2017/12/07 by Chen, Xuezhang, Lai, Mijia, Wang, Fang · 2 citations
#53C25 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1712.02540
Let (Xn,g+) (n≥ 3) be a Poincaré-Einstein manifold which is C3,α conformally compact with conformal infinity (∂ X, [g]). On the conformal compactification (X, g=ρ2g+) via some boundary defining function ρ, there are two types of Yamabe constants: Y(X,∂ X,[ g]) and Q(X,∂ X,[ g]). (See definitions (\refdef.type1) and (\refdef.type2)). In \citeGH, Gursky and Han gave an inequality between Y(X,∂ X,[ g]) and Y(∂ X,[g]). In this paper, we first show that the equality holds in Gursky-Han's theorem if and only if (Xn,g+) is isometric to the standard hyperbolic space (ℍn, gℍ). Secondly, we derive an inequality between Q(X,∂ X,[ g]) and Y(∂ X, [ g]), and show that the equality holds if and only if (Xn,g+) is isometric to (ℍn, gℍ). Based on this, we give a simple proof of the rigidity theorem for Poincaré-Einstein manifolds with conformal infinity being conformally equivalent to the standard sphere.