Rigidity versus flexibility for tight confoliations

Abstract

In their book "Confoliations" [Univ. Lecture Ser. 13, Amer. Math. Soc. (1998)], Y Eliashberg and W Thurston gave a definition of tight confoliations. We give an example of a tight confoliation $\xi$ on $T^3$ violating the Thurston–Bennequin inequalities. This answers a question from "Confoliations" negatively. Despite this, it is still possible to prove restrictions on homotopy classes of plane fields which contain tight confoliations.

The failure of the Thurston–Bennequin inequalities for tight confoliations is due to the presence of overtwisted stars. Overtwisted stars are particular configurations of Legendrian curves which bound a disc with finitely many punctures on the boundary. We prove that the Thurston–Bennequin inequalities hold for tight confoliations without overtwisted stars and that symplectically fillable confoliations do not admit overtwisted stars.