Metriques autoduales sur la boule
| dc.creator | Biquard, Olivier | |
| dc.date | 2000-10-19 | |
| dc.date | 2000-11-16 | |
| dc.date.accessioned | 2026-07-07T04:38:08Z | |
| dc.date.available | 2026-07-07T04:38:08Z | |
| dc.description | A conformal metric on a 4-ball induces on the boundary 3-sphere a conformal metric and a trace-free second fundamental form. Conversely, such a data on the 3-sphere is the boundary of a unique selfdual conformal metric, defined in a neighborhood of the sphere. In this paper we characterize the conformal metrics and trace-free second fundamental forms on the 3-sphere (close to the standard round metric) which are boundaries of selfdual conformal metrics on the whole 4-ball. When the data on the boundary is reduced to a conformal metric (the trace-free part of the second fundamental form vanishes), one may hope to find in the conformal class of the filling metric an Einstein metric, with a pole of order 2 on the boundary. We determine which conformal metrics on the 3-sphere are boundaries of such selfdual Einstein metrics on the 4-ball. In particular, this implies the Positive Frequency Conjecture of LeBrun. The proof uses twistor theory, which enables to translate the problem in terms of complex analysis; this leads us to prove a criterion for certain integrable CR structures of signature (1,1) to be fillable by a complex domain. Finally, we solve an analogous, higher dimensional problem: selfdual Einstein metrics are replaced by quaternionic-Kahler metrics, and conformal structures on the boundary by quaternionic contact structures (previously introduced by the author); in contrast with the 4-dimensional case, we prove that any small deformation of the standard quaternionic contact structure on the (4m-1)-sphere is the boundary of a quaternionic-Kahler metric on the 4m-ball. | |
| dc.description | 76 pages, 4 figures; new section 11 on higher dimensional case | |
| dc.identifier | https://arxiv.org/abs/math/0010188 | |
| dc.identifier | http://arxiv.org/abs/math/0010188 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/60163 | |
| dc.subject | Differential Geometry | |
| dc.subject | Complex Variables | |
| dc.subject | 53C25, 32G07, 53C28, 53C26 | |
| dc.title | Metriques autoduales sur la boule | |
| dc.type | text |