The Gauss equation on surfaces of bounded integral curvature

Based on some results of A. Huber and Y. G. Reshetnyak concerning Surfaces of Bounded Integral Curvature, we show that even in such a weak setting one can write down the ”Gauss” equation (which in the classical case reads −∆ρ = 2Ke ρ , where ∆ is the Eu- clidean Laplace operator), locally relating the Gaussian curvature K to the metric and its weak Laplacian. This approach naturally leads to the analysis of the regularity theory of singular Liouville type equations as developed by H. Brezis and F. Merle. Several examples are provided to illustrate the local regularity properties of this sort of singular surfaces.