Rectifiability and tangents in a rough Riemannian setting

Abstract: Characterizing rectifiability of Radon measures in Euclidean space has led to fundamental contributions to geometric measure theory. Conditions involving existence of principal values of certain singular integrals \cite{mattila1995rectifiable} and the existence of densities with respect to Euclidean balls \cite{preiss1987geometry} have given rise to major breakthroughs. We explore similar questions in a rough elliptic setting where Euclidean balls $B(a,r)$ are replaced by ellipses $B_{\Lambda}(a,r)$ whose eccentricity and principal axes depend on $a$. Precisely, given $\Lambda : \mathbb{R}^{n} \to GL(n,\mathbb{R})$, we first consider the family of ellipses $B_{\Lambda}(a,r) = a + \Lambda(a) B(0,r)$ and show that almost everywhere existence of the principal values $$ \lim_{\epsilon \downarrow 0} \int_{\mathbb{R}^{n} \setminus B_{\Lambda}(a,\epsilon)} \frac{\Lambda(a)^{-1}(y-a)}{|\Lambda(a)^{-1}(y-a)|^{m+1}} d \mu(y) \in (0, \infty) $$ implies rectifiability of the measure $\mu$ under a positive lower density condition. Second we characterize rectifiability in terms of the almost everywhere existence of $$ \theta^{m}_{\Lambda(a)}(\mu,a) = \lim_{r \downarrow 0} \frac{\mu(B_{\Lambda}(a,r))}{r^{m}} \in (0, \infty).$$