多重位势论对四元Monge-Ampère算子、复k-Hessian算子的应用

The complex Monge-Ampère operator plays a very important role in the investigating of many fields, such as several complex variables, complex geometry, algebraic geometry, number theory, complex dynamical system and mathematical physics. By using the method of pluripotential theory, this research project intend to develop the theory of closed positive currents for the complex k-Hessian operator and quaternionic Monge-Ampère operator respectively. The weak convergence of the complex k-Hessian operator will be extended to the unbounded case. By comparing with the classical Lelong number for the plurisubharmonic functions, this project intend to define the Lelong number and the generalized Lelong number for the complex k-convex functions, then apply it to the intersection theory of currents. In this project, the quaternionic version of Lelong-Jensen type formula will be established; the comparison theorem as well as an explicit formula for quaternionic boundary measure of the strictly pseudoconvex domain will be given; many properties in the pluripotential theory will be discussed. Furthermore, this project intend to prove the quasicontinuity theorem for the quaternionic plurisubharmonic functions, then study the C^{2,α} a priori estimate of domains with boundary on quaternionic space, which will lay the groundwork for the solving of quaternionic Calabi conjecture. These work will enrich the theory of closed positive currents and quaternionic Monge-Ampère operator, and will open up a new prospect of several quaternionic variables.

复Monge-Ampère算子已作为一种重要工具广泛应用于多复变、复几何、代数几何、数论、复动力系统及数学物理等领域。本项目拟用多重位势论方法对复k-Hessian算子、四元Monge-Ampère算子发展出相应的闭正流理论，将复k-Hessian测度的弱收敛性推广到无界函数上，对复k-凸函数定义Lelong数并将其与经典的Lelong数相比较，应用到流的交截理论；证明四元Lelong-Jensen公式，得到四元空间上严格拟凸域的边界测度的比较定理及确切表达式，讨论四元Monge-Ampère算子的容量、极集、可除集以及极大性等位势论方面的性质；证明四元多次下调和函数的拟连续性，得到四元空间带边区域上解的C^{2,α}先验估计,这将为四元Calabi猜想的解决奠下基础。这些问题的解决将丰富和完善闭正流理论及四元Monge-Ampère算子理论，将为四元分析打开一个全新的局面。

本项目将多复变函数论中关于复Monge-Ampère算子的多重位势理论推广到了复k-Hessian算子及四元Monge-Ampère算子，并发展出四元闭正流理论。复Monge-Ampère算子已作为一种重要工具广泛应用于多复变、复几何、代数几何、数论、复动力系统及数学物理等领域。我们将复k-Hessian测度的弱收敛性推广到了无界函数上，对复k-凸函数定义了Lelong数；证明了四元多次下调和函数的Lelong-Jensen公式，并推广了Bedford-Taylor理论，证明了四元 Monge-Ampère 方程的 Chern-Levine-Nirenberg 估计式，将四元Monge-Ampère算子的定义域推广到了局部有界的四元多次下调和函数类。作为应用我们还证明四元多次下调和函数的拟连续性，得到了四元Monge-Ampère算子的容量、极集、可除集以及极大性等位势论方面的性质。此外我们还得到了四元Monge-Ampère方程的Dirichlet问题的粘性解的结果。