Parabolic quaternionic Monge-Amp\`{e}re equation on compact manifolds with a flat hyperK\"ahler metric
J. Korean Math. Soc. 2022 Vol. 59, No. 1, 13-33
https://doi.org/10.4134/JKMS.j200626
Published online November 8, 2021
Printed January 1, 2022
Jiaogen Zhang
University of Science and Technology of China
Abstract : The quaternionic Calabi conjecture was introduced by \linebreak Alesker-Verbitsky, analogous to the K\"ahler case which was raised by Calabi. On a compact connected hypercomplex manifold, when there exists a flat hyperK\"ahler metric which is compatible with the underlying hypercomplex structure, we will consider the parabolic quaternionic \MA~equation. Our goal is to prove the long time existence and $C^{\infty}$ convergence for normalized solutions as $t\ri\infty$. As a consequence, we show that the limit function is exactly the solution of quaternionic \MA~equation, this gives a parabolic proof for the quaternionic Calabi conjecture in this special setting.
Keywords : A priori estimates, hyperK\"ahler manifold with torsion, parabolic quaternionic \MA~equation
MSC numbers : 35B45, 53C26, 35J96
Supported by : The authors are partially supported by NSF in China No.~11625106, 11801535 and 11721101. The research was partially supported by the project ``Analysis and Geometry on Bundle" of Ministry of Science and Technology of the People's Republic of China, No.~SQ2020YFA070080.
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