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November 2018 Navigating the space of symmetric CMC surfaces
Lynn Heller, Sebastian Heller, Nicholas Schmitt
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J. Differential Geom. 110(3): 413-455 (November 2018). DOI: 10.4310/jdg/1542423626

Abstract

In this paper we introduce a flow on the spectral data for symmetric CMC surfaces in the $3$-sphere. The flow is designed in such a way that it changes the topology but fixes the intrinsic (metric) and certain extrinsic (periods) closing conditions of the CMC surfaces. By construction the flow yields closed (possibly branched) CMC surfaces at rational times and immersed higher genus CMC surfaces at integer times. We prove the short time existence of this flow near the spectral data of (certain classes of) CMC tori and obtain thereby the existence of new families of closed (possibly branched) connected CMC surfaces of higher genus. Moreover, we prove that flowing the spectral data for the Clifford torus is equivalent to the flow of Plateau solutions by varying the angle of the fundamental piece in Lawson’s construction for the minimal surfaces $\xi_{g,1}$.

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Lynn Heller. Sebastian Heller. Nicholas Schmitt. "Navigating the space of symmetric CMC surfaces." J. Differential Geom. 110 (3) 413 - 455, November 2018. https://doi.org/10.4310/jdg/1542423626

Information

Received: 6 March 2015; Published: November 2018
First available in Project Euclid: 17 November 2018

zbMATH: 06982216
MathSciNet: MR3880230
Digital Object Identifier: 10.4310/jdg/1542423626

Rights: Copyright © 2018 Lehigh University

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Vol.110 • No. 3 • November 2018
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