Abstract
Centroaffine minimal surfaces are considered as an interesting class of surfaces from the viewpoint of not only variational problems in centroaffine differential geometry but also integrable systems. Typical examples of centroaffine minimal surfaces are proper affine spheres centered at the origin when we regard them as centroaffine surfaces. On the other hand, the study of projective minimal surfaces has a long history in projective differential geometry. Typical examples of projective minimal surfaces are proper affine spheres again, and so-called Demoulin surfaces or Godeaux-Rozet surfaces. In this paper, we shall regard centroaffine surfaces as projective surfaces and study projective minimality of centroaffine minimal surfaces. Using the fact that any centroaffine minimal surfaces have a one-parameter family of deformation known as associated surfaces, we shall give a classification of indefinite centroaffine minimal surfaces whose associated surfaces are all projective minimal, which includes centroaffine surfaces with vanishing Tchebychev operator and those found by the first author before. We shall also show that any indefinite centroaffine minimal surface whose associated surfaces are all Godeaux-Rozet surfaces is a proper affine sphere.
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To the memory of Takashi OKAYASU
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Fujioka, A., Furuhata, H. & Sasaki, T. Projective minimality for centroaffine minimal surfaces. J. Geom. 105, 87–102 (2014). https://doi.org/10.1007/s00022-013-0196-9
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DOI: https://doi.org/10.1007/s00022-013-0196-9