Wave equations and the LeBrun-Mason correspondence

A construction of Einstein-Weyl spaces via LeBrun-Mason type twistor correspondence

We construct infinitely many Einstein-Weyl structures on $S^2 \times R$ of signature (-++) which is sufficiently close to the model case of constant curvature, and whose space-like geodesics are all closed. Such structures are obtained from small perturbations of the diagonal of $CP^1 \times CP^1$ using the method of LeBrun-Mason type twistor theory. The geometry of constructed Einstein-Weyl space is well understood from the configuration of holomorphic disks. We also review Einstein-Weyl structures and their properties in the former half of this article.

Null-geodesics in complex conformal manifolds and the LeBrun correspondence

In the complex-Riemannian framework we show that a conformal manifold containing a compact, simply-connected, null-geodesic is conformally flat. In dimension 3 we use the LeBrun correspondence, that views a conformal 3-manifold as the conformal infinity of a selfdual four-manifolds. We also find a relation between the conformal invariants of the conformal infinity and its ambient.

On the uniqueness of certain families of holomorphic disks

A Zoll metric is a Riemannian metric whose geodesics are all circles of equal length. Via the twistor correspondence of LeBrun and Mason, a Zoll metric on the 2 dimensional sphere corresponds to a family of holomorphic disks in CP_2 with boundary in a totally real submanifold P. In this paper, we show that for a fixed totally real submanifold P, such a family is unique if it exists, implying that the twistor correspondence of LeBrun and Mason is injective. One of the key ingredients in the proof is the blow-up and blow-down constructions in the sense of Melrose.

论几乎Grassmann流形的扭子理论

On the Twistor Theory of Almost-Grassmannian Manifolds

本文研究实和复流形上半平坦几乎Grassmann结构的扭子对应，建立复情形局部理论及实2-平面Grassmann流形上的整体对应，并推广LeBrun-Mason的带边界曲线模空间方法。