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Search: MSC category 30D35 ( Distribution of values, Nevanlinna theory )

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1. CJM Online first

Kawakami, Yu
Function-theoretic Properties for the Gauss Maps of Various Classes of Surfaces
We elucidate the geometric background of function-theoretic properties for the Gauss maps of several classes of immersed surfaces in three-dimensional space forms, for example, minimal surfaces in Euclidean three-space, improper affine spheres in the affine three-space, and constant mean curvature one surfaces and flat surfaces in hyperbolic three-space. To achieve this purpose, we prove an optimal curvature bound for a specified conformal metric on an open Riemann surface and give some applications. We also provide unicity theorems for the Gauss maps of these classes of surfaces.

Keywords:Gauss map, minimal surface, constant mean curvature surface, front, ramification, omitted value, the Ahlfors island theorem, unicity theorem.
Categories:53C42, 30D35, 30F45, 53A10, 53A15

2. CJM 2004 (vol 56 pp. 1190)

Frank, Günter; Hua, Xinhou; Vaillancourt, Rémi
Meromorphic Functions Sharing the Same Zeros and Poles
In this paper, Hinkkanen's problem (1984) is completely solved, {\em i.e.,} it is shown that any meromorphic function $f$ is determined by its zeros and poles and the zeros of $f^{(j)}$ for $j=1,2,3,4$

Keywords:Uniqueness, meromorphic functions, Nevanlinna theory

3. CJM 2002 (vol 54 pp. 648)

Yuan, Wenjun; Li, Yezhou
Rational Solutions of Painlevé Equations
Consider the sixth Painlev\'e equation~(P$_6$) below where $\alpha$, $\beta$, $\gamma$ and $\delta$ are complex parameters. We prove the necessary and sufficient conditions for the existence of rational solutions of equation~(P$_6$) in term of special relations among the parameters. The number of distinct rational solutions in each case is exactly one or two or infinite. And each of them may be generated by means of transformation group found by Okamoto [7] and B\"acklund transformations found by Fokas and Yortsos [4]. A list of rational solutions is included in the appendix. For the sake of completeness, we collected all the corresponding results of other five Painlev\'e equations (P$_1$)--(P$_5$) below, which have been investigated by many authors [1]--[7].

Keywords:Painlevé differential equation, rational function, Bäcklund transformation
Categories:30D35, 34A20

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