Vitushkins Conjecture for Removable Sets

Vitushkins conjecture, a special case of Painlevés problem, states that a compact subset of the complex plane with finite linear Hausdorff measure is removable for bounded analytic functions if and only if it intersects every rectifiable curve in a set of zero arc length measure. Chapters 6-8 of this carefully written text present a major recent accomplishment of modern complex analysis, the affirmative resolution of this conjecture. Four of the five mathematicians whose work solved Vitushkins conjecture have won the prestigious Salem Prize in analysis.

Chapters 1-5 of this book provide important background material on removability, analytic capacity, Hausdorff measure, arc length measure, and Garabedian duality that will appeal to many analysts with interests independent of Vitushkins conjecture. The fourth chapter contains a proof of Denjoys conjecture that employs Melnikov curvature. A brief postscript reports on a deep theorem of Tolsa and its relevance to going beyond Vitushkins conjecture. Although standard notation is used throughout, there is a symbol glossary at the back of the book for the readers convenience.

This text can be used for a topics course or seminar in complex analysis. To understand it, the reader should have a firm grasp of basic real and complex analysis.