  We report here on a method to fabricate and characterize highly perfect,
periodically rippled graphene monolayers and islands, epitaxially grown on
single crystal metallic substrates under controlled UHV conditions. The
periodicity of the ripples is dictated by the difference in lattice parameters
of graphene and substrate, and, thus, it is adjustable. We characterize its
perfection at the atomic scale by means of STM and determine its electronic
structure in the real space by local tunnelling spectroscopy. There are
periodic variations in the geometric and electronic structure of the graphene
monolayer. We observe inhomogeneities in the charge distribution, i.e a larger
occupied Density Of States at the higher parts of the ripples. Periodically
rippled graphene might represent the physical realization of an ordered array
of coupled graphene quantum dots. The data show, however, that for rippled
graphene on Ru(0001) both the low and the high parts of the ripples are
metallic. The fabrication of periodically rippled graphene layers with
controllable characteristic length and different bonding interactions with the
substrate will allow a systematic experimental test of this fundamental
problem.
