On Page 62 of "canonical metrics in Kahler geometry" written by Gang Tian, the author pointed out that properness of Ding-functional $F_\omega$ is independent of $\omega$, without proof. What I want to know is how to solve this problem or some reference of this problem.
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The properness of the Ding functional is defined as \begin{equation} F_{\alpha}(\omega) > \varepsilon i_{\alpha}(\omega) - C ~;~ \forall \omega \in c^{+}_1(M) \end{equation} where $i_{\alpha}(\omega) = I_{\alpha}(\omega) - J_{\alpha}{\omega}$ is the Aubin-Yau functional. Now, the fact that this is independent of $\alpha$ follows from the following two facts. If $\alpha'$ is any other metric in $c_1(M)$
1) cocycle relation : $F_{\alpha'}(\omega) = F_{\alpha}(\omega) + F_{\alpha'}(\alpha)$
2) $\exists C>0$ such that $|i_{\alpha'}(\omega) - i_{\alpha}(\omega)| < C $ for all K\"ahler metrics $\omega \in c_1(M)$
The first is standard. The second one is also easy. For a proof refer to Lemma 2 in http://arxiv.org/pdf/0903.5504v1.pdf
