  Motivated by recent works on Hamiltonian circle actions satisfying certain
minimal conditions, in this paper, we consider Hamiltonian circle actions
satisfying an almost minimal condition. More precisely, we consider a compact
symplectic manifold $(M, \omega)$ admitting a Hamiltonian circle action with
fixed point set consisting of two connected components $X$ and $Y$ satisfying
$\dim(X)+\dim(Y)=\dim(M)$. Under certain cohomology conditions, we determine
the circle action, the integral cohomology rings of $M$, $X$ and $Y$, and the
total Chern classes of $M$, $X$, $Y$, and of the normal bundles of $X$ and $Y$.
The results show that these data are unique --- they are exactly the same as
those in the standard example $\Gt_2(\R^{2n+2})$, the Grassmannian of oriented
$2$-planes in $\R^{2n+2}$, which is of dimension $4n$ with (any) $n\in\N$,
equipped with a standard circle action. Moreover, if $M$ is K\"ahler and the
action is holomorphic, we can use a few different criteria to claim that $M$ is
$S^1$-equivariantly biholomorphic and $S^1$-equivariantly symplectomorphic to
$\Gt_2(\R^{2n+2})$.
