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Quasi-Fuchsian manifolds close to the Fuchsian locus are foliated by constant mean curvature surfaces

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Abstract

Even though it is known that there exist quasi-Fuchsian hyperbolic three-manifolds that do not admit any monotone foliation by constant mean curvature (CMC) surfaces, a conjecture due to Thurston asserts the existence of CMC foliations for all almost-Fuchsian manifolds, namely those quasi-Fuchsian manifolds that contain a closed minimal surface with principal curvatures in \((-1,1)\). In this paper we prove that there exists a (unique) monotone CMC foliation for all quasi-Fuchsian manifolds that lie in a sufficiently small neighborhood of the Fuchsian locus.

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Acknowledgements

The authors are indebted to Zeno Huang for many discussions related to CMC surfaces and quasi-Fuchsian manifolds and for useful comments on a previous version of this manuscript. The authors are grateful to Jean-Marc Schlenker for useful discussions and for his encouragement, and to an anonymous referee for several comments that improved the exposition of this paper. Finally, the second author would like to thank Gennady Uraltsev, for helpful conversations on some of analytic aspects of this work.

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Authors and Affiliations

  1. Department of Mathematics, University of Luxembourg, 6 avenue de la Fonte, 4364, Esch-Sur-Alzette, Luxembourg

    Diptaishik Choudhury

  2. Department of Mathematics, University of Virginia, 141 Cabell Drive, Charlottesville, VA, 22904-4137, USA

    Filippo Mazzoli

  3. Univ. Grenoble Alpes, CNRS, IF 38000, Grenoble, France

    Andrea Seppi

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  1. Diptaishik Choudhury
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  2. Filippo Mazzoli
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Choudhury, D., Mazzoli, F. & Seppi, A. Quasi-Fuchsian manifolds close to the Fuchsian locus are foliated by constant mean curvature surfaces. Math. Ann. 388, 3981–4010 (2024). https://doi.org/10.1007/s00208-023-02625-7

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