On the isometric version of Whitney's strong embedding theorem

Author:

Wentao Cao, László Székelyhidi Jr

Keyword:

Mathematics, Differential Geometry, Differential Geometry (math.DG), Analysis of PDEs (math.AP)

journal:

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date:

2023-06-21 16:00:00

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.