  In this paper, we study Mabuchi metrics on Fano manifolds. We prove that
Mabuchi metrics exist if the modified Ding functional is proper modulo a
reductive subgroup of its automorphism group. On the other hand, the inverse
that Mabuchi metrics implies the properness is obtained by using
Darvas-Rubinstein's properness principle. As an application, we establish a
criterion for the existence of Mabuchi metrics on Fano group compactifications.
