# Rectifiability and Fine Geometry of Sets, Radon Measures, Harmonic Functions, and Temperatures

### Principal Investigator: Matthew Badger

### Award Dates: August 2022 – July 2025

## Award Abstract

Geometric measure theory provides an analytical toolkit to detect hidden structure in high-dimensional data sets and to describe non-smooth phenomena such as the formation of corners and singularities in soap bubble clusters. The term *measure* refers to an abstract generalization of length, area, and volume, which assigns a notion of size to each mathematical set. The pervasiveness of measures within contemporary mathematics notwithstanding, there are presently only a few tools available to analyze measures in regimes with low regularity. The proposed investigation seeks to develop novel and robust quantitative methods to study the geometry of general sets and measures in the absence of traditional simplifying hypotheses. The project will explore applications of these tools and methods to the analysis of harmonic functions and temperatures (solutions to the heat equation) in domains with rough boundary, both in ideal settings and inside non-homogenous media. The project will also provide training to graduate research assistants at the University of Connecticut and to visiting Ph.D. students working in geometric measure theory, harmonic analysis, and partial differential equations.

The project will develop four interrelated threads of research on the fine geometry of sets, the structure and regularity of measures, and the solutions of partial differential equations. The first line of inquiry pursues questions about subsets of rectifiable curves in infinite dimensional Banach spaces, with a concrete goal of solving the Analyst's Traveling Salesman Problem in a non-Hilbert setting. A second line of inquiry concerns the parameterization problem for Lipschitz images of the plane and the classification of 2-rectifiable Radon measures. Additional work will focus on the structure of measures supported on the graphs of Hölder continuous functions. A third direction of study will involve improved upper bounds on the Hausdorff dimension of harmonic and caloric measures on general domains, along with the relationship between the static and time-dependent cases. The final strand of the research program will confront contemporary challenges and initiate new inquiries in the theory of non-variational free boundary problems for harmonic and elliptic measures with two or more phases.

## Research Products

**#26** Square packings and rectifiable doubling measures

(arXiv:2309.01283)
(*with* Raanan Schul)
[Click to Show/Hide Abstract]
We prove that for all integers 2 ≤ m ≤ d-1, there exists doubling measures on **R**^{d} with full support that are m-rectifiable and purely (m-1)-unrectifiable in the sense of Federer (i.e. without assuming μ≪H^{m}). The corresponding result for 1-rectifiable measures is originally due to Garnett, Killip, and Schul (2010). Our construction of higher-dimensional Lipschitz images is informed by a simple observation about square packing in the plane: N axis-parallel squares of side length s pack inside of a square of side length ⌈N^{1/2}⌉s. The approach is robust and when combined with standard metric geometry techniques allows for constructions in complete Ahlfors regular metric spaces. One consequence of the main theorem is that for each m=2,3,4 and s< there exist doubling measures μ on the Heisenberg group **H**^{1} and Lipschitz maps f:E⊂R^{m}→**H**^{1} such that μ≪H^^{s-ε} for all ε>, f(E) has Hausdorff dimension s, and μ(f(E))>0. This is striking, because H^{m}(f(E))=0 for every Lipschitz map f:E⊂**R**^{m}→**H**^{1} by a theorem of Ambrosio and Kirchheim (2000). Another application of the square packing construction is that every compact metric space **X** of Assouad dimension strictly less than m is a Lipschitz image of a compact set E⊂[0,1]^{m}. Of independent interest, we record the existence of doubling measures on complete Ahlfors regular metric spaces with prescribed lower and upper Hausdorff and packing dimensions.
**Status:** preprint, submitted.
**#24** Slowly vanishing mean oscillations: non-uniqueness of blow-ups in a two-phase free boundary problem

(arXiv:2210.17531)
(*with* Max Engelstein and Tatiana Toro)
*Dedicado a Carlos Kenig, un gran maestro y amigo en conmemoración de sus 70 años*
[Click to Show/Hide Abstract]
In Kenig and Toro's two-phase free boundary problem, one studies how the regularity of the Radon-Nikodym derivative h of exterior harmonic measure with respect to interior harmonic measure on complementary NTA domains controls the geometry of their common boundary. It is now known that log h being Holder continuous implies that pointwise the boundary has a unique blow-up, which is the zero set of a homogeneous harmonic polynomial. In this note, we give examples of domains with log h being continuous whose boundaries have points with non-unique blow-ups. Philosophically the examples arise from oscillating or rotating a blow-up limit by an infinite amount, but very slowly.
**Status:** accepted, to appear in Vietnam J. Math.
**#23** Subsets of rectifiable curves in Banach spaces II: universal estimates for almost flat arcs

(arXiv:2208.10288)
(*with* Sean McCurdy)
[Click to Show/Hide Abstract]
We prove that in any Banach space the set of windows in which a rectifiable curve resembles two or more straight line segments is quantitatively small with constants that are independent of the curve, the dimension of the space, and the choice of norm. Together with Part I, this completes the proof of the necessary half of the Analyst's Traveling Salesman theorem with sharp exponent in uniformly convex spaces.
**Status:** accepted, to appear in Illinois Journal of Mathematics
**#22** Lower bounds on Bourgain's constant for harmonic measure

(arXiv:2205.15101)
(*with* Alyssa Genschaw)
[Click to Show/Hide Abstract]
For every n≥2, Bourgain's constant b_{n} is the largest number such that the (upper) Hausdorff dimension of harmonic measure is at most n-b_{n} for every domain in R^{n} on which harmonic measure is defined. Jones and Wolff (1988) proved that b_{2}=1. When n≥3, Bourgain (1987) proved that b_{n} and Wolff (1995) produced examples showing b_{n}<1. Refining Bourgain's original outline, we prove that b_{n}≥c*n^{-2n(n-1)}/ln(n) for all n≥3, where c>0 is a constant that is independent of n. We further estimate b_{3}≥1×10^{-15} and b_{4}≥2×10^{-26}.
**Status:** preprint, submitted.
**#21** Identifying 1-rectifiable measures in Carnot groups

(arXiv:2109.06753)
(*with* Sean Li and Scott Zimmerman)
[Click to Show/Hide Abstract]
We continue to develop a program in geometric measure theory that seeks to identify how measures in a space interact with canonical families of sets in the space. In particular, extending a theorem of the first author and R. Schul in Euclidean space, for an arbitrary locally finite Borel measure in an arbitrary Carnot group, we develop tests that identify the part of the measure that is carried by rectifiable curves and the part of the measure that is singular to rectifiable curves. Our main result is entwined with an extension of the Analyst's Traveling Salesman Theorem, which characterizes subsets of rectifiable curves in R^{2} (P. Jones, 1990), in R^{n} (K. Okikolu, 1992), or in an arbitrary Carnot group (the second author) in terms of local geometric least squares data called Jones' β-numbers. In a secondary result, we implement the Garnett-Killip-Schul construction of a doubling measure in R^{n} that charges a rectifiable curve in an arbitrary complete, quasiconvex, doubling metric space.
**Status:** accepted, to appear in Anal. Geom. Metr. Spaces
**#18** Subsets of rectifiable curves in Banach spaces I: sharp exponents in traveling salesman theorems

(arXiv:2002.11878)
(*with* Sean McCurdy)
[Click to Show/Hide Abstract]
The Analyst's Traveling Salesman Problem is to find a characterization of subsets of rectifiable curves in a metric space. This problem was introduced and solved in the plane by Jones in 1990 and subsequently solved in higher-dimensional Euclidean spaces by Okikiolu in 1992 and in the infinite-dimensional Hilbert space ℓ_{2} by Schul in 2007. In this paper, we establish sharp extensions of Schul's necessary and sufficient conditions for a bounded set E ⊂ ℓ_{p} to be contained in a rectifiable curve from p=2 to 1<p<∞. While the necessary and sufficient conditions coincide when p=2, we demonstrate that there is a strict gap between the necessary condition and sufficient condition when p≠2. We also identify and correct technical errors in the proof by Schul. This investigation is partly motivated by recent work of Edelen, Naber, and Valtorta on Reifenberg-type theorems in Banach spaces and complements work of Hahlomaa and recent work of David and Schul on the Analyst's TSP in general metric spaces.
**Note:** While revising an earlier draft of the manuscript, we identified a mistake in Schul's 2007 proof of the necessary conditions in the traveling salesman theorem in infinite-dimensional Hilbert space. We show how to correct the error in a minimal way, leaving the outline of virtually all of the original proofs intact.
**Status:** accepted, to appear in Illinois Journal of Mathematics
Date of Freshest Content: September 6, 2023.