We consider the natural generalization of the parabolic Monge–Ampère equation to HKT geometry. We prove that in the compact case the equation has always a short-time solution and when the hypercomplex structure is locally flat and admits a compatible hyperkähler metric, then the equation has a long-time solution whose normalization converges to a solution of the quaternionic Monge–Ampère equation first introduced in Alekser and Verbitsky (Isr J Math 176:109–138, 2010). The result gives an alternative proof of a theorem of Alesker (Adv Math 241:192–219, 2013).
A parabolic approach to the Calabi–Yau problem in HKT geometry
Bedulli L.;Vezzoni L.
2022-01-01
Abstract
We consider the natural generalization of the parabolic Monge–Ampère equation to HKT geometry. We prove that in the compact case the equation has always a short-time solution and when the hypercomplex structure is locally flat and admits a compatible hyperkähler metric, then the equation has a long-time solution whose normalization converges to a solution of the quaternionic Monge–Ampère equation first introduced in Alekser and Verbitsky (Isr J Math 176:109–138, 2010). The result gives an alternative proof of a theorem of Alesker (Adv Math 241:192–219, 2013).File in questo prodotto:
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