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

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 S-3 that can be realized as an extremal set with respect to an inner metric on S-3 that has nonnegative curvature in the Alexandrov sense.