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.
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Semyon Alesker was partially supported by ISF grant 1369/04.
Misha Verbitsky is an EPSRC advanced fellow supported by CRDF grant RM1-2354-MO02 and EPSRC grant GR/R77773/01.
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Alesker, S., Verbitsky, M. Quaternionic Monge-Ampère equation and Calabi problem for HKT-manifolds. Isr. J. Math. 176, 109–138 (2010). https://doi.org/10.1007/s11856-010-0022-0
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DOI: https://doi.org/10.1007/s11856-010-0022-0