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Real and Complex Submanifolds pp 155–165Cite as

Some Geometric Aspects of the Hessian One Equation

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 106))

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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Authors and Affiliations

  1. University of Granada, 18071, Granada, Spain

    Antonio Martínez & Francisco Milán

Authors
  1. Antonio Martínez
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  2. Francisco Milán
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Corresponding author

Correspondence to Antonio Martínez .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Kyungpook National University, Daegu, Republic of Korea

    Young Jin Suh

  2. Department of Mathematics, King’s College London, London, UK

    Jürgen Berndt

  3. Department of Mathematics, Osaka City University, Osaka, Japan

    Yoshihiro Ohnita

  4. Department of Applied Mathematics, Kyung Hee University, Yongin, Republic of Korea

    Byung Hak Kim

  5. The Center for Geometry and its Applications, Pohang University of Science and Technology, Pohang, Republic of Korea

    Hyunjin Lee

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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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