Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The geometry of $C^{1,α}$ flat isometric immersions

Camillo De Lellis, Mohammad Reza Pakzad

Abstract

We show that any isometric immersion of a flat plane domain into $\mathbb R^3$ is developable provided it enjoys the little Hölder regulairty $c^{1,2/3}$. In particular, isometric immersions of local $C^{1,α}$ regularity with $α> 2/3$ belong to this class. The proof is based on the existence of a weak notion of second fundamental form for such immersions, the analysis of the Gauss-Codazzi-Mainardi equations in this weak setting, and a parallel result on the very weak solutions to the degenerate Monge-Ampère equation analyzed by Lewicka and the second author.

The geometry of $C^{1,α}$ flat isometric immersions

Abstract

We show that any isometric immersion of a flat plane domain into is developable provided it enjoys the little Hölder regulairty . In particular, isometric immersions of local regularity with belong to this class. The proof is based on the existence of a weak notion of second fundamental form for such immersions, the analysis of the Gauss-Codazzi-Mainardi equations in this weak setting, and a parallel result on the very weak solutions to the degenerate Monge-Ampère equation analyzed by Lewicka and the second author.

Paper Structure

This paper contains 19 sections, 22 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Recent developments
  4. The main result
  5. Acknowledgments.
  6. Preliminaries: Developable mappings
  7. Global vs. local developability
  8. Developability through test functions
  9. A Hölder continuity toolbox
  10. Notations
  11. Little Hölder spaces
  12. A distributional product and a criteria for Hölder continuity
  13. A second fundamental form for $c^{1,\alpha}$ isometries
  14. Definition of the second fundamental form $A$
  15. $A$ satisfies Gauss-Codazzi-Mainardi equations (in a weak sense)
  16. ...and 4 more sections

Key Result

Theorem 1

Let $\Omega\subset {\mathbb R}^2$ be a domain and $2/3 \le \alpha<1$. If $u \in c^ {1,\alpha}(\Omega, {\mathbb R}^3)$ is an isometric immersion, then $u$ is developable. In particular, if $\alpha>2/3$, we have $C^{1,\alpha}(\Omega) \subset c^{1,2/3} (\Omega)$ and therefore all isometric immersions $

Theorems & Definitions (58)

  • Definition 1
  • Remark 1.1
  • Theorem 1
  • Remark 1.2
  • Theorem 2
  • Remark 1.3
  • Proposition 2.1
  • Corollary 2.2
  • Remark 2.3
  • Remark 2.4
  • ...and 48 more