  We investigate the universal property of curvatures in surface models which
display a flat phase and a rough phase whose criticality is described by the
Gaussian model. Earlier we derived a relation between the Hessian of the free
energy and the Gaussian coupling constant in the six-vertex model. Here we show
its validity in a general setting using renormalization group arguments. The
general validity of the relation is confirmed numerically in the RSOS model by
comparing the Hessian of the free energy and the Gaussian coupling constant in
a transfer matrix finite-size-scaling study. The Hessian relation gives clear
understanding of the universal curvature jump at roughening transitions and
facet edges and also provides an efficient way of locating the phase
boundaries.
