Convex integration theory without integration

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 exhibit a self-similarity property. Kuiper relations also enable a uniform expression of the Corrugation Process that no longer involves integrals. We apply the Corrugation Process to build a new explicit immersion of \({\mathbb {R}}P^2\) inside \({\mathbb {R}}^3\).