Convex Integration and Legendrian Approximation of Curves

TL;DR

This paper presents a constructive method using convex integration to approximate any continuous curve in a contact 3-manifold by a Legendrian curve, providing a new proof of a classical result.

Contribution

It offers a novel constructive proof employing convex integration for Legendrian approximation in contact 3-manifolds.

Findings

Every continuous curve can be approximated by Legendrian curves.

Convex integration provides a constructive approach.

The method applies to contact 3-manifolds.

Abstract

Using convex integration we give a constructive proof of the well-known fact that every continuous curve in a contact $3$-manifold can be approximated by a Legendrian curve.