘totally geodesic’’ since geodesics of N

which

are tangent to M

at a point stay inside M

forever.)

The Penrose inequality (which allows for M

to have

general second fundamental form) is known as the

Riemannian Penrose inequality when the second

fundamental form is set to zero.

We also want to only consider (M

, g) that are

asymptotically flat at infinity, which means that for

some compact set K, the ‘‘end’’ M

nK is diffeo-

morphic to R

(0), where the metric g is

asymptotically approaching (with certain decay

conditions) the standard flat metric 

on R

infinity. The simplest example of an asymptotically

flat manifold is (R

, 

) itself. Other good examples

are the conformal metrics (R

, u(x)



), where u(x)

approaches a constant sufficiently rapidly at infinity.

(Also, sometimes it is convenient to allow (M

, g)to

have multiple asymptotically flat ends, in which

case each connected component of M

nK must

have the property described above.) A qualitative

picture of an asympotically flat 3-manifold is shown

in Figure 1.

The purpose of these assumptions on the asymp-

totic behavior of (M

, g) at infinity is that they imply

the existence of the limit

m ¼

16

lim

!1



i;j

ðg

ij;i



 g

ii;j



Þd

where S



is the coordinate sphere of radius ,  is the

unit normal to S



,andd is the area element of S



in the

coordinate chart. The quantity m is called the ‘ ‘total

mass’ ’ (or ADM mass) of (M

, g) and does not depend

on the choice of asymptotically flat coordinate chart.

The above equation is where many people would

stop reading an article like this. But before you do,

we will promise not to use this definition of the total

mass in this paper. In fact, it turns out that total mass

can be quite well understood with an example. Going

back to the example (R

, u(x)



), if we suppose that

u(x) > 0 has the asymptotics at infinity

uðxÞ¼a þ b=jxjþOð1=jxj

Þ½1

Figure 1 A qualitative picture of an asymptotically flat

3-manifold.

510 Geometric Flows and the Penrose Inequality

(and derivatives of the O(1=jxj

) term are O(1=jxj

)),

then the total mass of (M

, g)is

m ¼ 2ab ½2

Furthermore, suppose ( M

, g) is any metric whose

‘‘end’’ is isometric to (R

nK, u(x)



), where u(x)is

harmonic in the coordinate chart of the end (R

K, 

) and goes to a constant at infinity. Then

expanding u(x) in terms of spherical harmonics

demonstrates that u(x) satisfies condition [1].

We will call these Riemannian manifolds (M

, g)

‘‘harmonically flat at infinity,’’ and we note that the

total mass of these manifolds is also given by eqn [2].

A very nice lemma by Schoen and Yau is that,

given any >0, it is always possible to perturb an

asymptotically flat mani fold to become harmoni-

cally flat at infinity such that the total mass changes

less than  and the metric changes less than

 pointwise, all while maintaining non-negative

scalar curvature. Hence, it happens that to prove

the theorems in this paper, we only need to con sider

harmonically flat manifolds! Thus, we can use eqn

[2] as our definition of total mass. As an example,

note that (R

, 

) has zero total mass. Also, note

that, qualitatively, the total mass of an asymptoti-

cally flat or harmonically flat manifold is the 1=r

rate at which the metric becomes flat at infinity.

The Phenomenon of Gravitational Attraction

What do the above definitions of total mass have to

do with anything physical? That is, if the total mass

is the 1=r rate at which the metric beco mes flat at

infinity, what does this have to do with our real-

world intuitive idea of mass?

The answer to this question is very nice. Given a

Schwarzschild spacetime metric



; 1 þ

2jxj



þ dx





1  m=2jxj

1 þ m=2jxj





jxj > m=2, for example, note that the t = 0 slice

(which has zero-second fundamental form) is the

spacelike Schwarzschild metric



n B

m=2

ð0Þ; 1 þ

2jxj







(discussed more later). Note that according to eqn

[2], the parameter m is in fact the total mass of this

3-manifold.

On the other hand, suppose we wer e to release a

small test particle, initially at rest, a large distance r

from the center of the Schwarzschild spacetime. If

this particle is not acted upon by external forces,

then it should follow a geodesic in the spacetime. It

turns out that with respect to the asymptotically flat

coordinate chart, these geodesics ‘‘accelerate’’

towards the middle of the Schwarzschild metric

proportional to m=r

(in the limit as r goes to

infinity). Thus, our Newtonian notion of mass also

suggests that the total mass of the spacetime is m.

Local Energy Density

Another quantification of mass which is well under-

stood is local energy density. In fact, in this setting,

the local energy density at each point is

 ¼

16

where R is the scalar curvature of the 3-manifold

(which has zero-second fundamental form in the

spacetime) at each point. Note that (R

, 

) has zero

energy density at each point as well as zero total mass.

This is appropriate since (R

, 

)isinfacta‘‘t = 0’’

slice of Minkowski spacetime, which represents a

vacuum. Classically, physicists consider   0tobea

physical assumption. Hence, from this point on, we

will not only assume that (M

, g) is asymptotically flat,

but also that it has non-negative scalar curvature,

R  0

This notion of energy density also helps us

understand total mass better. After all, we can take

any asymptotically flat manifold and then change

the metric to be perfectly flat outside a large

compact set, thereby giving the new metric zero

total mass. However, if we introduce the physical

condition that both metrics have non-negative scalar

curvature, then it is a beautiful theorem that this is

in fact not possible, unless the original metric was

already (R

, 

)! (This theorem is actually a corollary

to the positive mass theorem discussed below.)

Thus, the curvature obstruction of having non-

negative scalar curvature at each point is a very

interesting condition.

Also, notice the indirect connection between the

total mass and local energy density. At this point,

there does not seem to be much of a connection at

all. The total mass is the 1=r rate at which the metric

becomes flat at infinity, and local energy density is

the scalar curvature at each point. Furthermore, if a

metric is changed in a compact set, local energy

density is changed, but the total mass is unaffected.

The reason for this is that the total mass is ‘‘not’’

the integral of the local energy density over the

manifold. In fact, this integral fails to take potential

energy into account (which woul d be expected to

Geometric Flows and the Penrose Inequality 511

contribute a negative energy) as well as gravitational

energy. Hence, it is not initially clear what we should

expect the relationship between total mass and local

energy density to be, so let us begin with an example.

Example Using Superharmonic Functions in R

Once again, let us return to the (R

, u(x)



)

example. The formula for the scalar curvature is

R ¼8uðxÞ

5

uðxÞ

Hence, since the physical assumption of non-

negative energy density implies non-negative scalar

curvature, we see that u(x) > 0 must be super-

harmonic (u  0). For simplicity, let us also

assume that u( x) is harmonic outside a bounded set

so that we can expand u(x) at infinity using

spherical harmonics. Hence, u(x) has the asympto-

tics of eqn [1]. By the maximum principle, it follo ws

that the minimum value for u(x) must be a, referring

to eqn [1]. Hence, b  0, which implies that m  0!

Thus, we see that the assumption of non-negative

energy density at each point of (R

, u(x)



) implies

that the total mass is also non-negative, which is

what one would hope.

The Positive Mass Theorem

Why would one hope this? What would be the

difference if the total mass were negative? This

would mean that a gravitational system of positive

energy density could collectively act as a net

negative total mass. This phenomenon has not

been observed experimentally, and so it is not a

property that we would hope to find in general

relativity.

More generally, suppose we have any asymptotically

flat manifold with non-negative scalar curvature, is it

true that the total mass is also non-negative? The

answer is yes, and this fact is know as the positive mass

theorem, first proved by Schoen and Yau (1979) using

minimal surface techniques and then by Witten (1981)

using spinors. In the zero-second fundamental form

case, the positive mass theorem is known as the

Riemannian positive mass theorem and is stated below.

Theorem 1 (Schoen, Yau). Let (M

, g) be any

asymptotically flat, complete Riemannian manifold

with non-negative scalar curvature. Then the total

mass m  0, with equality if and only if (M

, g) is

isometric to (R

, ).

Gravitational Energy

The previous example neglects to illustrate some of

the subtleties of the positive mass theorem. For

example, it is easy to construct asymptotical ly flat

manifolds (M

, g) (not conform al to R

) which have

zero scalar curvature everywhere and yet have

‘‘nonzero’’ total mass. By the positive mass theorem,

the mass of these manifolds is positive. Physically,

this corresponds to a spacetime with zero energy

density everywhere which still has positive total

mass. From where did this mass come? How can a

vacuum have positive total mass?

Physicists refer to this extra energy as gravita-

tional energy. There is no known local definition of

the energy densi ty of a gravitational field, and

presumably such a definition does not exist. The

curious phenomenon, then, is that for some reason,

gravitational energy always makes a non-negative

contribution to the total mass of the system.

Black Holes

Another very interesting and natural phenomenon in

general relativity is the existence of black holes.

Instead of thinking of black holes as singularities in

a spacetime, we will think of black holes in terms of

their horizons. For example, suppose we are explor-

ing the universe in a spacecraft capable of traveling

at any speed less than the speed of light. If we are

investigating a black hole, we would want to make

sure that we don’t get too close and get trapped by

the ‘‘gravitational forces’’ of the black hole. In fact,

we could imagine a ‘‘sphere of no return’’ beyond

which it is impossible to escape from the black hole.

This is called the event horizon of a black hole.

However, one limitation of the notion of an event

horizon is that it is very hard to determine its location.

One way is to let daredevil spacecraft see how close

they can get to the black hole and still escape from it

eventually. The only problem with this approach

(besides the cost in spacecraft) is that it is hard to

know when to stop waiting for a daredevil spacecraft

to return. Even if it has been 50 years, it could be that

this particular daredevil was not trapped by the black

hole but got so close that it will take it 1000 or more

years to return. Thus, to define the location of an event

horizon even mathematically, we need to know the

entire evolution of the spacetime. Hence, event

horizons can not be computed based only on the

local geometry of the spacetime.

This problem is solved (at least for the mathema-

tician) with the notion of apparent horizons of black

holes. Given a surface in a spacetime, suppo se that it

emits an outward shell of light. If the surface area of

this shel l of light is decreasing everywhere on the

surface, then this is called a trapped surface. The

outermost boundary of these trapped surfaces is

called the apparent horizon of the black hole.

Apparent horizons can be computed based on their

512 Geometric Flows and the Penrose Inequality

local geometry, and an apparent horizon always

implies the existence of an event horizon outside of

it (Hawking and Ellis 1973).

Now let us return to the case we are considering

in this paper where (M

, g)isa‘‘t = 0’’ slice of a

spacetime with zero-sec ond fundamental form. Then

it is a very nice geometric fact that apparent

horizons of black holes intersected with M

corres-

pond to the connected components of the outermost

minimal surface 

of (M

, g).

All of the surfaces we are considering in this paper

will be required to be smooth boundaries of open

bounded regio ns, so that outermost is well defined

with respect to a chosen end of the manifold.

A minimal surface in (M

, g) is a surface which is a

critical point of the area function with respect to any

smooth variation of the surface. The first variational

calculation implies that minimal surfaces have zero

mean curvature. The surface 

of (M

, g) is defined

as the boundary of the union of the open regions

bounded by all of the minimal surfaces in (M

, g). It

turns out that 

also has to be a minimal surface,

so we call 

the ‘‘outermost minimal surface.’’

A qualitative sketch of an outermost minimal

surface of a 3-manifold is shown in Figure 2.

We will also define a surface to be ‘‘(strictly) outer

minimizing’’ if every surfa ce which encloses it has

(strictly) greater area. Note that outermost minimal

surfaces are strictly outer minimizing. Also, we define

a ‘‘horizon’’ in our context to be any minimal surface

which is the boundary of a bounded open region.

It also follows from a stability argument (using

the Gauss–Bonnet theorem interestingly) that each

component of an outermost minimal surface (in a

3-manifold with non-negative scalar curvature) must

have the topology of a sphere. Furthermore, there is

a physical argument, based on Penrose (1973),

which suggests that the mass contributed by the

black holes (thought of as the conn ected compo-

nents of 

) should be defined to be

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

=16

where A

is the area of 

. Hence, the physical

argument that the total mass should be greater than

or equal to the mass contributed by the black holes

yields the following geometric statement.

The Riemannian Penrose Inequality Let (M

, g)be

a complete, smooth, 3-manifold with non-negative

scalar curvature which is harmonically flat at

infinity with total mass m and which has an

outermost minimal surface 

of area A

. Then,

m 

ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

16

½3

with equality if and only if (M

, g) is isometric to the

Schwarzschild metric



nf0g; 1 þ

2jxj







outside their respective outermost minimal surfaces.

The above statement has been proved by the

present author, and Huisken and Ilmanen proved it

when A

is defined instead to be the area of the

largest connected component of 

. We will discuss

both approaches in this paper, which are very

different, although they both involve flowing sur-

faces and/or metrics.

We also clarify that the above statement is with

respect to a chosen end of (M

, g), since both the

total mass and the definition of outermost refer to a

particular end. In fact, nothing very important is

gained by considering manifolds with more than one

end, since extra ends can always be compactified by

connect summing them (around a neighborhood of

infinity) with large spheres while still preserving non-

negative scalar curvature, for example. Hence, we

will typically consider manifolds with just one end. In

the case that the manifold has multiple ends, we will

require every surface (which could have multiple

connected components) in this paper to enclose all of

the ends of the manifold except the chosen end.

The Schwarzschild Metric

The Schwarzschild metric



nf0g; 1 þ

2jxj







referred to in the above statement of the Rieman-

nian Penrose inequality, is a particularly important

example to consider, and corresponds to a zero-

second fundamental form, spacelike slice of the

usual (3 þ 1)-dimensional Schwarzschil d metric

(which represents a spherically symmetric static

black hole in vacuum). The three-dimensional

Figure 2 A qualitative sketch of an outermost minimal surface

of a 3-manifold.

Geometric Flows and the Penrose Inequality 513

Schwarzschild metrics have total mass m > 0 and

are characterized by being the only spherically

symmetric, geodesically complete, zero scalar curva-

ture 3-metrics, other than (R

, 

). They can also be

embedded in four-dimensional Euclidean space

(x, y, z, w) as the set of points satisfying

jðx; y; zÞj ¼ ðw

=8mÞþ2m

which is a parabola rotated around an S

. This last

picture allows us to see that the Schwarzschild

metric, which has two ends, has a Z

symmetry

which fixes the sphere with w = 0 and j(x, y, z)j=

2m, which is clearly minimal. Furthermore, the area

of this sphere is 4(2m)

, giving equality in the

Riemannian Penrose inequali ty.

A Brief History of the Problem

The Riemannian Penrose inequality has a rich

history spanning nearly three decades and has

motivated much interesting mathematics and phy-

sics. In 1973, R Penrose in effect conjectured an

even more general version of inequality [3] using a

very clever physical argument, which we will not

have room to repeat here (Penrose 1973). His

observation was that a counterexample to inequality

[3] would yield Cauchy data for solving the Einstein

equations, the solution to which would likely violate

the cosmic censor conjecture (which says that

singularities generically do not form in a spacetime

unless they are inside a black hole).

Jang and Wald (1977), extending ideas of Geroch,

gave a heuristic proof of inequality [3] by defining a

flow of 2-surfaces in (M

, g) in which the surfaces

flow in the outward normal direction at a rate equal

to the inverse of their mean curvatures at each point.

The Hawking mass of a surface (which is supposed

to estimate the total amount of energy inside the

surface) is defined to be

Hawking

ðÞ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

jj

16

1 

16





(where jj is the area of  and H is the mean

curvature of  in (M

, g)) and, amazingly, is

nondecreasing under this ‘‘inverse mean curvature

flow.’’ This is seen by the fact that under inverse

mean curvature flow, it follows from the Gauss

equation and the second variation formula that

Hawking

ðÞ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

jj

16



16



þR  2K þ

ð

 



when the flow is smooth, where R is the scalar

curvature of (M

, g), K is the Gauss curvature of the

surface , and 

and 

are the eigenvalues of the

second fundamental form of , or principle curva-

tures. Hence,

R  0

and



K  4 ½4

(which is true for any con nected surface by the

Gauss–Bonnet theorem) imply

Hawking

ðÞ0 ½5

Furthermore,

Hawking

ð

Þ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

j

16

since 

is a minimal surface and has zero mean

curvature. In addition, the Hawking mass of suffi-

ciently round spheres at infinity in the asymptotically

flat end of (M

, g) approaches the total mass m. Hence,

if inverse mean curvature flow beginning with 

eventually flows to sufficiently round spheres at

infinity, inequality [3] follows from inequality [5].

As noted by Jang and Wald, this argument only

works when inverse mean curvature flow exists and

is smooth, which is generally not expected to be the

case. In fact, it is not hard to construct manifolds

which do not admit a smooth inverse mean

curvature flow. The problem is that if the mean

curvature of the evolving surface becom es zero or is

negative, it is not clear how to define the flow.

For 20 years, this heuristic argument lay dormant

until the work of Huisken and Ilmanen in 1997. With

a very clever new approach, Huisken and Ilmanen

discovered how to reformulate inverse mean curvature

flow using an energy minimization principle in such a

way that the new generalized inverse mean curvature

flow always exists. The added twist is that the surface

sometimes jumps outward. However, when the flow is

smooth, it equals the original inverse mean curvature

flow, and the Hawking mass is still monotone. Hence,

as will be described in the next section, their new flow

produced the first complete proof of inequality [3] for

a single black hole.

Coincidentally, the author found another proof of

inequality [3], submitted in 1999, which works for any

number of black holes. The approach involves flowing

the original metric to a Schwarzschild metric (outside

the horizon) in such a way that the area of the

outermost minimal surface does not change and the

514 Geometric Flows and the Penrose Inequality

total mass is nonincreasing. Then, since the Schwarzs-

child metric gives equality in inequality [3],the

inequality follows for the original metric.

Fortunately, the flow of metrics which is defined

is relatively simple, and in fact stays inside the

conformal class of the original metric. The outer-

most minimal surface flows outwards in this

conformal flow of metrics, and encloses any

compact set (and hence all of the topology of the

original metric) in a finite amount of time. Further-

more, this conformal flow of metrics preserves non-

negative scalar curvature. We will describe this

approach later in the paper.

Other contributions on the Penrose conjecture

have also been made by Herzlich using the Dirac

operator which Witten used to prove the positive

mass theorem, by Gibbons in the special case of

collapsing shells, by Tod, by Bartnik for quasi-

spherical metrics, and by the present author using

isoperimetric surfaces. There is also some interesting

work of Ludvigsen and Vickers using spinors and

Bergqvist, both concerning the Penrose inequality

for null slices of a spacetime.

Inverse Mean Curvature Flow

Geometrically, Huisken and Ilmanen’s idea can be

described as follows. Let (t) be the surface

resulting from inverse mean curvature flow for

time t beginning with the minimal surface 

Define



(t) to be the outermost minimal area

enclosure of (t). Typically,  (t) =



(t) in the flow,

but in the case that the two surfaces are not equal,

immediately replace (t ) with



(t) and then con-

tinue flowing by inverse mean curvature.

An immediate consequence of this modified flow is

that the mean curvature of



(t)isalwaysnon-negative

by the first variation formula, since otherwise



(t)

would be enclosed by a surface with less area. This is

becauseifweflowasurface in the outward

direction with speed , the first variation of the area



H,whereH is the mean curvature of .

Furthermore, by stability, it follows that in the

regions where



(t) has zero mean curvature, it is

always possible to flow the surface out slightly to

have positive mean curvature, allowing inverse mean

curvature flow to be defined, at least heuristically at

this point.

Furthermore, the Hawking mass is still monotone

under this new modified flow. Notice that when (t)

jumps outwards to



(t),



ðtÞ



ðtÞ

since



(t) has zero mean curvature where the two

surfaces do not touch. Furthermore,



ðtÞj ¼ jðtÞj

since (this is a neat argument) j



(t)jj(t)j (since



(t) is a minimal area enclosure of (t)) and we

cannot have j



(t)j < j(t)j since (t) would have

jumped outwards at some earlier time. This is only a

heuristic argument, but we can then see that the

Hawking mass is nondecreasing during a jump by

the above two equations.

This new flow can be rigorously defined, always

exists, and the Hawking mass is monotone. Huisken

and Ilmanen define (t) to be the level sets of a

scalar valued function u(x) defined on (M

, g) such

that u(x) = 0 on the original surface 

and satisfies

div

jruj



¼jruj½6

in an appropri ate weak sense. Since the left-hand

side of the above equation is the mean curvature of

the level sets of u(x) and the right-hand side is the

reciprocal of the flow rate, the above equation

implies inverse mean curvature flow for the level sets

of u(x) when jru(x)j 6¼ 0.

Huisken and Ilmanen use an energy minimization

principle to define weak solutions to eqn [6].

Equation [6] is said to be weakly satisfied in  by

the locally Lipschitz function u if for all locally

Lipschitz v with {v 6¼ u}  ,

ðuÞJ

ðvÞ

where

ðvÞ :¼



jrvjþvjruj

It can then be seen that the Euler–Lagrange equation

of the above energy functional yields eqn [6].

In order to prove that a solution u exists to the above

two equations, Huisken and Ilmanen regularize the

degenerate elliptic equation 6 to the elliptic equation

div

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

jruj

þ 

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

jruj

þ 

Solutions to the above equation are then shown to

exist using the existence of a subsolution, and then

taking the limit as  goes to zero yields a weak

solution to eqn [6]. There are many details which we

are skipping here, but these are the main ideas.

As it turns out, weak solutions u(x)toeqn [6]

often have flat regions where u(x) equals a

constant. Hence, the level sets (t)ofu(x) will be

Geometric Flows and the Penrose Inequality 515