# Graph manifolds as ends of negatively curved Riemannian manifolds

## TL;DR

This paper constructs negatively curved Riemannian manifolds with specific geometric ends modeled on graph manifolds with hyperbolic pieces, extending the understanding of possible ends of such manifolds.

## Contribution

It demonstrates that certain graph manifolds can serve as ends of 4-dimensional negatively curved Riemannian manifolds with finite volume, using warped cusp metrics.

## Key findings

- Existence of complete negatively curved metrics on R x M
- Graph manifolds with hyperbolic pieces can be realized as ends of finite volume manifolds
- Construction of warped cusp metrics with sectional curvature in [-1,0)

## Abstract

Let $M$ be a graph manifold such that each piece of its JSJ decomposition has the $\Bbb H^2 \times \Bbb R$ geometry. Assume that the pieces are glued by isometries. Then, there exists a complete Riemannian metric on $\Bbb R \times M$ which is an "eventually warped cusp metric" with the sectional curvature $K$ satisfying $-1 \le K <0$. A theorem by Ontaneda then implies that $M$ appears as an end of a 4-dimensional, complete, non-compact Riemannian manifold of finite volume with sectional curvature $K$ satisfying $-1 \le K <0$.