The Chern–Ricci flow on primary Hopf surfaces

Abstract

The Hopf surfaces provide a family of minimal non-Kähler surfaces of class VII on which little is known about the Chern–Ricci flow. We use a construction of Gauduchon–Ornea for locally conformally Kähler metrics on primary Hopf surfaces of class 1 to study solutions of the Chern–Ricci flow. These solutions reach a volume collapsing singularity in finite time, and we show that the metric tensor satisfies a uniform upper bound, supporting the conjecture that the Gromov-Hausdorff limit is isometric to a round\(S^1\). Uniform \(C^{1+\beta }\) estimates are also established for the potential. Previous results had only been known for the simplest examples of Hopf surfaces.