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تسجيل مستخدم جديدOn convexity of the regular set of conical Kahler-Einstein metrics
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Ved V. Datar
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In this note we prove convexity, in the sense of Colding-Naber, of the regular set of solutions to some complex Monge-Ampere equations with conical singularities along simple normal crossing divisors. In particular, any two points in the regular set can be joined by a smooth minimal geodesic lying entirely in the regular set. We show that as a result, the classical theorems of Myers and Bishop-Gromov extend almost verbatim to this singular setting.
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The existence of emph{weak conical Kahler-Einstein} metrics along smooth hypersurfaces with angle between $0$ and $2pi$ is obtained by studying a smooth continuity method and a emph{local Mosers iteration} technique. In the case of negative and zero
Ricci curvature, the $C^0$ estimate is unobstructed; while in the case of positive Ricci curvature, the $C^0$ estimate obstructed by the properness of the emph{twisted K-Energy}. As soon as the $C^0$ estimate is achieved, the local Moser iteration could improve the emph{rough bound} on the approximations to a emph{uniform $C^2$ bound}, thus produce a emph{weak conical Kahler-Einstein} metric. The method used here do not depend on the bound of any background conical Kahler metrics.
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