Mabuchi K\"ahler solitons versus extremal K\"ahler metrics and beyond

TL;DR

This paper establishes a correspondence between Mabuchi K"ahler solitons and extremal K"ahler metrics on Fano varieties, providing a new criterion for their existence based on scalar curvature bounds.

Contribution

It proves a new correspondence linking Mabuchi solitons and extremal K"ahler metrics, extending the understanding of their existence criteria on Fano manifolds.

Findings

Characterization of Mabuchi solitons via extremal K"ahler metrics with scalar curvature less than 2(n+1)

Extension of the correspondence to v-solitons

New insights into the existence conditions for K"ahler solitons

Abstract

Using the Yau-Tian-Donaldson type correspondence for $v$-solitons established by Han-Li, we show that a smooth complex $n$-dimensional Fano variety admits a Mabuchi soliton provided it admits an extremal K\"ahler metric whose scalar curvature is strictly less than $2(n+1)$. Combined with previous observations by Mabuchi and Nakamura in the other direction, this gives a characterization of the existence of Mabuchi solitons in terms of the existence of extremal K\"ahler metrics on Fano manifolds. An extension of this correspondence to $v$-solitons is also obtained.