On the $C^\alpha$-convergence of the Solution of the Chern-Ricci Flow on Elliptic Surfaces

Abstract

We will study the Chern-Ricci flow on non-Kähler properly elliptic surfaces. These surfaces are compact complex surfaces whose first Betti 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.