Comptes Rendus
no. 4
Differential geometry
Extremal metrics for the Q′-curvature in three dimensions

Presented by: Haïm Brézis

Jeffrey S. Case1; Chin-Yu Hsiao2; Paul Yang3
1 Department of Mathematics, McAllister Building, The Pennsylvania State University, University Park, PA 16802, United States
2 Institute of Mathematics, Academia Sinica, 6F, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan
3 Department of Mathematics, Princeton University, Princeton, NJ 08544, United States
Comptes Rendus. Mathématique, Volume 354 (2016) no. 4, pp. 407-410.

We construct contact forms with constant Q′-curvature on compact three-dimensional CR manifolds that admit a pseudo-Einstein contact form and satisfy some natural positivity conditions. These contact forms are obtained by minimizing the CR analogue of the II-functional from conformal geometry. Two crucial steps are to show that the P′-operator can be regarded as an elliptic pseudodifferential operator and to compute the leading-order terms of the asymptotic expansion of the Green's function for P′.

On construit des formes de contact à Q′-courbure constante sur les variétés de Cauchy–Riemann de dimension 3 qui admettent une pseudo-forme de contact d'Einstein et satisfont certaines conditions naturelles de positivité. Ces formes sont obtenues en minimisant l'analogue en CR-géométrie de la II-fonctionelle en géométrie conforme. Cette construction repose sur deux étapes cruciales. On montre que le P′-opérateur peut être vu comme un opérateur pseudo-differentiel elliptique et on calcule les termes dominants du développement asymtotique de la forme de Green pour P′.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2015.12.012

Jeffrey S. Case 1; Chin-Yu Hsiao 2; Paul Yang 3

1 Department of Mathematics, McAllister Building, The Pennsylvania State University, University Park, PA 16802, United States
2 Institute of Mathematics, Academia Sinica, 6F, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan
3 Department of Mathematics, Princeton University, Princeton, NJ 08544, United States 
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Jeffrey S. Case; Chin-Yu Hsiao; Paul Yang. Extremal metrics for the Q′-curvature in three dimensions. Comptes Rendus. Mathématique, Volume 354 (2016) no. 4, pp. 407-410. doi : 10.1016/j.crma.2015.12.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.12.012/

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