$C^{1,1-\epsilon}$ Isometric embeddings

TL;DR

This paper advances the understanding of isometric embeddings by establishing a local h-principle in the class C^{1,1- ext{epsilon}} using an enhanced convex integration method, applicable to n-dimensional manifolds in specific codimension.

Contribution

It introduces a novel convex integration approach with an extra iteration to prove a local h-principle for C^{1,1- ext{epsilon}} isometric embeddings.

Findings

Established local h-principle for C^{1,1- ext{epsilon}} embeddings

Extended convex integration techniques with an additional iteration

Applicable to n-dimensional manifolds in half codimension

Abstract

In this paper we use the convex integration technique enhanced by an extra iteration originally due to K\"all\'en and revisited by Kr\"oner to provide a local $h$-principle for isometric embeddings in the class $C^{1,1-\epsilon}$ for $n$-dimensional manifolds in codimension $\frac{1}{2}n(n+1)$.