Quaternionic monge-ampère equation and calabi problem for HKT-manifolds

S. Alesker*, M. Verbitsky

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

A quaternionic version of the Calabi problem on the Monge-Ampère equation is introduced, namely a quaternionic Monge-Ampère equation on a compact hypercomplex manifold with an HKT-metric. The equation is non-linear elliptic of second order. For a hypercomplex manifold with holonomy in SL(n,ℍ), uniqueness (up to a constant) of a solution is proven, aas well as the zero order a priori estimate. The existence of a solution is conjectured, similar to the Calabi-Yau theorem. We reformulate this quaternionic equation as a special case of the complex Hessian equation, making sense on any complex manifold.

Original languageEnglish
Pages (from-to)109-138
Number of pages30
JournalIsrael Journal of Mathematics
Volume176
Issue number1
DOIs
StatePublished - Mar 2010

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