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. 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.
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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. Conversely, if has finite -measure and has an approximate -tangent plane for -almost every , then is -rectifiable.
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. Besicovitch, and later extended to higher dimensions by H. Recently , B. White [a19] has shown how the higher-dimensional version of this theorem follows via an inductive argument from the planar version.
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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. The complementary notion of a measure being purely -unrectifiable is defined by requiring that is singular with respect to all -rectifiable measures cf.
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. 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. 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.
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. Of particular importance is the class of -rectifiable currents: this class consists of the currents that can be written as. 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. Many tools in geometric measure theory GMT have been developed to generalise the notion of a smooth surface in order to deal with Plateau's problem of finding surfaces of minimal area. More generally, GMT studies geometric properties of sets, measures, and generalised surfaces; its methods have proved useful in many areas of mathematics including Geometry, Calculus of Variations, PDEs, and their applications.
An overview of the subject can be found in the Encyclopedia of Mathematics . In the seminar, we will study rectifiable sets, currents, and varifolds. The focus will be on varifolds and their application to problems involving curvature such as the Willmore energy, mean curvature flow, models for phase transitions, or the Helfrich bending energy of biological membranes. Basic knowledge of measure theory measures, integration, convergence theorems is essential, knowledge of submanifolds in the Euclidean space is nice but not necessary.
Interested students please contact me by email. A preliminary meeting to finalise arrangements will be held end of September or early in October. Institute for Applied Mathematics.
Geometric Measure Theory and its Connections
Helmers Seminar sessions Tuesday in room 2. Synopsis Many tools in geometric measure theory GMT have been developed to generalise the notion of a smooth surface in order to deal with Plateau's problem of finding surfaces of minimal area. Prerequisites Basic knowledge of measure theory measures, integration, convergence theorems is essential, knowledge of submanifolds in the Euclidean space is nice but not necessary.
I have tried to keep the notes as brief as possible, subject to the constraint of covering the really important and central ideas. There have of course been omissions; in an expanded version of these notes which I hope to write in the near future , topics which would obviously have a high priority for inclusion are the theory of flat chains, further applications of G.
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Partial Differential Equations and Geometric Measure Theory
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