On Einstein, Hermitian 4-Manifolds

Let (M,h) be a compact 4-dimensional Einstein manifold, and suppose that h is Hermitian with respect to some complex structure J on M. Then either (M,J,h) is Kaehler-Einstein, or else, up to rescaling and isometry, it is one of the following two exceptions: the Page metric on CP2 # (-CP2), or the Einstein metric on CP2 # 2 (-CP2) constructed in Chen-LeBrun-Weber.Comment: 33 pages, 3 figure

Einstein Metrics, Harmonic Forms, and Conformally Kaehler Geometry

The author has elsewhere given a complete classification of those compact oriented Einstein 4-manifolds on which the self-dual Weyl curvature is everywhere positive in the direction of some self-dual harmonic 2-form. In this article, similar results are obtained when the self-dual Weyl curvature is everywhere non-negative in the direction of a self-dual harmonic 2-form that is transverse to the zero section of the bundle of self-dual 2-forms. However, this transversality condition plays an essential role in the story; dropping it leads one into wildly different territory where entirely different phenomena predominate.Comment: 26 pages, LaTeX2e. This version strengthens several technical results, and modifies some key terminology in order to agree with standard convention

Hyperbolic Manifolds, Harmonic Forms, and Seiberg-Witten Invariants

New estimates are derived concerning the behavior of self-dual hamonic 2-forms on a compact Riemannian 4-manifold with non-trivial Seiberg-Witten invariants. Applications include a vanishing theorem for certain Seiberg-Witten invariants on compact 4-manifolds of constant negative sectional curvature.Comment: 22 pages, LaTeX2

Einstein Metrics, Harmonic Forms, and Symplectic Four-Manifolds

If $M$ is the underlying smooth oriented $4$-manifold of a Del Pezzo surface, we consider the set of Riemannian metrics $h$ on $M$ such that $W^+(\omega , \omega )> 0$, where $W^+$ is the self-dual Weyl curvature of $h$, and $\omega$ is a non-trivial self-dual harmonic $2$-form on $(M,h)$. While this open region in the space of Riemannian metrics contains all the known Einstein metrics on $M$, we show that it contains no others. Consequently, it contributes exactly one connected component to the moduli space of Einstein metrics on $M$.Comment: in Annals of Global Analysis and Geometry (2015

Curvature, Covering Spaces, and Seiberg-Witten Theory

The Yamabe invariant Y(M) of a smooth compact manifold is roughly the supremum of the scalar curvatures of unit-volume constant-scalar-curvature Riemannian metrics g on M. (To be precise, one only considers those constant-scalar-curvature metrics which are Yamabe minimizers, but this technicality does not, e.g. affect the sign of the answer.) In this article, it is shown that many 4-manifolds M with Y(M) < 0 have have finite covering spaces \tilde{M} with Y(\tilde{M}) > 0.Comment: Source file for published version. Discussion expanded, minor errors corrected. 8 pages, LaTeX2

The Einstein-Maxwell Equations, Kaehler Metrics, and Hermitian Geometry

Any constant-scalar-curvature Kaehler (cscK) metric on a complex surface may be viewed as a solution of the Einstein-Maxwell equations, and this allows one to produce solutions of these equations on any 4-manifold that arises as a compact complex surface with b_1 even. It is shown, however, that not all solutions of the Einstein-Maxwell equations on such manifolds arise in this way; new examples can be constructed by means of conformally Kaehler geometry.Comment: 19 pages; 1 figure. With added references, improved notation, and many minor corrections. To appear in special issue of the Journal of Geometry and Physic

Calabi Energies of Extremal Toric Surfaces

We derive a formula for the L^2 norm of the scalar curvature of any extremal Kaehler metric on a compact toric manifold, stated purely in terms of the geometry of the corresponding moment polytope. The main interest of this formula pertains to the case of complex dimension 2, where it plays a key role in construction of Bach-flat metrics on appropriate 4-manifolds.Comment: 28 pages. Published version. Added section on Abreu formalism generalizes main result to higher dimension

Einstein Metrics, Four-Manifolds, and Differential Topology

This article presents a new and more elementary proof of the main Seiberg-Witten-based obstruction to the existence of Einstein metrics on smooth compact 4-manifolds. It also introduces a new smooth manifold invariant which conveniently encapsulates those aspects of Seiberg-Witten theory most relevant to the study of Riemannian variational problems on 4-manifolds.Comment: 21 pages, LaTeX2

Yamabe Constants and the Perturbed Seiberg-Witten Equations

Among all conformal classes of Riemannian metrics on ${\Bbb CP}_2$, that of the Fubini-Study metric is shown to have the largest Yamabe constant. The proof, which involves perturbations of the Seiberg-Witten equations, also yields new results on the total scalar curvature of almost-K\"ahler 4-manifolds.Comment: LaTeX file, 17 page

Weyl Curvature, Del Pezzo Surfaces, and Almost-Kaehler Geometry

If a smooth compact 4-manifold M admits a Kaehler-Einstein metric g of positive scalar curvature, Gursky showed that its conformal class [g] is an absolute minimizer of the Weyl functional among all conformal classes with positive Yamabe constant. Here we prove that, with the same hypotheses, [g] also minimizes of the Weyl functional on a different open set of conformal classes, most of which have negative Yamabe constant. An analogous minimization result is then proved for Einstein metrics g which are Hermitian, but not Kaehler.Comment: 34 pages, 2 figure