Complete Willmore Legendrian surfaces in $\mathbb{S}^5$ are minimal Legendrian surfaces

TL;DR

This paper proves that all complete Willmore Legendrian surfaces in the 5-sphere are minimal and provides new examples of csL Willmore surfaces, advancing understanding of their geometric properties.

Contribution

It establishes the minimality of complete Willmore Legendrian surfaces in $ ext{S}^5$ and constructs novel nontrivial csL Willmore surface examples.

Findings

Complete Willmore Legendrian surfaces are minimal.

Constructed nontrivial csL Willmore surfaces.

Enhanced understanding of Legendrian surface geometry.

Abstract

In this paper we continue to consider Willmore Legendrian surfaces and csL Willmroe surfaces in $\mathbb{S}^5$, notions introduced by Luo in \cite{Luo}. We will prove that every complete Willmore Legendrian surface in $\mathbb{S}^5$ is minimal and construct nontrivial examples of csL Willmore surfaces in $\mathbb{S}^5$.