Tohoku Mathematical Journal
2016
December
SECOND SERIES VOL. 68, NO. 4

Tohoku Math. J.
68 (2016), 621-638

Title WINTGEN IDEAL SUBMANIFOLDS OF CODIMENSION TWO, COMPLEX CURVES, AND M\"OBIUS GEOMETRY

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

(Received December 4, 2014, revised April 2, 2015)
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.

Mathematics Subject Classification. Primary 53C42; Secondary 53A30, 53C43.
Key words and phrases. Wintgen ideal submanifolds, Möbius geometry, mean curvature sphere, conformal Gauss map, minimal surfaces, holomorphic curves.

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