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

TL;DR

This paper explores the balance between flexibility and rigidity in low-codimension isometric embeddings of Riemannian manifolds, extending classical results to broader regularity classes and dimensions with analytic proofs.

Contribution

It provides comprehensive analytic proofs of the Nash-Kuiper and Borisov results for $C^{1,eta}$ embeddings across all dimensions and metrics.

Findings

Extension of Nash-Kuiper theorem to $C^{1,eta}$ embeddings for all dimensions.

Rigidity results for $C^{1,eta}$ embeddings with $eta>2/3$.

Analytic proofs applicable to general dimensions and metrics.

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.