The uniqueness of Poincaré type extremal Kähler metric

Let $D$ be a smooth divisor on a closed Kähler manifold $X$. Suppose that $Au{t}_0(D)=\{Id\}$. We prove that the Poincaré type extremal Kähler metric with a cusp singularity at $D$ is unique up to a holomorphic transformation on $X$ that preserves $D$. This generalizes Berman-Berndtson’s work on the uniqueness of extremal Kähler metrics from closed manifolds to some complete and noncompact manifolds.