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Rigidity and Flexibility of Isometric Extensions

Wentao Cao, Dominik Inauen

Abstract

In this paper we consider the rigidity and flexibility of $C^{1, θ}$ isometric extensions and we show that the Hölder exponent $θ_0=\frac12$ is critical in the following sense: if $u\in C^{1,θ}$ is an isometric extension of a smooth isometric embedding of a codimension one submanifold $Σ$ and $θ> \frac12$, then the tangential connection agrees with the Levi-Civita connection along $Σ$. On the other hand, for any $θ<\frac12$ we can construct $C^{1,θ}$ isometric extensions via convex integration which violate such property. As a byproduct we get moreover an existence theorem for $C^{1, θ}$ isometric embeddings, $θ<\frac12$, of compact Riemannian manifolds with $C^1$ metrics and sharper amount of codimension.

Rigidity and Flexibility of Isometric Extensions

Abstract

In this paper we consider the rigidity and flexibility of isometric extensions and we show that the Hölder exponent is critical in the following sense: if is an isometric extension of a smooth isometric embedding of a codimension one submanifold and , then the tangential connection agrees with the Levi-Civita connection along . On the other hand, for any we can construct isometric extensions via convex integration which violate such property. As a byproduct we get moreover an existence theorem for isometric embeddings, , of compact Riemannian manifolds with metrics and sharper amount of codimension.

Paper Structure

This paper contains 12 sections, 11 theorems, 212 equations.

Table of Contents

  1. Introduction
  2. The proof of Theorem \ref{['t:rigidity-flexibility']} (1): Rigidity part
  3. Preliminaries
  4. Hölder Norms and interpolation
  5. Mollification estimates
  6. Hölder norms and mollification on manifolds
  7. Matrix decomposition
  8. Existence of normal vectorfields
  9. Iteration proposition
  10. Proof of Theorem \ref{['t:rigidity-flexibility']} (2): Flexibility part
  11. Proof of Theorem \ref{['t:global']}
  12. Proof of Lemma \ref{['l:normal']}

Key Result

Theorem 1.1

Let $\Sigma$ be a codimension one oriented submanifold of the compact Riemannian manifold $(\mathcal{M}, g)$, where $g\in C^1$, and let $\nu$ be the unique unit normal vectorfield respecting the orientation. Suppose moreover that $f:\Sigma\to \mathbb R^{m}$ is an isometric embedding satisfying e:HW

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Proposition 4.1
  • Remark 4.1: Constants
  • proof
  • ...and 5 more