Existence of weak conical Kahler-Einstein metrics along smooth hypersurfaces

The existence of weak conical Kahler-Einstein metrics along smooth hypersurfaces with cone angle between and is obtained by studying a family of Aubin's (J Funct Anal 57:143-153, 1984) continuity paths and obtaining a uniform estimate by a local Moser's iteration technique. As soon as the estimate is achieved, the local Moser's iteration technique could improve the rough estimate in Chen et al. (J Am Math Soc 28:183-197, 2015) to a uniform estimate. Since in the cases of negative and zero Ricci curvature, the estimate is unobstructed, the weak conical Kahler-Einstein metrics are obtained; while in the case of positive Ricci curvature, the estimate is achieved under the assumption of the properness of the Twisted K-Energy. The method used in this paper does not depend on the bound of the holomorphic bisectional curvature of any global background conical Kahler metrics.