Spacelike CMC surfaces near null infinity of the Schwarzschild spacetime

2023/06/06 by Luen-Fai Tam, Tam, Luen-Fai
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Primary 53C44 #Secondary 83C30

paper · pdf · doi:10.48550/arxiv.2306.03353

openalex publication_date 2023/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Motivated by a result of Treibergs, given a smooth function f(y) on the standard sphere S², and any positive constant H₀, we construct a spacelike surface with constant mean curvature H₀ in the Schwarzschild spacetime, which is the graph of a function u(y, r) defined on r>r₀ for some r₀>0 in the standard coordinates exterior to the blackhole. Moreover, u has the following asymptotic behavior: |u(y,r)-r_*-(f(y)+r^-1ϕ(y)+1/2 r^-2ψ(y)|≤ Cr^-3 for some C>0, where r_*=r+2mlog(r/(2m)-1). Here ϕ, ψare functions determined by f and H₀. 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_* is uniformly Lipschitz near the future null infinity.

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