Addendum to the article Contact stationary Legendrian surfaces in $\mathbb{S}^5$

Yong Luo

Vol. 304 (2020), No. 2, 607–611

DOI: 10.2140/pjm.2020.304.607

Abstract

We proved (Pacific J. Math. 293:1 (2018), 101–120) that if $L$ is a contact stationary Legendrian surface in ${\mathbb{𝕊}}^5$ with the canonical Sasakian structure and the square length of its second fundamental form belongs to $\left[0,2\right]$, then $L$ is either totally umbilical or is a flat minimal Legendrian torus. In this addendum we further prove that if $L$ is a totally umbilical contact stationary Legendrian surface in ${\mathbb{𝕊}}^5$, then $L$ is totally geodesic.

Keywords

contact stationary surfaces, rigidity

Mathematical Subject Classification 2010

Primary: 53B25

Secondary: 53C42

Milestones

Received: 12 October 2018

Accepted: 4 September 2019

Published: 12 February 2020