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

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.