Working on differential geometry, partly with some flavor of function theory of several complex variables. I’ve been mainly studying geometry of “asymptotically complex hyperbolic spaces,” with emphasis on a partial differential equation called Einstein’s equation on them. While being a generalization of geometry of bounded strictly pseudoconvex domains in function theory of several complex variables, it’s beyond the scope of the field of complex geometry. Based on this experience, I’m now aiming toward some more general theory that applies to other “spaces that converge to ones with much symmetry,” which are called “asymptotically symmetric spaces.”
Asymptotically hyperbolic spaces, which are the most basic examples of asymptotically symmetric spaces, are lacking symmetries such as the homogeneity or the isotropicity of the genuine hyperbolic spaces in the strict sense. However, as a point in the space moves more away from a fixed one, its neighborhood looks more like an open set of the hyperbolic space. Recall the “natural” conformal structure on the sphere at infinity of the hyperbolic space—the boundary at infinity of an asymptotically hyperbolic space is equipped with a conformal structure in the same way. In the case of asymptotically complex hyperbolic spaces, whose model has little less isotropicity than that of the hyperbolic space, the associated geometric structure on the boundary at infinity is the CR (Cauchy–Riemann) structure.
The fundamental question of geometry of asymptotically symmetric spaces is the following: what property of the space reflects the geometric structure at infinity and the topology of the space, and in what way? This includes the question whether or not there is a space with some certain property under a given condition on the structure at infinity and the topology. The existence problem of Einstein metrics is a typical example.
Although asymptotically hyperbolic spaces have been studied for several decades, many fundamental questions remain unsolved. And, if we think of understandings from the general viewpoint of geometry of asymptotically symmetric spaces, our field still seems to be kind of a wilderness. I’d cultivate it by going back and forth between analyses on special cases and abstract considerations.