  In this note, we complete the classification of the geometry of non-compact
two-dimensional gradient Ricci solitons. As a consequence, we obtain two
corollaries: First, a complete two-dimensional gradient Ricci soliton has
bounded curvature. Second, we give examples of complete two-dimensional
expanding Ricci solitons with negative curvature that are topologically disks
and are not hyperbolic space.
