Four-dimensional Einstein manifolds, and beyond (Q2771297)

When looking for the ``best'' metric on a given smooth manifold, Einstein metrics, i.e., metrics for which the Ricci curvature tensor is proportional to the metric by a real constant \(\lambda\), are promising candidates. It is therefore natural to ask when a given smooth manifold \(M\) admits Einstein metrics, and, if so, to what extent these are unique. This leads to the study of the moduli space of Einstein metrics with unit volume on a compact manifold, which is the first theme in this survey paper. A second theme concerns the sign of the possible Einstein constants \(\lambda\). NEWLINENEWLINENEWLINEIn dimensions \(2\) and \(3\), these questions are readily answered, since Einstein spaces then have constant sectional curvature. In particular, the sign of \(\lambda\) is completely dictated by the underlying topology of \(M\). In dimension \(4\), the Einstein condition does not describe the full curvature tensor anymore, but obstructions to the existence of Einstein metrics or uniqueness results can still be derived, thanks mainly to the presence of self-dual and anti-self-dual \(2\)-forms. These results take into consideration both topological and diffeomorphism invariants and, in an essential way, the relationship between topology and curvature. For higher dimensions, no non-existence or uniqueness results are known at all. This paper aims to give a survey of the present state of the research on Einstein metrics on four-dimensional compact spaces, with an occasional digression to higher dimensions. NEWLINENEWLINENEWLINEThe author starts off with the classical Hitchin-Thorpe inequality, which gives a topological obstruction to the existence of Einstein metrics (Section 2). Seiberg-Witten theory allows estimates on the \(L_2\)-norm of the scalar curvature for particular classes of spaces which lead to improvements on the Hitchin-Thorpe inequality (Section 4). Moreover, also estimates on the self-dual Weyl curvature follow from the Seiberg-Witten equations and these give rise to a further strengthening of the results (Section 7). NEWLINENEWLINENEWLINEOther recent techniques which the author highlights are the use of surgery while controlling the behaviour of volume and scalar curvature (Section 5), the theory of minimal volumes for manifolds satisfying certain curvature restrictions (Section 8) and the link between volume entropy and the Ricci curvature (Section 9). Although these techniques are quite independent, they lead to analogous results concerning obstructions to the existence of Einstein metrics and the construction of new examples of closed four-manifolds without any Einstein metric. NEWLINENEWLINENEWLINEConcerning the second theme of the survey, the sign of possible Einstein constants on a given manifold, the author presents the class of Kähler-Einstein manifolds in Section~3 together with necessary and sufficient conditions for a compact complex surface to admit a Kähler-Einstein metric with positive, negative or zero Einstein constant~\(\lambda\). As an application of the Hitchin-Thorpe inequality and Seiberg-Witten theory, the sign of the Einstein constant is completely determined by the smooth structure (though not by the topological structure) of the complex surface. For higher-dimensional Kähler-Einstein spaces, this is no longer true (Section~6). Further, several results concerning a negative Einstein constant, \(\lambda<0\), are sprinkled throughout the text and the positive case, \(\lambda>0\), gets special attention in Section~10.NEWLINENEWLINEFor the entire collection see [Zbl 0961.00021].