Gaussian curvature on singular surfaces
We consider prescribing Gaussian curvature on surfaces with conical singularities in both critical and supercritical cases. First we prove a variant of Kazdan-Warner type necessary conditions. Then we obtain sufficient conditions for a function to be the Gaussian curvature of some pointwise conformai 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.
Critical and supercritical cases, Math Subject Classification: 35J20, 35J60, 53A30, nonlinear elliptic equations, prescribing gaussian curvature, surfaces with conical singularities, variational methods
Chen, Wenxiong, and Congming Li. "Gaussian curvature on singular surfaces." The Journal of Geometric Analysis 3, no. 4 (1993): 315-334.
The Journal of Geometric Analysis