Dual Thurston norm of Euler classes of foliations on negative curvature 3-Manifolds

TL;DR

This paper provides an upper bound estimate for the dual Thurston norm of the Euler class of one-dimensional smooth foliations on negatively curved closed 3-manifolds, depending on geometric and curvature parameters.

Contribution

It introduces a new upper bound estimate for the dual Thurston norm of Euler classes in the context of foliations on negatively curved 3-manifolds, linking topology and geometry.

Findings

Upper bound estimate depends on injectivity radius, volume, curvature, and mean curvature of leaves.

Establishes a quantitative relationship between foliation Euler classes and manifold geometry.

Bridges foliation theory with geometric invariants in hyperbolic 3-manifolds.

Abstract

In this paper we give an upper bound estimate on the dual Thurston norm of the Euler class of an arbitrary smooth foliation $\mathcal{F}$ of dimension one defined on a closed three-dimensional orientable manifold $M^3$ of negative curvature, which depends on the constants bounded the injectivity radius $inj(M^3)$, the volume $Vol(M^3)$, sectional curvature of the manifold $M^3$ and the mean curvature modulus of the leaves of the foliation $\mathcal{F}$.