On the regularity problem of complex Monge-Ampere equations with conical singularities

Abstract

In the category of metrics with conical singularities along a smooth divisor with angle in $(0,2\pi )$, we show that locally defined weak solutions ($C^{1,1}-$solutions) to the Kähler-Einstein equations actually possess maximum regularity, which means the metrics are actually Hölder continuous in the singular polar coordinates. This shows the weak Kähler-Einstein metrics constructed by Guenancia-Paun \cite{GP}, and independently by Yao \cite{GT}, are all actually strong-conical Kähler-Einstein metrics. The key step is to establish a Liouville-type theorem for weak-conical Kähler-Ricci flat metrics defined over $\C^{n}$, which depends on a Calderon-Zygmund theory in the conical setting.

32 pages, comments are welcome

References in corpus (4)

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• Kahler-Einstein metrics on Fano manifolds, II: limits with cone angle less than 2 π
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• Conical Kahler-Einstein metric revisited