Holonomic approximation through convex integration

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.

Convex integration with avoidance and hyperbolic (4,6) distributions

“…In[25], P. Massot and M. Theillière prove that convex integration can be used to prove holonomic approximation in spaces of 1-jets. This is a beautiful application of classic convex integration in which checking ampleness is highly non-trivial.…”