Scaling Method And Its Applications In Several Complex Variables

One of the important issues of several complex variables is to classify domains in C~n.Here a domain in C~nwe mean a connected open set.In complex plane,this issue has a good answer because of the Riemann mapping theorem:all proper simply connected domains in complex plane are holomorphically equivalent[69].But if n≥2,the situation is more complicated because Poincaréhas proved that the unit ball and the polydisc in C~nare not biholomorphically equivalent[54].In fact,Burns,Shnider,Wells[11]and Greene,Krantz[27]have proved that strongly pseudoconvex domains with smooth boundaries are not biholomorphically equivalent in general.So we need to classify domains in terms of geometric invariants.The work of Fefferman[19],Bell[7],Bell,Ligocka[6]shows that the holomorphic automorphisms of some type of pseudoconvex domains can be extended smoothly to the boundary,this make it possible to use Poincaré’s original idea to use the geometric invariants of the boundary of the domains to classify domains.Chern and Moser[14]have done a lot of work in this direction.Another direction,also motivated by Poincaré,is to study the holomorphic automorphism groups of domains.The automorphism groups are nature invariants of domains,and it reflect the Bergman and Levi geometry of domains in many ways.The scaling method is a powerful tool to study automorphism groups.Scaling method concentred in analysis the local geometry of the boundary of the domains.This method is not only used for studying the automorphism groups[63],[21],[28];but also used for studying kernels functions[53],[45];the boundary behavior of Bergman metrics[40],[36],[37],[38]and others.The maintain of this article are two applications of the scaling method in the analysis of multi-dimensional complex variables.The thesis comprises three main parts.The first part related to the basic concepts in several com-plex variables.In chapter one we give the definitions of the defining functions of domains,domain of holomorphy,plurisubharmonic functions.Also we give a simple description of invariant metrics,complex geodesics,complex analytic varieties,which will be used in the proofs of the two applications.In chapter two we introduce the scaling method.We use examples in dimension one and two to clear the key steps of this method,and prove the Wong-Rosay theorem in the last as its application.Secondly,we give the first application of scaling method.The study of weakly pseudoconvex domains plays an important role in several complex variables,especially the domains with smooth boundaries of finite type.These domains include:finite type domains in C~2;convex domains of finite type;domains in C~nwith Levi rank greater or equal than n-2;bounded linear convex domains and others.All these domains are included by a type of domains called h-extendible(or semi-regular domains)domains.First we give the definition of h-extendible models,and then use it to define h-extendible domains.Notice that these domains are not necessarily bounded domains.Then we explore the basic geometries of h-extendible models,such as Kobayashi hyperbolicity,completeness,the existence of local holomorphic peak functions.Next,we study the h-extendible domains with non-compact automorphism group.If the holomorphic automorphism group of a domain is non-compact,then by the theorem of Cartan,there exists a point in its boundary called the accumulation boundary point,and the geometry of the boundary near this point determines the whole geometry of this domain.Wong-Rosay theorem is a reflection of this idea.Gaussier[23]proved that if the accumulation boundary point is convex and finite type,then the domains is biholomorphic to a domain defined by a weighted homogeneous polynomial with degree less than the D’Angelo’s variety type of this boundary point.We proved that if the boundary accumulation point is h-extendible,then the domain is biholomorphic to its h-extendible model with an extra cone condition.Here the model is defined by a weighted homogeneous polynomial.Finally,we used scaling method to prove an Alexander type theorem on a type of convex domain.A continuous mapping is called proper if the pre-image of an arbitrary compact set is also a compact set.So if the mapping can be extended across the boundary,then it must map the boundary into the boundary.This makes it possible to know the characters of the map by study its behavior near boundary.Alexander[2]proved that every proper holomorphic self-mapping of the unit ball in C~n,n≥2 is necessarily biholomorphic.In the last chapter we introduce a bounded domain called generalized minimal ball,which is a convex domain having non-smooth boundary.It can be thought of as an generalization of the minimal ball[30].Every proper holomorphic self-mapping of the minimal ball is biholomorphic[59].The generalized minimal ball can be regarded as an interpolation between the minimal ball and the Euclidean ball.We study the properties of the complex geodesics of a type of generalized minimal ball,and then use them to compute the automorphism group of it.We use scaling method to prove that if f is a proper holomorphic self-mapping of this type of generalized minimal ball,then the branch locus of f must lies in the intersection of the closure of the branch locus and the boundary of the domain.Also we show f must map the non-smooth part of the boundary of the domain to the non-smooth part.Then by the factorizations of f by a subgroup of automorphism group,we get a contradiction,which show that f is unbranched.Thus f must be biholomorphic.This part can be thought of as an generalization of some conclusions in[59].