Generalizing Kähler Metrics of Poincaré Type

<p>We define and explore various properties of a generalization of Poincaré-type Kähler metrics defined on the complement of a complex hypersurface <i>X</i> embedded in an ambient Kähler manifold <i>N</i>. After motivating interest in a generalization, especially from the viewpoint of extremal Kähler geometry, we construct a distortion potential <i>ψ_{τ V}</i> christened the <i>gnarl</i> associated to the vector field <i>V</i> due to its simulation of flowing along level sets of <i>τ</i> in the direction of <i>V</i> upon approaching <i>X</i>. 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.</p><p>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 <i>N </i>\ <i>X</i>, 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 <i>X</i> in a compact Kähler manifold <i>M</i>, consider the case that <i>N</i> is the disk bundle of an Hermitian line bundle over <i>X</i>, and lastly proposing some open problems and avenues for further work using gnarls.</p>