Symplectic periodic flows with exactly three equilibrium points

TL;DR

This paper proves that a symplectic manifold with a circle action having exactly three fixed points is symplectomorphic to the complex projective plane, revealing a unique geometric structure under these conditions.

Contribution

It establishes a classification result for symplectic manifolds with a specific circle action fixed point count, showing they are all equivalent to P^2.

Findings

The only such manifold with three fixed points is P^2.

The circle action structure uniquely determines the manifold's symplectic type.

Provides a classification for symplectic manifolds with minimal fixed points.

Abstract

Let the circle act symplectically on a compact, connected symplectic manifold $M$. If there are exactly three fixed points, $M$ is equivariantly symplectomorphic to $\mathbb{CP}^2$.