Tohoku Mathematical Journal

Wintgen ideal submanifolds of codimension two, complex curves, and Möbius geometry

Tongzhu Li, Xiang Ma, Changping Wang, and Zhenxiao Xie

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Abstract

Wintgen ideal submanifolds in space forms are those ones attaining the equality pointwise in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the scalar normal curvature. Using the framework of Möbius geometry, we show that in the codimension two case, the mean curvature spheres of the Wintgen ideal submanifold correspond to a 1-isotropic holomorphic curve in a complex quadric. Conversely, any 1-isotropic complex curve in this complex quadric describes a 2-parameter family of spheres whose envelope is always a Wintgen ideal submanifold of codimension two at the regular points. Via a complex stereographic projection, we show that our characterization is equivalent to Dajczer and Tojeiro's previous description of these submanifolds in terms of minimal surfaces in the Euclidean space.

Article information

Source
Tohoku Math. J. (2), Volume 68, Number 4 (2016), 621-638.

Dates
First available in Project Euclid: 4 February 2017

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1486177219

Digital Object Identifier
doi:10.2748/tmj/1486177219

Mathematical Reviews number (MathSciNet)
MR3605451

Zentralblatt MATH identifier
1381.53104

Subjects
Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
Secondary: 53A30: Conformal differential geometry 53C43: Differential geometric aspects of harmonic maps [See also 58E20]

Keywords
Wintgen ideal submanifolds Möbius geometry mean curvature sphere conformal Gauss map minimal surfaces holomorphic curves

Citation

Li, Tongzhu; Ma, Xiang; Wang, Changping; Xie, Zhenxiao. Wintgen ideal submanifolds of codimension two, complex curves, and Möbius geometry. Tohoku Math. J. (2) 68 (2016), no. 4, 621--638. doi:10.2748/tmj/1486177219. https://projecteuclid.org/euclid.tmj/1486177219


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