Rigidity in dimension four of area-minimising Einstein manifolds

Abstract

AbstractThe aim of this paper is to prove a sharp inequality for the area of a four dimensional compact Einstein manifold (Σ,gΣ) embedded into a complete five dimensional manifold (M5,g) with positive scalar curvatureRand 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,gΣ) is isometric to a standard sphere ($\mathbb{S}$4,gcan) and in a neighbourhood of Σ, (M5,g) splits as ((-ϵ, ϵ) ×$\mathbb{S}$4,dt2+gcan) and the Riemannian covering of (M5,g) is isometric to$\Bbb{R}$×$\mathbb{S}$4.