Taut foliations

TL;DR

This paper explores different notions of tautness in various types of foliations and examines their implications, providing examples and discussing their impact on classical results in foliation theory.

Contribution

It introduces and compares notions of tautness across different foliation regularities and constructs examples illustrating their relationships with contact structures.

Findings

Different notions of tautness are not equivalent across foliation types.

Examples show smoothly taut foliations can be approximated by both tight and overtwisted contact structures.

Impacts classical results by highlighting distinctions in tautness definitions.

Abstract

We describe notions of tautness that arise in the study of $C^0$ foliations, $C^{1,0}$ or smoother foliations, and in geometry. We give examples to show that these notions are different, and discuss how these differences impact some classical foliation results. We construct examples of smoothly taut $C^{\infty,0}$ foliations that can be $C^0$ approximated by both weakly symplectically fillable, universally tight contact structures and by overtwisted contact structures.