The Continuity Method to Deform Cone Angle

The continuity method is used to deform the cone angle of a weak conical Kahler-Einstein metric with cone singularities along a smooth anti-canonical divisor on a smooth Fano manifold. This leads to an alternative proof of Donaldsons Openness Theorem on deforming cone angle cite{Don} by combining it with the regularity result of Guenancia-P$breve{text{a}}$un cite{GP} and Chen-Wang cite{CW}. This continuity method uses relatively less regularity of the metric (only weak conical Kahler-Einstein) and bypasses the difficult Banach space set up; it is also generalized to deform the cone angles of a emph{weak conical Kahler-Einstein metric} along a simple normal crossing divisor (pluri-anticanonical) on a smooth Fano manifold (assuming no tangential holomorphic vector fields).