Poincaré-Einstein 4-manifolds with cusps

Abstract

In this paper, we construct Poincaré-Einstein 4-manifolds with various kinds of cusps. In particular, we construct: (1) Infinite families of Einstein metrics on $(0,\infty )\times \mathcal{N}$, where $\mathcal{N}\to T^2$ is a principal ${\mathbb{S}}^1$-bundle over $T^2$, with one Poincaré-Einstein end and one end asymptotic to a real or complex hyperbolic cusp. (2) Infinite families of Einstein metrics on $(0,\infty )\times P$, where $P\to \hat{I}{\pounds }_{\mathtt{g}}$ is a principal ${\mathbb{S}}^1$-bundle over a closed Riemann surface $\hat{I}{\pounds }_{\mathtt{g}}$ of genus $\mathtt{g}\ge 2$, with one Poincaré-Einstein end and one end asymptotic to a bundle of two-dimensional hyperbolic cusps over hyperbolic $\hat{I}{\pounds }_{\mathtt{g}}$. Universal covers of (1) and (2) provide new complete negative Einstein metrics on ${\mathbb{R}}^4$. These Einstein metrics also exhibit interesting degeneration phenomena. With this construction, we give a negative answer to a question of Anderson concerning cusp formation for Poincaré-Einstein 4-manifolds.

35 pages. Comments are welcome