On Calabi's extremal metric and properness

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In this paper we extend recent breakthrough of Chen-Cheng CC1, CC2, CC3 on existence of constant scalar Kähler metric on a compact Kähler manifold to Calabi's extremal metric. Our argument follows CC3 and there are no new a prior estimates needed, but rather there are necessary modifications adapted to the extremal case. We prove that there exists an extremal metric with extremal vector V if and only if the modified Mabuchi energy is proper, modulo the action the subgroup in the identity component of automorphism group which commutes with the flow of V. We introduce two essentially equivalent notions, called reductive properness and reduced properness. We observe that one can test reductive properness/reduced properness only for invariant metrics. We prove that existence of an extremal metric is equivalent to reductive properness/reduced properness of the modified Mabuchi energy.