*Published Paper*

**Inserted:** 30 sep 2009

**Last Updated:** 2 jul 2013

**Journal:** Nonlinear Partial Differential Equations. Abel Symposia 7.

**Pages:** 83-116

**Year:** 2012

**Abstract:**

In this paper we study the embedding of Riemannian manifolds in low codimension. The well-known result of Nash and Kuiper says that any short embedding in codimension one can be uniformly approximated by $C^1$ isometric embeddings. This statement clearly cannot be true for $C^2$ embeddings in general, due to the classical rigidity in the Weyl problem. In fact Borisov extended the latter to embeddings of class $C^{1,\alpha}$ with $\alpha>2/3$. On the other hand he announced in that the Nash-Kuiper statement can be extended to local $C^{1,\alpha}$ embeddings with $\alpha<(1+n+n^2)^{-1}$, where $n$ is the dimension of the manifold, provided the metric is analytic. Subsequently a proof of the 2-dimensional case appeared. In this paper we provide analytic proofs of all these statements, for general dimension and general metric.

For the most updated version and eventual errata see the page

http:/www.math.uzh.ch*index.php?id=publikationen&key1=493
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**Keywords:**
isometric embeddings, convex integration