Convex Integration Theory without Integration

TL;DR

This paper introduces a Corrugation Process as an alternative to traditional convex integration, defines Kuiper differential relations, and applies these concepts to construct self-similar solutions and explicit immersions, notably of the real projective plane.

Contribution

It develops a new Corrugation Process framework and introduces Kuiper differential relations, simplifying the construction of solutions with self-similarity and explicit geometric embeddings.

Findings

Proves the totally real relation is Kuiper.

Constructs a self-similar totally real isometric embedding.

Builds a new explicit immersion of the real projective plane in R^3.

Abstract

We replace the usual Convex Integration formula by a Corrugation Process and introduce the notion of Kuiper differential relations. This notion provides a natural framework for the construction of solutions with self-similarity properties. We consider the case of the totally real relation, we prove that it is Kuiper and we state a totally real isometric embedding theorem. We then show that the totally real isometric embeddings obtained by the Corrugation Process exhibits a self-similarity property. Kuiper relations also enable a uniform expression of the Corrugation Process that no longer involves integrals. This expression generalizes the ansatz used in arXiv:0905.0370 to generate isometric maps. We apply it to build a new explicit immersion of the real projective plane inside R^3.