An Aubin continuity path for asymptotically conical toric shrinking gradient K\"ahler-Ricci solitons: openness and a solution for $t=0$

TL;DR

This paper develops a continuity method to solve complex Monge-Ampère equations for toric asymptotically conical shrinking gradient Kähler-Ricci solitons, establishing existence and openness results in the toric setting.

Contribution

It introduces an Aubin continuity path for these solitons and proves the openness of the initial parameter, extending the solution existence beyond the toric case.

Findings

Established a solution at the initial path parameter value in the toric case.

Proved openness of the initial parameter independent of toricity.

Set up a continuity framework for solving complex Monge-Ampère equations in this geometric context.

Abstract

We show that any toric asymptotically conical shrinking gradient K\"ahler-Ricci soliton on an anti-canonically polarised resolution of a K\"ahler cone satisfies a complex Monge-Amp\`ere equation. We then set up an Aubin continuity path to solve the resulting equation and show that it has a solution at the initial value of the path parameter in the toric case. This we do by implementing another continuity method. Finally, we prove openness of the initial value of the path parameter independent of the toricity.