Continuity method for the Mabuchi soliton on the extremal Fano manifolds

TL;DR

This paper provides an analytic proof for the existence of Mabuchi solitons on extremal Fano manifolds using the continuity method, emphasizing the boundedness of energy functionals without relying on the minimal model program.

Contribution

It introduces a new analytic approach to prove the existence of Mabuchi solitons, extending the method to general g-solitons and g-extremal metrics, avoiding the minimal model program.

Findings

Boundedness of energy functionals along the continuity method

Analytic proof of Mabuchi soliton existence on extremal Fano manifolds

Extension of the method to g-solitons and g-extremal metrics

Abstract

We run the continuity method for Mabuchi's generalization of K\"{a}hler-Einstein metrics, assuming the existence of an extremal K\"{a}hler metric. It gives an analytic proof (without minimal model program) of the recent existence result obtained by Apostolov, Lahdili and Nitta. Our key observation is the boundedness of the energy functionals along the continuity method. The same argument can be applied to general $g$-solitons and $g$-extremal metrics.