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February 2022 Constant mean curvature spheres in homogeneous three-spheres
William H. Meeks III, Pablo Mira, Joaquín Pérez, Antonio Ros
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J. Differential Geom. 120(2): 307-343 (February 2022). DOI: 10.4310/jdg/1645207520

Abstract

We give a complete classification of the immersed constant mean curvature spheres in a three-sphere with an arbitrary homogeneous metric, by proving that for each $H \in \mathbb{R}$, there exists a constant mean curvature $H$ sphere in the space that is unique up to an ambient isometry.

Funding Statement

The first author was supported in part by NSF Grant DMS-1004003. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the NSF.
The second author was supported by Project PID2020-118137GB-100 funded by MCIN/AEI/10.13039/501100011033.
The third and fourth authors were supported in part by MINECO-FEDERMICINN Grants, No. MTM2014-52368-P, MTM2017-89677-P and PID2020-117868GB-I00, and by the Maria de Maeztu Excellence Unit IMAG, reference CEX2020-001105-M, funded by MCIN/AEI/10.13039/501100011033/.

Citation

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William H. Meeks III. Pablo Mira. Joaquín Pérez. Antonio Ros. "Constant mean curvature spheres in homogeneous three-spheres." J. Differential Geom. 120 (2) 307 - 343, February 2022. https://doi.org/10.4310/jdg/1645207520

Information

Received: 2 October 2018; Accepted: 28 January 2020; Published: February 2022
First available in Project Euclid: 23 February 2022

Digital Object Identifier: 10.4310/jdg/1645207520

Subjects:
Primary: 53A10
Secondary: 49Q05 , 53C42

Keywords: $H$-potential , constant mean curvature , curvature estimates , homogeneous three-manifold , Hopf uniqueness , index of stability , left invariant Gauss map , left invariant metric , metric Lie group , minimal surface , nullity of stability , stability

Rights: Copyright © 2022 Lehigh University

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Vol.120 • No. 2 • February 2022
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