On surfaces that are intrinsically surfaces of revolution

Minimal Lagrangian surfaces in $\mathbb{C}P^2$ via the loop group method Part I: the contractible case

“…In [22] the authors have considered surfaces in R 3 which are intrinsically surfaces of revolution. By definition this means that the induced metric is conformal with metric factor only depending on r. Under the additional assumption that a) the principal curvatures only depend on the radius, b) the principal curvature directions only depend on the angle of rotation (but the radius), they proved that these surfaces have constant mean curvature and are Smyth surfaces (for a precise formulation see loc.cit Theorem 1.3).…”

“…Smyth had assumed radially symmetric and constant mean curvature. In this regard the paper [22] makes an additional statement about the minimal case.…”