We develop geometric versions of Rademacher and Calderón type differentiability theorems in two categories. A special case of our results is that for any Lipschitz or continuous $W^{1,p}$ Sobolev map $f$ from ${[0,1]}^n$ into a Euclidean space with $p>n$, the image $f\left({[0,1]}^n\right)$ has a unique tangent set (Attouch-Wets convergence) at almost every point with respect to the $n$-dimensional Hausdorff measure. In the analogous case when $f$ is a continuous $N^{1,p}$ map from ${[0,1]}^n$ into a metric space, we show that the image $f\left({[0,1]}^n\right)$ has a unique metric tangent (Gromov-Hausdorff convergence) almost everywhere. These results complement, but are distinct from, Federer's theorem on existence and uniqueness of approximate tangents of $n$-rectifiable sets in ${\mathbb{R}}^d$.

We investigate the minimum and maximum number of nodal domains across all time-dependent homogeneous caloric polynomials of degree $d$ in ${\mathbb{R}}^n\times \mathbb{R}$ (space × time), i.e., polynomial solutions of the heat equation satisfying ${\partial }_{t}p\not\equiv 0$ and $p(\lambda x,{\lambda }^{2}t)={\lambda }^{d}p(x,t)$ for all $x\in {\mathbb{R}}^n$, $t\in \mathbb{R}$, and $\lambda >0$. When $n=1$, the number of nodal domains is classically precisely $\lceil d/2\rceil$. When $n=2$, we prove that the minimum number of nodal domains is $2$ if $d\not\equiv 0\left(\mathrm{mod}4\right)$ and is $3$ if $d\equiv 0\left(\mathrm{mod}4\right)$. When $n\ge 3$, we prove that the minimum number of nodal domains is $2$ for all $d$. Finally, the maximum number of nodal domains is $\mathrm{\Theta }\left(d^n\right)$ as $d\to \infty$ and lies between ${\lfloor d/n\rfloor }^n$ and $\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)$ for all $n$ and $d$.

We prove that for all integers $2\le m\le d-1$, there exist doubling measures on ${\mathbb{R}}^d$ with full support that are $m$-rectifiable and purely $(m-1)$-unrectifiable in the sense of Federer (i.e., without assuming $\mu \ll {\mathcal{H}}^m$). Our construction is informed by a simple observation about square packing in the plane: $N$ axis-parallel squares of side length $s$ pack inside a square of side length $\lceil N^{1/2}\rceil s$. One consequence is that for each $m=2,3,4$ and $s<\infty$, there exist doubling measures $\mu$ on the Heisenberg group ${\mathbb{H}}^1$ and Lipschitz maps $f:E\subset {\mathbb{R}}^m\to {\mathbb{H}}^1$ such that $\mu \ll {\mathcal{H}}^s$, $f\left(E\right)$ has Hausdorff dimension $s$, and $\mu \left(f\right(E\left)\right)>0$. This is striking because ${\mathcal{H}}^m\left(f\right(E\left)\right)=0$ for every Lipschitz map $f:E\subset {\mathbb{R}}^m\to {\mathbb{H}}^1$ by a theorem of Ambrosio and Kirchheim (2000).

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 Hölder 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. The examples arise from oscillating or rotating a blow-up limit by an infinite amount, but very slowly.

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.

For every $n\ge 2$, 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 ${\mathbb{R}}^n$ on which harmonic measure is defined. Jones and Wolff (1988) proved that $b_2=1$. When $n\ge 3$, Bourgain (1987) proved that $b_n>0$ and Wolff (1995) produced examples showing $b_n<1$. Refining Bourgain's original outline, we prove that $b_n\ge c^{*}{n}^{-2n(n-1)/\ln \left(n\right)}$ for all $n\ge 3$, where $c>0$ is a constant independent of $n$. We further estimate $b_3\ge 1\times {10}^{-15}$ and $b_4\ge 2\times {10}^{-26}$.

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. 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 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 ${\mathbb{R}}^2$ (P. Jones, 1990), in ${\mathbb{R}}^n$ (K. Okikiolu, 1992), or in an arbitrary Carnot group (the second author) in terms of local geometric least squares data called Jones' $\beta$-numbers. In a secondary result, we implement the Garnett-Killip-Schul construction of a doubling measure in ${\mathbb{R}}^n$ that charges a rectifiable curve in an arbitrary complete, quasiconvex, doubling metric space.

We examine caloric measures on general domains in ${\mathbb{R}}^{n+1}={\mathbb{R}}^n\times \mathbb{R}$ (space × time) from the perspective of geometric measure theory. We give a direct proof that the lower parabolic Hausdorff dimension of caloric measure is at least $n$ and is absolutely continuous with respect to the $n$-dimensional parabolic Hausdorff measure. We also prove that the upper parabolic Hausdorff dimension of caloric measure is at most $n+2-{\beta }_n$, where ${\beta }_n>0$ depends only on $n$. Analogous bounds for harmonic measures were first shown by Nevanlinna (1934) and Bourgain (1987). In the course of the proof, we give a caloric measure analogue of Bourgain's alternative involving cubes $Q$, subcubes $F$ and $Q^{*}$, and closed sets $E\subset {\mathbb{R}}^n$: either the part of $E$ in $Q$ has relatively large caloric measure in $Q\setminus E$, or the part of $E$ in $Q^{*}$ has relatively small $\rho$-dimensional parabolic Hausdorff content for every $n<\rho \le n+2$.

For all $1\le m\le n-1$, we investigate the interaction of locally finite measures in ${\mathbb{R}}^n$ with the family of $m$-dimensional Lipschitz graphs. We characterize Radon measures $\mu$ that are carried by Lipschitz graphs in the sense that there exist graphs ${\Gamma }_1,{\Gamma }_2,\dots$ such that $\mu ({\mathbb{R}}^n\setminus \bigcup {\Gamma }_i)=0$, using only countably many evaluations of the measure. This problem was classically studied within smaller classes of measures, e.g., for the restrictions of $m$-dimensional Hausdorff measure ${\mathcal{H}}^m$ to sets $E$ with $0<{\mathcal{H}}^m\left(E\right)<\infty$. To develop a characterization for arbitrary Radon measures, we look at the behavior of coarse doubling ratios of the measure on dyadic cubes intersecting conical annuli.

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 (1990) and subsequently solved in higher-dimensional Euclidean spaces by Okikiolu (1992) and in the infinite-dimensional Hilbert space ${\ell }^2$ by Schul (2007). We establish sharp extensions of Schul's necessary and sufficient conditions for a bounded set $E\subset {\ell }^p$ to be contained in a rectifiable curve from $p=2$ to $1<p<\infty$. 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 $p\ne 2$. We also identify and correct technical errors in the proof by Schul.

We investigate the Hölder geometry of curves generated by iterated function systems (IFS) in a complete metric space. We prove that every connected attractor of an IFS is $1/s$-Hölder path-connected, where $s$ is the similarity dimension of the IFS, and is parameterized by a $1/a$-Hölder curve for all $a>s$. At the endpoint $a=s$, a theorem of Remes (1998) established that connected self-similar sets in Euclidean space satisfying the open set condition are parameterized by $1/s$-Hölder curves. An interesting phenomenon emerges in the self-affine setting: while the optimal parameter $s$ for a self-similar curve in ${\mathbb{R}}^n$ is at most $n$, the optimal parameter for a self-affine curve in ${\mathbb{R}}^n$ may be strictly greater than $n$.

In non-variational two-phase free boundary problems for harmonic measure, we examine how the relationship between the interior and exterior harmonic measures of a domain in $n$-dimensional Euclidean space influences the geometry of its boundary. The focus is on the singular set in the free boundary, where the boundary looks infinitesimally like zero sets of homogeneous harmonic polynomials of degree at least 2. We prove that if the Radon-Nikodym derivative of the exterior harmonic measure with respect to the interior harmonic measure has a Hölder continuous logarithm, then the free boundary admits unique geometric blowups at every singular point and the singular set can be covered by countably many $C^{1,\beta }$ submanifolds of dimension at most $n-3$. This is obtained in part by adapting Garofalo and Petrosyan's Weiss type monotonicity formula and an epiperimetric inequality from the variational to the non-variational setting.

We investigate the geometry of sets in Euclidean and infinite-dimensional Hilbert spaces. We establish sufficient conditions that ensure a set of points is contained in the image of a $(1/s)$-Hölder continuous map $f:[0,1]\to {\ell }^2$, with $s>1$. Our results generalize the "sufficient half" of the Analyst's Traveling Salesman Theorem, which characterizes subsets of rectifiable curves in ${\mathbb{R}}^N$ or ${\ell }^2$ in terms of a quadratic sum of linear approximation numbers called Jones' $\beta$-numbers. We show how to construct parameterizations of intermediate approximating curves $f_k\left(\right[0,1\left]\right)$ and find conditions in terms of tube approximations ensuring the approximating curves converge to a Hölder curve. As an application, we provide sufficient conditions to guarantee fractional rectifiability of pointwise doubling measures in ${\mathbb{R}}^N$.

One goal of geometric measure theory is to understand how measures in the plane or higher dimensional Euclidean space interact with families of lower dimensional sets. An important dichotomy arises between the class of rectifiable measures, which give full measure to a countable union of the lower dimensional sets, and the class of purely unrectifiable measures, which assign measure zero to each distinguished set. In this paper, we describe a framework for generalized rectifiability, review classical results on rectifiable measures in this context, and survey recent advances on the identification problem for Radon measures that are carried by Lipschitz, Hölder, or $C^{1,\alpha }$ images of Euclidean subspaces, including theorems of Azzam-Tolsa, Badger-Schul, Badger-Vellis, Edelen-Naber-Valtorta, Ghinassi, and Tolsa-Toro.

We investigate the influence that $s$-dimensional lower and upper Hausdorff densities have on the geometry of a Radon measure in ${\mathbb{R}}^n$ when $s$ is a real number between $0$ and $n$. When $0<s<1$, we prove that measures with almost everywhere positive lower density and finite upper density are carried by countably many bi-Lipschitz curves. When $1\le s<n$, we identify conditions on the lower density that ensure the measure is either carried by or singular to $(1/s)$-Hölder curves. We also introduce Hölder and bi-Lipschitz parameterization theorems for Euclidean sets with "small" Assouad dimension.

A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures in $n$-dimensional Euclidean space for all $n\ge 2$ in terms of positivity of the lower density and finiteness of a geometric square function, which records in an $L^2$ gauge the extent to which the measure admits approximate tangent lines along its support. In contrast with the classical theorems of Besicovitch, Morse and Randolph, and Moore, we do not assume an a priori relationship between the measure and $1$-dimensional Hausdorff measure. We also characterize purely 1-unrectifiable Radon measures, i.e., locally finite measures that give measure zero to every finite length curve. Along the way, we develop an $L^2$ variant of Jones' traveling salesman construction.

The zero sets of harmonic polynomials play a crucial role in the study of the free boundary regularity problem for harmonic measure. We study how "degree $k$ points" sit inside zero sets of harmonic polynomials in ${\mathbb{R}}^n$ of degree $d$ (for all $n\ge 2$ and $1\le k\le d$) and inside sets admitting arbitrarily good local approximations by such zero sets. We obtain a general structure theorem including sharp Hausdorff and Minkowski dimension estimates on the singular set of "degree $k$ points" ( $k\ge 2$) without proving uniqueness of blowups or using PDE methods such as monotonicity formulae. In the presence of a topological separation condition, the sharp dimension estimates improve and depend on the parity of $k$. An application is given to the two-phase free boundary regularity problem for harmonic measure below the continuous threshold introduced by Kenig and Toro.

Consider a 1-sided NTA domain (a uniform domain) in ${\mathbb{R}}^{n+1}$, $n\ge 2$, whose boundary is $n$-dimensional Ahlfors-David regular. We characterize the rectifiability of the boundary in terms of absolute continuity of surface measure with respect to harmonic measure. We show these are equivalent to: the boundary being covered ${\mathcal{H}}^n$-a.e. by boundaries of bounded chord-arc subdomains, and to the boundary possessing exterior corkscrew points in a qualitative sense ${\mathcal{H}}^n$-a.e. Our methods apply to harmonic measure and also to elliptic measures associated with real symmetric second order divergence form elliptic operators with locally Lipschitz coefficients whose derivatives satisfy a natural Carleson condition.

We identify two sufficient conditions for locally finite Borel measures on ${\mathbb{R}}^n$ to give full mass to a countable family of Lipschitz maps of ${\mathbb{R}}^m$. The first condition, extending a prior result of Pajot, is a sufficient test in terms of $L^p$ affine approximability for a locally finite Borel measure $\mu$ satisfying ${lim\; sup}_{r\downarrow 0}\mu \left(B\right(x,r\left)\right)/r^m<\infty$ at $\mu$-almost every $x$ to be $m$-rectifiable. The second condition is an assumption on the growth rate of the 1-density that ensures a locally finite Borel measure $\mu$ with ${lim}_{r\downarrow 0}\mu \left(B\right(x,r\left)\right)/r=\infty$ at $\mu$-almost every $x$ is 1-rectifiable.

We investigate the interplay between the local and asymptotic geometry of a set $A$ in ${\mathbb{R}}^n$ and the geometry of model sets that approximate $A$ locally uniformly on small scales. The framework for local set approximation developed here unifies and extends ideas of Jones, Mattila-Vuorinen, Reifenberg, and Preiss. We indicate several applications to variational problems in geometric measure theory and PDE. For instance, we show that the singular part of the support of an $(n-1)$-dimensional asymptotically optimally doubling measure in ${\mathbb{R}}^n$ ($n\ge 4$) has upper Minkowski dimension at most $n-4$.

A quasiplane is the image of an $n$-dimensional Euclidean subspace of ${\mathbb{R}}^N$ ($1\le n\le N-1$) under a quasiconformal map of ${\mathbb{R}}^N$. We give sufficient conditions in terms of the weak quasisymmetry constant of the underlying map for a quasiplane to be a bi-Lipschitz $n$-manifold and for a quasiplane to have big pieces of bi-Lipschitz images of ${\mathbb{R}}^n$. One main novelty is the analysis of quasiplanes in arbitrary codimension $N-n$. To establish the big pieces criterion, we prove new extension theorems for "almost affine" maps. This work is related to investigations by Tukia and Väisälä on extensions of quasisymmetric maps with small distortion.

We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in ${\mathbb{R}}^n$, $n>2$. To each locally finite Borel measure $\mu$, we associate a function $t{J}_2(\mu ,x)$ which uses a weighted sum to record how closely the mass of $\mu$ is concentrated on a line in the triples of dyadic cubes containing $x$. We show that $t{J}_2(\mu ,x)<\infty$ $\mu$-a.e. is a necessary condition for $\mu$ to give full mass to a countable family of rectifiable curves. This confirms a conjecture of Peter Jones from 2000. No assumption is made on the upper Hausdorff density of the measure, allowing analysis of generic 1-rectifiable measures that are mutually singular with ${\mathcal{H}}^1$.

For each $1\le p<\infty$, we formulate a necessary and sufficient condition for an admissible metric to be extremal for the Fuglede $p$-modulus of a system of measures. When $p=2$, this characterization generalizes Beurling's criterion, a sufficient condition for an admissible metric to be extremal for the extremal length of a planar curve family. In addition, we prove that every non-negative Borel function in Euclidean space with positive and finite $p$-norm is extremal for the $p$-modulus of some curve family.

We obtain Dini conditions with "exponent 2" that guarantee that an asymptotically conformal quasisphere is rectifiable. We also establish estimates for the weak quasisymmetry constant of a global $K$-quasiconformal map in neighborhoods where the maximal dilatation is close to 1.

We obtain quantitative estimates of local flatness of zero sets of harmonic polynomials. There are two alternatives: at every point either the zero set stays uniformly far from a hyperplane in the Hausdorff distance at all scales, or the zero set becomes locally flat on small scales with arbitrarily small constant. An application is given to a free boundary problem for harmonic measure from two sides, where blow-ups of the boundary are zero sets of harmonic polynomials.

We show the David-Jerison construction of big pieces of Lipschitz graphs inside a corkscrew domain does not require its surface measure to be upper Ahlfors regular. Thus we can study absolute continuity of harmonic measure and surface measure on NTA domains of locally finite perimeter using Lipschitz approximations. A partial analogue of the F. and M. Riesz Theorem for simply connected planar domains is obtained for NTA domains in space. As a consequence, every Wolff snowflake has infinite surface measure.

We show that on an NTA domain, if each tangent measure to harmonic measure at a point is a polynomial harmonic measure, then the associated polynomials are homogeneous. More precisely, if $\omega$ is harmonic measure on an NTA domain $\Omega$ and at a point $q\in \partial \Omega$ every tangent measure to $\omega$ is of the form ${\left|p\right|}^2{\mathcal{H}}^{n-1}\lfloor p^{-1}\left(0\right)\rfloor$ for some harmonic polynomial $p$, then $p$ must be homogeneous. Geometric information for solutions of a two-phase free boundary problem studied by Kenig and Toro is derived.