Degenerations of Negative K\"ahler-Einstein Surfaces

TL;DR

This paper investigates the degeneration behavior of negative Kähler-Einstein surfaces, constructing a model for their limiting geometry by analyzing special degenerating families and interpolating metrics.

Contribution

It introduces a new construction of a Kähler-Einstein neck region for degenerating families of negative Kähler-Einstein surfaces, extending previous work in the Calabi-Yau case.

Findings

Constructed a Kähler-Einstein neck region for degenerating surfaces

Provided a model for the limiting geometry of degenerations

Extended techniques from Calabi-Yau degenerations to negative Kähler-Einstein surfaces

Abstract

Every compact K\"ahler manifold with negative first Chern class admits a unique metric $g$ such that $\text{Ric}(g) = -g$. Understanding how families of these metrics degenerate gives insight into their geometry and is important for understanding the compactification of the moduli space of negative K\"ahler-Einstein metrics. I study a special class of such families in complex dimension two. Following the work of Sun and Zhang (2019) in the Calabi-Yau case, I construct a K\"ahler-Einstein neck region interpolating between canonical metrics on components of the central fiber. This provides a model for the limiting geometry of metrics in the family.