  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.
