Plurisubharmonic functions of logarithmic growth
Abstract (Summary)Restricted Item. Print thesis available in the University of Auckland Library or available through Inter-Library Loan. The main body of this thesis is divided into three parts. Part 1 of this thesis, consisting of the first five chapters, is a study of the plurisubharmonic functions VK(n)*, n=l,2,...., associated to a compact set K ? CN. These functions are defined in chapter 1; they are one-variable in nature and increase quasi-everywhere to VK*, the Siciak-Zaharjuta extremal function. We mainly study VK(1); we show that it induces a metric on a subclass of the lineally convex sets, and is related to a capacity (the projection capacity) that is defined in chapter 4. Part 2 of this thesis, consisting of chapters 6 and 7, is a study of the complex Monge-Ampere operator. After developing some background material, we derive a formula for the Monge.Ampere mass of a plurisubharmonic function which is the maximum of 3 pluriharmonic functions in C2, then use the formula to compute the Monge-Ampere mass of such a function. We also show how our formula is related to the problem of determining whether certain real submanifolds of C2 are complex analytic. Part 3 of this thesis, consisting of chapters 8-10, is concerned with the existence of complex ellipses on which the Siciak-Zaharjuta extremal function of a compact set K in RN ? CN is harmonic. We summarize the work of Lundin and Baran who proved that existence holds for symmetric bodies and some special nonsymmetric bodies. Following an idea of Burns, we then prove the existence of such ellipses for all convex bodies in RN, and present two applications. First, we show that the sets K for which VK(1) is not ? VK holds form a dense subspace of the collection of convex bodies in R2 with respect to two metrics. For the second application, we verify a conjecture of Lundin on the polynomial approximation of harmonic functions in R2.
School Location:New Zealand
Source Type:Master's Thesis
Date of Publication:01/01/2003