Boundary behaviors of spacelike constant mean curvature surfaces in Schwarzschild spacetime

In this work, we will study the boundary behaviors of a spacelike positive constant mean curvature surface \(\Sigma \) $\mathrm{\Sigma }$ in the Schwarzschild spacetime exterior to the black hole. We consider two boundaries: the future null infinity \(\mathcal {I}^+\) ${\mathcal{I}}^{+}$ and the horizon. Suppose near \(\mathcal {I}^+\) ${\mathcal{I}}^{+}$ , \(\Sigma \) $\mathrm{\Sigma }$ is the graph of a function \(-P(\textbf{y},s)\) $-P(\mathbf{y},s)$ in the form \(\overline{v}=-P\) $\overline{v}=-P$ , where \(\overline{v}\) $\overline{v}$ is the retarded null coordinate with \(s=r^{-1}\) $s=r^{-1}$ and \(\textbf{y}\in \mathbb {S}^2\) $\mathbf{y}\in {\mathbb{S}}^2$ . Suppose the boundary value of \(P(\textbf{y},s)\) $P(\mathbf{y},s)$ at \(s=0\) $s=0$ is a smooth function f on the unit sphere \(\mathbb {S}^2\) ${\mathbb{S}}^2$ . If P is \(C^4\) $C^4$ at \(\mathcal {I}^+\) ${\mathcal{I}}^{+}$ , then f must satisfy a fourth order PDE on \(\mathbb {S}^2\) ${\mathbb{S}}^2$ . If P is \(C^3\) $C^3$ , then all the derivatives of P up to order three can be expressed in terms of f and its derivatives on \(\mathbb {S}^2\) ${\mathbb{S}}^2$ . For the extrinsic geometry of \(\Sigma \) $\mathrm{\Sigma }$ , under certain conditions we obtain decay rate of the trace-free part of the second fundamental forms \(\mathring{A}\) $\mathring{A}$ . In case \(\mathring{A}\) $\mathring{A}$ decays fast enough, some further restrictions on f are given. For the intrinsic geometry, we show that under certain conditions, \(\Sigma \) $\mathrm{\Sigma }$ is asymptotically hyperbolic in the sense of Chruściel–Herzlich (Pac J Math 212(2):231–264, 2003). Near the horizon, we prove that under certain conditions, \(\Sigma \) $\mathrm{\Sigma }$ can be expressed as the graph of a function u which is smooth in \(\eta =\left( 1-\frac{2m}{r}\right) ^{\frac{1}{2}}\) $\eta ={\left(1,-,\frac{2m}{r}\right)}^{\frac{1}{2}}$ and \(\textbf{y}\in \mathbb {S}^2\) $\mathbf{y}\in {\mathbb{S}}^2$ , and all its derivatives are determined by the boundary value u at \(\eta =0\) $\eta =0$ . In particular, a Neumann-type condition is obtained. This may be related to a remark of Bartnik (in: Proc Centre Math Anal Austral Nat Univ, 1987). As for intrinsic geometry, we show that under certain conditions the inner boundary of \(\Sigma \) $\mathrm{\Sigma }$ given by \(\eta =0\) $\eta =0$ is totally geodesic.