多重位势论的一些研究
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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。
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.
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。
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.
引文
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[3]S. Alesker, Quaternionic Monge-Ampere equations, J. Geom. Anal.13 (2) (2003) 205-238.
[4]S. Alesker, Valuations on convex sets, non-commutative determinants, and pluripotential theory, Adv. Math.195 (2) (2005) 561-595.
[5]S. Alesker, Plurisubharmonic functions on the octonionic plane and Spin(9)-invariant valuations on convex sets, J. Geom. Anal.18 (3) (2008) 651-686.
[6]S. Alesker, Pluripotential theory on quaternionic manifolds, J. Geom. Phys. 62 (2012) 1189-1206.
[7]S. Alesker and M. Verbitsky, Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry, J. Geom. Anal.16 (2006) 375-399.
[8]S. Alesker and M. Verbitsky, Quaternionic Monge-Ampere equation and Calabi problem for HKT-manifolds, Israel J. Math.176 (2010) 109-138.
[9]R. J. Baston, Quaternionic complexes, J. Geom. Phys.8 (1992) 29-52.
[10]E. Bedford, Survey of pluripotential theory, in Several Complex Variables (Stockholm,1987/88), Math. Notes,38 Princeton Univ. Press, Princeton, N J, (1993) 48-95.
[11]E. Bedford and B. A. Taylor, The Dirichlet problem for a complex Monge-Ampere equation, Invent. Math.37 (1976) 1-44.
[12]E. Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math.149 (1982) 1-41.
[13]E. Bedford and B. A. Taylor, Fine topology, silov boundary and (ddc)n, J. Funct. Anal.72 (2) (1987) 225-251.
[14]Z. Blocki, Estimates for the complex Monge-Ampere operator, Bull. Pol. Acad. Sci. Math.41 (1993) 151-157.
[15]Z. Blocki, The C1,1 regularity of the pluricomplex Green function, Michigan Math. J.47 (2000) 211-215.
[16]Z. Blocki, Regularity of the pluricomplex Green function with several poles, Indiana Univ. Math. J.50 (1) (2001) 335-351.
[17]Z. Blocki, The Bergman metric and the pluricomplex Green function, Trans. Amer. Math. Soc.357 (2005) 2613-2625.
[18]Z. Blocki, Weak solutions to the complex Hessian equation, Ann. Inst. Fourier, (Grenoble) 55 (2005) 1735-1756.
[19]Z. Blocki, The domain of definition of the complex Monge-Ampere operator, Amer. J. Math.128 (2006) 519-530.
[20]Z. Blocki, Remark on the definition of the complex Monge-Ampere operator, Functional analysis and complex analysis,17-21, Contemp. Math.481 Amer. Math. Soc., Providence, RI,2009.
[21]H. J. Bremermann, On a generalized Dirichlet problem for plurisubharmonic functions and pseudo-convex domains. Characterization of silov boundaries, Trans. Amer. Math. Soc.91 (1959) 246-276.
[22]M. Carlehed, U. Cegrell and F. Wikstrom, Jensen measures, hyperconvexity and boundary behavior of the pluricomplex Green function, Ann. Polon. Math.71 (1999)87-103.
[23]H. Cartan, Theorie du potentiel Newtonien:energie, capacite, suites de potentiels(French) Bull. Soc. Math. France 73 (1945) 74-106.
[24]U. Cegrell, Removable singularities for plurisubharmonic functions and re-lated problems, Proc. London Math. Soc.36 (3) (1978) 310-336.
[25]U. Cegrell, On the Dirichlet problem for the complex Monge-Ampere oper-ator, Math. Z.185 (1984) 247-251.
[26]U. Cegrell, Sums of continuous plurisubharmonic functions and the complex Monge-Ampere operator in Cn, Math. Z.193 (1986) 373-380.
[27]U. Cegrell, Capacities in complex analysis, Aspects of Mathematics, E14. Friedr. Vieweg and Sohn, Braunschweig,1988.
[28]U. Cegrell, Pluricomplex energy, Acta Math.180 (1998) 187-217.
[29]U. Cegrell, The general definition of the complex Monge-Ampere operator, Ann. Inst. Fourier.54 (2004) 159-179.
[30]U. Cegrell, Maximal plurisubharmonic functions, Uzbek Math. J.1(2009) 10-16.
[31]U. Cegrell, Approximation of plurisubharmonic functions in hyperconvex domains,In:Complex Analysis and Digital Geometry. Acta Universitatis Upsaliensis Skr. Uppsala Univ. C. Organ. Hist.86. Uppsala Universitet,Up-psala,2009.125-129.
[32]U. Cegrell, Potentials with respect to the pluricomplex Green function, Ann. Polon. Math.106 (2012) 107-111.
[33]U. Cegrell and L. Persson, An energy estimate for the complex Monge-Ampere operator, Ann. Polon. Math.67 (1997) 95-102.
[34]S. S. Chern, H. I. Levine and L. Nirenberg, Intrinsic norms on a complex manifold. Global Analysis (Papers in Honor of K. Kodaira) Univ. Tokyo Press, Tokyo 1969.119-139.
[35]R. Czyz, Convergence in capacity of the pluricomplex Green function, Univ. Iagel. Acta Math.43 (2005) 41-44.
[36]J. P. Demailly, Mesures de Monge-Ampere et caracterisation geometrique des varietes algebriques affines, Mem. Soc. Math. France (N.S.) 19 (1985) 1-124.
[37]J. P. Demailly, Mesures de Monge-Ampere et mesures pluriharmoniques, Math. Z.194 (1987) 519-564.
[38]J. P. Demailly, Potential theory in several complex variables, Ecole d'eted'Analyse complexe du CIMPA, Nice,1989.
[39]J. P. Demailly, Monge-Ampere operators, Lelong numbers and intersection theory, Complex analysis and geometry,115-193, Univ. Ser. Math.,Plenum, NewYork (1993).
[40]T.-C. Dinh and N. Sibony, Dynamics in several complex variables:endomor-phisms of projective spaces and polynomial-like mappings, Holomorphic dy-namical systems,165-294, Lecture Notes in Math.,1998, Springer, Berlin, 2010.
[41]L. Garding, An inequality for hyperbolic polynomials, J. Math. Mech.8 (1959) 957-965.
[42]R. Harvey, Holomorphic chains and their boundaries, in Several Complex Variables, part 1,309-382, American Mathematical Society,1977.
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