Calculus of Variations and Geometric Measure Theory

S. Conti - C. De Lellis - L. J. Székelyhidi

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

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


Published Paper

Inserted: 30 sep 2009
Last Updated: 2 jul 2013

Journal: Nonlinear Partial Differential Equations. Abel Symposia 7.
Pages: 83-116
Year: 2012


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


Keywords: isometric embeddings, convex integration