Cosmic no-hair conjecture and conformal vector fields

Abstract

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