A discrete uniformization theorem for decorated piecewise Euclidean metrics on surfaces

In this paper, we introduce a new discretization of the Gaussian curvature on surfaces, which is defined as the quotient of the angle defect and the area of some dual cell of a weighted triangulation at the conic singularity. A discrete uniformization theorem for this discrete Gaussian curvature is established on surfaces with non-positive Euler number. The main tools are Bobenko-Lutz’s discrete conformal theory for decorated piecewise Euclidean metrics on surfaces and variational principles with constraints.