Two-Dimensional Gradient Ricci Solitons Revisited

In this note, we complete the classification of the geometry of noncompact 2D gradient Ricci solitons. As a consequence, we obtain two corollaries: First, a complete 2D gradient Ricci soliton has bounded curvature. Secondly, we give examples of complete 2D expanding Ricci solitons with negative curvature that are topologically disks and are not isometric to hyperbolic space.