Poincar\'e-Einstein 4-manifolds with cusps

Abstract

In this paper, we construct Poincar\'e-Einstein 4-manifolds with various kinds of cusps. In particular, we construct: (1) Infinite families of Einstein metrics on $(0,\infty)\times \mathscr{N}$, where $\mathscr{N}\to T^2$ is a principal $\mathbb{S}^1$-bundle over $T^2$, with one Poincar\'e-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 $P\to \Sigma_{\mathtt{g}}$ is a principal $\mathbb{S}^1$-bundle over a closed Riemann surface $\Sigma_{\mathtt{g}}$ of genus $\mathtt{g}\geq 2$, with one Poincar\'e-Einstein end and one end asymptotic to a bundle of two-dimensional hyperbolic cusps over hyperbolic $\Sigma_{\mathtt{g}}$. 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 Poincar\'e-Einstein 4-manifolds.