An Aubin continuity path for shrinking gradient Kähler-Ricci solitons

Let be a toric Kähler-Einstein Fano manifold. We show that any toric shrinking gradient Kähler-Ricci soliton on certain toric blowups of C × D \mathbb{C}\times D satisfies a complex Monge-Ampère equation. We then set up an Aubin continuity path to solve this equation and show that it has a solution at the initial value of the path parameter. This we do by implementing another continuity method.