Abstract
We give new examples of closed smooth 4-manifolds which support singular metrics of nonpositive curvature, but no smooth ones, thereby answering affirmatively a question of Gromov. The obstruction comes from patterns of incompressible 2-tori sufficiently complicated to force branching of geodesics for nonpositively curved metrics.
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Stadler, S. An obstruction to the smoothability of singular nonpositively curved metrics on 4-manifolds by patterns of incompressible tori. Geom. Funct. Anal. 25, 1575–1587 (2015). https://doi.org/10.1007/s00039-015-0341-8
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DOI: https://doi.org/10.1007/s00039-015-0341-8