Projective invariants of linear 3-webs and Gronwall's Conjecture

TL;DR

This paper develops a projectively invariant framework for planar linear 3-webs, introduces a family of Cartan connections, and proposes an algorithm to resolve Gronwall's conjecture, successfully proving it for specific cases.

Contribution

It introduces a new invariant description of linear 3-webs, a family of Cartan connections, and an algorithmic approach to prove Gronwall's conjecture.

Findings

Proved Gronwall's conjecture for 3-webs with two pencils of lines.

Established a web linearization criterion based on projective invariants.

Developed a method to analyze non-hexagonal 3-webs using Cartan connections.

Abstract

We present a projectively invariant description of planar linear 3-webs. For a non-hexagonal 3-web, we introduce family of projective torsion-free Cartan connections, the web leaves being geodesics for each member of the family, and give a web linearization criterion. Finally, we propose an algorithm for resolving the Gronwall conjecture and illustrate this approach by proving the conjecture for 3-webs whose 2 foliations are 2 pencils of lines.