The L2-Norm of the Euler Class for Foliations on Closed Irreducible Riemannian 3-Manifolds

An upper bound for the L 2 -norm of the Euler class e(F) of an arbitrary transversely orientable foliation 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 only finitely many cohomological classes of the group H 2 (M 3 ) that can be realized by the Euler class e(F) of a twodimensional transversely oriented foliation F whose leaves have the modulus of mean curvature which is bounded above by the fixed constant H 0 .