Wintgen ideal submanifolds of codimension two, complex curves, and Moebius geometry

TL;DR

This paper characterizes Wintgen ideal submanifolds of codimension two using Moebius geometry, linking their mean curvature spheres to 1-isotropic holomorphic curves in a complex quadric, and explores their relation to minimal surfaces.

Contribution

It establishes a correspondence between Wintgen ideal submanifolds and 1-isotropic holomorphic curves in a complex quadric within Moebius geometry, providing a new geometric characterization.

Findings

Mean curvature sphere corresponds to 1-isotropic holomorphic curve in Q

Any such curve describes a family of spheres forming a Wintgen ideal submanifold

Connections with minimal surfaces and prior work are discussed

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. Using the framework of Moebius geometry, we show that in the codimension two case, the mean curvature sphere of the Wintgen ideal submanifold corresponds to an 1-isotropic holomorphic curve in a complex quadric Q. Conversely, any 1-isotropic complex curve in Q describes a 2-parameter family of m-dimensional spheres whose envelope is always a m-dimensional Wintgen ideal submanifold at the regular points. The relationship with Dajczer and Tojeiro's work on the same topic as well as the description in terms of minimal surfaces in the Euclidean space is also discussed.