# h-Principle and Rigidity for $C^{1,\alpha}$ Isometric Embeddings

created by delellis on 30 Sep 2009
modified on 02 Jul 2013

[BibTeX]

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.

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http:/www.math.uzh.chindex.php?id=publikationen&key1=493

Keywords: isometric embeddings, convex integration

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