MATH 522: Topics in Real Analysis, Spring 2003
Intoduction to Geometric Measure Theory
Geometric Measure Theory treats measure-theoretic properties
of geometrically defined sets of various dimensions. Some of
the critical notions are Hausdorff measure,
rectifiable sets, and rectifiable currents. The
k dimensional Hausdorff (outer) measure H k(A) gives,
for every nonnegative number k , a precise notion
of the k dimensional size of A . Suppose that in R 3 , C
is a smooth embedded curve, S is a smooth embedded surface,
and U is an open region. Then H 1(C) is
the length of C , H2 (S) is the area of S
, and H 3(U) is the volume or Lebesgue measure
of U . Also H2 (C)=H 2 (C)=H
3(S)=0 while H1(S)=H
1(U)=H2(U)= oo. Noninteger Hausdorff measures
are useful for some fractals, e.g.
0 < Hlog2/log3(Cantor set) < oo . For integer
k , a subset M is of R n called
k rectifiable if M is H k
almost the subset of a countable union of continuously differentiable
manifolds. Blowing up homothetically about H
k almost every point a in M gives
(under measure convergence) a multiplicity 1 "tangent k plane"
An important result is the Structure Theorem
which associates with any finite Hk measure
set A a rectifiable subset M so that the difference
A \ M (which may still have positive Hk measure) orthogonally
projects to almost every k plane onto a Lebesgue null set.
By assigning, for Hk almost every a
in a k dimensional rectifiable set M, an integer
multiplicity m(a) and an orientation v(a) for Tan(M,a)
, one obtains a k dimensional rectifiable current T. One
may integrate smooth k forms of R n on
M . The boundary of T is then given simply by
d T(f) = T(df) for a k-1 form f , and the
mass of T is the is J
M m(a)dH ka . The Compactness Theorem
of Federer-Fleming for rectifiable currents implies the existence, for
a given k-1 dimensional rectifiable boundary, a k dimensional
rectifiable current of minimum mass.
Both of these results have enjoyed new, more accessible proofs
in the nineties based on properties of slices of sets or currents by
general affine n-k planes. Also some of these notions have
remarkably been carried over to more general metric spaces.
We will try to present some of these new ideas in the course
as well as many of the standard analytic and geometric constructions for
rectifiable sets and currents.
Prerequesites for the course include some knowledge of basic
analysis and measure theory as in Math 425, 426 and beginning topology
as in Math 443.
We will meet initially MWF 12-12:50 in Herman Brown 438, Math.
Commons Room, and discuss possible changes of time then.
BV Compactness for Maps to a Metric Space
Compactness of Rectifiable Currents in a Metric Space
F. Morgan, Geometric Measure Theory, A Beginner's
Guide, Academic Press, 1988.
H. Federer, Geometric Measure Theory, Springer-Verlag,
P. Mattilla, Lecture notes on geometric measure theory
.Universidad de Extramadura, 1986.
L. Simon, Lectures on geometric measure theory. Australian
National University, 1983
B. White, A new proof of Federer's structure theorem for
k-dimensional subsets of RN. J.A.M.S. 11
L. Ambrosio & B. Kirchheim, Rectifiable sets in metric
and Banach spaces. Math. Ann.318(2000), 527--555.
L. Ambrosio & B. Kirchheim, Currents in metric
spaces. Acta Math. 185 (2000), 1--80.
- Robert Hardt Office: Herman Brown 430; Office hours:
11-12 MWF (and others by appt.),
- Email: firstname.lastname@example.org, Telephone: ext 3280
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