  A projective manifold is algebraically hyperbolic if the degree of any curve
is bounded from above by its genus times a constant, which is independent from
the curve. This is a property which follows from Kobayashi hyperbolicity. We
prove that hyperkahler manifolds are non algebraically hyperbolic when the
Picard rank is at least 3, or if the Picard rank is 2 and the SYZ conjecture on
existence of Lagrangian fibrations is true. We also prove that if the
automorphism group of a hyperkahler manifold is infinite then it is
algebraically non-hyperbolic.
