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In this paper, we study a natural discretization of the smooth Gaussiancurvature on surfaces, which is defined as the quotient of the angle defect andthe area of a geodesic disk at a vertex of a polyhedral surface. It is provedthat each decorated piecewise Euclidean metric on surfaces with nonpositiveEuler number is discrete conformal to a decorated piecewise Euclidean metricwith this discrete curvature constant. We further investigate the prescribingcombinatorial curvature problem for a parametrization of this discretecurvature and prove some Kazdan-Warner type results. The main tools areBobenko-Lutz's discrete conformal theory for decorated piecewise Euclideanmetrics on surfaces and variational principles with constraints.

A discrete uniformization theorem for decorated piecewise Euclidean metrics on surfaces, II