Non-Measure Hyperbolicity of Complex K3 Surfaces

We show that the non-measure hyperbolicity of K3 surfaces – which M. Green and P. Griffiths verified for certain cases in 1980 – holds for all K3 surfaces. As a byproduct, we prove the non-measure hyperbolicity of any Hilbert schemes of points on K3 surfaces. We also obtain a new proof of the non-measure hyperbolicity of any Enriques surface.