Rectifiability; a survey

Abstract

This is a survey on rectifiability. I discuss basic properties of rectifiable sets, measures, currents and varifolds and their role in complex and harmonic analysis, potential theory, calculus of variations, PDEs and some other topics.

arXiv comment129 pages, several small changes and additions

References in corpus (24)

• Rectifiable-Reifenberg and the Regularity of Stationary and Minimizing Harmonic Maps
• Uniform rectifiability, Calderon-Zygmund operators with odd kernel, and quasiorthogonality
• Structure of measures in Lipschitz differentiability spaces
• Second order rectifiability of integral varifolds of locally bounded first variation
• A quantitative version of the Besicovitch projection theorem via multiscale analysis
• Characterizations of rectifiable metric measure spaces
• Rectifiability and approximate differentiability of higher order for sets
• Uniform measures and uniform rectifiability
• Boundedness of the square function and rectifiability
• C^1 Hypersurfaces of the Heisenberg Group are N-Rectifiable
• $L^2$ Curvature Bounds on Manifolds with Bounded Ricci Curvature
• Characterising the big pieces of Lipschitz graphs property using projections
• A rectifiability result for finite-perimeter sets in Carnot groups
• Geometric conditions for the $L^2$-boundedness of singular integral operators with odd kernels with respect to measures with polynomial growth in ${\mathbb{R}}^d$
• Local regularity properties of almost- and quasiminimal sets with a sliding boundary condition
• Necessary condition for rectifiability involving Wasserstein distance $W_2$
• Conjectures and Open Questions on the Structure and Regularity of Spaces with Lower Ricci Curvature Bounds
• On uniform measures in the Heisenberg group
• Second order rectifiability of varifolds of bounded mean curvature
• Dimension rigidity in conformal structures
• A sharp bound on the Hausdorff dimension of the singular set of an n-uniform measure
• Cones, rectifiability, and singular integral operators
• Pointwise differentiability of higher order for distributions
• Dynamics of the scenery flow and conical density theorems