The Moebius geometry of Wintgen ideal submanifolds

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 Moebius invariant objects. The mean curvature sphere defines a conformal Gauss map into a Grassmann manifold. We show that any Wintgen ideal submanifold 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.Comment: This is a survey of our recent work on the Moebius geometry of Wintgen ideal submanifolds, which also include two new important results. Submitted to the the conference "ICM 2014 Satellite Conference on Real and Complex Submanifolds