## Tohoku Mathematical Journal

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

#### 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

#### 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

#### References

• R. Bryant, A duality theorem for Willmore surfaces, J. Differential Geom. 20 (1984), 20–53.
• R. Bryant, Surfaces in Conformal Geometry, The mathematical heritage of Hermann Weyl (Durham, NC, 1987), 227–240, Proc. Sympos. Pure Math. 48, Amer. Math. Soc., Providence, RI, 1988.
• R. Bryant, Some remarks on the geometry of austere manifolds, Bol. Soc. Brasil. Mat. (N.S.) 21 (1991), 133–157.
• B. Chen, Classification of Wintgen ideal surfaces in Euclidean 4-space with equal Gauss and normal curvatures, Ann. Global Anal. Geom. 38 (2010), 145–160.
• M. Dajczer and R. Tojeiro, A class of austere submanifolds, Illinois J. Math. 45 (2001), no. 3, 735–755.
• M. Dajczer and R. Tojeiro, All superconformal surfaces in $R^4$ in terms of minimal surfaces, Math. Z. 261 (2009), no. 4, 869–890.
• M. Dajczer and R. Tojeiro, Submanifolds of codimension two attaining equality in an extrinsic inequality, Math. Proc. Cambridge Philos. Soc. 146 (2009), no. 2, 461–474.
• P. J. De smet, F. Dillen, L. Verstraelen and L. Vrancken, A pointwise inequality in submanifold theory, Arch. Math. 35 (1999), 115–128.
• F. Dillen, J. Fastenakels and J. Van Der Veken, Remarks on an inequality involving the normal scalar curvature, Pure and applied differential geometry PADGE 2007, 83–92, Ber. Math., Shaker Verlag, Aachen, 2007.
• J. Ge and Z. Tang, A proof of the DDVV conjecture and its equality case, Pacific J. Math. 237 (2008), 87–95.
• I. Guadalupe and L.Rodríguez, Normal curvature of surfaces in space forms, Pacific J. Math. 106 (1983), 95–103.
• T. Li, X. Ma and C. Wang, Wintgen ideal submanifolds with a low-dimensional integrable distribution (I), Front. Math. China 10 (2015), no. 1, 111–136.
• T. Choi and Z. Lu, On the DDVV conjecture and the comass in calibrated geometry (I), Math. Z. 260 (2008), 409–429.
• Z. Lu, Normal Scalar Curvature Conjecture and its applications, J. Funct. Anal. 261 (2011), 1284–1308.
• X. Ma and Z. Xie, The Moebius geometry of Wintgen ideal submanifolds, Proceedings of the ICM 2014 Satellite Conference on Real and Complex Submanifolds (Daejeon, 2014), 411–425, Springer, 2014.
• M. Petrovié-torgašev and L. Verstraelen, On Deszcz symmetries of Wintgen ideal Submanifolds, Arch. Math. 44 (2008), 57–67.
• B. Rouxel, Harmonic spheres of a submanifold in Euclidean space, Proceedings of the 3rd Congress of Geometry (Thessaloniki, 1991), 357–364, Aristotle Univ. Thessaloniki, 1992.
• C. Wang, Möbius geometry of submanifolds in $S^n$, Manuscripta Math. 96 (1998), 517–534.
• P. Wintgen, Sur l'inégalité de Chen-Willmore, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), 993–995.
• Z. Xie, T. Li, X. Ma and C. Wang, Möbius geometry of three-dimensional Wintgen ideal submanifolds in $\mathbb{S}^5$, Sci. China Math. 57 (2014), 1203–1220.