Complex structures on Riemann surfaces

Author:
Garo Kiremidjian

Journal:
Trans. Amer. Math. Soc. **169** (1972), 317-336

MSC:
Primary 32G15; Secondary 30A46

MathSciNet review:
0310296

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Abstract: Let be a Riemann surface (compact or noncompact) with the property that the length of every closed geodesic is bounded away from zero. Then we show that sufficiently small complex structures on can be described without making use of Schwarzian derivatives or the theory of quasiconformal mappings. Instead, we use methods developed in Kuranishi's work on the existence of locally complete families of deformations of compact complex manifolds.

We introduce norms ( a positive integer) on the space of -forms with values in the tangent bundle on , which are similar to the usual Sobolev -norms. (In the compact case is equivalent to .) Then we prove that certain properties of , crucial for Kuranishi's approach, are also satisfied by .

**[1]**Lars V. Ahlfors,*The complex analytic structure of the space of closed Riemann surfaces.*, Analytic functions, Princeton Univ. Press, Princton, N.J., 1960, pp. 45–66. MR**0124486****[2]**-,*Lectures in quasiconformal mappings*, Van Nostrand Math. Studies, no. 10, Van Nostrand, Princeton, N. J., 1966. MR**34**#336.**[3]**Aldo Andreotti and Edoardo Vesentini,*Carleman estimates for the Laplace-Beltrami equation on complex manifolds*, Inst. Hautes Études Sci. Publ. Math.**25**(1965), 81–130. MR**0175148****[4]**Lipman Bers,*Spaces of Riemann surfaces*, Proc. Internat. Congress Math. 1958, Cambridge Univ. Press, New York, 1960, pp. 349–361. MR**0124484****[5]**-,*On moduli of Riemann surfaces*, Lecture notes, Eidgenössische Technische Hochschule, Zürich, 1964.**[6]**K. Kodaira and D. C. Spencer,*On deformations of complex analytic structures. I, II*, Ann. of Math. (2)**67**(1958), 328–466. MR**0112154****[7]**M. Kuranishi,*On the locally complete families of complex analytic structures*, Ann. of Math. (2)**75**(1962), 536–577. MR**0141139****[8]**M. Kuranishi,*New proof for the existence of locally complete families of complex structures*, Proc. Conf. Complex Analysis (Minneapolis, 1964) Springer, Berlin, 1965, pp. 142–154. MR**0176496****[9]**-,*Deformations of complex analytic structures on compact manifolds*, Seminar on Global Analysis, Montreal, 1969 (to appear).

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1972-0310296-X

Keywords:
Complex manifold,
Riemann surface,
almost complex and complex structures,
deformations of complex structures,
holomorphic vector bundles,
Laplace operator,
-ellipticity,
Green's operator,
-norms

Article copyright:
© Copyright 1972
American Mathematical Society