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 Σ in the Schwarzschild spacetime exterior to the black hole. We consider two boundaries: the future null infinity I + and the horizon. Suppose near I + , Σ is the graph of a function - P (y , s) in the form v ¯ = - P , where v ¯ is the retarded null coordinate with s = r - 1 and y ∈ S 2 . Suppose the boundary value of P (y , s) at s = 0 is a smooth function f on the unit sphere S 2 . If P is C 4 at I + , then f must satisfy a fourth order PDE on 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 S 2 . For the extrinsic geometry of Σ , under certain conditions we obtain decay rate of the trace-free part of the second fundamental forms A ˚ . In case A ˚ decays fast enough, some further restrictions on f are given. For the intrinsic geometry, we show that under certain conditions, Σ 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, Σ can be expressed as the graph of a function u which is smooth in η = 1 - 2 m r 1 2 and y ∈ S 2 , and all its derivatives are determined by the boundary value u at η = 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 Σ given by η = 0 is totally geodesic. [ABSTRACT FROM AUTHOR]