Holonomic approximation through convex integration

TL;DR

This paper demonstrates how convex integration can be used to prove the holonomic approximation theorem for first order jets, linking two key concepts in differential topology and geometry.

Contribution

It introduces a new perspective by reducing the holonomic approximation theorem to a problem of flexibility of a specific relation, solvable via convex integration.

Findings

Holonomic approximation theorem can be proved using convex integration.

The specific relation involved is open and ample, ensuring flexibility.

Convex integration provides a straightforward proof method for the theorem.

Abstract

Convex integration and the holonomic approximation theorem are two well-known pillars of flexibility in differential topology and geometry. They may each seem to have their own flavor and scope. The goal of this paper is to bring some new perspective on this topic. We explain how to prove the holonomic approximation theorem for first order jets using convex integration. More precisely we first prove that this theorem can easily be reduced to proving flexibility of some specific relation. Then we prove this relation is open and ample, hence its flexibility follows from off-the-shelf convex integration.