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}dinger’s 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.