Properly embedded minimal annuli in $\mathbb{S}^2 \times \mathbb{R}

Abstract We prove that every properly embedded minimal annulus in $\mathbb{S}^2\times\mathbb{R}$ is foliated by circles. We show that such minimal annuli are given by periodic harmonic maps $\mathbb{C} \to \mathbb{S}^2$ of finite type. Such harmonic maps are parameterized by spectral data, and we sh...

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Bibliographic Details
Published in:Journal of Integrable Systems 2020-01, Vol.5 (1)
Main Authors: Hauswirth, L, Kilian, M, Schmidt, M U
Format: Article
Language:English
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Summary:Abstract We prove that every properly embedded minimal annulus in $\mathbb{S}^2\times\mathbb{R}$ is foliated by circles. We show that such minimal annuli are given by periodic harmonic maps $\mathbb{C} \to \mathbb{S}^2$ of finite type. Such harmonic maps are parameterized by spectral data, and we show that continuous deformations of the spectral data preserve the embeddedness of the corresponding annuli. A curvature estimate of Meeks and Rosenberg is used to show that each connected component of spectral data of embedded minimal annuli contains a maximum of the flux of the third coordinate. A classification of these maxima allows us to identify the spectral data of properly embedded minimal annuli with the spectral data of minimal annuli foliated by circles.
ISSN:2058-5985
2058-5985
DOI:10.1093/integr/xyaa005