We display explicit half-conformally-flat metrics on the connected sum of any number of copies of the complex projective plane. These metrics are obtained from magnetic monopoles in hyperbolicâ€¦ (More)

It is shown that there are infinitely many compact simply connected smooth 4-manifolds which do not admit Einstein metrics, but nevertheless satisfy the strict Hitchin-Thorpe inequality 2Ï‡ > 3|Ï„ |.â€¦ (More)

Using Seiberg-Witten theory, it is shown that any KÃ¤hler metric of constant negative scalar curvature on a compact 4-manifold M minimizes the L-norm of scalar curvature among Riemannian metricsâ€¦ (More)

The classical uniformization theorem provides a complete translation dictionary for the etymologically unrelated languages of complex 1-manifolds and constant curvature Riemannian 2-manifolds. Inâ€¦ (More)

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 absolutelyâ€¦ (More)

We derive new, sharp lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact 4manifold with a non-trivial Seiberg-Witten invariant. These allow one, forâ€¦ (More)

We classify compact surfaces with torsion-free affine connections for which every geodesic is a simple closed curve. In the process, we obtain completely new proofs of all the major results [4]â€¦ (More)

The Yamabe invariant of a smooth compact manifold is by definition the supremum of the scalar curvatures of unit-volume Yamabe metrics on the manifold. For an explicit infinite class of 4-manifolds,â€¦ (More)

Let Z be a compact complex (2n+1)-manifold which carries a complex contact structure, meaning a codimension-1 holomorphic sub-bundle D âŠ‚ TZ which is maximally non-integrable. If Z admits aâ€¦ (More)

Let (M,J) be a compact complex manifold, and let E âŠ‚ H1,1(M,R) be the set of all cohomology classes which can be represented by KÃ¤hler forms of extremal KÃ¤hler metrics, in the sense of Calabi [3].â€¦ (More)