I have a question I have been thinking for a while and feel like it should be true but have no idea of how to prove it. First let me introduce a piece of notation.

Consider $(\mathbb{CP}^n,\omega)$ where $\omega$ is the standard Fubini form. We say that a function $f:\mathbb{CP}^n\to \mathbb{R}$ is height-like if $$f([z_0:\ldots:z_n])=g(\|z_0\|,\ldots,\|z_n\|),$$ for $g:\mathbb{R}^{n+1}\to \mathbb{R}$ having a critical point only at the origin.

Obs: This is a generalization of the notion of height fuction on $S^2\cong \mathbb{CP}^1$, given by $f([z_0:z_1])=\frac{\|z_1\|^2-\|z_0\|^2}{\|z_1\|^2+\|z_0\|^2}$. In higher dimensions we could take for example: $$f([z_0:\ldots:z_n])=\sum_{i=0}^n\frac{i\|z_i\|^2}{\|z_1\|^2+\ldots+\|z_n\|^2}$$

Height-like functions in $\mathbb{CP}^n$ have exactly $n+1$ critical points, they are: $$[1:0:\ldots:0],\ldots,[0:0:\ldots:1].$$

So, my question is, given a Morse function $f$ in $\mathbb{CP}^n$ with exactly $n+1$ critical points (notice $f$ is not necessarily height-like), does there exists a symplectomorphism $\Phi:\mathbb{CP}^n\to\mathbb{CP}^n$ s.t. $g:=f\circ\Phi$ is hight-like?

I think I have a proof for $n=1$, roughly speaking, in dimension 2 $\Phi$ is symplectomorphic if and only if takes loops bounding a disk of area $A$ to loops bounding a disk of area $A$. Such a map coul be ''relatively" easy to accomplish for $n=1$. Inhiger dimensions this idea is completely useless.

I would appreciate any help I could get, either a negative answer to my question or a positive answer, hopefully with a proof.

I have a question I have been thinking for a while and feel like it should be true but have no idea of how to prove it. First let me introduce a piece of notation.

Consider $(\mathbb{CP}^n,\omega)$ where $\omega$ is the standard Fubini form. We say that a function $f:\mathbb{CP}^n\to \mathbb{R}$ is height-like if $$f([z_0:\ldots:z_n])=g(\|z_0\|,\ldots,\|z_n\|),$$ for $g:\mathbb{R}^{n+1}\to \mathbb{R}$ having a critical point only at the origin.

Obs: This is a generalization of the notion of height fuction on $S^2\cong \mathbb{CP}^1$, given by $f([z_0:z_1])=\frac{\|z_1\|^2-\|z_0\|^2}{\|z_1\|^2+\|z_0\|^2}$. In higher dimensions we could take for example: $$f([z_0:\ldots:z_n])=\sum_{i=0}^n\frac{i\|z_i\|^2}{\|z_1\|^2+\ldots+\|z_n\|^2}$$

Height-like functions in $\mathbb{CP}^n$ have exactly $n+1$ critical points, they are: $$[1:0:\ldots:0],\ldots,[0:0:\ldots:1].$$

So, my question is, given a Morse function $f$ in $\mathbb{CP}^n$ with exactly $n+1$ critical points (notice $f$ is not necessarily height-like), does there exists a symplectomorphism $\Phi:\mathbb{CP}^n\to\mathbb{CP}^n$ s.t. $g:=f\circ\Phi$ is hight-like?

I think I have a proof for $n=1$, roughly speaking, in dimension 2 $\Phi$ is symplectomorphic if and only if takes loops bounding a disk of area $A$ to loops bounding a disk of area $A$. Such a map could be ''relatively" easy to accomplish for $n=1$. In higer dimensions this idea is completely useless.

I would appreciate any help I could get, either a negative answer to my question or a positive answer, hopefully with a proof.