A knot characterization and 1-connected nonnegatively curved 4-manifolds with circle symmetry

TL;DR

This paper classifies certain 4-manifolds with nonnegative curvature and circle symmetry, and explores which knots can appear as extremal sets in nonnegatively curved metrics on S^3.

Contribution

It provides a classification of nonnegatively curved 4-manifolds with circle symmetry and characterizes knots realizable as extremal sets in such geometries.

Findings

Classification of nonnegatively curved 4-manifolds with circle symmetry

Identification of knots that can be extremal sets in nonnegatively curved S^3

Extension of classification to knotted curves in the singular set

Abstract

We classify nonnegatively curved simply connected 4-manifolds with circle symmetry up to equivariant diffeomorphisms. The main problem is rule out knotted curves in the singular set of the orbit space. As an extension of this work we classify all knots in S^3 which can be realized as an extremal set with respect to an inner metric on S^3 which has nonnegative curvature in the Alexandrov sense.