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

The ISGRI CdTe gamma camera In-flight behavior

The INTEGRAL Soft Gamma-Ray Imager (ISGRI) is the first large CdTe gamma camera ever built. It provided faultless operations in space since the launch of INTEGRAL in October 2002. A general presentation of the system is given with particular attention to the noisy pixel handling. The observed effect of charge particles on the detectors and their integrated electronics is presented. The in-flight detector evolution is detailed and the camera performance is reviewed.Comment: 5 pages, 8 figures, IEEE-NSS, RTSD conf. 2004, submitted to Trans. Nucl. S

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

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

Twistors, Holomorphic Disks, and Riemann Surfaces with Boundary

Moduli spaces of holomorphic disks in a complex manifold Z, with boundaries constrained to lie in a maximal totally real submanifold P, have recently been found to underlie a number of geometrically rich twistor correspondences. The purpose of this paper is to develop a general Fredholm regularity criterion for holomorphic curves-with-boundary, and then show how this applies, in particular, to various moduli problems of twistor-theoretic interest.Comment: 19 pages, LaTeX2e. To appear in the Proceedings of the CRM short program in Differential Geometry, Montreal, 200

Curvature Functionals, Optimal Metrics, and the Differential Topology of 4-Manifolds

This paper investigates the question of which smooth compact 4-manifolds admit Riemannian metrics that minimize the L2-norm of the curvature tensor. Metrics with this property are called OPTIMAL; Einstein metrics and scalar-flat anti-self-dual metrics provide us with two interesting classes of examples. Using twistor methods, optimal metrics of the second type are constructed on the connected sums kCP_2 for k > 5. However, related constructions also show that large classes of simply connected 4-manifolds do not admit any optimal metrics at all. Interestingly, the difference between existence and non-existence turns out to delicately depend on one's choice of smooth structure; there are smooth 4-manifolds which carry optimal metrics, but which are homeomorphic to infinitely many distinct smooth 4-manifolds on which no optimal metric exists.Comment: To appear in "Different Faces of Geometry," Donaldson, Eliashberg, and Gromov, editors; Kluwer/Plenum, 200