Refinements of the holonomic approximation lemma

TL;DR

This paper refines the holonomic approximation lemma by incorporating quantitative geometric controls, enhancing its applications in flexible symplectic and contact topology through convex integration techniques.

Contribution

It introduces new refinements of the holonomic approximation lemma with quantitative geometric estimates, leveraging Gromov's convex integration idea for improved applications.

Findings

Refined holonomic approximation lemma with quantitative bounds

Applications to flexible symplectic topology

Applications to flexible contact topology

Abstract

The holonomic approximation lemma of Eliashberg and Mishachev is a powerful tool in the philosophy of the $h-$principle. By carefully keeping track of the quantitative geometry behind the holonomic approximation process, we establish several refinements of this lemma. Gromov's idea from convex integration of working one pure partial derivative at a time is central to the discussion. We give applications of our results to flexible symplectic and contact topology.