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 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.