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Lipschitz maps with prescribed local Lipschitz constants

Aidan Backus, Ng Ze-An

Abstract

Let $Γ$ be a closed subset of a complete Riemannian manifold $M$ of dimension $\geq 2$, let $f: M \to N$ be a Lipschitz map to a complete Riemannian manifold $N$, and let $ψ$ be a continuous function which dominates the local Lipschitz constant of $f$. We construct a Lipschitz map which agress with $f$ on $Γ$ and whose local Lipschitz constant is $ψ$.

Lipschitz maps with prescribed local Lipschitz constants

Abstract

Let be a closed subset of a complete Riemannian manifold of dimension , let be a Lipschitz map to a complete Riemannian manifold , and let be a continuous function which dominates the local Lipschitz constant of . We construct a Lipschitz map which agress with on and whose local Lipschitz constant is .
Paper Structure (12 sections, 16 theorems, 70 equations)

This paper contains 12 sections, 16 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Idea of the proof
  3. Acknowledgements
  4. Preliminaries
  5. Metric geometry
  6. Maxima of upper-semicontinous functions
  7. Lipschitz maps
  8. Proof of Theorem \ref{['eikonal extension']}
  9. Modifying the Lipschitz constant on a packing
  10. Iterating the packing procedure
  11. Taking the limit of the packings
  12. Removing the geometric assumptions

Key Result

Theorem 1.1

Let $M$ be a complete Riemannian manifold such that $\dim M \geq 2$, and let $N$ be a complete Riemannian manifold. Let $\Gamma \subseteq M$ be a closed set and $\psi: M \setminus \Gamma \to \mathbf{R}_+$ be a continuous function. Then for every Lipschitz map $f: M \to N$ such that $Lf \leq \psi$ on

Theorems & Definitions (38)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 28 more