论文标题
Schwarzschild时空中空间状常数平均曲率表面的边界行为
Boundary behaviors of spacelike constant mean curvature surfaces in Schwarzschild spacetime
论文作者
论文摘要
我们证明，空中式球形对称常数平均曲率（SSCMC）表面和一般的间距类似常数均值曲率（CMC）表面具有某些边界状态，在未来的schwarzschild spacetime中未来的无界限是wang \ cite {wang2001} and chrich的意义上是渐进的副周期。 \ cite {chruscielherzlich}。接近未来的零污染度（$ s = 0 $），我们得出的是，可以将空格型CMC表面的边界数据表示为$ \ m athbb {s}^{2} $上的三个订单，并获得第四阶衍生物接近$ s = 0 $的兼容性条件。我们还表明，如果第二个基本形式的无跟踪部分$ \ MATHING足够快地将此空间型CMC表面衰减快速降低，则限制其副函数$ p $（有关定义，请参见Null-Infinity上的\ eqref {defofp}），必须是$ \ null-Infinction in Null-Infinity of null-Infinity of null-Infinity的eigenfunction null-Infinction null-infinction null-infinction null-infinction null-infinction null-Infinction n laplace on laplace。尤其是在Minkowski时空中，证明了$ s = 0 $的独特结果和空间般的CMC表面的构造。另外，我们表明某些空间类CMC表面的内部边界完全是大地测量的。
We prove that a spacelike spherical symmetric constant mean curvature (SSCMC) surface and a general spacelike constant mean curvature (CMC) surface with certain boundary condition at the future null-infinity in Schwarzschild spacetime are asymptotically hyperbolic in the sense of Wang \cite{Wang2001} and Chruściel-Herzlich \cite{ChruscielHerzlich} respectively. Near the future null-infinity ($s=0$), we derive that the boundary data of spacelike CMC surfaces can be expressed as those on $\mathbb{S}^{2}$ 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 \eqref{defofp} ) on the null-infinity must be a first eigenfunction of the Laplace on $\mathbb{S}^2$ 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.
