Nonlinear equations on a Lie group
Hauser and Ernst proved that (the germ of) any stationary, axially symmetric Einstein metric may be obtained from the flat metric by means of so-called Kinnersley-Chitre transformations: this is the Geroch conjecture [see: \textit{I. Hauser} and \textit{F. J. Ernst}, J. Math. Phys. 22, 1051-1063 (1981; Zbl 0496.70001) and the references therein]. It is known that these transformations are equivalent to an action of the loop group of SU(1,1) on the space of germs of Einstein metrics, i.e. the group of real-analytic maps \(S^ 1\to SU(1,1):\) theGeroch conjecture asserts that this action is transitive. The same authors generalized this result to include N abelian gauge fields, which is related to the loop group of \(SU(N+1,1)\) [\textit{I. Hauser} and \textit{F. J. Ernst}, Proof of a generalized Geroch conjecture, Galaxies, axisymmetric systems and relativity, Essays presented to W. B. Bonnor on his 65th birthday, ed. by M. A. H. MacCallum, Cambridge University Press (Cambridge, 1985)]. The present author gives a general group theoretic formulation of these results and proves analogues of them in which \(SU(N+1,1)\) is replaced by any real semi-simple Lie group which arises as the fixed points of the action on GL(n,\({\mathbb{C}})\) of a conjugation of inner type or a holomorphic automorphism of outer type.
scientific article; zbMATH DE number 4069771