  Let $M$ be a hyperk\"ahler manifold with $b_2(M)\geq 5$. We improve our
earlier results on the Morrison-Kawamata cone conjecture by showing that the
Beauville-Bogomolov square of the primitive MBM classes (i.e. the classes whose
orthogonal hyperplanes bound the K\"ahler cone in the positive cone, or, in
other words, the classes of negative extremal rational curves on deformations
of $M$) is bounded in absolute value by a number depending only on the
deformation class of $M$. The proof uses ergodic theory on homogeneous spaces.
