Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Tangents to Lipschitz and Sobolev images

Matthew Badger, Jared Krandel, Vyron Vellis

TL;DR

The paper advances geometric differentiability by proving that images of Lipschitz and Sobolev maps possess unique tangents in Attouch-Wets (Euclidean targets) or Gromov-Hausdorff (metric targets) senses when the Sobolev exponent exceeds the dimension, i.e. $p>n$. The authors build a cohesive framework combining AW and GH tangents, conical tangents, and Newtonian-Sobolev theory for metric targets, and they show that finite $n$-packing content is key to upgrading approximate tangents to true tangents. A central methodological innovation is the use of expansive subsets to certify infinite packing content unless tangents stabilize, enabling sharp upgrade criteria. The results illuminate the infinitesimal structure of Sobolev images, extend rectifiability concepts to metric spaces, and offer tools for analyzing tangents in both Euclidean and non-Euclidean targets with potential applications in geometric measure theory and analysis on metric spaces.

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.

Tangents to Lipschitz and Sobolev images

TL;DR

The paper advances geometric differentiability by proving that images of Lipschitz and Sobolev maps possess unique tangents in Attouch-Wets (Euclidean targets) or Gromov-Hausdorff (metric targets) senses when the Sobolev exponent exceeds the dimension, i.e. . The authors build a cohesive framework combining AW and GH tangents, conical tangents, and Newtonian-Sobolev theory for metric targets, and they show that finite -packing content is key to upgrading approximate tangents to true tangents. A central methodological innovation is the use of expansive subsets to certify infinite packing content unless tangents stabilize, enabling sharp upgrade criteria. The results illuminate the infinitesimal structure of Sobolev images, extend rectifiability concepts to metric spaces, and offer tools for analyzing tangents in both Euclidean and non-Euclidean targets with potential applications in geometric measure theory and analysis on metric spaces.

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 Sobolev map from into a Euclidean space with , the image has a unique tangent set (Attouch-Wets convergence) at almost every point with respect to the -dimensional Hausdorff measure. In the analogous case when is a continuous map from into a metric space, we show that the image 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 -rectifiable sets in . We show that approximate tangents to Sobolev images can be upgraded to Attouch-Wets or Gromov-Hausdorff tangents by first proving that the -packing content of Sobolev images is finite, then proving that the inability to upgrade on a set of positive measure implies infinite packing content.
Paper Structure (21 sections, 33 theorems, 150 equations, 2 figures)

This paper contains 21 sections, 33 theorems, 150 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tangent Sets and Metric Tangents
  4. Tangents in metric spaces
  5. Attouch-Wets tangents
  6. Conical tangents
  7. Gromov-Hausdorff versus Attouch-Wets tangents
  8. Sobolev maps into metric spaces
  9. Expansive subsets of metric spaces
  10. Preliminaries on expansive subsets
  11. Expansive subsets give infinite packing content
  12. Tangents and approximate tangents
  13. Approximate Attouch-Wets tangents
  14. Approximate metric tangents
  15. An Application to Hölder images
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $n\geq 1$ and $p\in (n,\infty]$. For every continuous map $f \in W^{1,p}(\mathbb{B}^n,\mathbb{R}^d)$, the image $f(\mathbb{B}^n)$ has finite $\mathcal{H}^n$ measure, and at $\mathcal{H}^n$-a.e. $x \in f(\mathbb{B}^n)$ there exists an $n$-plane $V_x\subset\mathbb{R}^d$ such that $\mathop{\mathrm{

Figures (2)

  • Figure 1: The logarithmic spiral $\mathcal{S}$.
  • Figure 2: If $E=[0,1]^2\times\{0\} \cup \{(0,0)\}\times[0,1]$, then $E$ has an approximate tangent plane at the origin but not a "true" tangent plane. Here $\tilde{E} = \{(0,0)\}\times[0,1]$.

Theorems & Definitions (86)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: Rectifiability
  • Definition 2.2: Finite packing content
  • Definition 2.3: Packing measure
  • Definition 2.4: Packing content
  • Remark 2.5
  • Remark 2.6
  • Lemma 3.1
  • proof
  • ...and 76 more