Some remarks on conic degeneration and bending of Poincar\'e-Einstein metrics

TL;DR

This paper explores a family of Poincaré-Einstein metrics on the total space of the canonical bundle over a compact Kähler-Einstein manifold, revealing complex degeneration and singularity behaviors as parameters vary.

Contribution

It introduces a multi-parameter family of Poincaré-Einstein metrics exhibiting novel degeneration patterns and singularities, enriching understanding of the moduli space of such metrics.

Findings

Convergence to PE metrics with conic singularities as a parameter tends to zero.

Existence of complete Ricci-flat Kähler metrics as limits of scaled metrics.

Presence of edge singularities with variable cone angles along the zero section.

Abstract

Let $(M,g)$ be a compact K\"ahler-Einstein manifold with $c_1 > 0$. Denote by $K\to M$ the canonical line-bundle, with total space $X$, and $X_0$ the singular space obtained by blowing down $X$ along its zero section. We employ a construction by Page and Pope and discuss an interesting multi-parameter family of Poincar\'e--Einstein metrics on $X$. One 1-parameter subfamily $\{g_t\}_{t>0}$ has the property that as $t\searrow 0$, $g_t$ converges to a PE metric $g_0$ on $X_0$ with conic singularity, while $t^{-1}g_t$ converges to a complete Ricci-flat K\"ahler metric $\hat{g}_0$ on $X$. Another 1-parameters subfamily has an edge singularity along the zero section of $X$, with cone angle depending on the parameter, but has constant conformal infinity. These illustrate some unexpected features of the Poincar\'e-Einstein moduli space.