Geometric Measure Theory Recent Applications Tatiana Toro GMT Introduction questions that gave rise to the field, briefly mention some Geometric Measure Theory (GMT) provides a framework of the milestones, and then focus on some of the recent to address questions in very different areas of mathemat- developments at the intersection of GMT, potential theory, ics, including calculus of variations, geometric analysis, po- and harmonic analysis. tential theory, free boundary regularity, harmonic analy- The origins of the field can be traced to the following sis, and theoretical computer science. Progress in different question: do the infinitesimal properties of a measure deter- branches of GMT has led to the emergence of new chal- mine the structure of its support? lenges, making it a very vibrant area of research. In this In the late 1920s and early ’30s Besicovitch was inter- 2 note we will provide a historic background to some of the ested in understanding the structure of a set 퐸 ⊂ ℝ satis- fying 0 < ℋ1(퐸) < ∞ and such that for ℋ1-a.e. 푥 ∈ 퐸, Tatiana Toro is the Craig McKibben & Sarah Merner Professor in Mathematics ℋ1(퐵(푥, 푟) ∩ 퐸) at the University of Washington. She was partially supported by NSF grant lim = 1, (1) + DMS-1664867. Her email address is [email protected]. 푟→0 2푟 Communicated by Notices Associate Editor Chikako Mese. where ℋ1 denotes the 1-dimensional Hausdorff measure. For permission to reprint this article, please contact: (The formulation above is a modern version of the prob- [email protected]. lem. Besicovitch, who most likely was unaware of the exis- DOI: https://doi.org/10.1090/noti1853 tence of the Hausdorff measure, formulated the question 474 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 4 in terms of the linear measure.) For 푛 ≥ 1, the 푛-dimen- 휇 = ℋ푛 퐸 by a general Radon measure 휇? Initially 푛 푚 sional Hausdorff measure ℋ in ℝ generalizes the no- progress on these questions was slow. tions of length of a curve (푛 = 1), surface area (푛 = In 1944, Morse and Randolph [52] proved when 푚 = 푚 2), and volume (푛 = 3) to subsets of ℝ . Moreover 2 that if 휇 is a Radon measure on ℝ푚 for which the 1- 푛 푛 푛 푛 ℒ ℝ = ℋ ℝ density exists 휇-a.e., then 휇 is 1-rectifiable. In 1950, Moore Besicovitch showed that if 퐸 satisfies the hypothesis (1), [51] showed that this result holds for any 푚. In 1961, then 퐸 is 1-rectifiable; that is, 퐸 is contained in a countable Marstrand [48] showed that if 퐸 ⊂ ℝ3 and the 2-density union of Lipschitz images of ℝ union a set of 1-Hausdorff exists for ℋ2 퐸, ℋ2-a.e 푥 ∈ ℝ3, then 퐸 is 2-rectifiable. measure 0 (see [16], [17]). In GMT the notion of rectifia- In 1975, Mattila [49] proved that if 퐸 ⊂ ℝ푚 and the 푛- bility is used to describe the structure (also the regularity) density exists for ℋ푛 퐸, ℋ푛-a.e 푥 ∈ ℝ푚, then 퐸 is 푛- of a set or a measure in a way similar to how the degree of rectifiable, completing the study of the problem formea- differentiability of charts is used to describe the smooth- sures that were defined as the restriction of Hausdorff mea- 푚 ness of a manifold in differential geometry. A set 퐸 ⊂ ℝ sure to a subset of Euclidean space. This still left open the is 푛-rectifiable if case of a general Radon measure. ∞ In 1987, in a true masterpiece, Preiss [56] showed that 푛 푚 퐸 ⊂ ⋃ 푓푗(ℝ ) ∪ 퐸0, if 휇 is a Radon measure on ℝ for which the 푛-density 푖=1 exists for 휇-a.e 푥 ∈ ℝ푚, then 휇 is 푛-rectifiable. Preiss 푛 푚 푛 where 푓푗 ∶ ℝ → ℝ is a Lipschitz map and ℋ (퐸0) = introduced a number of new tools and ideas whose appli- 0. Recall that 푓푗 is Lipschitz if there exists 퐿푗 > 0 s.t. for cations are still being unraveled and play a central role in 푥, 푦 ∈ ℝ푛 the results to be discussed later in this article. The question of rectifiability of a measure carries information about the |푓푗(푥) − 푓푗(푦)| ≤ 퐿푗|푥 − 푦|. fine structure of its measure-theoretic support. Motivated In 1947, Federer [30] proved a general converse of Besicov- by this perspective, Preiss introduced the notion of tangent 푚 itch’s theorem: if 푛 < 푚 and 퐸 ⊂ ℝ is 푛-rectifiable, then measures, which play the role that derivatives do when an- 푛 for ℋ -a.e. 푥 ∈ 퐸 alyzing the regularity of a function. They are obtained by ℋ푛(퐵(푥, 푟) ∩ 퐸) a limiting process of rescaled multiples of the initial mea- lim = 1, 푛 (2) 푟→0+ 휔푛푟 sure. Preiss’s argument includes a number of major steps, some of which have given rise to very interesting questions. where 휔푛 denotes the Lebesgue measure of the unit ball in ℝ푛. Roughly speaking, a blow up procedure shows that when 푛 휇 휇 휇 We introduce some terminology that will help us set the -density of exists -a.e., then at -a.e. point all tan- 푚 푛 휈 푛 the framework. Let 휇 be a Radon measure in ℝ (i.e. a gent measures are -uniform. A measure is -uniform 퐶 > 0 푟 > 0 푥 Borel regular measure that is finite on compact sets). The if there is a constant so that for and in Λ = spt 휈 = {푦 ∈ ℝ푚 ∶ 휈(퐵(푦, 푠)) > 푛-density of 휇 at 푥 the support 0 for every 푠 > 0} of 휈, we have 휇(퐵(푥, 푟)) 휃푛(휇, 푥) ∶= lim 푛 (3) 푛 푟→0 휔푛푟 휈(퐵(푥, 푟)) = 퐶푟 . (4) exists if the limit exists and 휃푛(휇, 푥) ∈ (0, ∞) 푚 His argument now requires a detailed understanding of A locally finite measure 휇 on ℝ is 푛-rectifiable if 휇 is the structure and geometry of the support of 푛-uniform absolutely continuous with respect to ℋ푛 (휇 ≪ ℋ푛, i.e 푛 measures. By work of Kirchheim and Preiss [43], the sup- ℋ (퐹) = 0 implies 휇(퐹) = 0) and port of an 푛-uniform measure 휈 is an analytic variety. Thus, ∞ using the fact that at 휇-a.e point, tangent measures to tan- 푚 푛 휇(ℝ \ ⋃ 푓푗(ℝ )) = 0, gent measures to 휇 are tangent measures to 휇, he showed 푗=1 that at 휇-a.e. point there is an 푛-flat tangent measure; that 푛 푚 where each 푓푗 ∶ ℝ → ℝ is Lipschitz. Recasting the re- is, a measure that is a multiple of the 푛-dimensional Haus- sults above in this light, we have in the case when 푚 = 2 dorff measure restricted to an 푛-plane. Then he showed and 푛 = 1, that by Besicovitch’s work if 퐸 ⊂ ℝ푚 such that that either an 푛-uniform measure 휈 is an 푛-flat measure 0 < ℋ푛(퐸) < ∞, and for ℋ푛-a.e. 푥 ∈ 퐸 the density of or its support is very far away from any 푛-plane. Using a 휇 = ℋ푛 퐸 exists and is 1, then 퐸 is 푛-rectifiable. By Fed- deep result about the “cones” of measures, he proves that erer’s work the converse in any dimension is true, that is, necessarily at 휇-a.e point all tangent measures are 푛-flat. if 퐸 ⊂ ℝ푚 is 푛-rectifiable then the density of 휇 = ℋ푛 퐸 Then modulo showing that this implies that the measure- exists and is 1 for ℋ푛-a.e. 푥 ∈ 퐸. Two natural questions theoretic support of 휇 satisfies the hypothesis of the Mar- arise at this point: 1) does Besicovitch’s result hold for any strand-Mattila rectifiability criterion, one concludes that 휇 푛, 푚 ∈ ℕ with 푛 < 푚? 2) what happens if we replace is 푛-rectifiable. APRIL 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 475 Harmonic Measure To illustrate how ideas from one field can have a profound impact in another, we will focus on some of the recent applications of Preiss’s work to harmonic analysis and po- tential theory. The harmonic measure is a canonical mea- sure associated to the Laplacian (see definition below). It plays a fundamental role in potential theory, constitutes the main building block for the solutions of the classical Dirichlet problem, and in non-smooth domains is the ob- ject that allows us to describe boundary regularity of the so- lutions to Laplace’s equation. We recall some of the back- ground. Let Ω ⊂ ℝ푛+1 be a bounded domain, let 푓 be a continuous function on the boundary of Ω, i.e. 푓 ∈ 퐶(휕Ω). The classical Dirichlet problems asks whether there exists a function 푢 ∈ 퐶(Ω) ∩ 푊1,2(Ω) such that Δ푢 = 0 in Ω { (5) 푢 = 푓 on 휕Ω. 1,2 Figure 1. Kowalski-Preiss cone Here 푢 ∈ 푊 (Ω) means that 푢 and its weak deriva- tives are in 퐿2(Ω) and Δ푢 = 0 is interpreted in the weak 휁 ∈ 퐶1(Ω) A natural and easy-to-state question derived from this sense; that is, for any 푐 , work is: what does the support of an 푛-uniform measure on ℝ푚 really looks like? Clearly the restriction of 푛-Haus- ∫⟨∇푢, ∇휁⟩ = 0. dorff measure to an 푛-plane is 푛-uniform, but are there other examples? Work of Preiss [56] shows that if 푛 = The questions here are whether a solution 푢 of (5) exists, if 1, 2 flat measures are the only examples of 푛-uniform mea- so how regular it is, and whether there is a formula in terms sures.