Boundary behaviors of spacelike constant mean curvature surfaces in Schwarzschild spacetime

TL;DR

This paper investigates the boundary behaviors of spacelike constant mean curvature surfaces in Schwarzschild spacetime, establishing asymptotic properties, boundary data expressions, and conditions for uniqueness and geometric features.

Contribution

It provides new insights into the asymptotic behavior, boundary conditions, and uniqueness of spacelike CMC surfaces in Schwarzschild spacetime, including Minkowski space as a special case.

Findings

Spacelike CMC surfaces are asymptotically hyperbolic near null-infinity.

Boundary data can be expressed up to third order on the sphere.

Under certain decay conditions, associated functions are eigenfunctions of the Laplacian.

Abstract

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 Chru\'sciel-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.