$C^{1,0}$ Foliation Theory

TL;DR

This paper extends classical results in codimension one foliation theory to less smooth cases using flow box decompositions, with potential applications to higher-dimensional manifolds.

Contribution

It introduces methods to generalize smoothing, blowup, and approximation results for $C^{1,0}$ foliations, beyond the $C^2$ setting.

Findings

Extended smoothing of leaves to $C^{1,0}$ foliations

Generalized Denjoy blowup techniques

Approximations by fibrations in lower regularity settings

Abstract

Transverse one dimensional foliations play an important role in the study of codimension one foliations. In \cite{KR2}, the authors introduced the notion of flow box decomposition of a 3-manifold $M$. This is a decomposition of $M$ that reflects both the structure of a given codimension one foliation and that of a given transverse flow. In this paper, flow box decompositions are used to extend some classical foliation results to foliations that are not $C^2$. Enhancements of well-known results of Calegari on smoothing leaves, Dippolito on Denjoy blowup of leaves, and Tischler on approximations by fibrations are obtained. The methods developed are not intrinsically 3-dimensional techniques, and should generalize to prove corresponding results for codimension one foliations in $n$-dimensional manifolds.