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http://dx.doi.org/10.4134/JKMS.j200626

PARABOLIC QUATERNIONIC MONGE-AMPÈRE EQUATION ON COMPACT MANIFOLDS WITH A FLAT HYPERKÄHLER METRIC  

Zhang, Jiaogen (School of Mathematical Sciences University of Science and Technology of China)
Publication Information
Journal of the Korean Mathematical Society / v.59, no.1, 2022 , pp. 13-33 More about this Journal
Abstract
The quaternionic Calabi conjecture was introduced by Alesker-Verbitsky, analogous to the Kähler case which was raised by Calabi. On a compact connected hypercomplex manifold, when there exists a flat hyperKähler metric which is compatible with the underlying hypercomplex structure, we will consider the parabolic quaternionic Monge-Ampère equation. Our goal is to prove the long time existence and C∞ convergence for normalized solutions as t → ∞. As a consequence, we show that the limit function is exactly the solution of quaternionic Monge-Ampère equation, this gives a parabolic proof for the quaternionic Calabi conjecture in this special setting.
Keywords
A priori estimates; hyperKahler manifold with torsion; parabolic quaternionic Monge-Ampere equation;
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