We proved (Pacific J. Math.293:1 (2018), 101–120) that if
is a contact stationary
Legendrian surface in
with the canonical Sasakian structure and the square length of its second fundamental form
belongs to
,
then
is
either totally umbilical or is a flat minimal Legendrian torus. In this addendum we further
prove that if
is a totally umbilical contact stationary Legendrian surface in
, then
is
totally geodesic.