Calabi symmetry and the continuity method

Abstract

We study the convergence and curvature blow up of the continuity method on a generalized Hirzebruch surface. We show that the Gromov–Hausdorff convergence is similar to that of the Kähler–Ricci flow and obtain curvature estimates. We also show that a general solution to the continuity method either exists at all times, or the scalar curvature blows up. This behavior is known to be exhibited by the Kähler–Ricci flow.