I'm trying to understand the proof of the following theorem [DELZANT]
Theorem: Let $(M,w)$ be a compact symplectic manifold of dimension $2n$ and $H$ a 1-periodic hamiltonian in $M$. If $H$ only has two critical values and one is non-degenerate(in usual sense), then $M$ is isomorphic to $(\mathbb{CP}^n, \lambda \omega_{FS})$ for a certain $\lambda$.
During the proof, it is supposed that the critical points are $0$ and $\lambda$ where $H(M)= [0,\lambda]$. Then, for regular value $\epsilon$ , it says that $H^{-1}(\epsilon)$ is an sphere ( $S^{2n-1}$). Why is that? I can' t understand it.
Let $(M, \omega)$ be a compact connected symplectic manifold. (We have to assume connectedness, otherwise any disjoint union of $\mathbb{C}P^{n}$'s would work.) Let $H : M \to \mathbb{R}$ be a 1-periodic Hamiltonian and suppose that is only has two critical values; As $M$ is compact, $H$ has a minimum (say w.l.o.g. $0$) and a maximum (say $\lambda \ge 0$), so the assumption on $H$ implies that the two critical values are precisely the maximal value $\lambda > 0$ and the minimal value $0$ of $H$.
Assume that one of the two critical values is nondegenerate (w.l.o.g., possibly considering $\tilde{H} := \lambda - H$ instead, say $0$), that is, for any critical point $p \in H^{-1}(\{0\})$, the Hessian of $H$ at $p$ is nondegenerate. Put differently, the function $H : M \setminus H^{-1}(\{\lambda \}) \to [0, \lambda)$ is Morse. As $M$ is compact, it implies that $H^{-1}(\{0\})$ consists of finitely many isolated points.
With the aid of Morse lemma, one observes that given $p \in H^{-1}(\{0\})$ and a 'Morse chart' $(U, \phi : U \to \mathbb{R}^{2n})$ for $H$ about $p$, there exists $\epsilon > 0$ sufficiently small such that the intersection of a level set $H^{-1}(\{c \})$ for $c \in (0,0 + \epsilon)$ with $U$ is diffeomorphic to $S^{2n-1}$. According to Morse theory, the topology of level sets of a Morse function changes only at critical values. It follows that $M \setminus H^{-1}(\{\lambda \})$ is diffeomorphic to $\sqcup_{p \in H^{-1}(\{0\})} B^{2n}$, where $B^{2n}$ denotes the open 2n-dimensional ball.
It seems at least intuitively plausible that 'the Morse flow' of $H$ glues each $\overline{B^{2n}}$ (or rather, its boundary) to (a connected component of) $H^{-1}(\{\lambda \})$. The gluing has to be done in such a way that we recover the smooth manifold $M$; It is not a trivial result, even though I guess it is intuitively plausible, that this forces each connected component $H^{-1}(\{\lambda\})$ to be glued with precisely one of the $B^{2n}$'s. As we assume $M$ to be connected, this implies that $H^{-1}(\{\lambda \})$ is connected and also that there is only one $B^{2n}$ (i.e. $H^{-1}(\{0\})$) has cardinality 1). This 'proves' the claim.
As the gluing has to be done in an equivariant way with respect to the $S^1$-action induced by $H$, the (surjective) gluing map $\partial B^{2n} = S^{2n-1} \to H^{-1}(\{\lambda \})$ has to factorize through the Hopf fibration $S^{2n-1} \to \mathbb{C}P^{n-1}$. So $M$ would be diffeomorphic to a quotient of $\mathbb{C}P^n$ by an equivalence relation which at most identifies points in the hyperplane $\mathbb{C}P^{n-1}$; It appears difficult (and according to the theorem, impossible) for such a (nontrivial) quotient to be smooth and symplectic (one could collapse $\mathbb{C}P^{n-1}$ to a point, so $M$ would be the smooth manifold $S^{2n}$, but it cannot be symplectic if $n > 1$).