Classification of Moebius homogeneous Wintgen ideal submanifolds

TL;DR

This paper classifies Moebius homogeneous Wintgen ideal submanifolds in real space forms, revealing three main classes related to well-known minimal surfaces and geometric structures.

Contribution

It provides a complete classification of higher-dimensional Moebius homogeneous Wintgen ideal submanifolds, connecting them to classical minimal surfaces and geometric constructions.

Findings

Three classes of non-trivial examples identified

Connections established with minimal surfaces in spheres and projective spaces

Classification extends understanding of Moebius invariant submanifolds

Abstract

A submanifold in a real space form attaining equality in the DDVV inequality at every point is called a Wintgen ideal submanifold. They are invariant objects under the Moebius transformations. In this paper, we classify those Wintgen ideal submanifolds of dimension m>3 which are Moebius homogeneous. There are three classes of non-trivial examples, each related with a famous class of homogeneous minimal surfaces in $S^n$ or $CP^n$: the cones over the Veronese surfaces $S^2$ in $S^n$, the cones over homogeneous flat minimal surfaces in $S^n$, and the Hopf bundle over the Veronese embeddings of $CP^1$ in $CP^n$.