关于复Monge-Ampère算子的复多重位势论目前已广泛应用于复几何、代数几何、动力系统以及物理等领域。本项目拟将多复变的方法应用于四元结构，在四元流形上研究四元Monge-Ampère算子并发展出其相应的多重位势论。具体研究四元流形上的闭正流理论及其应用、四元Monge-Ampère算子的变分性质、四元流形上非退化四元Monge-Ampère方程Dirichlet问题解的存在性及正则性，并探讨四元Monge-Ampère算子在四元流形上的各种收敛性，最大自然定义域、像以及稳定性等。四元结构在理论物理中有着重要应用，特别地，近年来四元流形上的Calabi猜想吸引了许多数学物理学家们的关注和研究。这个猜想在物理超弦理论中有重要应用，本项目对四元流形上四元Monge-Ampère算子的研究将为四元Calabi猜想的解决奠定必要的基础。

The pluripotential theory for the complex Monge-Ampère operator has many important applications in many fields, such as complex geometry, algebraic geometry, complex dynamical system and mathematical physics. By using the method of several complex variables, this research project intend to study the quaternionic Monge-Ampère operator and its pluripotential theory on quaternionic manifolds, and to develop the theory of closed positive currents. Quaternionic analysis has many important applications in physics, in particular the quaternionic version of Calabi conjecture, which has applications in super string, has attracted many mathematicians and physicians to study on it. Relating to this problem, this project also intend to study the variational approach to the quaternionic Monge-Ampère operator, and study the existence and regularity of the solutions to the Dirichlet problem of the non-degenerate quaternionic Monge-Ampère equation. Moreover, we will also study the convergence results、biggest definition domain、the range and stability results for the quaternionic Monge-Ampère operator on quaternionic manifolds.