Journal of the
Korean Mathematical Society JKMS

We show that an umbilic-free minimal surface in $\mathbb{R}^3$ belongs to the associate family of the catenoid if and only if the geodesic curvatures of its lines of curvature have a constant ratio. As a corollary, the helicoid is shown to be the unique umbilic-free minimal surface whose lines of curvature have the same geodesic curvature. A similar characterization of the deformation family of minimal surfaces with planar lines of curvature is also given.

Keywords: Liouville's equation, geodesic curvature, associate minimal surfaces, helicoid, catenoid, Enneper surface

MSC numbers: 53A10, 49Q05, 53C42