Quaternionic Monge-Ampère equation and Calabi problem for HKT-manifolds
نویسنده
چکیده
A quaternionic version of the Calabi problem on the MongeAmpère equation is introduced, namely a quaternionic MongeAmpère 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.
منابع مشابه
Balanced HKT metrics and strong HKT metrics on hypercomplex manifolds
A manifold (M, I, J,K) is called hypercomplex if I, J,K are complex structures satisfying quaternionic relations. A quaternionic Hermitian hypercomplex manifold is called HKT (hyperkähler with torsion) if the (2,0)-form Ω associated with the corresponding Sp(n)-structure satisfies ∂Ω = 0. A Hermitian metric ω on a complex manifold is called balanced if d∗ω = 0. We show that balanced HKT metrics...
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