2022/02/02 by Li, Caiyan, Shi, Yuguang, Tam, Luen-Fai
#Differential Geometry (math.DG) #FOS: Mathematics #Primary 53C44 #Secondary 83C30
paper · doi:10.48550/arxiv.2202.00926
We prove that a spacelike spherical symmetric constant mean curvature (SSCMC) surface and a general spacelike constant mean curvature (CMC) surface with certain boundary condition at the future null-infinity in Schwarzschild spacetime are asymptotically hyperbolic in the sense of Wang \citeWang2001 and Chruściel-Herzlich \citeChruscielHerzlich respectively. Near the future null-infinity (s=0), we derive that the boundary data of spacelike CMC surfaces can be expressed as those on \mathbbS2 up to three order and obtain a compatibility condition for fourth order derivatives near s=0. We also show that if the trace free part of the second fundamental forms \mathring A of this spacelike CMC surface decay fast enough then the restriction of its associate function P (for definition, see \eqrefdefofp ) on the null-infinity must be a first eigenfunction of the Laplace on \mathbbS2 or constant. In particular in Minkowski spacetime, a uniqueness result and constructions of spacelike CMC surfaces near s=0 are proved. Also, we show that the inner boundary of certain spacelike CMC surfaces are totally geodesic.