ebook9781441967091d4325

Description

Vitushkin’s conjecture, a special case of Painlev√©’s 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 Vitushkin’s 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 Vitushkin’s conjecture.¬† The fourth chapter contains a proof of Denjoy’s conjecture that employs Melnikov curvature.¬† A brief postscript reports on a deep theorem of Tolsa and its relevance to going beyond Vitushkin’s conjecture.¬† Although standard notation is used throughout, there is a symbol glossary at the back of the book for the reader’s 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.