Journal of the
Korean Mathematical Society

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008



J. Korean Math. Soc. 2022; 59(1): 13-33

Online first article January 1, 2022      Printed January 1, 2022

Copyright © The Korean Mathematical Society.

Parabolic quaternionic Monge-Amp\`{e}re equation on compact manifolds with a flat hyperK\"ahler metric

Jiaogen Zhang

University of Science and Technology of China


The quaternionic Calabi conjecture was introduced by 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 Monge-Amp\`{e}re equation.  Our goal is to prove the long time existence and $C^{\infty}$ convergence for normalized solutions  as $t\rightarrow\infty$. As a consequence, we show that the limit function is exactly the solution of quaternionic Monge-Amp\`{e}re 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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