On the isometric version of Whitney's strong embedding theorem

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 Hölder class $C^{1,\theta }$ with $\theta <1/3$ for $n=2$ and $\theta <{(n+2)}^{-1}$ for $n\ge 3$. 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.