The Moebius geometry of Wintgen ideal submanifolds

TL;DR

This paper studies Wintgen ideal submanifolds, which are Moebius invariant and attain equality in the DDVV inequality, revealing their geometric structure and properties of their conformal Gauss maps.

Contribution

It establishes that Wintgen ideal submanifolds have a Riemannian submersion over a Riemann surface with spherical fibers and characterizes their conformal Gauss maps as super-conformal harmonic maps.

Findings

Wintgen ideal submanifolds admit a Riemannian submersion over a Riemann surface.

The conformal Gauss map is super-conformal and harmonic.

The paper surveys previous related results.

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.