Open Access
July, 2012 Value distribution of the Gauss map of improper affine spheres
J. Math. Soc. Japan 64(3): 799-821 (July, 2012). DOI: 10.2969/jmsj/06430799


We give the best possible upper bound for the number of exceptional values of the Lagrangian Gauss map of complete improper affine fronts in the affine three-space. We also obtain the sharp estimate for weakly complete case. As an application of this result, we provide a new and simple proof of the parametric affine Bernstein problem for improper affine spheres. Moreover, we get the same estimate for the ratio of canonical forms of weakly complete flat fronts in hyperbolic three-space.


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Yu KAWAKAMI. Daisuke NAKAJO. "Value distribution of the Gauss map of improper affine spheres." J. Math. Soc. Japan 64 (3) 799 - 821, July, 2012.


Published: July, 2012
First available in Project Euclid: 24 July 2012

zbMATH: 1252.53009
MathSciNet: MR2965428
Digital Object Identifier: 10.2969/jmsj/06430799

Primary: 53A15
Secondary: 30D35 , 53A35 , 53C42

Keywords: Bernstein type theorem , complete , exceptional value , flat front , improper affine sphere , Lagrangian Gauss map , Liouville property , weakly complete

Rights: Copyright © 2012 Mathematical Society of Japan

Vol.64 • No. 3 • July, 2012
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