I am looking for a known upper bound on the number of monotone regions of a Morse function by the number of its critical points in the interior of the manifold and on its boundary.
Here I try to give a motivation
Given a Morse function $f:\mathbb{R}^N \to \mathbb{R}$, I look at the simple integral $\int_{(0,t)^N} e^{-f(x)^2}\nabla(f(x))dx$. I would like to make a change of variables, thus I consider the number of monotonous regions of $f$ in $(0,t)^N$.
Suppose then that $\{A\}_i^{M}$ is the number of such a regions, then I have
$\int_{(0,t)^N} e^{-f(x)^2}\nabla(f(x))dx = \sum_{\{A_i\}}\int_{A_i} e^{-x^2}dx$
Having done that I am looking to get an upper bound on this sum by, say, $T(p,g)$
$\sum_{\{A_i\}}\int_{A_i} e^{-x^2}dx \leq T(p,g)$
Where $p$ is the number of the critical points in the interior of $(0,t)^N$ and $g$ is the number of the critical points on the boundary $[0,t]^N$ (that is the critical point of the restriction of $f$ to the boundary).
Any suggestions are welcome.