Corrugation Process and $\epsilon$-isometric maps

TL;DR

This paper introduces a new Corrugation Process for convex integration, demonstrating its effectiveness in constructing epsilon-isometric maps with specific properties, especially in Kuiper problems.

Contribution

It extends convex integration techniques by replacing the classical formula with Corrugation Process, applied to epsilon-isometric maps and proving Kuiper property in codimension 1.

Findings

Proves epsilon-isometric maps are Kuiper in codimension 1.

Constructs epsilon-isometric maps from short maps with conical singularities.

Introduces a new Corrugation Process for convex integration.

Abstract

Convex Integration is a theory developed in the '70s by M. Gromov. This theory allows to solve families of differential problems satisfying some convex assumptions. From a subsolution, the theory iteratively builds a solution by applying a series of convex integrations. In a previous paper arXiv:1909.04908, we proposed to replace the usual convex integration formula by a new one called Corrugation Process. This new formula is of particular interest when the differential problem under consideration has the property of being of Kuiper. In this paper, we consider the differential problem of $\epsilon$-isometric maps and we prove that it is Kuiper in codimension 1. As an application, we construct $\epsilon$-isometric maps from a short map having a conical singularity.