Abstract
Working in the Nash-Moser category, it is shown that the harmonic and holomorphic differentials and the Weierstrass points on a closed Riemann surface depend smoothly on the complex structure. It is also shown that the space of complex structures on any compact surface forms a principal bundle over the Teichmüller space and hence that the uniformization maps of the closed disk and the sphere depend smoothly on the complex structure.
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References
C. J. Eearle and J. Eells, Fibre Bundle Description of Teichmüller Theory, J. Differential Geometry3 (1969), 19–43.
C. J. Earle and A. Schatz, Teichmüller Theory for Surfaces With Boundary, J. Differential Geometry4 (1970), 169–185.
D. G. Ebin, The Manifold of Riemannian Metrics, in: Global Analysis, Proceedings of Symposia in Pure Mathematics 15, Amer. Math. Soc., Providence 1970.
H. M. Farkas and I. Kra, Riemann Surfaces, Springer Verlag, New York 1980.
J. Gravesen, Geometrical Methods in Quantum Field Theory, D. Phil. Thesis, Oxford, 1987; see also: On the topology of holomorphic maps, Acta Math.162 (1989), 247–286.
R. S. Hamilton, The Inverse Function Theorem of Nash and Moser, Bull. Amer. Math. Soc.7 (1982), 65–222.
T. N. Subramaniam, Slices for Actions of Infinite Dimensional Groups, in: Differential Analysis in Infinite Dimensional Spaces, Contemporary Mathematics 54, Amer. Math. Soc., Providence 1986, pp. 65–77.
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Gravesen, J. Complex structures in the Nash-Moser category. Ann Glob Anal Geom 7, 155–161 (1989). https://doi.org/10.1007/BF00127865
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DOI: https://doi.org/10.1007/BF00127865