Rigidity and Flexibility of Isometric Extensions

TL;DR

This paper investigates the rigidity and flexibility of $C^{1, heta}$ isometric extensions, revealing a critical H"older exponent at $rac{1}{2}$ that determines whether the tangential connection aligns with the Levi-Civita connection.

Contribution

It establishes the critical exponent $ heta_0=1/2$ for isometric extensions and constructs examples via convex integration for $ heta<1/2$, also providing existence results for low-regularity embeddings.

Findings

For $ heta>1/2$, the tangential connection matches Levi-Civita.

For $ heta<1/2$, convex integration produces flexible isometric extensions.

Existence of $C^{1, heta}$ isometric embeddings for $ heta<1/2$ with sharper codimension.

Abstract

In this paper we consider the rigidity and flexibility of $C^{1, \theta}$ isometric extensions and we show that the H\"older exponent $\theta_0=\frac12$ is critical in the following sense: if $u\in C^{1,\theta}$ is an isometric extension of a smooth isometric embedding of a codimension one submanifold $\Sigma$ and $\theta> \frac12$, then the tangential connection agrees with the Levi-Civita connection along $\Sigma$. On the other hand, for any $\theta<\frac12$ we can construct $C^{1,\theta}$ isometric extensions via convex integration which violate such property. As a byproduct we get moreover an existence theorem for $C^{1, \theta}$ isometric embeddings, $\theta<\frac12$, of compact Riemannian manifolds with $C^1$ metrics and sharper amount of codimension.