Geodesics inMinimal Surfaces

In this paper, we consider connected minimal surfaces in $\mathbb{R}^3$ with isothermal coordinates and with a family of geodesic coordinates curves, these surfaces will be called GICM-surfaces. We give a classification of the GICM-surfaces. This class of minimal surfaces includes the catenoid, the helicoid and Enneper's surface. Also, we show that one family of this class of minimal surfaces has at least one closed geodesic and one $1$-periodic family of this class has finite total curvature. As application we show other characterization of catenoid and helicoid. Finally, we show that the class of GICM-surfaces coincides with the class of minimal surfaces whose the geodesic curvature $k_g^1$ and $k_g^2$ of the coordinates curves satisfy $\alpha k_g^1+\beta k_g^2=0$, $\alpha$, $\beta \in \mathbb{R}$.