THE GRADUATE STUDENT SECTION

WHAT IS…

a Rectifiable Set?Editors

A rectifiable set is our best definition of a generalized𝑘-dimensional surface inℝ𝑛, perhaps with infinitely manysingularities and infinite topological type, but still niceenough tododifferential geometry.While smoothsurfacescan be defined locally as images of nice smooth functionsfrom the unit ball inℝ𝑘 intoℝ𝑛, rectifiable sets are definedas countable unions of images of Lipschitz functions fromarbitrary subsets of ℝ𝑘 into ℝ𝑛. A Lipschitz function 𝑓 bydefinition satisfies

|𝑓(𝑦) − 𝑓(𝑥)| ≤ 𝐶|𝑦− 𝑥|.There is also a condition that a rectifiable set have finite𝑘-dimensional measure (technically given by Hausdroffmeasure).

The simplest nonsmooth example of a Lipschitz func-tion from ℝ1 to ℝ2 is 𝑓(𝑥) = (𝑥, |𝑥|) as in Figure 1; itsimage has a singularity at the origin. A 1-dimensionalrectifiable set is a countable union of continuous curvesof finite total length. A 2-dimensional rectifiable set canhave dense singularities and infinite topological type, asin Figure 2.

Figure 1. A rectifiable set, as the image of a Lipschitzfunction, can have sharp corner singularities.

Rademacher’s Theoremsays that a Lipschitz function isdifferentiable almost everywhere. The corresponding factabout rectifiable sets is the existence of an approximatetangent plane at almost every point𝑝.Approximatemeans

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Figure 2. Adding infinitely many handles of finitetotal area to the sphere can produce a rectifiable set𝑆 with infinite topological type and a singular denseCantor set of positive measure. Nevertheless, 𝑆 hasan approximate tangent plane at almost all points.

ignoring sets of density 0 at 𝑝. This fact means thatrectifiable sets admit a measure theoretic generalizationof differentiable geometry. It also means that you canintegrate smooth differential forms over rectifiable setsand thus view them as the so-called rectifiable currentsof geometric measure theory.

Besicovitch and Federer proved an amazing structuretheorem, which says that every set of finite 𝑘-dimensionalmeasure can be decomposed into a rectifiable set anda purely unrectifiable set. The purely unrectifiable sethas the property that it is invisible from almost everydirection, i.e., that its projection onto almost every 𝑘-plane

1086 Notices of the AMS Volume 65, Number 9

THE GRADUATE STUDENT SECTIONhas measure 0. Cantor sets provide examples of purelyunrectifiable sets, as in Figure 3.

Figure 3. The pictured Cantor set, obtained bysuccessively removing all but the three small cornersof every triangle, is a purely unrectifiable set of unit1-dimensional Hausdorff measure. Its projection ontoalmost every line has measure 0; a horizontal line isone exception.

Federer andFlemingprovidedanamazing compactnesstheorem for nice rectifiable sets with multiplicity in a ballinℝ𝑛 in the context of geometricmeasure theory. By “nice”one means that in an appropriate sense the boundariesalso are rectifiable. The only other hypothesis is a boundon the measures of the sets and of their boundaries.Then every sequence has a convergent subsequence inan appropriate weak topology, given by the so-called flatnorm. For example, the flat distance between two closeconcentric spheres is the volume between them. In thisnorm, the exotic surface of Figure 4 and the flat diskare close together, because the region between them hassmall volume. This compactness theorem can be used toshow that many problems have solutions. For example,every smooth closed curve in ℝ3 is the boundary of asurface of least area.

Figure 4.This exotic surface with its long thintentacles is close to the planar disc in the flat norm,because the region between them has small volume.

All of these notions and results have generalizationsfrom ℝ𝑛 to nice Riemannian manifolds. Although solu-tions to geometric problems need not always be smooth,much current regularity theory focuses on proving themrectifiable.

References[1] Morgan, Frank, Geometric Measure Theory: A Beginner’s

Guide, 5th ed., Academic Press, 2016.

Image CreditsFigures by James F. Bredt, copyright Frank Morgan. All rightsreserved.

October 2018 Notices of the AMS 1087