Description
Book SynopsisCounterexamples are remarkably effective for understanding the meaning, and the limitations, of mathematical results. This title looks at some of the major ideas of several complex variables by considering counterexamples to what might seem like reasonable variations or generalizations.
Table of ContentsSome notations and definitions Holomorphic functions Holomorphic convexity and domains of holomorphy Stein manifolds Subharmonic/Plurisubharmonic functions Pseudoconvex domains Invariant metrics Biholomorphic maps Counterexamples to smoothing of plurisubharmonic functions Complex Monge Ampere equation $H^\infty$-convexity CR-manifolds Pseudoconvex domains without pseudoconvex exhaustion Stein neighborhood basis Riemann domains over $\mathbb{C}^n$ The Kohn-Nirenberg example Peak points Bloom's example D'Angelo's example Integral manifolds Peak sets for A(D) Peak sets. Steps 1-4 Sup-norm estimates for the $\bar{\partial}$-equation Sibony's $\bar{\partial}$-example Hypoellipticity for $\bar{\partial}$ Inner functions Large maximum modulus sets Zero sets Nontangential boundary limits of functions in $H^\infty(\mathbb{B}^n$ Wermer's example The union problem Riemann domains Runge exhaustion Peak sets in weakly pseudoconvex boundaries The Kobayashi metric Bibliography.
