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 C1,θ with θ<1/3 for n=2 and θ<(n+2)−1 for n≥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.