On Gronwall conjecture

TL;DR

This paper investigates the Gronwall conjecture, providing a partial affirmative answer for certain planar 3-webs with specific curvature properties, and introduces an explicit criterion for unique linearization.

Contribution

It offers a partial proof of the Gronwall conjecture for webs with curvature vanishing to third order and derives an explicit differential criterion for linearization.

Findings

Partial confirmation of the Gronwall conjecture for specific 3-webs.

Derived an explicit differential criterion for web linearization.

Identified conditions under which web curvature determines linearizability.

Abstract

Gronwall conjecture states that a planar 3-web which admits more than one distinct linearization is locally equivalent to an algebraic web. We give a partial answer to the conjecture in the affirmative for the class of planar 3-webs with the web curvature that vanishes to order three at a point. The differential relation on the third order jet of web curvature provides an explicit criterion for unique linearization.