GAUSSIAN CURVATURE ON SINGULAR SURFACES

We consider prescribing Gaussian curvature on surfaces with conical singularities in both Then we obtain sufficient conditions for a function to be the Gaussian curvature of some pointwise conformal singular metric. We only require that the values of the function are not too large at singular points of the metric with the smallest angle, say, less or equal to 0, or less than its average value. To prove the results, we apply some new ideas and techniques. One of them is to estimate the total curvature along a certain minimizing sequence by using the ''Distribution of Mass Principle'' and the behavior of the critical points at infinity.