# MATH 522: Topics in Real Analysis, Spring 2003

### Description:

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"  Tan(M,a) .

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.

### Meets:

We will meet initially MWF 12-12:50 in Herman Brown 438, Math. Commons Room, and discuss possible changes of time then.

### Handouts:

BV Compactness for Maps to a Metric Space
Compactness of Rectifiable Currents in a Metric Space

### Some References:

F. Morgan, Geometric Measure Theory, A Beginner's Guide,   Academic Press, 1988.
H. Federer, Geometric Measure Theory, Springer-Verlag, 1970.
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 (1998),693- 701.
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.

### Instructor:

Robert Hardt    Office: Herman Brown 430; Office hours: 11-12 MWF (and others by appt.),
Email: hardt@rice.edu, Telephone: ext 3280

### Homepage:

http://math.rice.edu/~hardt/522S03/