Rigidity in dimension four of area-minimising Einstein manifolds

Source abstract

Abstract The aim of this paper is to prove a sharp inequality for the area of a four dimensional compact Einstein manifold (Σ, g Σ ) embedded into a complete five dimensional manifold ( M 5 , g ) with positive scalar curvature R and nonnegative Ricci curvature. Under a suitable choice, we have $area(\Sigma)^{\frac{1}{2}}\inf_{M}R \leq 8\sqrt{6}\pi$ . Moreover, occurring equality we deduce that (Σ, g Σ ) is isometric to a standard sphere ( $\mathbb{S}$ 4 , g can ) and in a neighbourhood of Σ, ( M 5 , g ) splits as ((-ϵ, ϵ) × $\mathbb{S}$ 4 , dt 2 + g can ) and the Riemannian covering of ( M 5 , g ) is isometric to $\Bbb{R}$ × $\mathbb{S}$ 4 .