Quaternionic Monge-Ampere equation and Calabi problem for HKT-manifolds
Complex Variables
2010-11-03 v3 Analysis of PDEs
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.
Cite
@article{arxiv.0802.4202,
title = {Quaternionic Monge-Ampere equation and Calabi problem for HKT-manifolds},
author = {Semyon Alesker and Misha Verbitsky},
journal= {arXiv preprint arXiv:0802.4202},
year = {2010}
}
Comments
32 pages, minor corrections