Lets consider a Morse function $f:\mathbb{S}^2\rightarrow \mathbb{R}$ such that it has two maximal points, one minumun and one saddle at $c$. Notice that $f^{-1}(-\infty,c)$ is topologically a disk.
Can we use $f$ as a coordinate to parametrize $f^{-1}(-\infty,c)$?
Similarly $f^{-1}(c,+\infty)$ consist (topologically) of two disks.
Can we use $f$ as a coordinate to parametrize each of the connected componets of $f^{-1}(c,+\infty,)$?
In general I am interested in knowing if something like this can be done in a somewhat confortable way. I thought about this for a while but I could not find a solution myself. I also would like to know if there is a way to consider the oposite problem:
Given three function Morse $f_1,f_2,f_3$ on the disks each with only one maximum each (one minimun and two maximum). Can we construct a Morse function on the sphere $f:\mathbb{S}^2\rightarrow \mathbb{R}$ such that the the respective restriction to the disk are $f_1,f_2,f_3$? Or do we need some conditions?
Thanks for the answers!