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Graph manifolds as ends of negatively curved Riemannian manifolds

Koji Fujiwara and Takashi Shioya

Geometry & Topology 24 (2020) 2035–2074
Abstract

Let M be a graph manifold such that each piece of its JSJ decomposition has the 2 × geometry. Assume that the pieces are glued by isometries. Then there exists a complete Riemannian metric on × M which is an “eventually warped cusp metric” with the sectional curvature K satisfying 1 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 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