Beijing Journal of Pure and Applied Mathematics

Volume 1 (2024)

Number 2

Spacelike CMC surfaces near null infinity of the Schwarzschild spacetime

Pages: 853 – 884

DOI: https://dx.doi.org/10.4310/BPAM.2024.v1.n2.a9

Author

Luen-Fai Tam (The Institute of Mathematical Sciences and the Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China)

Abstract

Motivated by a result of Treibergs, given a smooth function $f(y)$ on the standard sphere $S2, \mathbf{y} \in S^2$, and any positive constant $H_0$, we construct a spacelike surface with constant mean curvature $H_0$ in the Schwarzschild spacetime, which is the graph of a function $u(\mathbf{y}, r)$ defined on $r \gt r_0$ for some $r_0 \gt 0$ in the standard coordinates exterior to the blackhole. Moreover, $u$ has the following asymptotic behavior:$\def\y{\mathbf{y}}$\[\Biggl|{u (\y,r) - r_\ast - \left( f(\y)+r^{-1} \phi (\y) + \dfrac{1}{2} r^{-2} \psi (y) \right)}\Biggr|\leq C_r^{-3}\]for some $C \gt 0$, where $r_{\ast} = r + 2m \log(\frac{r}{2m} - 1)$. Here $\phi, \psi$ are functions on $\mathbb{S}^2$ given by\[\begin{cases}\phi = \frac{1}{2} (H^{-2}_0 + {\lvert \nabla_{\mathbb{S}^2} f \rvert}^2_{\mathbb{S}^2}) \: , \\\psi = \frac{1}{2} (H^{-2}_0 \Delta_{\mathbb{S}^2} f + {\langle \nabla_{\mathbb{S}^2} \vert \nabla_{\mathbb{S}^2} f {\rvert}^2_{\mathbb{S}^2}, \nabla_{\mathbb{S}^2} f \rangle}_{\mathbb{S}^2}) \;.\end{cases}\]In particular, the surface intersects the future null infinity with the cut given by the function $f$. In addition, we prove that the function $u - r_\ast$ is uniformly Lipschitz near the future null infinity.

Keywords

Schwarzschild spacetime, spacelike constant mean curvature surface, null-infinity

2010 Mathematics Subject Classification

Primary 53C44. Secondary 83C30.

Received 3 July 2023

Accepted 9 January 2024

Published 17 July 2024