zbMATH — the first resource for mathematics

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.

58D15 Manifolds of mappings
58D29 Moduli problems for topological structures
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
Full Text: DOI EuDML
[1] [AB82] M.F. Atiyah, R. Bott: The Yang-Mills equations over Riemann surfaces. Phil. Trans. R. Soc. Lond. A308, 523-615 (1982) · Zbl 0509.14014
[2] [BGS85] W. Ballman, M. Gromov, V. Schroeder: Manifolds of Nonpositive Curvature. Volume 61 of Progress in Math. Birkhäuser, Basel Boston, 1985 · Zbl 0591.53001
[3] [BO69] R.L. Bishop, B. O’Neill: Manifolds of negative curvature. Trans. of the A.M.S.143, 1-49 (1969) · doi:10.1090/S0002-9947-1969-0251664-4
[4] [Bry] J.-L. Brylinski (Private communication)
[5] [Bry92] J.-L. Brylinski: Loop Spaces, Characteristic Classes and Geometric Quantization. Volume 107 of Progress in Math. Birkhäuser, Basel Boston, 1992
[6] [Cor88] K. Corlette: FlatG-bundles with canonical metrics. J. Differ. Geom.28, 361-382 (1988)
[7] [DE86] A. Douady, C. Earle: Conformally natural extensions of homeomorphisms of the circle. Acta Math.157, 23-48 (1986) · Zbl 0615.30005 · doi:10.1007/BF02392590
[8] [DM86] P. Deligne, G.D. Mostow: Monodromy of hypergeometric functions and non-lattice integral monodromy. Publ. Math. I.H.E.S.63, 5-90 (1986) · Zbl 0615.22008
[9] [Don83] S.K. Donaldson: A new proof of a theorem of Narasimhan and Seshadri. J. Differ. Geom.18, 269-277 (1983) · Zbl 0504.49027
[10] [DS88] N. Dunford, J.T. Schwartz: Linear Operators Part I: General Theory. Wiley, New York, 1988 · Zbl 0635.47001
[11] [Ham82] R.S. Hamilton: The inverse function theorem of Nash and Mosher. Bull. Am. Math. Soc.7, 65-221 (1982) · Zbl 0499.58003 · doi:10.1090/S0273-0979-1982-15004-2
[12] [Hel81] S. Helgason: Topics in Harmonic Analysis on Homogeneous Spaces. Volume 13 of Progress in Math. Birkhäuser, Basel Boston, 1981 · Zbl 0467.43001
[13] [Kem78] G. Kempf: Instability in invariant theory. Ann. Math.108, 299-316 (1978) · Zbl 0406.14031 · doi:10.2307/1971168
[14] [Kir84] F. Kirwan: Cohomology of Quotients in Symplectic and Algebraic Geometry. Volume 31 of Mathematical Notes. Princeton University Press, 1984 · Zbl 0553.14020
[15] [Kly92] A. Klyachko: Spatial polygons and stable configurations of points in the projective line, Algebraic Geometry and its Applications. pp. 67-84. Yaroslavl, 1992
[16] [KM] M. Kapovich, J.J. Millson: The symplectic geometry of polygons in Euclidean space (Pre-print)
[17] [KN78] G. Kempf, L. Ness: The length of vectors in representation spaces. Vol. 732, pp. 233-244. Springer Lecture Notes in Math., 1978
[18] [Kos65] J.L. Koszul: Lectures on Groups of Transformations, Lectures on Mathematics 32. Tata Institute of Fundamental Research, 1965
[19] [Lem93] L. Lempert: Loop spaces as complex manifolds. J. Differ. Geom.38, 519-543 (1993) · Zbl 0792.58008
[20] [LM] B. Leeb, J.J. Millson: The non-euclidean center of mass and geometric invariant theory (In preparation)
[21] [LP] J. Langer, R. Perline: The Poisson geometry of the filament equation (Pre-print)
[22] [MZ] J.J. Millson, B. Zombro (In preparation)
[23] [Nes84] L. Ness: A stratification of the null cone via the moment map. Am. J. Math.106, 1281-1329 (1984) · Zbl 0604.14006 · doi:10.2307/2374395
[24] [Thu] W. Thurston: Shapes of polyhedra (Pre-print)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.