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Let the circle act in a Hamiltonian fashion on a connected compact symplecticmanifold $(M, \omega)$ of dimension $2n$. Then the $S^1$-action has at least$n+1$ fixed points. In a previous paper, we study the case when the fixed pointset consists of precisely $n+1$ isolated points. In this paper, we study thecase when the fixed point set consists of exactly $n+2$ isolated points. Weshow that in this case $n$ must be even. We find equivalent conditions on thefirst Chern class of $M$ and a particular weight of the $S^1$-action. We alsoshow that the particular weight can completely determine the integralcohomology ring and the total Chern class of $M$, and the sets of weights ofthe $S^1$-action at all the fixed points. We will see that all these data areisomorphic to those of known examples, $\widetilde{G}_2(\mathbb{R}^{n+2})$ with$n\geq 2$ even, equipped with standard circle actions.

Hamiltonian circle actions with almost minimal isolated fixed points