Non-measure hyperbolicity of K3 surfaces and Hilbert schemes of points on K3 surfaces

Abstract 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 all Hilbert schemes of points on K3 surfaces. We also obtain a new proof of the non-measure hyperbolicity of any Enriques surface.