A subset M of Rn
is a k rectifiable set if M is
contained in a
countable union M0 U
M1 U M2
where M0 has k dimensional Hausdorff measure 0 and each Mk for k=1,2,..., is a continuously differentiable submanifold of Rn . Here the k dimensional Hausdorff (outer) measure may be defined for any subset A of a metric space and any nonnegative real number k by using sums of the kth powers of diameters of efficient countable covers of A by small sets. In the case k=0 , H0 is counting measure and M is 0 rectifiable iff it is countable. In the case k=n, Hn is n dimensional Lebesgue measure and all subsets are n rectifiable. But the really interesting cases are where k = 2,3,…, n-1 .
Though rectifiable sets may have possibly have
"holes", they enjoy, and are even characterized, by important density
and tangential properties. The k dimensional density
of A at a point a is the limit (if
it exists) as r ->0 of
Hk[A intersect Br(a)] / wk rk for a suitable constant wk (which equals the Lebesgue measure of the unit ball in Rk in case k is an integer). For a rectifiable set M the density function coincides Hk a.e. with the characteristic function XM . Also larger and larger homothetic expansions of M about Hk almost every point a of M converge weakly or measure-theoretically to a $k$ dimensional affine plane (the approximate tangent space at a ). A few of the striking theorems about a compact set A of finite positive Hk measure that we will prove are:
(1) (Marstrand) If the k dimensional density exists and is positive and finite at Hk almost all a in A , then k is an integer.
(2) (Besicovitch, Federer) If k is an integer , then A = M U N for some k rectifiable set M and a set N whose projection onto almost all k dimensional subspaces has k dimensional Lebesgue measure 0 .
(3) (Matilla, Preiss) If k is an integer and the k dimensional density =1 for Hk almost all a in A , then A is k rectifiable.
One may alternately characterize a rectifiable set as an Hk almost subset of a countable union of images Lipschitz maps of subsets of Rk. This can be used to define a k rectifiable subset of a metric space, although some interesting metric spaces, like the set N above may be totally unrectifiable in the sense that they do not contain any rectifiable subset of positive Hk measure. For example the square of a ½ dimensional Cantor set is 1 unrectifiable and the Heisenberg group is 2 unrectifiable. Nevertheless many of the basic properties of rectifiable sets do carry over to metric spaces, following work of Ambrosio, Kirchheim, and others over the last 10 years.
Rectifiable sets have proven useful for various geometric variational problems involving area minimization or image processing. Rectifiable sets with additional structures or properties enjoy compactness results. For example, rectifiable sets (possibly with integer multiplicities) of uniformly bounded measures with bounds on their first variations (weakly defined mean curvatures) have weakly convergent subsequences. Adding Hk measurable orientations of the approximate tangent spaces gives rectifiable currents and these become compact when there are bounds on mass and boundary mass. This allows for the solvability of a k dimensional Plateau problem of mass-minimization with a given boundary. Singular sets of Plateau solutions have been shown to be rectifiable in the appropriate dimensions by L. Simon.
Also the solutions themselves, such as “soap-bubbles” inside a very dirty medium enjoy an extra uniform
rectifiability property following works of David and Semmes.
In the course we will first review Hausdorff measures, basic density properties, and the calculus-type area and co-area formulas. Much of the course will involve only basic measure theory combined with numerous geometric constructions.
We will frequently refer to the recent book by F.Lin and X.Yang Geometric Measure Theory—An Introduction,
Science Press (
Tues.Thurs in Herman Brown 453
Robert Hardt Office: Herman Brown 430; Office hours: 1-2 MWF (and others by appt.),
Email: email@example.com, Telephone: ext 3280
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