TY - JOUR

T1 - Quaternionic monge-ampère equation and calabi problem for HKT-manifolds

AU - Alesker, S.

AU - Verbitsky, M.

N1 - Funding Information:
∗ 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. Received March 23, 2008 and in revised form June 18, 2008

PY - 2010/3

Y1 - 2010/3

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=77953518225&partnerID=8YFLogxK

U2 - 10.1007/s11856-010-0022-0

DO - 10.1007/s11856-010-0022-0

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AN - SCOPUS:77953518225

SN - 0021-2172

VL - 176

SP - 109

EP - 138

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 1

ER -