An Aubin continuity path for shrinking gradient K\"ahler-Ricci solitons

TL;DR

This paper develops a continuity method approach to solve complex Monge-Ampère equations for toric shrinking gradient K"ahler-Ricci solitons on specific blowups, advancing understanding of their existence.

Contribution

It introduces an Aubin continuity path combined with a second continuity method to establish solutions for these solitons on toric blowups.

Findings

Existence of solutions at initial path parameter.

Implementation of a new continuity method.

Application to toric K"ahler-Ricci solitons.

Abstract

Let $D$ be a toric K\"ahler-Einstein Fano manifold. We show that any toric shrinking gradient K\"ahler-Ricci soliton on certain toric blowups of $\mathbb{C}\times D$ satisfies a complex Monge-Amp\`ere 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.