#### Vol. 304, No. 2, 2020

 Recent Issues Vol. 307: 1  2 Vol. 306: 1  2 Vol. 305: 1  2 Vol. 304: 1  2 Vol. 303: 1  2 Vol. 302: 1  2 Vol. 301: 1  2 Vol. 300: 1  2 Online Archive Volume: Issue:
 The Journal Subscriptions Editorial Board Officers Contacts Submission Guidelines Submission Form Policies for Authors ISSN: 1945-5844 (e-only) ISSN: 0030-8730 (print) Special Issues Author Index To Appear Other MSP Journals
Addendum to the article Contact stationary Legendrian surfaces in $\mathbb{S}^5$

### Yong Luo

Vol. 304 (2020), No. 2, 607–611
##### 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
Primary: 53B25
Secondary: 53C42