多重位势论的一些研究

【作者】 万东睿；

【导师】 王伟；

【作者基本信息】 浙江大学 ， 基础数学， 2013， 博士

【摘要】 本文的主要思想是将复多重位势论中的方法应用到其他结构,这里我们分别应用到了实k-凸函数以及四元多次下调和函数上,将关于复Monge-Ampere算子的一些成果分别做到了实k-Hessian算子以及四元Monge-Ampere算子上,得到了全新的有意思的结果,主要包括边界测度、Lelong-Jensen公式、Lelong数、格林函数以及闭正流理论等方面的内容。第一章介绍了复Monge-Ampere测度、多次下调和函数、闭正流、实k-Hessian测度以及四元Monge-Ampere测度的历史背景和研究的近现状,并介绍了本文的研究思想和主要结论。第二章研究了k-Hessian算子与k-凸函数,介绍了他们的一些基本性质以及k-Hessian测度的弱收敛性定理。并且我们通过对相对极函数的研究给出了k-凸函数在k-超凸域上的一个整体逼近,并得到了关于混合k-Hessian测度的几个不同类型的估计式。第三章研究了关于k-凸函数的Lelong-Jensen型公式以及Lelong数。我们给出了k-Hessian边界测度的具体表达式,得到了关于k-凸函数的Lelong-Jensen型公式,此公式可以看成是k-凸函数版本的Poisson积分公式。另外我们证明了k-Hessian极限边界测度的比较原理,并对k-凸函数定义了Lelong数以及广义的Lelong数。第四章研究了单极点以及多极点的k-格林函数。我们用不同方法证明了二者的连续性,说明了他们即是Dirichlet问题的唯一解,并研究了其在超凸域边界的收敛性。第五章研究了四元空间Hn上的闭正流及四元Monge-Ampere算子。我们先给出了Baston算子△的性质及具体表达式,用O-Cauchy-Fueter复形的第二个算子D给出了闭的流的定义,并发展了一套闭正流的理论。我们对无界的多次下调和函数u1,…,up以及闭正流T将△u1∧…∧△%∧T定义为一个闭正流,并得到了其收敛性,最后研究了四元的Lelong-Jensen型公式以及Lelong数。更多还原

【Abstract】 In this thesis, we generalize the method of pluripotential theory for complex Monge-Ampere operator to that for the k-Hessian operator and quaternionic Monge-Ampere operator respectively. We get some interesting and completely new results, such as boundary measure, Lelong-Jensen type formula, Lelong num-ber, Green function, closed positive current and so on.In Chapter1, we give a comprehensive survey of the backgrounds and modern developments of complex Monge-Ampere operator, plurisubharmonic functions, closed positive current, k-Hessian measure and quaternionic Monge-Ampere operator. And then we introduce some idea and concepts referring to this thesis and the main results of the subject.In Chapter2, we mainly discuss the k-Hessian operator and k-convex func-tions. We show some basic properties and the weak convergence theorem for k-Hessian measure. Then by studying the relative extremal function, we establish an global approximation of negative k-convex functions on some k-hyperconvex domain. Moreover, we give several estimates for the mixed k-Hessian operator.Chapter3is devoted to the study of Lelong-Jensen type formula and Lelong number for k-convex functions. We find an explicit formula for k-Hessian bound-ary measure, and establish Lelong-Jensen type formula, which can be regarded as a k-convex version of Poisson integral formula. We also show the comparison theorem for k-Hessian boundary measure and introduce Lelong number and the generalized Lelong number for k-convex functions.In Chapter4, we study the k-Green functions with single pole and with several poles respectively. We prove their continuity by using different methods, and show that the the k-Green function is the unique solution to the Dirich-let problem. Their behavior on the boundary of k-hyperconvex domain is also studied.In the last chapter, we mainly discuss closed positive current on quaternionic space and the quaternionic Monge-Ampere operator. We show some properties and an explicit formula for Baston operator Δ. We say a current T is close if DT=0, where D is the second operator in the O-Cauchy-Fueter complex. Then we define Δu1∧...∧Δuk∧T as a closed positive current also in the case when the plurisubharmonic functions u1,..., uk are not bounded below, and prove the weak convergence theorem. Finally, we establish Lelong-Jensen type formula and Lelong number for quaternionic plurisubharmonic functions.更多还原