Rectifiability and tangents in a rough Riemannian setting · arXivDesk
2311.00589Nov 1, 2023Substantial re-work and additions to previous version of the paper, including but no limited to: adding a full converse to the second main theorem that principal values existing implies rectifiability, as well as weakening the previous hypotheses on this theorem to no longer require a principal value to exist to deduce rectifiability
Rectifiability and tangents in a rough Riemannian setting
Emily Casey, Max Goering, Tatiana Toro, Bobby Wilson
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 mattila1995rectifiable and the existence of densities with respect to Euclidean balls preiss1987geometry have given rise to major breakthroughs. We study similar questions in a rough elliptic setting where Euclidean balls B(a,r) are replaced by ellipses BΛ(a,r) whose eccentricity and principal axes depend on a
Nearby in the stack
. Given
Λ:Rn→GL(n,R)
, consider the family of ellipses
BΛ(a,r)=a+Λ(a)B(0,r)
. We characterize
m
-rectifiability in terms of the almost everywhere existence of the densities
θΛ(a)m(μ,a)=r↓0limrmμ(BΛ(a,r))∈(0,∞).
We characterize
m
-rectifiable measures in terms of the existence of the principal values-- and even under the weaker assumptions that