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Flexibility of Codimension One $C^{1,θ}$ Isometric Immersions

Dominik Inauen

Abstract

We study the problem of constructing $C^{1,θ}$ isometric immersions of Riemannian metrics on $n$-dimensional domains into $\mathbb{R}^{n+1}$. While the classical Nash--Kuiper theorem established the flexibility of $C^1$ isometries, subsequent work has extended this to $C^{1,θ}$ isometries for certain $θ$, though the optimal exponent remains unknown. In this work we show that any short immersion can be uniformly approximated by $C^{1,θ}$ isometric immersions for $θ< 1/(1+2(n-1))$, improving upon the previously known exponent for $n\geq 3$. The improvement is obtained via a convex integration scheme incorporating a refined iterative integration by parts procedure resting on a detailed structural analysis of error terms and the interaction of multiple frequency scales.

Flexibility of Codimension One $C^{1,θ}$ Isometric Immersions

Abstract

We study the problem of constructing isometric immersions of Riemannian metrics on -dimensional domains into . While the classical Nash--Kuiper theorem established the flexibility of isometries, subsequent work has extended this to isometries for certain , though the optimal exponent remains unknown. In this work we show that any short immersion can be uniformly approximated by isometric immersions for , improving upon the previously known exponent for . The improvement is obtained via a convex integration scheme incorporating a refined iterative integration by parts procedure resting on a detailed structural analysis of error terms and the interaction of multiple frequency scales.
Paper Structure (21 sections, 16 theorems, 279 equations)

This paper contains 21 sections, 16 theorems, 279 equations.

Table of Contents

  1. Introduction
  2. The Nash iteration
  3. $C^{1,\theta}$ convergence
  4. Iterative integration by parts
  5. Novelties
  6. Structure of the paper
  7. Notation
  8. Preliminaries: Decomposition and iterative integration by parts
  9. Decomposition
  10. Integration by parts
  11. Step perturbation
  12. Preliminaries
  13. Step perturbation
  14. Iterative Proposition: Stage
  15. Substages
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $n\geq2$ and let $g\in C^2(\bar{\Omega},\mathrm{Sym}_n^+)$ be a Riemannian metric on a smooth, bounded open set $\Omega\subset\mathbb{R}^n$. Then for any short immersion $\underline u\in C^1(\bar{\Omega},\mathbb{R}^{n+1})$, any $\varepsilon>0$ and any Hölder exponent there exists an isometric immersion $u\in C^{1,\theta}(\bar{\Omega},\mathbb{R}^{n+1})$ such that

Theorems & Definitions (29)

  • Theorem 1.1
  • Remark 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 3.1: Normal vectorfields
  • ...and 19 more