Some Rigidity Phenomena for Einstein Metrics

Abstract

Abstract.In this note we study the following problem: When must a complete Einstein metric g on an «-manifold with Ric = (n -X)Xg be a constant curvature metric of sectional curvature I ?

Rigidity theorems for nonpositive Einstein metrics

“…[A2,Ba,S], etc.). Roughly speaking, if a complete Ricci-flat metric g has sufficiently small total curvature, i.e., there is a small e = £(aa) > 0 such that if Theorem 3 ( [Y]).…”

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.…”