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discontinuous in t in this case, which corresponds

to the ‘‘jumping out’’ phenomenon referred to at

the beginning of this section.

We also note that since the Hawking mass of the

level sets of u(x) is monotone, this inverse mean

curvature flow technique not only proves the

Riemannian Penrose inequality, but also gives a

new proof of the positive mass theorem in dimen-

sion 3. This is seen by letting the initial surface be a

very small, round sphere (which will have approxi-

mately zero Hawking mass) and then flowing by

inverse mean curvature, thereby proving m  0.

The Huisken and Ilmanen inverse mean curvat ure

flow also seems ideally suited for proving Penrose

inequalities for 3-manifolds which have R 6 and

which are asymptotically hyperbolic. This situation

occurs if (M

, g) is chosen to be a constant mean

curvature slice of the spacetime or if the spacetime is

defined to solve the Einstein equation with nonzero

cosmological constant. In these cases, there exists a

modified Hawking mass which in monotone under

inverse mean curvature flow which is the usual

Hawking mass plus 4(jj=16)

3=2

. However, because

the monotonicity of the Hawking mass relies on the

Gauss–Bonnet theorem, these arguments do not work

in higher dimensions, at least so far. Also, because

of the need for eqn [4], inverse mean curvature

flow only proves the Riemannian Penrose inequality

for a single black hole. In the next section, we

present a technique which proves the Riemannian

Penrose inequality for any number of black holes,

and which can likely be generalized to higher

dimensions.

The Conformal Flow of Metrics

Given any initial Riem annian manifold (M

, g

)

which has non-negative scalar curvature and which

is harmonically flat at infinity, we will define a

continuous, one-parameter family of metrics (M

, g

0  t < 1. This family of metrics will converge to a

three-dimensional Schwarzschild metric and will have

other special properties which will allow us to prove

the Riemannian Penrose inequality for the original

metric (M

, g

In particular, let 

be the outermost minimal

surface of (M

, g

)withareaA

. Then, we will also

defineafamilyofsurfaces(t)with(0) =

such

that (t) is minimal in (M

, g

). This is natural since

as the metric g

changes, we expect that the location

of the horizon (t)willalsochange.Then,the

interesting quantities to keep track of in this flow are

A(t), the total area of the horizon (t)in(M

, g

and m(t), the total mass of (M

, g

) in the chosen end.

In addition to all of the metrics g

having non-

negative scalar curvature, we will also have the very

nice properties that

ðtÞ¼0

ðtÞ0

for all t  0. Then, since (M

, g

) converges to a

Schwarzschild metric (in an appropriate sense)

which gives equality in the Riemannian Penrose

inequality as described in the intro duction,

mð0Þmð1Þ ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Að1Þ

16

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Að0Þ

16

½7

which proves the Riemannian Penrose inequality for

the original metric (M

, g

). The hard part, then, is

to find a flow of metrics which preserves non-

negative scalar curvature and the area of the

horizon, decreases total mass, and converges to a

Schwarzschild metric as t goes to infinity.

The Definition of the Flow

In fact, the metrics g

will all be conform al to g

This conformal flow of metrics can be thought of as

the solution to a first-order ODE in t defined by

eqns [8]–[11]. Let

¼ u

ðxÞ

½8

and u

(x)  1. Given the metri c g

, define

ðtÞ¼the outermost minimal area

enclosure of 

in ðM

; g

Þ½9

where 

is the original outer minimizing horizon in

, g

). In the cases in which we are interested, (t)

will not touch 

, from which it follows that (t)is

actually a strictly outer minimizing horizon of (M

, g

Then given the horizon (t), define v

(x) such that



ðxÞ0 outside ðtÞ

ðxÞ¼0onðtÞ

lim

x!1

ðxÞ¼e

t

½10

and v

(x)  0 inside (t). Finally, given v

(x), define

ðxÞ¼1 þ

ðxÞds ½ 11

so that u

(x) is continuous in t and has u

(x)  1.

Note that eqn [11] implies that the first-order rate

of change of u

(x)isgivenbyv

(x). Hence, the first-

order rate of change of g

is a function of itself, g

and v

(x) which is a function of g

, t,and(t)which

is in turn a function of g

and 

. Thus, the first-order

rate of change of g

is a function of t, g

, g

,and

516 Geometric Flows and the Penrose Inequality

Theorem 2 Taken together, eqns [8]–[11] define a

first-order ODE in t for u

(x) which has a solution

which is Lipschitz in the t variable, C

in the x

variable everywhere, and smooth in the x variable

outside (t). Furthermore, (t) is a smooth, strictly

outer minimizing horizon in (M

, g

) for all t  0,

and (t

) encloses but does not touch (t

) for all

> t

 0.

Since v

(x) is a superharmonic function in (M

, g

)

(harmonic everywhere except on (t), where it is

weakly superharmonic), it follows that u

(x) is super-

harmonic as well. Thus, from eqn [11] we see that

lim

x !1

(x) = e

t

and consequently that u

(x) > 0

for all t by the maximum principle. Then, since

Rðg

Þ¼u

ðxÞ

5

ð8

þ Rðg

ÞÞu

ðxÞ½12

it follows that ( M

, g

) is an asymptotically flat

manifold with non-negative scalar curvature.

Even so, it still may not seem like g

is particularly

naturally defined since the rate of change of g

appears

to depend on t and the original metric g

in eqn [10].

We would prefer a flow where the rate of change of g

can be defined purely as a function of g

(and 

perhaps), and interestingly enough this actually does

turn out to be the case! The present author has proved

this very important fact and defined a new equivalence

class of metrics called the harmonic conformal class.

Then, once we decide to find a flow of metrics which

stays inside the harmonic conformal class of the

original metric (outside the horizon) and keeps the

area of the horizon (t) constant, then we are basically

forced to choose the particular conformal flow of

metrics defined above.

Theorem 3 The function A(t) is constant in t and

m(t) is nonincreasing in t, for all t  0.

The fact that A

(t) = 0 follows from the fact that

to first order the metric is not changing on (t)

(since v

(x) = 0 there) and from the fact that to first

order the area of (t) does not change as it moves

outward since (t) is a critical point for area in

, g

). Hence, the interesting part of Theorem 3 is

proving that m

(t)  0. Curiously, this follows from

a nice trick using the Riemannian positive mass

theorem, which we describe later.

Another important aspect of this conformal flow of

themetricisthatoutsidethehorizon(t), the manifold

, g

) becomes more and more spherically sym-

metric and ‘‘approaches’’ a Schwarzschild manifold

n{0}, s) in the limit as t goes to 1. More precisely,

Theorem 4 For sufficiently large t, there exists a

diffeomorphism 

between (M

, g

) outside the

horizon (t) and a fixed Schwarzschild manifold

n{0}, s) outside its horizon. Furthermore, for all

>0, there exists a T such that for all t > T, the

metrics g

and 



(s)(when determining the lengths

of unit vectors of (M

, g

)) are within  of each other

and the total masses of the 2- manifolds are within

 of each other. Hence,

lim

t!1

mðt Þ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

AðtÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

16

Theorem 4 is not that surprising really although a

careful proof is reasonably long. However, if one is

willing to believe that the flow of metrics converges

to a spherically symmetric metric outside the

horizon, then Theorem 4 follows from two facts.

The first fact is that the scalar curvature of (M

, g

)

eventually becomes identically zero outside the

horizon (t) (assuming (M

, g

) is harmonically

flat). This follows from the facts that (t) encloses

any compac t set in a finite amount of time, that

harmonically flat manifolds have zero scalar curva-

ture outside a compact set, that u

(x) is harmonic

outside (t), and eqn [12] . The second fact is that

the Schwarzschild metrics are the only complete,

spherically symmetric 3-manifolds with zero scalar

curvature (except for the flat metric on R

The Riemannian Penrose inequal ity, inequality

[3], then follows from eqn [7] using Theorems 2–4,

for harmonically flat manifolds. Since asym ptoti-

cally flat manifolds can be approximated arbitrarily

well by harmonically flat manifolds while changing

the relevant quantities arbitrarily little, the asymp-

totically flat case also follows. Finally, the case of

equality of the Penrose inequality follows from a

more careful analysis of these same arguments.

Qualitative Discussion

Figures 3 and 4 are meant to help illustrate some of the

properties of the conformal flow of the metric. Figure 3

is the original metric which has a strictly outer

minimizing horizon 

.Ast increases, (t)moves

outwards, but never inwards. In Figure 4,wecan

observe one of the consequences of the fact that

A(t) = A

is constant in t. Since the metric is not

changing inside (t), all of the horizons (s), 0  s  t

Σ(0) = Σ

Σ(t )

, g

)

Figure 3 Original metric having a strictly outer minimizing

horizon 

Geometric Flows and the Penrose Inequality 517

have area A

in (M

, g

). Hence, inside (t), the

manifold (M

, g

) becomes cylinder-like in the sense

that it is laminated (i.e., foliated but with some gaps

allowed) by all of the previous horizons which all have

the same area A

with respect to the metric g

Now let us suppose that the original horizon 

of (M

, g) had two components, for example. Then

each of the components of the horizon will move

outwards as t increases, and at some point before

they touch they will suddenly jump outwards to

form a horizon with a single component enclosing

the previous horizon with two components. Even

horizons with only one component will sometimes

jump outwards, but no more than a countable

number of times. It is interesting that this phenom-

enon of surfaces jumping is also found in the

Huisken–Ilmanen approach to the Penrose conjec-

ture using their generalized 1=H flow.

Proof that m

(t)  0

The most surprising aspect of the flow defined

earlier is that m

(t)  0. As mentioned in that

section, this important fact follows from a nice

trick using the Riemannian positive mass theorem.

The first step is to realize that while the rate of

change of g

appears to depend on t and g

, this is in

fact an illusion. As described in detail by Bray, the

rate of change of g

can be described purely in terms

of g

(and 

). It is also true that the rate of change

of g

depends only on g

and (t). Hence, there is no

special value of t, so proving m

(t)  0 is equivalent

to proving m

(0)  0. Thus, without loss of genera l-

ity, we take t = 0 for convenience.

Now expand the harmonic function v

(x), defined

in eqn [10], using spherical harmonics at infinity, to get

ðxÞ¼1 þ

jxj

þO

jxj

½13

for some constant c. Since the rate of change of the

metric g

at t = 0 is given by v

(x) and since the total

mass m(t) depends on the 1/r rate at which the

metric g

becomes flat at infinity (see eqn [2]), it is

not surprising that direct calculation gives us that

ð0Þ¼2ðc  mð0ÞÞ

Hence, to show that m

(0)  0, we need to show

that

c  mð0Þ½14

In fact, cou nterexamples to eqn [14] can be found

if we remove either of the requirements that (0)

(which is used in the definition of v

(x)) be a

minimal surface or that (M

, g

) have non-negative

scalar curvature. Hence, we quickly see that eqn

[14] is a fairly deep conjecture which says something

quite interesting about manifold with non-negative

scalar curvature. Well, the Riemannian positive

mass theorem is also a deep conjecture which says

something quite interesting abo ut manifolds with

non-negative scalar curvature. Henc e, it is natural to

try to use the Riemannian positive mass theorem to

prove eqn [14].

Thus, we want to create a manifold whose total

mass depends on c from eqn [13]. The idea is to use

a reflection trick similar to one used by Bunt ing and

Masood-ul-Alam (1987) for another purpose. First,

remove the region of M

inside (0) and then reflect

the remainder of (M

, g

) through (0). Define the

resulting Riemannian manifold to be (



) which

has two asymptotically flat ends since (M

, g

) has

exactly one asymptotically flat end not contained by

(0). Note that (



) has non-negative scalar

curvature everywhere except on (0) where the

metric has corners. In fact, the fact that (0) has

zero mean curvature (since it is a minimal surface)

implies that (



) has ‘‘distributional’’ non-

negative scalar curvature everywhere, even on (0).

This notion is made rigorous by Bray. Thus, we have

used the fact that (0) is minimal in a critical way.

Recall from eqn [10] that v

(x) was defined to be

the harmonic function equal to zero on (0) which

goes to 1 at infinity. We want to reflect v

(x)tobe

defined on all of (



). The trick here is to define

(x)on(



) to be the harmoni c function which

goes to 1 at infinity in the original end and goes to

1 at infinity in the reflect end. By symmetry, v

(x)

equals 0 on (0) and so agrees with its original

definition on (M

, g

The next step is to compactify one end of (



By the maximum principle, we know that v

(x) > 1

and c > 0, so the new Riemannian manifold (



(x) þ 1)



) does the job quite nicely and compac-

tifies the original end to a point. In fact, the

compactified point at infinity and the metric there

Σ(0) = Σ

Σ(t )

, g

)

Figure 4 Metric after time t.

518 Geometric Flows and the Penrose Inequality

can be filled in smoothly (using the fact that (M

, g

)is

harmonically flat). It then follows from eqn [12] that

this new compactified manifold has non-negative

scalar curvature since v

(x) þ 1isharmonic.

The last step is simply to apply the Riemannian

positive mass theorem to (



,(v

(x) þ 1)



). It is

not surprising that the total mass

m(0) of this

manifold involves c, but it is quite lucky that direct

calculation yields

mð0Þ¼4ðc  mð0ÞÞ

which must be positive by the Riemannian positive

mass theorem. Thus, we have that

ð0Þ¼2ðc  mð0ÞÞ ¼ 

mð0Þ0

Open Questions and Applications

Now that the Riemannian Penrose conjecture has been

proved, what are the next interesting directions? What

applications can be found? Is this subject only of

physical interest, or are there possibly broader

applications to other problems in mathematics?

Clearly, the most natural open problem is to find a

way to prove the general Penrose inequality in which

is allowed to have any second fundamental form in

the spacetime. There is good reason to think that this

may follow from the Riemannian Penrose inequality,

although this is a bit delicate. On the other hand, the

general positive mass theorem followed from the

Riemannian positive mass theorem as was originally

shown by Schoen and Yau using an idea due to Jang.

For physicists, this problem is definitely a top priority

since most spacetimes do not even admit zero-second

fundamental form spacelike slices.

Another interesting question is to ask these same

questions in higher dimensions. The author is currently

working on a paper to prove the Riemannian Penrose

inequality in dimensions <8. Dimension 8 and higher

are harder because of the surprising fact that minimal

hypersurfaces (and hence apparent horizons of black

holes) can have codimension 7 singularities (points

where the hypersurface is not smooth). This curious

technicality is also the reason that the positive mass

conjecture is still open in dimensions 8 and higher for

manifolds which are not spin.

Naturally, it is harder to tell what the applications

of these techniques might be to other problems, but

already there have been some. One application is to

the famous Yamabe problem: given a compact

3-manifold M

, define E(g) =

where g is

scaled so that the total volume of (M

, g)is1,R

the scalar curvature at each point, and dV

is the

volume form. An idea due to Yamabe was to try to

construct canonical metrics on M

by finding critical

points of this energy functional on the space of metrics.

Define C(g) to be the infimum of E(



g) over all metrics



conformal to g. Then the (topological) Yamabe

invariant of M

, denoted here as Y(M

), is defined to

be the supremum of C(g) over all metrics g. Y(S

) = 6 

(2

)

2=3

 Y

is known to be the largest possible value

for Yamabe invariants of 3-manifolds. It is also known

that Y(T

) = 0andY(S

 S

) = Y

= Y(S

S

where S

S

is the nonorientable S

bundle over S

The author, working with Andre Neves on a

problem suggested by Richard Schoen, recently was

able to compute the Yamabe invariant of RP

using

inverse mean curvature flow techniques and found

that Y(RP

) = Y

2=3

 Y

. A corollary is Y(RP



) = Y

as well. These techniques also yield the

surprisingly strong result that the only prime 3-mani-

folds with Yamabe invariant larger than RP

are

, S

 S

,andS

S

. The Poincare conjecture for

3-manifolds with Yamabe invariant greater than RP

is therefore a corollary. Furthermore, the problem of

classifying 3-manifolds is known to reduce to the

problem of classifying prime 3-manifolds. The

Yamabe approach then would be to make a list of

prime 3-manifolds ordered by Y. The first five prime

3-manifolds on this list are therefore S

, S



, S

S

, RP

,andRP

 S

Acknowledgments

This research was supported in part by NSF

grants #DMS-9706006 and #DMS-9971960.

See also: Black Hole Mechanics; General Relativity:

Experimental Tests; General Relativity: Overview;

Geometric Analysis and General Relativity; Holomorphic

Dynamics; Mirror Symmetry: A Geometric Survey;

Spinors and Spin Coefficients; Stationary Black Holes.

Further Reading

Bray HL (1997) The Penrose Inequality in General Relativity and

Volume Comparison Theorems Involving Scalar Curvature.

Thesis, Stanford University.

Bray HL (2001) Proof of the Riemannian Penrose inequality using

the positive mass theorem. Journal of Differential Geometry.

59: 177–267.

Bray HL and Neves A (2004) Classification of Prime 3-Manifolds

with Yamabe Invariant Greater than RP

. Annals of Mathe-

matics 159: 407–424.

Bray HL and Schoen RM (1999) Recent proofs of the Riemannian

Penrose conjecture. In: Yau S-T (ed.) Current Developments in

Mathematics.

Bunting and Masood-ul-Alam (1987) Non-existence of multiple

black holes in asymptotically euclidean static vacuum space-

time. General Relativity and Gravitation 19(2).

Christodoulou D and Yau S-T (1988) Some remarks on the quasi-

local mass. Contemporary Mathematics 71: 9–14.

Geometric Flows and the Penrose Inequality 519

Hawking SW and Ellis GFR (1973) The Large-Scale Structure of

Space-Time.Cambridge: Cambridge University Press.

Huisken G and Ilmanen T (2001) The inverse mean curvature

flow and the Riemannian Penrose inequality. Journal of

Differential Geometry 59: 353–437.

Huisken G and Ilmanen T (1997a) A Note on Inverse Mean

Curvature Flow. Proceedings of the Workshop on Nonlinear

Partial Differential Equations, Saitama University, Sept. 1997

(available from Saitama University).

Huisken G and Ilmanen T (1997b) The Riemannian Penrose

inequality. International Mathematics Research Notices

20: 1045–10 58.

Jang PS and Wald RM (1977) The positive energy conjecture and

the cosmic censor hypothesis. Journal of Mathematical Physics

18: 41–44.

Parker TH and Taubes CH (1982) On Witten’s proof of the

positive energy theorem. Communications in Mathematical

Physics 84: 223–238.

Penrose R (1973) Naked singularities. Annals of the New York

Academy of Sciences 224: 125–134.

Schoen R and Yau S-T (1979a) On the proof of the positive mass

conjecture in general relativity. Communications in Mathema-

tical Physics 65: 45–76.

Schoen R and Yau S-T (1979b) Positivity of the total mass of a

general space-time. Physical Review Letters 43: 1457–1459.

Schoen R and Yau S-T (1981) Proof of the positive mass

theorem II. Communications in Mathematical Physics 79:

231–260.

Witten E (1981) A new proof of the positive energy theorem.

Communications in Mathematical Physics 80: 381–402.

Geometric Measure Theory

G Alberti, Universita

di Pisa, Pisa, Italy

Introduction

The aim of these pages is to give a brief, self-

contained introduction to that part of geometric

measure theory which is more directly related to the

calculus of variations, namely the theory of currents

and its applications to the solution of Plateau

problem. (The theory of finite-perimeter sets, which

is closely related to currents and to the Plateau

problem, is treated in the article Free Interfaces and

Free Discontinuities: Variational Problems in the

Encyclopedia.)

Named after the Belgian physicist J A F Plateau

(1801–1883), this problem was originally formulated

as follows: find the surface of minimal area spanning

a given curve in the space. Nowadays, it is mostly

intended in the sense of developing a mathematical

framework where the existence of k-dimensional

surfaces of minimal volume that span a prescribed

boundary can be rigorously proved. Indeed, several

solutions have been proposed in the last century,

none of which is completely satisfactory.

One difficulty is that the infimum of the area

among all smooth surfaces with a certain boundary

may not be attained. More precisely, it may happen

that all minimizing sequences (i.e., sequences of

smooth surfaces whose area approaches the infimum)

converge to a singular surface. Therefore, one is

forced to consider a larger class of admissible surfaces

than just smooth ones (in fact, one might want to do

this also for modeling reasons – this is indeed the case

with soap films, soap bubbles, and other capillarity

problems). But what does it mean that a set ‘‘spans’’ a

given curve? and what should we intend by area of a

set which is not a smooth surface?

The theory of integral currents developed by

Federer and Fleming (1960) provides a class of

generalized (oriented) surfaces with well-defined

notions of boundary and area (called mass) where

the existence of minimizers can be proved by direct

methods. More precisely, this class is large enough

to have good compactness properties with respect

to a topology that makes the mass a lower-

semicontinuous functional. This approach turned

out to be quite powerful and flexible, and in the

last decades the theory of currents has found

applications in several different areas, from dyna-

mical systems ( in parti cular, Mather theory) to the

theory of foliations, to optim al trans port probl ems.

Hausdorff Measures, Dimension,

and Rectifiability

The volume of a smooth d-dimensional surface in

is usually defined using parametrizations by

subsets of R

. The notion of Hausdorff measure

allows to compute the d-dimensional volume using

coverings instead of parametrizations, and, what is

more important, applies to all sets in R

, and makes

sense even if d is not an integer. Attached to

Hausdorff measure is the notion of Hausdorff

dimension. Again, it can be defined for all sets in

and is not necessarily an integer. The last

fundamental notion is rectifiability: k-rectifiable

sets can be roughly understood as the largest class

of k-dimensional sets for which it is still possible to

define a k-dimensional tangent bundle, even if only

in a very weak sense. They are essential to the

construction of integral currents.

520 Geometric Measure Theory

Hausdorff Measure

Let d  0 be a positive real number. Gi ven a set E in

, for every >0 we set



ðEÞ:¼

inf



ðdiamðE

ÞÞ



½1

where !

is the d-dimensional volume of the unit

ball in R

whenever d is an integer (there is no

canonical choice for !

when d is not an integer;

a convenient one is !

= 2

), and the infimum is

taken over all countable families of sets {E

} that

cover E and whose diameters satisfy diam(E

)  .

The d-dimensional Hausdorff measure of E is

ðEÞ:¼ lim

!0



ðEÞ½2

(the limit exists because H



(E) is decreasing in ).

Remarks

(i) H

is called d-dimensional because of its

scaling behavior: if E



is a copy of E scaled

homothetically by a factor , then

ðE



Þ¼

ðEÞ

Thus, H

scales like the length, H

scales like the

area, and so on.

(ii) The measure H

is clearly invariant under

rigid motions (translations and rotat ions). This

implies that H

agrees on R

with the Lebesgue

measure up to some constant factor; the renorma-

lization constant !

in [1] makes this factor

equal to 1. Thus, H

(E) agrees with the usual

d-dimensional volume for every set E in R

,and

the area formula shows that the same is true if

E is (a subset of) a d-dimensional surface of class C

in R

(iii) Besides the Hausdorff measure, there are

several other, less popular notions of d-dimensional

measure: all of them are invariant under rigid motion,

scale in the expected way, and agree with H

for sets

contained in R

or in a d-dimensional surface of class

, and yet they differ for other sets (for further

details, see Federer (1996, section 2.10)).

(iv) The definition of H

(E) uses only the notion

of diameter, and therefore makes sense when E is a

subset of an arbitrary metric space. Note that H

(E)

depends only on the restriction of the metric to E,

and not on the ambient space.

(v) The measure H

is countably additive on

the -algebra of Borel sets in R

, but not on all sets ;

to avoid pathological situations, we shall always

assume that sets and maps are Borel measurable.

Hausdorff Dimension

According to intuition, the length of a surface

should be infinite, while the area of a curve should

be null. These are indeed particular cases of the

following implications:

ðEÞ > 0 ) H

ðEÞ¼1 for d

< d

ðEÞ < 1)H

ðEÞ¼0 for d

> d

Hence, the infimum of all d such that H

(E) = 0 and

the supremum of all d such that H

(E) = 1

coincide. This number is called Hausdorff dimension

of E, and denoted by dim

(E). For surface of class

, the notion of Hausdorff dimension agrees with

the usual one. Example of sets with nonintegral

dimension are described in the next subsection.

Remarks

(i) Note that H

(E) may be 0 or 1 even for

d = dim

(E).

(ii) The Hausdor ff dimension of a set E is strictly

related to the metric on E, and not just to the

topology. Indeed, it is preserved under diffeomorph-

isms but not under homeomorphisms, and it does

not always agree with the topological dimension.

For instance, the Hausdorff dimension of the graph

of a continuous function f : R !R can be any

number between 1 and 2 (included).

(iii) For nonsmooth sets, the Hausdorff dimension

does not always conform to intuition: for example,

the dimension of a Cartesian product E  F of

compact sets does not agree in general with the sum

of the dimensions of E and F.

(iv) There are many other notions of dimension

besides Hausdorff and topological ones. Among

these, packing dimension and box-counting dimen-

sion have interesting applications (see Falconer

(2003, chapters 3 and 4)).

Self-Similar Fractals

Interesting examples of sets with nonintegral dimen-

sion are self-similar fractals. We present here a

simplified version of a construction due to Hutchinson

(Falconer 2003,chapter9).Let{

} be a finite set of

similitudes of R

with scaling factor 

< 1, and

assume that there exists a bounded open set V such

that the sets V

:=

(V) are pairwise disjoint and

contained in V. The self-similar fractal associated with

the system {

} is the compact set C that satisfies

C ¼

[



ðCÞ½3

The term ‘‘self-similar’’ follows by the fact that C

can be written as a union of scaled copies of itself.

Geometric Measure Theory 521

The existence (and uniqueness) of such a C follows

from a standard fixed-point argument applied to the

map C 7!



(C). The dimension d of C is the

unique solution of the equation



¼ 1 ½4

Formula [4] can be easily justified: if the sets 

(C)

are disjoint – and the assumption on V implies that

this almost the case – then [3] impli es H

H

(

(C)) =

(C), and therefore H

be positive and finite if and only if d satisfies [4].

An example of this construction is the usual

Cantor set in R, which is given by the similitudes



ðxÞ :¼

x and 

ðxÞ :¼

By [4], its dimension is d = log 2= log 3. Other

examples are described in Figure s 1–3.

Rectifiable Sets

Given an integer k = 1, ..., n, we say that a set E in

is k-rectifiable if it can be covered by a countable

family of sets {S

} such that S

is H

-negligible (i.e.,

) = 0) and S

is a k-dimensional surface of class

for j = 1, 2, ... Note that dim

(E)  k because

each S

has dimension k.

A k-rectifiable set E bears little resemblance to

smooth surfaces (it can be everywhere dense!), but it

still admits a suitably weak notion of tangent bundle.

More precisely, it is possible to associate with every

x 2 E a k-dimensional subspace of R

, denoted by

Tan(E, x), so that for every k-dimensional surface S

of class C

in R

there holds

TanðE; xÞ¼TanðS; xÞ for H

-a.e. x 2 E \ S ½5

where Tan(S, x) is the tangent space to S at x

according to the usual definition.

It is not difficult to see that Tan(E, x) is uniquely

determined by [5] up to an H

-negligible amount of

points x 2 E,andifE is a surface of class C

, then it

agrees with the usual tangent space for H

-almost

all points of E.

Remarks

(i) In the original definition of rectifiability, the

sets S

with j > 0 are Lipschitz images of R

, that is,

:= f

), where f

: R

is a Lipschitz map.

It can be shown that this definition is equivalent to

the one above.

(ii) The construction of the tangent bundle is

straightforward: Let {S

} be a covering of E as earlier,

and set Tan(E, x):= Tan(S

, x), where j is the smallest

positive integer such that x 2 S

.Then[5] is an

immediate corollary of the following lemma: if S and

are k-dimensional surfaces of class C

in R

,then

Tan(S, x) = Tan(S

, x)forH

-almost every x 2 S \ S

(iii) A set E in R

is called purely k-unrectifiable

if it contains no k-rectifiable subset with posi-

tive k-dimensional measure, or, equivalently, if

(E \ S) = 0 for every k-dimensional surface S of

class C

. For instance, every product E := E

 E

where E

and E

are H

-negligible sets in R is a

purely 1-unrectifiable set in R

(it suffices to show

that H

(E \ S) = 0 whenever S is the graph of a

function f : R !R of class C

, and this follows by

the usual formula for the length of the graph). Note

that the Hausdorff dimension of such product sets

can be any number between 0 and 2, hence

rectifiability is not related to dimension. The self-

similar fractals described in Figures 1 and 3 are both

purely 1-unrectifiable.

Rectifiable Sets with Finite Measure

If E is a k-rectifiable set with finite (or locally finite)

k-dimensional measure, then Tan(E, x) can be

related to the behavior of E close to the poin t x.

Let B(x, r) be the open ball in R

with center x

and radius r, and let C(x, T, a) be the cone with

center x,axisT –ak-dimensional subspace of R

–

and amplitude  = arcsin a, that is,

Cðx; T; aÞ:¼fx

2 R

: distðx

 x; TÞajx

 xjg

Figure 1 The maps 

, i = 1, ..., 4, take the square V into the

squares V

at the corners of V. The scaling factor is  for all i,

hence dim

number between 0 and 2, including 1.

....

Figure 2 A self-similar fractal with more complicated topology.

The scaling factor is 1/4 for all twelve similitudes, hence

dim

Figure 3 The von Koch curve (or snowflake). The scaling

factor is 1/3 for all four similitudes, hence dim

522 Geometric Measure Theory

For H

-almost every x 2 E, the measure of E \

B(x, r) is asymptotically equivalent, as r !0, to the

measure of a flat disk of radius r, that is,

E \ Bðx; rÞðÞ!

Moreover, the part of E contained in B(x, r)is

mostly located close to the tangent plane Tan(E, x),

that is,

E \ Bðx; rÞ\Cðx; TanðE; xÞ; aÞðÞ!

for every a > 0

When this condition holds, Tan(E, x) is called the

approximate tangent space to E at x (see Figure 4).

The Area Formula

The area formula allows to comput e the measure

((E)) of the image of a set E in R

as the

integral over E of a suitably defined Jacobian

determinant of .When is injective and takes

values in R

, we recover the usual change of

variable formula for multiple integrals.

We consider first the linear case. If L is a linear

map from R

to R

with m  k, the volume ratio

 := H

(L(E))=H

(E) does not depend on E, and

agrees with jdet (PL)j, where P is any linear

isometry from the image of L into R

, and det (PL)

is the determinant of the k  k matrix associated

with PL. The volume ratio  can be computed using

one of the following identities:

 ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

detðL



LÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ðdet MÞ

½6

where L



is the adjoint of L (thus, L



L is a linear

map from R

into R

), and the sum in the last term

is taken over all k  k minors M of the matrix

associated with L.

Let  : R

be a map of class C

with m  k,

and E a set in R

. Then

ðEÞ

#ð

1

ðyÞ\EÞdH

ðyÞ¼

JðxÞdH

ðxÞ½7

where #A stands for the number of elements of A,

and the Jacobian J is

JðxÞ:¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det rðxÞ



rðxÞðÞ

½8

Note that the left- hand side of [7] is H

((E)) when

 is injective.

Remark

Formula [7] holds even if E is a k-rectifiable set in R

In this case, the gradient r(x)in[8] should be

replaced by the tangential derivative of  at x (viewed

as a linear map from Tan(E, x)intoR

). No version of

formula [7] is available when E is not rectifiable.

Vectors, Covectors, and Differential

Forms

In this section, we review some basic notions of

multilinear algebra. We have chosen a definition of

k-vectors and k-covectors in R

, and of the corres-

ponding exterior products, which is quite convenient

for computations, even though not as satisfactory from

the formal viewpoint. The main drawback is that it

depends on the choice of a standard basis of R

,and

therefore cannot be used to define forms (and currents)

when the ambient space is a general manifold.

k-Vectors and Exterior Product

Let { e

, ..., e

} be the standard basis of R

. Given an

integer k  n, I(n, k) is the set of all multi-indices

i = (i

, ..., i

) with 1  i

< i

< < i

 n, and

for every i 2 I(n, k) we introduce the expression

¼ e

^ e

^^e

A k-vector in R

is an y formal linear combination



with 

2 R for every i 2 I(n, k). The space of

k-vectors is denoted by ^

); in particular,

) = R

. For reasons of formal convenience,

we set ^

):= R and ^

):= {0} for k > n.

We denote by jjthe Euclidean norm on ^

The exterior product v ^ w 2^

kþh

) is defined

for every v 2^

) and w 2^

), and is

completely determined by the following properties:

(1) associativity, (2) linearity in both arguments, and

(3) e

^ e

= e

^ e

for every i 6¼ j and e

^ e

= 0 for

every i.

Simple Vectors and Orientation

A simple k-vector is any v in ^

) that can be

written as a product of 1-vectors, that is,

v ¼ v

^ v

^^v

It can be shown that v is null if and only if the

vectors {v

} are linearly dependent. If v is not null,

T = Tan(E, x)

C(x, T, a)

Figure 4 A rectifiable set E close to a point x of approximate

tangency. The part of E contained in the ball B(x , r) but not in the

cone C(x, T , a) is not empty, but only small in measure.

Geometric Measure Theory 523

then it is uniquely determined by the following

objects: (1) the k-dimensional space M spanned

by {v

}; (2) the orientation of M associated with the

basis {v

}; (3) the euclidean norm jvj. In particular,

M does not depend on the choice of the vectors v

Note that jvj is equal to the k-dimensional volume of

the parallelogram spanned by {v

Hence, the map v 7!M is a one-to-one correspon-

dence between the class of simple k-vectors with

norm jvj= 1 and the Grassmann manifold of

oriented k-dimensional subspaces of R

This remark paves the way to the following definition:

if S is a k-dimensional surface of class C

in R

, possibly

with boundary, an orientation of S is a continuous map



: S !^

) such that 

(x)isasimplek-vector with

norm 1 that spans Tan(S, x) for every x.Withevery

orientation of S (if any exists) is canonically associated

the orientation of the boundary @S that satisfies



ðxÞ¼ðxÞ^

ðxÞ for every x 2 @S ½9

where (x) is the inner normal to @S at x.

k-Covectors

The standard basis of the dual of R

{dx

, ..., dx

}, where dx

: R

!R is the linear

functional that takes every x = (x

, ..., x

) into the

ith component x

. For every i 2 I(n, k) we set

¼ dx

^ dx

^^dx

and the space ^

)ofk-covectors consists of all

formal linear combinations 

. The exterior

product of covectors is defined as that for vectors.

The space ^

) is dual to ^

) via the duality

pairing h; i defined by the relations hdx

; e

i:= 

(that is, 1 if i = j and 0 otherwise).

Differential Forms and Stokes Theorem

A differential form of order k on R

is a map

! : R

). Using the canonical basis of

), we can write ! as

!ðxÞ¼

i2Iðn;kÞ

ðxÞdx

where the coordinates !

are real functions on R

The exterior derivative of a k-form ! of class C

the (k þ 1)-form

d!ð xÞ:¼

i2Iðn;kÞ

ðxÞ^dx

where, for every scalar function f, df is the 1-form

df ðxÞ:¼

i¼1

ðxÞdx

If S is a k-dimensional oriented surface, the

integral of a k-form ! on S is naturally defined by

! :¼

h!ðxÞ; 

ðxÞidH

ðxÞ

Stokes theorem states that for every (k  1)-form !

of class C

there holds

! ¼

d! ½10

provided that @S is endowed with the orientation 

that satisfies [9].

Currents

The definition of k-dimensional currents closely

resembles that of distributions: they are the dual of

smooth k-forms with compact support. Since every

oriented k-dimensional surface defines by integration

a linear functional on forms, currents can be regarded

as generalized oriented surfaces. As every distribution

admits a derivative, so every current admits a

boundary. Indeed, many other basic notions of

homology theory can be naturally extended to

currents – this was actually one of the motivations

behind the introduction of currents, due to de Rham.

For the applications to variational problems,

smaller classes of currents are usually considered;

the most relevant to the Plateau problem is that of

integral currents. Note that the definitions of the

spaces of normal, rectifiable, and integral currents

and the symbols used to denote them vary, some-

times more than slightly, depending on the author.

Currents, Boundary, and Mass

Let n, k be integers with n  k. The space of

k-dimensional currents on R

, denoted by D

is the dual of the space D

) of smooth k-forms

with compact support in R

. For k  1, the

boundary of a k-current T is the (k  1)-current @T

defined by

h@T; !i:¼hT; d!i for every ! 2 D

k1

ðR

Þ½11

while the boundary of a 0-current is set equal to 0.

The mass of T is the number

MðTÞ:¼ sup hT; !i: ! 2 D

ðR

Þ; j!j1

½12

Fundamental examples of k-currents are oriented

k-dimensional surfaces: with each oriented surface

S of class C

is canonically associated the current

hT;d!i: =

! (in fact, S is completely determined

by the action on forms, i.e., by the associated

524 Geometric Measure Theory

current). By Stok es theorem, the b oundary of T is

the current associated with th e boundary of S;thus,

the notion of boundary for c urrents is compatible

with the classical one for o riented surfaces.

A simple computation shows that M(T) = H

(S);

therefore, the mass provides a natural extension

of the notion of k-dimensional volume to

k-currents.

Remarks

(i) Not all k-currents look like k-dimensional

surfaces. For example, every k-vectorfield v : R

) defines by duality the k-current

hT; !i:¼

h!ðxÞ; vðxÞi dH

ðxÞ

The mass of T is

jvjdH

, and the boundary is

represented by a similar integral formula involving

the partial derivatives of v (in particular, for

1-vectorfields, the boundary is the 0-current asso-

ciated with the diver gence of v). Note that the

dimension of such T is k because k-vectorfields act

on k-forms, and there is no relation with the

dimension of the support of T, which is n.

(ii) To be precise, D

) is a locally convex

topological vector space, and D

) is its topolo-

gical dual. As such, D

) is endowed with a dual

(or weak



) topology. We say that a sequenc e of

k-currents (T

) converge to T if they converge in the

dual topology, that is,

; !i!hT; !i for every ! 2 D

ðR

Þ½13

Recalling the definition of mass, it is easy to show

that it is lower-semicontinuous with respect the dual

topology, and in particular

lim inf MðT

ÞMðTÞ½14

Currents with Finite Mass

By definition, a k-current T with finite mass is a

linear functional on k-forms which is bounded with

respect to the supremum norm, and by Riesz

theorem it can be represented as a bounded measure

with values in ^

). In other words, there exist a

finite positive measure  on R

and a density

function  : R

) such that j(x)j= 1 for

every x and

hT; !i¼

h!ðxÞ; ðxÞi dðxÞ

The fact that curren ts are the dual of a separable

space yields the following compactness result: a

sequence of k-currents (T

) with uniformly bounded

masses M(T

) admits a subsequence that converges

to a current with finite mass.

Normal Currents

A k-current T is called normal if both T and @T

have finite mass. The compactness result stated in

the previous paragraph implies the following com-

pactness theorem for normal currents: a sequence of

normal currents (T

) with M(T

) and M(@T

)

uniformly bounded admits a subsequence that

converges to a normal current.

Rectifiable Currents

A k-current T is called rectifiable if it can repre-

sented as

hT; !i¼

h!ðxÞ; ðxÞiðxÞdH

ðxÞ

where E is a k-rectifiable set E,  is an orientation of

E – that is, (x) is a simple unit k-vector that spans

Tan(E, x) for H

-almost every x 2 E – and  is a real

function such that

jjdH

is finite, called multi-

plicity. Such T is denoted by T = [E, , ]. In

particular, a rectifiable 0-current can be written as

hT; !i=

!(x

), where E = {x

} is a countable set in

and {

} is a sequence of real numbers with

j

j < þ1.

Integral Currents

If T is a rectifiable current and the multiplicity 

takes integral values, T is called an integer multi-

plicity rectifiable current. If both T and @T are

integer multiplicity rectifiable currents, then T is an

integral current.

The first nontrivial result is the bounda ry rectifia-

bility theorem: if T is an integer multiplicity

rectifiable current and @T has finite mass, then @T

is an integer multiplicity rectifiable current, too, and

therefore T is an integral current.

The second fundamental result is the compactness

theorem for integral currents: a sequence of integral

currents (T

) with M(T

)andM(@T

) uniformly

bounded admits a subsequence that converges to

an integral current.

Remarks

(i) The point of the compactness theorem for

integral currents is not the existence of a converging

subsequence – that being alre ady established by the

compactness theorem for normal currents – but the

fact that the limit is an integral current. In fact, this

result is often referred to as a ‘‘closure theorem’’

rather than a ‘‘compactness theorem.’’

Geometric Measure Theory 525