Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral
The question of characterization of removable sets for bounded analytic functions is known as the Painlevé problem and has attracted the interest of mathematicians for more than 115 years. In a reformulation due to L. Ahlfors, the point is to describe for which sets the analytic capacity vanishes. In 1977, A. P. Calderón proved the Denjoy Conjecture that one-dimensional rectifiable curves are removable if and only if their Hausdorff length is zero. The proof was based on estimates of singular integrals, namely of the Cauchy operator. The version of the Vitushkin conjecture that a purely unrectifiable set of finite Hausdorff length is removable has been proved in 1998 by G.David, after significant steps forward by M. Melnikov, V. Verdera, P. Mattila, M. Christ and others. In this research the Menger curvature appeared to be relevant. In 2003, X. Tolsa characterized removable sets in terms of the Menger curvature. In this volume, the author presents the above mentioned results and related developments. The first two chapters are devoted to the Hausdorff measure and rectifiability, including beta numbers and uniform rectifiability. The Menger curvature is explained in Chapter 3. In Chapter 4, the theory of singular integrals is applied to the Cauchy operator. The Painlevé problem and the analytic capacity are discussed in Chapter 5. In Chapter 6, the proofs of the Denjoy conjecture and of the Vitushkin conjecture as well as related results are presented. Some very recent results including the Tolsa theorem are given in Chapter 7. The book excellently explains this beautiful theory, which is a subtle mix of complex analysis, harmonic analysis and geometric measure theory. The text is almost self-contained. The history of this development is nicely reviewed. At the end, main open problems of the theory are listed.