Tangents to Lipschitz and Sobolev images

TL;DR

This paper establishes geometric differentiability theorems for Lipschitz and Sobolev maps, showing that their images have unique tangent sets or spaces almost everywhere, extending classical rectifiability results.

Contribution

It introduces new geometric tangent theorems for Sobolev and metric space maps, linking packing content to tangent uniqueness, and extends differentiability concepts to more general settings.

Findings

Lipschitz and Sobolev images have unique tangent sets almost everywhere.

Finite packing content of Sobolev images allows upgrading to metric tangents.

Inability to upgrade tangents implies infinite packing content.

Abstract

We develop geometric versions of Rademacher and Calderon 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 $f([0,1]^n)$ 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 $f([0,1]^n)$ 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 show that approximate tangents to Sobolev images can be upgraded to Attouch-Wets or Gromov-Hausdorff tangents by first proving that the $n$-packing content of Sobolev images is finite, then proving that the inability to upgrade on a set of positive measure implies infinite packing content.