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We solve spacelike spherically symmetric constant mean curvature (SS-CMC)hypersurfaces in Schwarzschild spacetimes and analyze their asymptotic behaviornear the coordinate singularity r = 2M. Furthermore, we join SS-CMChypersurfaces in the Kruskal extension to obtain complete ones and discuss thesmooth properties.

Spacelike spherically symmetric CMC hypersurfaces in Schwarzschild spacetimes (I): Construction