Abstract
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.
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W. C. partially supported by NSF Grant DMS-9116949. C. L. partially supported by NSF Grants DMS-9003694 and DMS-9100383.
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Chen, W., Li, C. Gaussian curvature on singular surfaces. J Geom Anal 3, 315–334 (1993). https://doi.org/10.1007/BF02921316
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DOI: https://doi.org/10.1007/BF02921316