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

Alesker, S. and Verbitski, M. (2010) Quaternionic Monge-Ampère equation and Calabi problem for HKT-manifolds. Israel Journal of Mathematics, 176(1), pp. 109-138. (doi:10.1007/s11856-010-0022-0)

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Publisher's URL: http://dx.doi.org/10.1007/s11856-010-0022-0

Abstract

A quaternionic version of the Calabi problem on the Monge–Ampere equation is introduced, namely 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 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.

Item Type:Articles
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Verbitski, Dr Mikhail
Authors: Alesker, S., and Verbitski, M.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Israel Journal of Mathematics
ISSN:0021-2172
ISSN (Online):1565-8511

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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
316541Hyperkaehler manifoldsMikhail VerbitskiEngineering & Physical Sciences Research Council (EPSRC)GR/R77773/01Mathematics