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

Abstract

An upper bound for the $L^2$- norm of the Euler class $e(\mathcal{F})$ of an arbitrary transversally orientable foliation $\mathcal{F}$ 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 $e(\mathcal{F})$ of a two-dimensional transversely oriented foliation $\mathcal{F}$ whose leaves have the modulus of mean curvature which is bounded above by the fixed constant $H_0$.

This version uses the language of norms on the cohomologies in the proof of the main result