Electronic structure of periodic curved surfaces -- continuous surface versus graphitic sponge

We investigate the band structure of electrons bound on periodic curved surfaces. We have formulated Schr{o}dingers equation with the Weierstrass representation when the surface is minimal, which is numerically solved. Bands and the Bloch wavefunctions are basically determined by the way in which the ``pipes are connected into a network, where the Bonnet(conformal)-transformed surfaces have related electronic strucutres. We then examine, as a realisation of periodic surfaces, the tight-binding model for atomic networks (``sponges), where the low-energy spectrum coincides with those for continuous curved surfaces.