Continuity method for the Mabuchi soliton on the extremal Fano manifolds

TL;DR

The article develops a direct analytic proof for the existence of Mabuchi solitons on extremal Fano manifolds by employing a continuity path for the $g$-twisted Monge–Ampère equation and establishing crucial a priori bounds. Central to the approach is the boundedness of the $g$-twisted Mabuchi energy $M_g$ along the path, together with a compactness framework in the finite-energy class to pass to the limit and recover a solution at $t=1$. The authors also connect the coercivity of $M_g$ to the ability to deduce closedness of the path and prove the equivalence between the existence of a $g$-soliton, $g$-extremal metric, and the Mabuchi invariant condition $m_X<1$, thereby providing a variant of the Yau–Tian–Donaldson correspondence in this weighted setting. The method extends to general $g$-solitons and $g$-extremal metrics, offering an analytic alternative to approaches relying on the minimal model program. Overall, the work advances the understanding of canonical metrics on Fano manifolds through energy functionals and continuity methods without invoking MMP techniques.

Abstract

We run the continuity method for Mabuchi's generalization of Kähler-Einstein metrics, assuming the existence of an extremal Kähler 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.

Theorems & Definitions (29)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Definition 2.4
• Theorem 2.5