Cosmic no‐hair conjecture and conformal vector fields

In this paper, we investigate cosmic no‐hair properties mathematically when a given Riemannian manifold admits a nontrivial closed conformal vector field. Let (Mn,g)$(M^n, g)$ be a compact Riemannian n$n$‐manifold with connected non‐empty boundary ∂M$\partial M$. Assume that there exists a smooth function f$f$ on M$M$ with f>0$f>0$ in M∖∂M$M \setminus \partial M$ and ∂M=f−1(0)$\partial M = f^{-1}(0)$ satisfying the static vacuum equation. We prove that if (Mn,g)$(M^n, g)$ admits a nontrivial closed conformal vector field, then M$M$ must be isometric to a hemisphere S+n${\mathbb {S}}_+^n$. We also discuss a static triple (Mn,g,f)$(M^n, g, f)$ admitting a nontrivial conformal vector field which is not necessarily closed.