  A rigid cohomology class on a complex manifold is a class that is represented
by a unique closed positive current. The positive current representing a rigid
class is also called rigid. For a compact Kahler manifold $X$ all eigenvectors
of hyperbolic automorphisms acting on $H^{1,1}(X)$ that have non-unit
eigenvalues are rigid classes. Such classes are always parabolic, namely, they
belong to the boundary of the Kahler cone and have vanishing volume. We study
parabolic $(1,1)$-classes on compact hyperkahler manifolds with $b_2 \geq 7$.
We show that a parabolic class is rigid if it is not orthogonal to a rational
vector with respect to the BBF form. This implies that a general parabolic
class on a hyperkahler manifold is rigid.
