Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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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)
  • Received : 2020.11.14
  • Accepted : 2021.09.09
  • Published : 2022.01.01
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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

Acknowledgement

The author is very grateful to his thesis advisor Professor Xi Zhang for his constant support and countless advice. The author is also grateful to the anonymous referees and the editor for their careful reading and helpful suggestions which greatly improved the article. 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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