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Compact embedded surfaces with constant mean curvature H>0 in
S^2\times R are symmetric bigraphs over domains of
the 2-sphere S^2 as an application of Alexandrov reflection
principle. Therefore, such surfaces coincide with solutions to the
overdetermined elliptic problem associated with the constant mean
curvature equation, with a capillarity condition and zero values along
the boundary. In this talk, we will give a sharp bound for the
curvature of the boundaries of the aforesaid domains (as spherical
curves), and then apply it to obtain examples of compact embedded
surfaces with arbitrary genus and constant mean curvature
0
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