Willmore orbits for isometric Lie actions

TL;DR

This paper introduces criteria for identifying Willmore orbits in homogeneous spaces under Lie group actions, providing new examples and estimates for Willmore submanifolds, simplifying the complex Euler-Lagrange approach.

Contribution

It offers a simplified criterion for finding Willmore orbits in Lie group actions, expanding the catalog of known Willmore submanifolds and providing sharp estimates for their quantities.

Findings

Existence of Willmore orbits in each stratified subset under certain conditions

Classical examples like Willmore torus and Veronese surface are special cases of Willmore orbits

Provides sharp estimates for the number of Willmore submanifolds in classical cases

Abstract

In this work, we study the Willmore submanifolds in a closed connected Riemannian manifold which are orbits for the isometric action of a compact connected Lie group. We call them homogeneous Willmore submanifolds or Willmore orbits. The criteria for these special Willmore submanifolds is much easier than the general theory which requires a complicated Euler-Lagrange equation. Our main theorem claims, when the orbit type stratification for the group action satisfies certain conditions, then we can find a Willmore orbit in each stratified subset. Some classical examples of special importance, like Willmore torus, Veronese surface, etc., can be interpreted as Willmore orbits and easily verified with our method. Our theorems provide a large number of new examples for Willmore submanifolds, as well as estimates for their numbers which are sharp in some classical cases.