#### Volume 18, issue 5 (2014)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement Author Index To Appear ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Other MSP Journals
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 ${\mathbb{S}}^{3}$ that can be realized as an extremal set with respect to an inner metric on ${\mathbb{S}}^{3}$ 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
##### Publication
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