On the isometric version of Whitney's strong embedding theorem

TL;DR

This paper extends Whitney's strong embedding theorem to isometric embeddings, demonstrating that smooth compact manifolds can be embedded into Euclidean space with specific regularity, using convex integration techniques.

Contribution

It establishes the existence of infinitely many isometric embeddings of compact manifolds into Euclidean space with Hölder regularity below 1/3, applying Nash-Kuiper convex integration and gluing methods.

Findings

Infinitely many isometric embeddings exist for compact manifolds.

Embeddings have Hölder regularity less than 1/3 for surfaces.

The method combines convex integration with a gluing technique.

Abstract

We prove a version of Whitney's strong embedding theorem for isometric embeddings within the general setting of the Nash-Kuiper h-principle. More precisely, we show that any $n$-dimensional smooth compact manifold admits infinitely many global isometric embeddings into $2n$-dimensional Euclidean space, of H\"older class $C^{1,\theta}$ with $\theta<1/3$ for $n=2$ and $\theta<(n+2)^{-1}$ for $n\geq3$. The proof is performed by Nash-Kuiper's convex integration construction and applying the gluing technique of the authors on short embeddings with small amplitude.