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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
A knot characterization and $1$–connected nonnegatively curved $4$–manifolds with circle symmetry

Karsten Grove and Burkhard Wilking

Geometry & Topology 18 (2014) 3091–3110
Abstract

We classify nonnegatively curved simply connected 4–manifolds with circle symmetry up to equivariant diffeomorphisms. The main problem is to rule out knotted curves in the singular set of the orbit space. As an extension of this work we classify all knots in S3 that can be realized as an extremal set with respect to an inner metric on S3 that has nonnegative curvature in the Alexandrov sense.

Keywords
nonnegative curvature, circle actions, knots, Alexandrov geometry
Mathematical Subject Classification 2010
Primary: 53C23
Secondary: 57M25, 57M60
References
Publication
Received: 10 December 2013
Revised: 13 June 2014
Accepted: 12 July 2014
Published: 1 December 2014
Proposed: Tobias H Colding
Seconded: John Lott, Peter Teichner
Authors
Karsten Grove
Department of Mathematics
University of Notre Dame
Notre Dame, IN 46556-4618
USA
Burkhard Wilking
Mathematisches Institut
Universität Münster
Einsteinstrasse 62
48149 Münster
Germany