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.