New examples of Willmore submanifolds in the unit sphere via isoparametric functions,II

TL;DR

This paper proves that focal submanifolds of isoparametric hypersurfaces with four principal curvatures are Willmore, and classifies which are Einstein, advancing understanding of submanifold geometry in spheres.

Contribution

It provides a unified geometric proof for Willmore property of focal submanifolds and classifies Einstein focal submanifolds in spheres with four principal curvatures.

Findings

Focal submanifolds of isoparametric hypersurfaces with four principal curvatures are Willmore.

Most focal submanifolds are Einstein, except for one case.

Unified geometric proof established for Willmore property.

Abstract

This paper is a continuation of a paper with the same title of the last two authors. In the first part of the present paper, we give a unified geometric proof that both focal submanifolds of every isoparametric hypersurface in spheres with four distinct principal curvatures are Willmore. In the second part, we completely determine which focal submanifolds are Einstein except one case.