Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Variation of conformal spheres by simultaneous sewing along several arcs

Author: T. L. McCoy
Journal: Trans. Amer. Math. Soc. 231 (1977), 65-82
MSC: Primary 30A30
MathSciNet review: 0444922
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let M be a closed Riemann surface of genus zero, $ \Gamma $ a tree on M with branches $ {\Gamma _j}$, and $ {p_0}$ a point of $ M - \Gamma $. A family of neighboring topological surfaces $ M(\varepsilon )$ is formed by regarding each $ {\Gamma _j}$ as a slit with edges $ \Gamma _j^ - $ and $ \Gamma _j^ + $, and re-identifying p on $ {\Gamma ^{{ - _j}}}$ with $ p + \varepsilon {\chi _j}(p,\varepsilon )$ on $ \Gamma _j^ + $, with $ {\chi _j}$ vanishing at the endpoints of $ {\Gamma _j}$. We assume the $ {\Gamma _j}$ and $ {\chi _j}$ are such that, under a certain natural choice of uniformizers, the $ M(\varepsilon )$ are closed Riemann surfaces of genus zero. Then there exists a unique function $ f(p,\varepsilon ;{p_0})$ mapping $ M(\varepsilon )$ conformally onto the complex number sphere, with normalization $ f({p_0},\varepsilon ;{p_0}) = 0,f'({p_0},\varepsilon ;{p_0}) = 1$. Under appropriate smoothness hypotheses on $ \Gamma $ and the $ {\chi _j}$, we find the first variation of f as a function of $ \varepsilon $. Further, we obtain smoothness results for f as a function of $ \varepsilon $. The problem is connected with the study of the extremal schlicht functions; that is, the schlicht mappings of the unit disc corresponding to boundary points of the coefficient bodies.

References [Enhancements On Off] (What's this?)

  • [1] R. Courant, Dirichlet's principle, conformal mapping, and minimal surfaces, Interscience, New York, 1950. MR 12, 90. MR 0036317 (12:90a)
  • [2] N. I. Mushelišvili, Singular integral equations. Boundary problems of function theory and their application to mathematical physics, OGIZ, Moscow, 1946; 2nd ed., Fizmatgiz, Moscow, 1962; English transl., Noordhoff, Groningen, 1953. MR 8, 586; 15, 434. MR 0355494 (50:7968)
  • [3] Zeev Nehari, Conformal mapping, International Series, McGraw-Hill, New York, 1952. MR 13, 640. MR 0045823 (13:640h)
  • [4] A. C. Schaeffer and D. C. Spencer, Coefficient regions for schlicht functions, Amer. Math. Soc. Colloq. Publ., vol. 35, Amer. Math. Soc., Providence, R. I., 1950. MR 12, 326. MR 0037908 (12:326c)
  • [5] M. M. Schiffer and D. C. Spencer, Functionals of finite Riemann surfaces, Princeton Univ. Press, Princeton, N. J., 1954. MR 16, 461. MR 0065652 (16:461g)
  • [6] George Springer, Introduction to Riemann surfaces, Addison-Wesley, Reading, Mass., 1957. MR 19, 1169. MR 0092855 (19:1169g)
  • [7] A. E. Taylor, Advanced calculus, Blaisdell, Waltham, Mass., 1955.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 30A30

Retrieve articles in all journals with MSC: 30A30

Additional Information

Article copyright: © Copyright 1977 American Mathematical Society

American Mathematical Society