Abstract
We give a complete list of affine minimal surfaces inA 3 with Euclidean rotational symmetry, completing the treatise given in [1] and prove that these surfaces have maximal affine surface area within the class of all affine surfaces of rotation satisfying suitable boundary conditions. Besides we show that for rotationally symmetric locally strongly convex affine minimal hypersurfaces inA n,n≥4, the second variation of the affine surface area is negative definite under certain conditions on the meridian.
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References
Blaschke, W.:Affine Differentialgeometrie, Springer, Berlin, 1923.
Calabi, E.: Hypersurfaces with maximal affinely invariant area,Amer. J. Math. 104(1) (1982), 91–126.
Chern, S. S.: Affine minimal hypersurfaces, inSelected Papers, Vol. III, Springer, New York, 1989, pp. 425–438.
Krauter, P.: Extremaleigenschaften von Affinminimalhyperflächen, Doktorarbeit, Universität Stuttgart, 1993.
Manhart, F.: Uneigentliche Relativsphären im dreidimensionalen euklidischen Raum, welche Drehflächen sind,Sitzungsber. Öster. Akad. Wiss. Math. Natur. Kl. 195 (1986), 281–289.
Schirokow, P. A. and Schirokow, A. P.:Affine Differentialgeometrie, Teubner, Leipzig, 1962.
Su, B.: Affine moulding surfaces and affine surfaces of revolution,Japan. J. Math. 5 (1928), 185–210.
Süss, W.: Ein affingeometrisches Gegenstück zu den Rotationsflächen,Math. Ann. 98 (1928), 684–696.
Verstraelen, L. and Vrancken, L.: Affine variation formulas and affine minimal surfaces,Michigan Math. J. 36 (1989), 77–93.
Yang, W.-M. and Nie, J.: On affine minimal rotation surfaces and conoid inA 3,J. Math. (PRC),7(2) (1987), 205–210.
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Krauter, P. Affine minimal hypersurfaces of rotation. Geom Dedicata 51, 287–303 (1994). https://doi.org/10.1007/BF01263997
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DOI: https://doi.org/10.1007/BF01263997