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The Chern–Ricci flow on primary Hopf surfaces

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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.

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Notes

  1. The primary Hopf surfaces consist of both those of class 1, and those of class 0 which are defined as quotients of \(\mathbb C^2 {\setminus } \{0\}\) of the form \((z_1,z_2) \mapsto (\beta ^m z_1 + \lambda z_2^m, \beta z_2)\) for some positive integer m and \(\beta ,\lambda \in \mathbb C\) with \(1 < |\beta |\) and \(\lambda \ne 0\).

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Acknowledgements

Research for this paper began at the American Institute of Mathematics workshop: Nonlinear PDEs in real and complex geometry in San Jose, CA August 2018. The author thanks AIM for their hospitality. The author also extends their thanks to Casey Kelleher, Valentino Tosatti, Yury Ustinovskiy, and Ben Weinkove for helpful discussions at the AIM workshop. The author was supported by the NSF grant RTG: Geometry and Topology at the University of Notre Dame.

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  1. University of Notre Dame, Notre Dame, IN, USA

    Gregory Edwards

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Edwards, G. The Chern–Ricci flow on primary Hopf surfaces. Math. Z. 299, 1689–1702 (2021). https://doi.org/10.1007/s00209-021-02735-5

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