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arXiv · 0802.4202

Quaternionic Monge-Ampere equation and Calabi problem for HKT-manifolds

Semyon Alesker·Misha Verbitsky
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Abstract

A quaternionic version of the Calabi problem on Monge-Ampere equation is introduced. It is a quaternionic Monge-Ampere 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, as well as the zero order a priori estimate. The existence of solution is conjectured, similar to Calabi-Yau theorem. We reformulate this quaternionic equation as a special case of a complex Hessian equation, making sense on any complex manifold.

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Semyon Alesker, Misha Verbitsky. 2008-07-27. Quaternionic Monge-Ampere equation and Calabi problem for HKT-manifolds. https://arxiv.org/abs/0802.4202

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