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Value distribution of the Gauss map of improper affine spheres

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.

preprint2012arXivOpen access
Yu KawakamiDaisuke Nakajo
math.CVmath.DG
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Yu KawakamiDaisuke Nakajo

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