  Let $X$ be a Hyperk\"ahler manifold, and let $H$ be an ample divisor on $X$.
We give a lower bound in terms of the Beauville-Bogomolov form $q(H)$ for the
twisted cotangent bundle $\Omega_X \otimes H$ to be pseudoeffective. If $X$ is
deformation equivalent to the Hilbert scheme of a K3 surface the lower bound
can be written down explicitly and we study its optimality.
