Rigidity theorems for nonpositive Einstein metrics

Abstract

Abstract.In this paper we study the following problem: When must a complete Einstein metric g on an «-manifold with Ric = (n -\)Xg , X < 0 , be a metric of constant curvature XI

Some old and new results about rigidity of critical metric

“…There are many other ✏-rigidity results that rely on a priori functional inequalities (such as a Sobolev inequality or as the above Hardy inequality) and integral bounds on the curvature (cf. for instance [5,13,20,21,25,27,29,30,32,33], [35, Theorem 7.1], [38]). Such results have been shown recently for critical metrics by G. Tian and J. Viaclovsky in dimension 4 and by X-X.…”

Some old and new results about rigidity of critical metric

“…There are many other ǫ-rigidity results that relies on a priori functional inequality (such as a Sobolev inequality or as the above Hardy inequality) and a integral bounds on the curvature (cf. for instance [5], [30], [27], [28], [24], [19], [26], [32,Theorem 7.1], [35], [20] ). Such a result has been shown recently for critical metric by G.Tian and J.Viaclovsky in dimension 4 and by X-X.…”