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QGM seminar by Gregory Edwards (University of Notre Dame)

Title: The Chern-Ricci flow on primary Hopf surfaces

2019.06.14 | Jane Jamshidi

Date Thu 27 Jun
Time 09:15 10:15
Location 1531-215 (Aud-D3)


The Chern-Ricci flow is a flow of Hermitian metrics which generalizes the Kähler-Ricci flow to non-Kähler metrics. While solutions of the flow have been classified on many families of complex non-Kähler surfaces, the Hopf surfaces provide a family of non-Kähler surfaces on which little is known about the Chern-Ricci flow. We use a construction of locally conformally Kähler metrics of Gauduchon-Ornea to study solutions of the Chern-Ricci flow on primary Hopf surfaces of class 1. 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 established for the potential. Previous results had only been known for the simplest examples of Hopf surfaces.