h-Principle and Rigidity for C ^1,α Isometric Embeddings

In this paper we study the embedding of Riemannian manifolds in low codimension. The well-known result of Nash and Kuiper (Nash in Ann. Math. 60:383–396, 1954; Kuiper in Proc. Kon. Acad. Wet. Amsterdam A 58:545–556, 1955; Kuiper in Proc. Kon. Acad. Wet. Amsterdam A 58:683–689, 1955) 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,α with α>2/3 in (Borisov in Vestn. Leningr. Univ. 14(13):20–26, 1959; Borisov in Vestn. Leningr. Univ. 15(19):127–129, 1960). On the other hand he announced in (Borisov in Doklady 163:869–871, 1965) that the Nash–Kuiper statement can be extended to local C ^1,α embeddings with α<(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 (Borisov in Sib. Mat. Zh. 45(1):25–61, 2004). In this paper we provide analytic proofs of all these statements, for general dimension and general metric.