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 [CDLS12] to generate isometric maps. We apply it to build a new explicit immersion of RP 2 inside R 3 .

Holonomic approximation through convex integration

“…Also, the traditional proof of the holonomic approximation theorem is rather monolithic and requires great care to get all details and constructions right at the same time. By contrast, convex integration, especially the implementation in [The19], is built out of a series of very cleanly encapsulated steps. So there is hope this version of the story will be easier to formalize.…”