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Hermitian metrics, (n-1, n-1) forms and Monge-Amp\`ere equations

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arxiv 1310.6326 v2 pith:TWRFYBX6 submitted 2013-10-23 math.DG math.CV

Valentino Tosatti , Ben Weinkove This is my paper · ORCID
classification math.DGmath.CV
keywords gauduchonmanifoldsmetricscalabi-yauequationhermitianmonge-ampereanother
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We show existence of unique smooth solutions to the Monge-Ampere equation for (n-1)-plurisubharmonic functions on Hermitian manifolds, generalizing previous work of the authors. As a consequence we obtain Calabi-Yau theorems for Gauduchon and strongly Gauduchon metrics on a class of non-Kahler manifolds: those satisfying the Jost-Yau condition known as Astheno-Kahler. Gauduchon conjectured in 1984 that a Calabi-Yau theorem for Gauduchon metrics holds on all compact complex manifolds. We discuss another Monge-Ampere equation, recently introduced by Popovici, and show that the full Gauduchon conjecture can be reduced to a second order estimate of Hou-Ma-Wu type.

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  1. Second order estimates for complex Hessian equations on Hermitian manifolds

    math.AP 2019-08 conditional novelty 6.0 of 10

    For chi-plurisubharmonic solutions of complex Hessian equations with gradient-dependent right-hand sides on compact Hermitian manifolds, the second covariant derivative of the solution is uniformly bounded in terms of...

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