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A Kähler structure on the moduli space of isometric maps of a circle into Euclidean space. (English) Zbl 0859.58007
The authors prove that the spaces $$\mathcal M$$ and $${\mathcal M}_{\text{Lip}}'$$ of smooth (resp. nondegenerate Lipschitz) isometric maps of a circle into Euclidean space modulo orientation preserving Euclidean motions, have the structure of infinite-dimensional Kähler manifolds. In particular, they are complex Fréchet (resp. Banach) manifolds. This is proved by an infinite-dimensional version of a theorem of Kirwan, Kempf and Ness relating symplectic quotients to holomorphic quotients, applied to the action of $$\text{PSL}_2(\mathbb{C})$$ on the free loop space of the 2-sphere. A key role is played by the conformal center of mass of A. Douady and C. J. Earle [Acta Math. 157, 23-48 (1986; Zbl 0615.30005)], of which notion the authors give a self-contained treatment in the present paper.

##### MSC:
 58D15 Manifolds of mappings 58D29 Moduli problems for topological structures 58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
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