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Journal of Zhejiang University SCIENCE A 2005 Vol.6 No.7 P.747~749

http://doi.org/10.1631/jzus.2005.A0747


Riemann surface with almost positive definite metric


Author(s):  CHEN Zhi-guo

Affiliation(s):  Department of Mathematics, Zhejiang University, Hangzhou 310027, China

Corresponding email(s):   zgchen@zju.edu.cn

Key Words:  Quasiconformal mapping, &mu, (z)-homeomorphisms, Beltrami equation, Isothermal coordinates


CHEN Zhi-guo. Riemann surface with almost positive definite metric[J]. Journal of Zhejiang University Science A, 2005, 6(7): 747~749.

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author="CHEN Zhi-guo",
journal="Journal of Zhejiang University Science A",
volume="6",
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year="2005",
publisher="Zhejiang University Press & Springer",
doi="10.1631/jzus.2005.A0747"
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T1 - Riemann surface with almost positive definite metric
A1 - CHEN Zhi-guo
J0 - Journal of Zhejiang University Science A
VL - 6
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PB - Zhejiang University Press & Springer
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DOI - 10.1631/jzus.2005.A0747


Abstract: 
In this paper, we consider and resolve a geometric problem by using &mu;(z)-homeomorphic theory, which is the generalization of quasiconformal mappings. A sufficient condition is given such that a C1-two-real-dimensional connected orientable manifold with almost positive definite metric can be made into a Riemann surface by the method of isothermal coordinates. The result obtained here is actually a generalization of Chern’s work in 1955.

Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article

Reference

[1] Brakalova, M., Jenkins, J., 1998. On solution of the Beltrami equation. Journal d’Analyse Mathematique, 76:67-92.

[2] Chen, Z.G., 2003. μ(z)-homeomorphisms in the plane. Mich. Math. Journal, 51(3):547-556.

[3] Chern, S.S., 1955. An elementary proof of the existence of isothermal parameters on a surface. Proc. Amer. Math. Soc., 6:771-782.

[4] David, G., 1988. Solution de l’equation de Beltrami avec ||μ||(=1. Ann. Acad. Sci. Fenn. Ser. AI. Math., 13:25-69.

[5] Gutlyanskii, V., Martio, O., Sugawa, T., Vuorinen, M., 2005. On the degenerate Beltrami equation. Trans. of Amer. Math. Soc., 357(3):875-900.

[6] Hartman, P., Wintner, A., 1953. On the existence of Riemannian manifolds which can not carry analytic or harmonic functions in the small. Amer. J. Math., 75:260-276.

[7] Iwaniec, T., Martin, G., 2001. Geometric Function and Non-linear Analysis. Oxford Univ. Press, New York.

[8] Koskela, P., Malý, J., 2003. Mappings of finite distortion: the zero set of the Jacobian. J. Eur. Math. Soc. (JEMS), 5(2):95-105.

[9] Koskela, P., Onninen, J., 2003. Mappings of finite distortion: the sharp modulus of continuity. Trans. Amer. Math. Soc., 355(5):1905-1920 (electronic).

[10] Koskela, P., Rajala, K., 2003. Mappings of finite distortion: removable singularities. Israel J. Math., 136:269-283.

[11] Lehto, O., Virtanen, K., 1973. Quasiconformal Mappings in the Plane. Springer-Verlag, Berlin.

[12] Martio, O., Miklyukov, V., 2004. On existence and uniqueness of degenerate Beltrami equation. Complex Var. Theory Appl., 49(7-9):647-656.

[13] Petersen, C.L., Zakeri, S., 2004. On the Julia set of a typical quadratic polynomial with a Siegel disk. Ann. of Math. (2), 159(1):1-52.

[14] Rajara, K., 2004. Mappings of finite dilatation. Ann. Acad. Sci. Fenn. Ser. AI. Math., 29(2):269-281.

[15] Ryazanov, V., Srebro, U., Yakubov, E., 2001a. Plane mappings with dilatation dominated by functions of bounded mean oscillation. Siberian Adv. Math., 11(2):94-130.

[16] Ryzanov, V., Srebro, U., Yakubov, E., 2001b. BMO-quasiconformal mappings. J. Anal. Math., 83:1-20.

[17] Ryazanov, V., Srebro, U., Yakubov, E. Degenerate Beltrami Equation and Radial Q-homeomorphisms (in preprint).

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