An extension of a theorem by K. Jörgens and a maximum principle at infinity for parabolic affine spheres
An extension of a theorem by K. Jörgens and a maximum principle at infinity for parabolic affine spheres
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DOI：
10.1007/pl00004700
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发表时间：
1999-03
影响因子：
0.8
通讯作者：
L. Ferrer;Antonio Martínez;F. Milán
中科院分区：
文献类型：
--
作者：
L. Ferrer;Antonio Martínez;F. Milán
这个方程出现在仿射微分几何问题的上下文中，作为单模仿射真实的3-空间中的抛物仿射球（简称PA-球）的方程（参见[C1]，[C2]，[CY]和[LSZ]）。与光滑有界凸域的情况相反，当域是无界的时，对（1）的解知之甚少。这里，我们回顾K. Jörgens断言，任何解决方案（1）对R = R2是一个二次多项式（见[J]），我们还提到以前的文件（见[FMM]），其中作者研究解决方案（1）对外部的一个平面域是经常在无穷大。由于（1）的基本几乎复结构是可积的，人们期望PA-球面（及其正则共形结构）可以方便地用亚纯函数来描述。读者将在第2节中找到PA球的复杂表示，特别是对（1）的解的复杂描述。
This equation arises in the context of an affine differential geometric problem as the equation of a parabolic affine sphere (in short PA-sphere) in the unimodular affine real 3-space (see [C1],[C2],[CY] and [LSZ]). Contrary to the case of smooth bounded convex domains, little is known about solutions of (1) when the domain is unbounded. Here, we recall a famous result by K. J örgens which asserts that any solution of (1) onΩ= R2 is a quadratic polynomial (see [J]) and we also mention a previous paper (see [FMM]) where the authors study solutions of (1) on the exterior of a planar domain that are regular at infinity. Since the underlying almost-complex structure of (1) is integrable, one expects PA-spheres (with their canonical conformal structure) to be conveniently described in terms of meromorphic functions. The reader will find in Sect. 2 a complex representation of PA-spheres and, particularly, a complex description for the solutions of (1).