Generalizing Kähler Metrics of Poincaré Type

We define and explore various properties of a generalization of Poincaré-type Kähler metrics defined on the complement of a complex hypersurface X embedded in an ambient Kähler manifold N. After motivating interest in a generalization, especially from the viewpoint of extremal Kähler geometry, we construct a distortion potential ψ_{τ V} christened the gnarl associated to the vector field V due to its simulation of flowing along level sets of τ in the direction of V upon approaching X. Key subexponential estimates are derived to relate the gnarled metric to a starting Poincaré-type metric, allowing us to prove statements about the volume and integrals of the curvatures of the gnarled metric.

To relate the gnarling construction to the extremal setting, we prove a local perturbation result showing the existence of cscK gnarled metrics in Kähler classes near to that of a standard product metric on N \ X, providing a significant step towards developing more general openness properties for extremal gnarled metrics. We discuss the challenges of adapting the gnarl to the global situation of embedding X in a compact Kähler manifold M, consider the case that N is the disk bundle of an Hermitian line bundle over X, and lastly proposing some open problems and avenues for further work using gnarls.