The continuity method on minimal elliptic K\"ahler surfaces

We prove that, on a minimal elliptic K\"ahler surface of Kodaira dimension one, the continuity method introduced by La Nave and Tian in \cite{LT} starting from any initial K\"ahler metric converges in Gromov-Hausdorff topology to the metric completion of the generalized K\"ahler-Einstein metric on its canonical model constructed by Song and Tian in \cite{ST06}.