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Motivated by recent works on Hamiltonian circle actions satisfying certainminimal conditions, in this paper, we consider Hamiltonian circle actionssatisfying an almost minimal condition. More precisely, we consider a compactsymplectic manifold $(M, \omega)$ admitting a Hamiltonian circle action withfixed point set consisting of two connected components $X$ and $Y$ satisfying$\dim(X)+\dim(Y)=\dim(M)$. Under certain cohomology conditions, we determinethe circle action, the integral cohomology rings of $M$, $X$ and $Y$, and thetotal 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 asthose 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 theaction is holomorphic, we can use a few different criteria to claim that $M$ is$S^1$-equivariantly biholomorphic and $S^1$-equivariantly symplectomorphic to$\Gt_2(\R^{2n+2})$.

Hamiltonian circle actions with fixed point set almost minimal