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Navigating the space of symmetric CMC surfaces. (English) Zbl 1403.53051
The authors introduce a flow on the spectral data for symmetric constant mean curvature (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. They 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, they 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 $$\xi_{g,1}$$.

##### MSC:
 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
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