## Variation of conformal spheres by simultaneous sewing along several arcs

HTML articles powered by AMS MathViewer

- by T. L. McCoy
- Trans. Amer. Math. Soc.
**231**(1977), 65-82 - DOI: https://doi.org/10.1090/S0002-9947-1977-0444922-6
- PDF | Request permission

## 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

- R. Courant,
*Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces*, Interscience Publishers, Inc., New York, N.Y., 1950. Appendix by M. Schiffer. MR**0036317** - N. I. Muskhelishvili,
*Singular integral equations*, Wolters-Noordhoff Publishing, Groningen, 1972. Boundary problems of functions theory and their applications to mathematical physics; Revised translation from the Russian, edited by J. R. M. Radok; Reprinted. MR**0355494** - Zeev Nehari,
*Conformal mapping*, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1952. MR**0045823** - A. C. Schaeffer and D. C. Spencer,
*Coefficient Regions for Schlicht Functions*, American Mathematical Society Colloquium Publications, Vol. 35, American Mathematical Society, New York, N. Y., 1950. With a Chapter on the Region of the Derivative of a Schlicht Function by Arthur Grad. MR**0037908** - Menahem Schiffer and Donald C. Spencer,
*Functionals of finite Riemann surfaces*, Princeton University Press, Princeton, N. J., 1954. MR**0065652** - George Springer,
*Introduction to Riemann surfaces*, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1957. MR**0092855**
A. E. Taylor,

*Advanced calculus*, Blaisdell, Waltham, Mass., 1955.

## Bibliographic Information

- © Copyright 1977 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**231**(1977), 65-82 - MSC: Primary 30A30
- DOI: https://doi.org/10.1090/S0002-9947-1977-0444922-6
- MathSciNet review: 0444922