Calabi Symmetry and the Continuity Method

TL;DR

This paper investigates the behavior of the continuity method on generalized Hirzebruch surfaces, demonstrating convergence properties, curvature estimates, and conditions for scalar curvature blow-up, paralleling known results for the Kahler-Ricci flow.

Contribution

It provides new insights into the convergence and curvature behavior of the continuity method, extending understanding to generalized Hirzebruch surfaces.

Findings

Gromov-Hausdorff convergence similar to Kahler-Ricci flow

Curvature estimates for the continuity method

Either solution exists for all times or scalar curvature blows up

Abstract

We study the convergence and curvature blow up of La Nave and Tian's continuity method on a generalised Hirzebruch surface. We show that the Gromov-Hausdorff convergence is similar to that of the Kahler-Ricci flow and obtain curvature estimates. We also show that a general solution to the continuity method either exist or all times, or the scalar curvature blows up. This behavior is known to be exhibited by the Kahler-Ricci flow.