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

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.