Graph manifolds as ends of negatively curved Riemannian manifolds

Fujiwara, Koji; Shioya, Takashi

doi:10.2140/gt.2020.24.2035

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Graph manifolds as ends of negatively curved Riemannian manifolds

Koji Fujiwara and Takashi Shioya

Geometry & Topology 24 (2020) 2035–2074

DOI: 10.2140/gt.2020.24.2035

Abstract

Let $M$ be a graph manifold such that each piece of its JSJ decomposition has the $ℍ^{2} \times ℝ$ geometry. Assume that the pieces are glued by isometries. Then there exists a complete Riemannian metric on $ℝ \times M$ which is an “eventually warped cusp metric” with the sectional curvature $K$ satisfying $- 1 \leq K < 0$ .

A theorem by Ontaneda then implies that $M$ appears as an end of a $4$ –dimensional, complete, noncompact Riemannian manifold of finite volume with sectional curvature $K$ satisfying $- 1 \leq K < 0$ .

Dedicated to Professor Kenji Fukaya on his 60th birthday

Keywords

ends of manifolds, negative curvature, graph manifold, cusp

Mathematical Subject Classification 2010

Primary: 53C20

Secondary: 57M50, 57N10

References

Publication

Received: 3 April 2019

Revised: 7 November 2019

Accepted: 7 December 2019

Published: 10 November 2020

Proposed: Tobias H Colding

Seconded: Bruce Kleiner, Dmitri Burago

Authors

Koji Fujiwara
Department of Mathematics Kyoto University Kyoto Japan

Takashi Shioya
Mathematics Institute Tohoku University Sendai Japan

Volume 24, issue 4 (2020)