We study the rigidity of polyhedral surfaces and the moduli
space of polyhedral surfaces using variational principles. Curvaturelike
quantities for polyhedral surfaces are introduced and are shown
to determine the polyhedral metric up to isometry. The action
functionals in the variational approaches are derived from the cosine
law. They can be considered as 2-dimensional counterparts of
the Schlaefli formula.
A discrete conformality for hyperbolic polyhedral surfaces is introduced
in this paper. This discrete conformality is shown to be computable.
It is proved that each hyperbolic polyhedral metric on a closed surface is discrete
conformal to a unique hyperbolic polyhedral metric with a given discrete
curvature satisfying Gauss-Bonnet formula. Furthermore, the hyperbolic polyhedral
metric with given curvature can be obtained using a discrete Yamabe
flow with surgery. In particular, each hyperbolic polyhedral metric on a closed
surface with negative Euler characteristic is discrete conformal to a unique
A discrete conformality for polyhedral metrics on surfaces is introduced in this paper which generalizes
earlier work on the subject. It is shown that each polyhedral metric on a surface is discrete
conformal to a constant curvature polyhedral metric which is unique up to scaling. Furthermore, the
constant curvature metric can be found using a discrete Yamabe flow with surgery.