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We investigate the universal property of curvatures in surface models whichdisplay a flat phase and a rough phase whose criticality is described by theGaussian model. Earlier we derived a relation between the Hessian of the freeenergy and the Gaussian coupling constant in the six-vertex model. Here we showits validity in a general setting using renormalization group arguments. Thegeneral validity of the relation is confirmed numerically in the RSOS model bycomparing the Hessian of the free energy and the Gaussian coupling constant ina transfer matrix finite-size-scaling study. The Hessian relation gives clearunderstanding of the universal curvature jump at roughening transitions andfacet edges and also provides an efficient way of locating the phaseboundaries.Comment: 19 pages, RevTex, 3 Postscript Figures, To appear in Phys. Rev.

Extended universality of the surface curvature in equilibrium crystal shapes