The $L^2$-norm of the Euler class for Foliations on closed irreducible Riemannian 3-Manifolds

TL;DR

This paper establishes an upper bound for the $L^2$-norm of the Euler class of transversally oriented foliations on closed irreducible Riemannian 3-manifolds, linking it to geometric constants and showing finiteness of realizable cohomology classes.

Contribution

It provides a new upper bound for the Euler class norm in terms of geometric parameters and proves finiteness of cohomology classes realizable by such foliations.

Findings

Bound on the $L^2$-norm of the Euler class in terms of geometric constants.

Finiteness of cohomology classes realizable by foliations with bounded mean curvature.

Explicit relation between foliation geometry and topological invariants.

Abstract

An upper bound for the $L^2$- norm of the Euler class $e(\cal F)$ of an arbitrary transversally orientable foliation $\cal F$ of codimension one, defined on a three-dimensional closed irreducible orientable Riemannian 3-manifold $M^3$ is given in terms of constants bounding the volume, the radius of injectivity, the sectional curvature of $M^3$ and the modulus of mean curvature of the leaves. As a consequence we get that only finitely many cohomolo\-gical classes of the group $H^2(M^3)$ that can be realized by the Euler class $e(\cal F)$ of a two-dimensional transversely oriented foliation $\cal F $ whose leaves have the modulus of mean curvature which is bounded above by the fixed constant $H_0$.