Four-Manifolds without Einstein Metrics

Abstract

It is shown that there are infinitely many compact orientable smooth 4-manifolds which do not admit Einstein metrics, but nevertheless satisfy the strict Hitchin-Thorpe inequality 2χ > 3|τ |. The examples in question arise as non-minimal complex algebraic surfaces of general type, and the method of proof stems from Seiberg-Witten theory. * Supported in part by NSF grant DMS-9505744.

Ricci curvature, minimal volumes, and Seiberg-Witten theory

“…Details are left to the interested reader. ♦ Remark Corollary 3.4 is the direct descendant of an analogous result in [28], where, using only scalar curvature estimates, a similar conclusion was proved for k ≥ 2 3 c 2 1 (X). It was later pointed out by Kotschick [22] that such a result alone suffices to imply the existence of homeomorphic pairs consisting of an Einstein manifold and a 4-manifold which does not admit Einstein metrics; however, the examples that arise by this method are quite complicated, and have huge c 2 1 .…”

“…It was later pointed out by Kotschick [22] that such a result alone suffices to imply the existence of homeomorphic pairs consisting of an Einstein manifold and a 4-manifold which does not admit Einstein metrics; however, the examples that arise by this method are quite complicated, and have huge c 2 1 . The intermediate step between [28] and Corollary 3.4 may be found in [30], where Seiberg-Witten estimates of Weyl curvature were first introduced. While crude by present standards, the method used there did lead to an obstruction when k ≥ 25 57 c 2 1 (X), or about two-thirds of the way to the present result.…”

“…for large classes of 4-manifolds M; for example, if M is a complex surface of general type, and if X is its minimal model, then [28] …”

“…Examples of 4-manifolds with |τ (M)|/χ(M) < 2/3 which do not admit Einstein metrics were first constructed by Gromov [15], using his simplicial volume invariant; however, this method only works in the presence of an infinite fundamental group. In [28], simply connected examples were first constructed, using Seiberg-Witten estimates for the scalar curvature. Shortly thereafter, Sambusetti [35] showed that the entropy estimates of Besson-Courtois-Gallot [7] allow one to construct examples of arbitrary Euler characteristic and signature, but again with huge fundamental group; see Petean [33] and del Rio [11] for similar results.…”

“…Seiberg-Witten theory naturally leads to non-trivial lower bounds for I s which, amazingly, are often sharp. Using this, the author has elsewhere [28,29] computed I s (M) for all complex surfaces M with even first Betti number; it turns out that I s (M) is positive exactly for the surfaces of general type, and for these it is given by the formula…”