An area of analysis concerned with solving
geometric problems via measure-theoretic techniques. The canonical
motivating physical problem is probably that investigated
experimentally by
J. Plateau
in the
nineteenth century
[a4]:
Given a boundary wire, how does one find
the (minimal) soap film which spans it?
Slightly more mathematically:
Given a
boundary curve, find the surface of minimal area spanning it. (Cf. also
Plateau problem.)
The many different approaches to solving this problem
have found utility in most areas
of modern mathematics and geometric measure theory is no exception:
techniques and ideas from geometric measure theory have been found
useful in the study of partial differential equations,
the calculus of variations, harmonic analysis, and fractals.
Successes in the field include: classifying the structure of
singularities in soap films
(see
[a18],
together with the fine descriptive
article
[a3]);
showing that the standard
"double bubble"
is the optimal shape for enclosing two prescribed volumes in
space
[a13],
and developing powerful
computer software for modelling the evolution of surfaces under the
action of physical
forces
[a7].
The main reference text for the subject is
[a12].
It is very densely written and
[a15]
serves
as a useful guide through it;
[a11]
provides a comprehensive overview of the subject and contains a
summary of its main results. For suitable introductions, see also
[a17],
which contains an introduction to the theory of varifolds and
Allard's regularity theorem,
and
[a14],
which includes information about tangent measures and their uses.
For a slightly different slant,
[a9]
discusses applications of some of the ideas of
geometric measure theory in the
theory of Sobolev spaces and functions of bounded variation.
Many variational problems (cf. also
Variational calculus)
are solved by enlarging the allowed class of
solutions, showing that in this enlarged class a solution exists, and
then
showing that the solution possesses more regularity than an arbitrary
element of the enlarged class. Much of the work in geometric measure
theory
has been directed towards placing this informal description on a
formal
footing appropriate for the study of surfaces.
Rectifiability for sets.
The key concept underlying the whole theory is that of rectifiability,
a
measure-theoretic notion of smoothness
(cf. also
Rectifiable curve).
A set
in Euclidean
-space
is (countably)
-rectifiable
if there is a sequence of
mappings,
,
such that
It is
purely
-unrectifiable
if for all
mappings
,
(Here,
denotes the
-dimensional
Hausdorff (outer) measure, defined by
where
denotes the diameter and the constant
is chosen so that, when
,
Hausdorff measure
is just the usual
Lebesgue measure.)
A basic
decomposition theorem
states that any set
of finite
-dimensional
Hausdorff measure may be written as the union of an
-rectifiable
set and a purely
-unrectifiable
set, with the intersection necessarily having
-measure
zero.
In practice, the definition of rectifiability is commonly used with
Lipschitz mappings
replacing
mappings: it may be shown that this does not change anything,
see
[a14],
Thm. 15.21.
A standard
example of a
-rectifiable set
in the plane is a
countable
union of circles whose centres are dense in the unit square and with
radii having a finite sum; the closure of the
resulting set contains the unit square, and
yet, as indicated below, the set itself still has
"tangents"
at
-almost
every point.
An
example of a purely
-unrectifiable set
is given by taking
the cross-product
of the
-Cantor set
with itself. (The
-Cantor
set is formed by removing
intervals of diameter
,
rather than
as for the plain
Cantor set,
at each stage of its construction.)
Approximate tangents.
The main importance of the class of rectifiable sets is that it
possesses many of the
nice properties of the smooth surfaces which one is seeking to
generalize.
For example, although, in general, classical tangents may not exist
(consider the circle example above), an
-rectifiable
set will possess a unique approximate tangent at
-almost
every point: An
-dimensional
linear subspace
of
is an
approximate
-tangent plane
for
at
if
and for all
,
Conversely, if
has finite
-measure
and has an approximate
-tangent
plane for
-almost
every
,
then
is
-rectifiable.
Besicovitch–Federer projection theorem.
Often, one is faced with the task of showing that some set, which is a
solution
to the problem under investigation, is in fact rectifiable, and hence
possesses
some smoothness. A major concern in geometric measure theory is
finding criteria which guarantee rectifiability. One of the most
striking
results in this direction is the
Besicovitch–Federer projection theorem,
which
illustrates the stark difference between rectifiable and
unrectifiable sets.
A basic version of it states that if
is a purely
-unrectifiable
set of finite
-dimensional
Hausdorff measure, then for almost every orthogonal projection
of
onto an
-dimensional
linear subspace,
.
(It is not particularly difficult to show that in contrast,
-rectifiable
sets have projections of positive
measure for almost every projection.) This deep result was first
proved for
-unrectifiable
sets in the plane by
A.S. Besicovitch,
and later extended to
higher dimensions by
H. Federer.
Recently
(1998),
B. White
[a19]
has shown how
the higher-dimensional version of this theorem follows via an
inductive argument from the planar version.
Rectifiability for measures.
It is also possible (and useful) to define a notion of rectifiability
for
Radon (outer) measures: A
Radon measure
is said to be
-rectifiable
if it is absolutely continuous (cf. also
Absolute continuity)
with respect to
-dimensional
Hausdorff measure and there is an
-rectifiable
set
for which
.
The complementary notion of a measure
being
purely
-unrectifiable
is defined by requiring that
is singular with respect to all
-rectifiable
measures (cf. also
Mutually-singular measures).
Thus, in particular, a set
is
-rectifiable
if and only if
(the restriction of
to
)
is
-rectifiable;
this allows one to study rectifiable sets through
-rectifiable
measures.
It is common in analysis to construct measures as solutions to
equations, and
one would like to be able to deduce something about the structure of
these
measures (for example, that they are rectifiable).
Often, the only a priori information available is some limited
metric information about the measure, perhaps
how the mass of small balls grows with radius. Probably the strongest
known
result in this direction is
Preiss' density theorem
[a16]
(see
also
[a14]
for a lucid sketch of the proof). This states that if
is a Radon measure on
for which
exists and is positive and finite for
-almost
every
,
then
is
-rectifiable.
Preiss' main tool in proving this result
was the notion of tangent measures. A non-zero Radon measure
is a
tangent measure
of
at
if there are sequences
and
such that for all continuous real-valued functions with
compact support,
Thus, an
-rectifiable
measure will, for almost-every point, have tangent
measures which are multiples of
-dimensional
Hausdorff measure
restricted to the approximate tangent plane at that point; for
unrectifiable
measures, the set of tangent measures will usually be much richer.
The utility of the notion lies in the fact that tangent measures
often
possess more regularity than the original measure, thus allowing a
wider
range of analytical techniques to be used upon them.
Currents.
A natural approach to solving a minimal surface problem would be
to take a sequence of approximating sets whose areas are decreasing
and
finally extract a convergent subsequence with the hope that the limit
would
possess the required properties. Unfortunately, the usual notions of
convergence for sets in Euclidean spaces are not suited to this. The
theory of
currents, introduced by
G. de Rham
and
extensively developed by Federer and
W.H. Fleming
in
[a10]
(see
[a11]
for a comprehensive outline of the theory
and
[a12]
for details),
was developed as a way around this obstacle for oriented surfaces.
In essence, currents are
generalized surfaces, obtained by viewing an
-dimensional
(oriented) surface as
defining a continuous linear functional on the space of differential
forms with compact support of degree
(cf. also
Current).
Using the duality with differential
forms, it is then possible to define many natural operations on
currents. For example, the
boundary of an
-current
can be defined to be the
-current,
,
which is given via the
exterior derivative
for differential forms
(cf. also
Exterior algebra)
by setting
 |
for a
differential form
of degree
.
Of particular importance is the class of
-rectifiable currents:
this class consists of the currents that can be written as
where
is an
-rectifiable
set with
,
is a positive integer-valued function with
and
can be written as
with
forming an orthonormal basis for the approximate tangent space of
at
for
-almost
every
.
(That is,
is a unit simple
-vector
whose associated
-dimensional
vector space is the approximate tangent space of
at
for
-almost
every
.)
The
mass
of a current given in this
way is defined by
.
If the boundary of an
-rectifiable
current is itself an
-rectifiable
current, then the
-current
is said to be an
integral current.
These are the class of currents suitable for investigating
Plateau's problem. The celebrated
Federer–Fleming closure theorem
says that on a not too
wild
compact
domain (it should be a
Lipschitz retract
of some open neighbourhood of itself), those integral
currents
on the domain which all have the same boundary
,
an
-current
with finite mass, and for which
is bounded above by some constant
,
form a compact set. (The topology is that generated by the
integral flat distance,
defined for
-integral
currents
,
by
where the infimum is over
and
such that
is an
-rectifiable
current on
and
is an
-rectifiable
current on
.)
In particular, if the constant
is chosen large enough so that this
set is non-empty, then one can deduce the existence of a
mass-minimizing
current with the given boundary
.
Varifolds.
The theory of currents is ideally suited for investigating oriented
surfaces,
but for unoriented surfaces problems arise. The theory of varifolds
was
initiated by
F.J. Almgren
and extensively developed by
W.K. Allard
[a1]
(see
also
[a2]
for a nice survey) as an alternative notion of
surface which did not require an orientation. An
-varifold
on an open subset
of
is a Radon measure on
.
(Here,
denotes the
Grassmann manifold
of
-dimensional
linear subspaces of
.)
The space of
-varifolds
is equipped with the
weak topology
given by saying that
if and only if
for all compactly supported, continuous
real-valued functions on
.
Given an
-varifold
,
one associates a Radon measure on
,
,
by setting
for
.
As a partial converse, to an
-rectifiable
measure
one can associate an
-rectifiable varifold
by defining for
,
where
is the approximate tangent plane at
.
The
first variation of an
-varifold
is a mapping from the space of smooth compactly supported vector fields on
to
,
defined by
If
,
then the varifold is said to be
stationary.
The idea is that the variation measures the rate of change in the
"size"
of
the varifold if it is perturbed slightly. A key result in the theory
of
varifolds is
Allard's regularity theorem,
which states that stationary
varifolds which satisfy a growth condition (detailed below) are
supported on
a smooth manifold. More precisely: For all
there are constants
,
such that whenever
,
,
and
is an
-dimensional
stationary varifold on the open ball
with
1)
;
2)
existing and equal to at least one for
-almost
every
;
and
3)
,
then
is a continuously differentiable embedded
-submanifold
of
,
and
for points in this
submanifold. (The distance between the tangent spaces is given by the
distance between their corresponding orthogonal projections.)
This is a theorem which gives much more than just rectifiability; it
gives
information about the degree of smoothness as well. See
[a17]
for some variants and a proof of this result.
Generalization.
Given the success of the theory in Euclidean spaces, it is natural to
ask
whether a similar theory holds in more general
spaces
[a8].
There are many difficulties to be overcome, but
[a5],
[a6]
suggest that it may be possible.