Abstract
Wintgen ideal submanifolds in space forms are those ones attaining equality pointwise in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the scalar normal curvature. They are Möbius invariant objects. The mean curvature sphere defines a conformal Gauss map into a Grassmann manifold. We show that any Wintgen ideal submanifold of dimension greater than or equal to 3 has a Riemannian submersion structure over a Riemann surface with the fibers being round spheres. Then the conformal Gauss map is shown to be a super-conformal and harmonic map from the underlying Riemann surface. Some of our previous results are surveyed in the final part.
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Notes
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The notion of the mean curvature sphere can be traced back to Blaschke [1] in 1920s.
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References
Blaschke, W.: Vorlesungen über Differentialgeometrie III: Differentialgeometrie der Kreise und Kugeln. Springer Grundlehren XXIX, Berlin (1929)
Bryant, R.: A duality theorem for Willmore surfaces. J. Differ. Geom. 20, 20–53(1984)
Bryant, R.: Some remarks on the geometry of austere manifolds. Bol. Soc. Bras. Mat. 21, 122–157 (1991)
Chen, B. Y.: Classification of Wintgen ideal surfaces in Euclidean 4-space with equal Gauss and normal curvatures. Ann. Glob. Anal. Geom. 38, 145–160 (2010)
Choi, T., Lu, Z.: On the DDVV conjecture and the comass in calibrated geometry (I). Math. Z. 260, 409–429 (2008)
Dajczer, M., Tojeiro, R.: A class of austere submanifolds. Illinois J. Math. 45(3), 735–755 (2001)
Dajczer, M., Tojeiro, R.: All superconformal surfaces in R 4 in terms of minimal surfaces. Math. Z. 261(4), 869–890 (2009)
Dajczer, M., Tojeiro, R.: Submanifolds of codimension two attaining equality in an extrinsic inequality. Math. Proc. Cambridge Philos. Soc. 146(2), 461–474 (2009)
De Smet, P. J., Dillen, F., Verstraelen, L., Vrancken, L.: A pointwise inequality in submanifold theory. Arch. Math. 35, 115–128 (1999)
Dillen, F., Fastenakels, J., Van Der Veken, J.: Remarks on an inequality involving the normal scalar curvature. In: Proceedings of the International Congress on Pure and Applied Differential Geometry-PADGE Brussels, pp. 83–92. Shaker, Aachen (2007)
Ejiri, N.: Willmore surfaces with a duality in S N(1). Proc. Lond. Math. Soc. (3) 57, 383–416 (1988)
Ge, J., Tang, Z.: A proof of the DDVV conjecture and its equality case. Pac. J. Math., 237, 87–95 (2008)
Guadalupe, I., Rodríguez, L.: Normal curvature of surfaces in space forms. Pac. J. Math. 106, 95–103 (1983)
Li, T., Ma, X., Wang, C.: Wintgen ideal submanifolds with a low-dimensional integrable distribution (I). http://arxiv.org/abs/1301.4742
Li, T., Ma, X., Wang, C., Xie, Z.: Wintgen ideal submanifolds of codimension two, complex curves, and Moebius geometry. http://arxiv.org/abs/1402.3400
Li, T., Ma, X., Wang, C., Xie, Z.: Classification of Moebius homogeneous Wintgen ideal submanifolds. http://arxiv.org/abs/1402.3430
Lu, Z.: Normal Scalar Curvature Conjecture and its applications. J. Funct. Anal. 261, 1284–1308 (2011)
Petrovié-torgas̆ev, M., Verstraelen, L.: On Deszcz symmetries of Wintgen ideal Submanifolds. Arch. Math. 44, 57–67 (2008)
Wang, C. P.: Möbius geometry of submanifolds in S n. Manuscr. Math. 96, 517–534 (1998)
Wintgen, P.: Sur l’inégalité de Chen-Willmore. C. R. Acad. Sci. Paris 288, 993–995 (1979)
Xie, Z., Li, T., Ma, X., Wang, C.: Möbius geometry of three dimensional Wintgen ideal submanifolds in \(\mathbb{S}^{5}\). Sci. China Math. doi:10.1007/s11425-013-4664-3. See also http://arxiv.org/abs/1402.3440
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Ma, X., Xie, Z. (2014). The Möbius Geometry of Wintgen Ideal Submanifolds. In: Suh, Y.J., Berndt, J., Ohnita, Y., Kim, B.H., Lee, H. (eds) Real and Complex Submanifolds. Springer Proceedings in Mathematics & Statistics, vol 106. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55215-4_37
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DOI: https://doi.org/10.1007/978-4-431-55215-4_37
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