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Willmore surfaces of constant Möbius curvature

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Abstract

We study Willmore surfaces of constant Möbius curvature \({\mathcal{K}}\) in \({\mathbb{S}}^4\) . It is proved that such a surface in \({\mathbb{S}}^3\) must be part of a minimal surface in \({\mathbb{R}}^3\) or the Clifford torus. Another result in this paper is that an isotropic surface (hence also Willmore) in \({\mathbb{S}}^4\) of constant \({\mathcal{K}}\) could only be part of a complex curve in \({\mathbb{C}}^2 \cong {\mathbb{R}}^4\) or the Veronese 2-sphere in \({\mathbb{S}}^4\) . It is conjectured that they are the only possible examples. The main ingredients of the proofs are over-determined systems and isoparametric functions.

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Authors and Affiliations

  1. School of Mathematical Sciences, Peking University, Beijing, 100871, P.R. China

    Xiang Ma & Changping Wang

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  1. Xiang Ma
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  2. Changping Wang
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Correspondence to Xiang Ma.

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This work is partially supported by RFDP (No. 20040001034).

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Ma, X., Wang, C. Willmore surfaces of constant Möbius curvature. Ann Glob Anal Geom 32, 297–310 (2007). https://doi.org/10.1007/s10455-007-9065-9

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  • DOI: https://doi.org/10.1007/s10455-007-9065-9

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