On the Cα-convergence of the Solution of the Chern-Ricci Flow on Elliptic Surfaces

We will study the Chem-Ricci flow on non-Kahler properly elliptic surfaces. These surfaces are compact complex surfaces whose first Beth number is odd, Kodaira dimension is equal to 1 and admit an elliptic fibration to a smooth compact curve. We will show that a solution of the Chern-Ricci flow converges in C-alpha-topology on these elliptic surfaces by choosing a special initial metric.