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A Hopf theorem for non-constant mean curvature and a conjecture of A. D. Alexandrov

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Abstract

We prove a uniqueness theorem for immersed spheres of prescribed (non-constant) mean curvature in homogeneous three-manifolds. In particular, this uniqueness theorem proves a conjecture by A. D. Alexandrov about immersed spheres of prescribed Weingarten curvature in \(\mathbb {R}^3\) for the special but important case of prescribed mean curvature. As a consequence, we extend the classical Hopf uniqueness theorem for constant mean curvature spheres to the case of immersed spheres of prescribed antipodally symmetric mean curvature in \(\mathbb {R}^3\).

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Notes

  1. If \(g(p)=\infty \), the condition \(g_z(p)\ne 0\) should be interpreted as \(\lim _{z\rightarrow p} g_z(p)/g(p)^2 \ne 0\).

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Acknowledgments

The authors were partially supported by MICINN-FEDER, Grant No. MTM2013-43970-P, Junta de Andalucía Grant No. FQM325, Junta de Andalucía, reference P06-FQM-01642 and Programa de Apoyo a la Investigación, Fundación Séneca-Agencia de Ciencia y Tecnología Región de Murcia, reference 19461/PI/14.

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Authors and Affiliations

  1. Departamento de Geometría y Topología, Universidad de Granada, 18071, Granada, Spain

    José A. Gálvez

  2. Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, 30203, Cartagena, Murcia, Spain

    Pablo Mira

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  1. José A. Gálvez
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  2. Pablo Mira
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Correspondence to José A. Gálvez.

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Gálvez, J.A., Mira, P. A Hopf theorem for non-constant mean curvature and a conjecture of A. D. Alexandrov. Math. Ann. 366, 909–928 (2016). https://doi.org/10.1007/s00208-015-1351-4

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