Capacité analytique et le problème de Painlevé

Pajot, Hervé

Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Exposé no. 936, pp. 301-328.

Le problème de Painlevé consiste à trouver une caractérisation géométrique des sous-ensembles du plan complexe qui sont effaçables pour les fonctions holomorphes bornées. Ce problème d'analyse complexe a connu ces dernières années des avancées étonnantes, essentiellement grâce au développement de techniques fines d'analyse réelle et de théorie de la mesure géométrique. Dans cet exposé, nous allons présenter et discuter une solution proposée par X. Tolsa en termes de courbure de Menger au problème de Painlevé.

The Painlevé problem consists in finding a geometric characterization of removable sets for bounded analytic functions in the complex plane. This problem of complex analysis has known very striking results in the last years. These progress are based on recent developments in real analysis and geometric measure theory. In this talk, we will present and discuss a solution to the Painlevé problem proposed by X. Tolsa in terms of Menger curvature.

Classification : 28A75, 30C85, 42B20
Mot clés : capacité analytique, rectifiabilité, intégrale de Cauchy, courbure de Menger, ensembles uniformément rectifiables
Keywords: analytic capacity, rectifiability, Cauchy integral, Menger curvature, uniformly rectifiable sets

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