Abstract
The Hessian one equation and its complex resolution provides an important tool in the study of improper affine spheres in \(\mathbb{R}^{3}\) with some kind of singularities. The singular set can be characterized and, in most of the cases, it determines the surface. Here, we show how to obtain improper affine spheres with a prescribed singular set and construct some global examples with the desired singularities. We also classify improper affine spheres admitting a planar singular set.
Research partially supported by Ministerio de Educación Grant No. MTM2013-43970-P, Junta de Andalucía Grants No. FQM325, No. P06-FQM-01642.
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Martínez, A., Milán, F. (2014). Some Geometric Aspects of the Hessian One Equation. In: Suh, Y.J., Berndt, J., Ohnita, Y., Kim, B.H., Lee, H. (eds) Real and Complex Submanifolds. Springer Proceedings in Mathematics & Statistics, vol 106. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55215-4_14
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