Pluricomplex Green functions and the complex Monge-Ampere operator.

The topics discussed in this thesis are in the field of potential theory in several complex variables or, briefly, pluripotential theory. This theory deals with the study of plurisubharmonic functions and closed positive currents; in particular, the currents arising from plurisubharmonic functions are of special interest. The central role in pluripotential theory is played by the complex Monge-Ampere operator. This operator associates positive Borel measures to suitable plurisubharmonic functions, and is a generalization to $\doubc\sp{\rm n}$ of the Laplace operator in $\doubc$. In Chapter 1 we state some of the main open problems in pluripotential theory, to be addressed in later chapters. Chapter 2 contains a brief presentation of the basic notions used in the sequel. Much of the work using currents depends on formulas of integration by parts. In Chapter 3 we prove such formulas and use them to generalize a formula of Demailly regarding Lelong numbers with weights. We also obtain some estimates for the Monge-Ampere capacity; this capacity is suitable for the study of plurisubharmonic functions. Chapter 4, which contains the main results of this thesis, is dedicated to the study of pluricomplex Green functions. These functions are solutions of the degenerate Monge-Ampere equation with finitely many singularities. We obtain the formula for the pluricomplex Green function of the unit ball of $\doubc\sp{\rm n}$ with two poles, thus solving a previously open problem. We also examine the behavior of the pluricomplex Green function as its pole approaches the boundary of its domain of definition. In Chapter 5 we study some regularity properties of the solution to the Dirichlet problem for the homogeneous Monge-Ampere equation. This solution is often called the Perron-Bremermann function, since it was constructed in analogy to the Perron method in classical potential theory. We show that if D is a bounded domain in $\doubc\sp{\rm n}$ such that the Perron-Bremermann function of D is Holder continuous on $\bar D$ whenever its boundary data is Holder continuous, then D is necessarily pseudoconvex of finite type. We also prove that the converse of this result holds in $\doubc\sp2$.Ph.D.MathematicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/130451/2/9732060.pd