The Chern-Ricci flow on primary Hopf surfaces

2019/05/30 by Edwards, Gregory · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1905.13114

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 S¹. Uniform C^1+β estimates are also established for the potential. Previous results had only been known for the simplest examples of Hopf surfaces.

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• Kählerness of compact Hermitian surfaces under semi-definite Strominger-Bismut-Ricci curvatures

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