Géométrie conforme en dimension $4$ : ce que l’analyse nous apprend

Margerin, Christophe

Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Exposé no. 950, pp. 415-468.

Cet article présente les idées, les outils et les résultats qui ont permis à Chang S.-Y. A., M. Gursky et Yang P. de donner une caractérisation intégrale conforme de la sphère standard en dimension 4. Nous démarrons avec une généralisation à cette dimension de la formule de Polyakov pour les déterminants régularisés, que nous utilisons ensuite pour résoudre des problèmes du type “Yamabe” pour des polynômes quadratiques en la courbure de Ricci. Nous introduisons au passage le concept de paire conforme, en particulier l’opérateur (du quatrième ordre) de Paneitz et sa courbure $Q$ associée, et nous discutons leurs relations à la géométrie conforme classique. On trouvera aussi une preuve d’un esprit différent du théorème principal : beaucoup plus courte et naturelle, elle généralise un argument dû à M. Gursky et J. Viaclovsky qui l’a largement inspirée. On y donne enfin quelques constructions de métriques de courbure $Q$ constante, conséquence des arguments développés précédemment.

Starting at a $4$-dimensional generalization of Polyakov’s formula for regularized determinants, we present the ideas, tools, and results which led Chang S.-Y. A., M. Gursky and Yang P. to a sharp $L^2$ conformal sphere theorem. This includes the solution of Yamabe type problems for quadratic polynomials in the Ricci curvature. On our way we introduce “Conformal Pairs”, in particular the (4th order) Paneitz operator and its associated $Q$-curvature, and discuss how they relate to more classical $4$-dimensional conformal geometry. Elaborating on an argument due to M. Gursky and J. Viaclovsky, we also give a completely different, more direct and natural, proof of the main sphere theorem, and discuss further some related constructions of metrics with constant $Q$-curvature.

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