Circle actions and scalar curvature

Positive Scalar Curvature and Surgery

Taming Free Circle Actions

It is shown that an arbitrary free action of the circle group on a closed manifold of dimension at least six is concordant to a "tame" action (so that the orbit space is a manifold). A consequence is that the concordance classification of arbitrary free actions of the circle on a simply connected manifold is the same as the equivariant homeomorphism classification of free tame actions. Consider...

Positive Scalar Curvature

One of the striking initial applications of the Seiberg-Witten invariants was to give new obstructions to the existence of Riemannian metrics of positive scalar curvature on 4– manifolds. The vanishing of the Seiberg–Witten invariants of a manifold admitting such a metric may be viewed as a non-linear generalization of the classic conditions [12, 11] derived from the Dirac operator. If a manifo...

Diffeomorphisms of the Circle and Hyperbolic Curvature

The trace Tf of a smooth function f of a real or complex variable is defined and shown to be invariant under conjugation by Möbius transformations. We associate with a convex curve of class C2 in the unit disk with the Poincaré metric a diffeomorphism of the circle and show that the trace of the diffeomorphism is twice the reciprocal of the geodesic curvature of the curve. Then applying a theor...

On Nearly Semifree Circle Actions

Recall that an effective circle action is semifree if the stabilizer subgroup of each point is connected. We show that if (M,ω) is a coadjoint orbit of a compact Lie group G then every element of π1(G) may be represented by a semifree S-action. A theorem of McDuff– Slimowitz then implies that π1(G) injects into π1(Ham(M,ω)), which answers a question raised by Weinstein. We also show that a circ...