Abstract
We study the spaces ℳ and ℳ and ℳ′Lip of smooth (resp. non-degenerate Lipschitz) isometric maps of a circle into Euclidean space modulo orientation preserving Euclidean motions. We prove that ℳ and ℳ′Lip are infinite dimensional Kähler manifolds. In particular, they are complex Fréchet (resp. Banach) manifolds. This is proved by an infinite dimensional version of the Kirwan, Kempf-Ness Theorem [Kir84], [KN78], [Nes84] relating symplectic quotients to holomorphic quotients, applied to the action ofPSL 2(ℂ) on the free loop space ofS 2.
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Oblatum 15-X-1994 & 5-VII-1995
This research was supported in part by NSF grant DMS-92-05154.
This research was partially supported by AFOSR grant F49620-92-J-0093.
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Millson, J.J., Zombro, B. A Kähler structure on the moduli space of isometric maps of a circle into Euclidean space. Invent Math 123, 35–59 (1996). https://doi.org/10.1007/BF01232366
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DOI: https://doi.org/10.1007/BF01232366