Using pluripotential theory on degenerate Sasakian manifolds, we show that a locally bounded conical Calabi-Yau potential on a Fano cone is actually smooth on the regular locus. This work is motivated by a similar result obtained by R. Berman in the case where the cone is toric. Our proof is purely pluripotential and independent of any extra symmetry imposed on the cone.
On the $C^{1,1}$ regularity of geodesics in the space of Kähler metrics
$n$-harmonic coordinates and the regularity of conformal mappings
Regularity of Solutions of the Nonlinear Sigma Model with Gravitino
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