Métriques autoduales sur la boule |
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Authors: | Olivier Biquard |
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Institution: | (1) Institut de Recherche Mathématique Avancée, Université Louis Pasteur et CNRS, 7, rue René Descartes, 67084 Strasbourg Cedex, France (e-mail: olivier.biquard@math.u-strasbg.fr), FR |
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Abstract: | 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-K?hler 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-K?hler metric on the (4m)-ball.
Oblatum 29-XI-2000 & 7-XI-2001?Published online: 1 February 2002 |
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Keywords: | Mathematical Subject Classification (2000): 53C25 32G07 53C28 53C26 |
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