Graph manifolds as ends of negatively curved Riemannian manifolds

Abstract

Let $M$ be a graph manifold such that each piece of its JSJ decomposition has the ${\mathbb{H}}^2\times \mathbb{R}$ geometry. Assume that the pieces are glued by isometries. Then, there exists a complete Riemannian metric on $\mathbb{R}\times M$ 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 $-1\le K<0$.

arXiv commentAdded Lemma 2.4 and Lemma 3.8 as a piece of the argument for the curvature estimate that was missing from the previous versions

References in corpus (2)

• 3-manifolds with(out) metrics of nonpositive curvature
• Initialization of lattice Boltzmann models with the help of the numerical Chapman-Enskog expansion