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1. CJM Online first
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)
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 Category:30D35 |
3. CJM 2002 (vol 54 pp. 648)
Rational Solutions of PainlevÃ© 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:PainlevÃ© differential equation, rational function, BÃ¤cklund transformation Categories:30D35, 34A20 |