Boundary behaviors of spacelike constant mean curvature surfaces in Schwarzschild spacetime

Abstract

In this work, we will study the boundary behaviors of a spacelike positive constant mean curvature surface $\Sigma$ in the Schwarzschild spacetime exterior to the black hole. We consider two boundaries: the future null infinity $\mathcal {I}^+$ and the horizon. Suppose near $\mathcal {I}^+$, $\Sigma$ is the graph of a function $-P(\textbf{y},s)$ in the form $\overline{v}=-P$, where $\overline{v}$ is the retarded null coordinate with $s=r^{-1}$ and $\textbf{y}\in \mathbb {S}^2$. Suppose the boundary value of $P(\textbf{y},s)$ at $s=0$ is a smooth function f on the unit sphere $\mathbb {S}^2$. If P is $C^4$ at $\mathcal {I}^+$, then f must satisfy a fourth order PDE on $\mathbb {S}^2$. If P is $C^3$, then all the derivatives of P up to order three can be expressed in terms of f and its derivatives on $\mathbb {S}^2$. For the extrinsic geometry of $\Sigma$, under certain conditions we obtain decay rate of the trace-free part of the second fundamental forms $\mathring{A}$. In case $\mathring{A}$ decays fast enough, some further restrictions on f are given. For the intrinsic geometry, we show that under certain conditions, $\Sigma$ 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, $\Sigma$ can be expressed as the graph of a function u which is smooth in $\eta =\left( 1-\frac{2m}{r}\right) ^{\frac{1}{2}}$ and $\textbf{y}\in \mathbb {S}^2$, and all its derivatives are determined by the boundary value u at $\eta =0$. 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 $\Sigma$ given by $\eta =0$ is totally geodesic.