Zoll Metrics, Branched Covers, and Holomorphic Disks

We strengthen our previous results regarding the moduli spaces of Zoll metrics and Zoll projective structures on S^2. In particular, we describe a concrete, open condition which suffices to guarantee that a totally real embedding of RP^2 in CP_2 arises from a unique Zoll projective structure on the 2-sphere. Our methods ultimately reflect the special role such structures play in the initial value problem for the 3-dimensional Lorentzian Einstein-Weyl equations.Comment: 21 pages, LaTeX2

Zoll Manifolds and Complex Surfaces

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 concerning the Riemannian case. In contrast to previous work, our approach is twistor-theoretic, and depends fundamentally on the fact that, up to biholomorphism, there is only one complex structure on CP2

Nonlinear Gravitons, Null Geodesics, and Holomorphic Disks

We develop a global twistor correspondence for pseudo-Riemannian conformal structures of signature (++--) with self-dual Weyl curvature. Near the conformal class of the standard indefinite product metric on S^2 x S^2, there is an infinite-dimensional moduli space of such conformal structures, and each of these has the surprising global property that its null geodesics are all periodic. Each such conformal structure arises from a family of holomorphic disks in CP_3 with boundary on some totally real embedding of RP^3 into CP_3. An interesting sub-class of these conformal structures are represented by scalar-flat indefinite K\"ahler metrics, and our methods give particularly sharp results in this more restrictive setting.Comment: 56 pages, LaTeX2

The Einstein-Weyl Equations, Scattering Maps, and Holomorphic Disks

We show that conformally compact, globally hyperbolic, Lorentzian Einstein-Weyl 3-manifolds are in natural one-to-one correspondence with orientation-reversing diffeomorphisms of the 2-sphere. The proof hinges on a holomorphic-disk analog of Hitchin's mini-twistor correspondence.Comment: 11 pages, LaTeX2e. Revised version strengthens result and completes proo

Geometric realization and K-theoretic decomposition of C*-algebras

Suppose that A is a separable C*-algebra and that G_* is a (graded) subgroup of K_*(A). Then there is a natural short exact sequence 0 \to G_* \to K_*(A) \to K_*(A)/G_* \to 0. In this note we demonstrate how to geometrically realize this sequence at the level of C*-algebras. As a result, we KK-theoretically decompose A as 0 \to A\otimes \Cal K \to A_f \to SA_t \to 0 where K_*(A_t) is the torsion subgroup of K_*(A) and K_*(A_f) is its torsionfree quotient. Then we further decompose A_t : it is KK-equivalent to \oplus_p A_p where K_*(A_p) is the p-primary subgroup of the torsion subgroup of K_*(A). We then apply this realization to study the Kasparov group K^*(A) and related objects.Comment: 9 pages.To appear in International J. Mat