Tangents to Lipschitz and Sobolev images

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.