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We prove that a spacelike spherical symmetric constant mean curvature (SSCMC)surface and a general spacelike constant mean curvature (CMC) surface withcertain boundary condition at the future null-infinity in Schwarzschildspacetime are asymptotically hyperbolic in the sense of Wang \cite{Wang2001}and Chru\'sciel-Herzlich \cite{ChruscielHerzlich} respectively. Near the futurenull-infinity ($s=0$), we derive that the boundary data of spacelike CMCsurfaces can be expressed as those on $\mathbb{S}^{2}$ up to three order andobtain a compatibility condition for fourth order derivatives near $s=0$. Wealso show that if the trace free part of the second fundamental forms$\mathring A$ of this spacelike CMC surface decay fast enough then therestriction of its associate function $P$ (for definition, see \eqref{defofp} )on the null-infinity must be a first eigenfunction of the Laplace on$\mathbb{S}^2$ or constant. In particular in Minkowski spacetime, a uniquenessresult and constructions of spacelike CMC surfaces near $s=0$ are proved. Also,we show that the inner boundary of certain spacelike CMC surfaces are totallygeodesic.

Boundary behaviors of spacelike constant mean curvature surfaces in Schwarzschild spacetime