On Einstein four-manifolds
Topological obstructions to the existence of Einstein metrics on a given compact and oriented 4-manifold were found by \textit{N. Hitchin} [J. Differ. Geom. 9, 435--441 (1974; Zbl 0281.53039)] and by \textit{M. J. Gursky} and \textit{C. LeBrun} [Ann. Global Anal. Geom. 17, No. 4, 315--328 (1999; Zbl 0967.53029)]. In the paper under review the author gives several improvements of these results. One of the results states that under the assumption \(\text{inf\,}K\geq 2\rho/3\) and \(\rho< 0\) the inequality \(\chi\geq 15|\tau|/8\) holds where \(K\) denotes the sectional curvature, \(\rho\) is the Ricci curvature and \(\tau\) the signature. In a related result, a characterization of the standard metrics on the 4-sphere, \(\mathbb{R} \mathbb{P}^4\) and \(\mathbb{C} \mathbb{P}^2\) is given. Here the assumption is \(\sup K\leq 2\rho/3\) and \(\rho > 0\). Further results are obtained concerning rigidity of Einstein metrics under similar assumptions on the sectional curvature.
scientific article; zbMATH DE number 2182242
scientific article