Let $F_s : M \to \mathbb{R}$, $s \in [0,1]$ be a family of smooth functions on a compact manifold $M$ with boundary $\partial M$. Suppose that for any $s \in [0,1]$, $0$ is a regular value of the function $F_s$ as well as its restriction $\partial F_s$ to the boundary $\partial M$.
I would like to prove that there exists a homotopy equivalence of pairs $$ (\{F_0 \leq 0\}, \{\partial F_0 \leq 0\}) \simeq (\{F_1 \leq 0\}, \{\partial F_1 \leq 0\}). $$
I understand how to obtain homotopy equivalences $$ \{F_0 \leq 0\} \simeq \{F_1 \leq 0\}, \quad \{\partial F_0 \leq 0 \} \simeq \{ \partial F_1 \leq 0 \} $$ separately, but I don't know how to build a homotopy equivalence of pairs. Here is what I understand from the problem:
Let $\epsilon > 0$ be such that for any $x \in F_s^{-1}((-\epsilon, \epsilon))$, we have $\nabla_x F_s \neq 0$. Define $$ \chi : \mathbb{R} \to \mathbb{R}_{\geq 0}, \quad \chi(x) = \begin{cases} 1 \text{ on } [-\frac{\epsilon}{2}, \frac{\epsilon}{2}]\\ 0 \text{ on } )-\infty, -\epsilon] \cup [\epsilon, \infty(. \end{cases} $$ Let $X : M \times [0,1] \to \Gamma(TM)$ be the following vector field: $$ X(x,s) := -\frac{\nabla_x F_s}{||\nabla_x F_s||^2} \chi(F_s(x)) \frac{\partial}{\partial s} F_s(x), $$ where $\nabla$ and $||.||$ are respectively the gradient and the norm defined by means of any given Riemannian metric on $M$. Then $X$ is well defined, and its flow $\phi_s$ satisfies $$ \phi_s(F_0^{-1}(0)) \simeq F_s^{-1}(0). $$ By continuity, we have as a consequence $$ \phi_s(\{ F_0 \leq 0\}) = \{F_s \leq 0 \}. $$
A similar argument holds for the boundary, but the function $\chi$ (more precisely the positive number $\epsilon$ and the vector field $X$ are defined in terms of the restriction $\partial F_s$. In particular, they might not by the restrictions to $\partial M$ of the function and vector field above.
Idea of the solution: an idea would be to define a suitable metric $g_s$ dependent on $s \in [0,1]$ such that the gradient $\nabla_x F_s$ at any point $x \in F_s^{-1}(0) \cap \partial M$ always stays tangent to $\partial M$. In this case.
Here are my questions:
- How to define such a metric ?
- Another case which I would like to study is when the function $F : M \to \mathbb{R}$ is fixed, but we consider its restrictions to a family of sub manifolds $\Gamma_s$, $s \in [0,1]$ such that $0$ is always a regular value of $F_{| \Gamma_s}$. It seems very similar, but I cannot find a way to relate both cases. Is there a way to do both at the same time ?
Thanks all in advance for your help.
UPDATE WITH A POSSIBLE SOLUTION:
As said above, the idea could be to define a metric such that the gradient $\nabla_x F_s$ is tangent to $\partial M$ at any point $x \in F_s^{-1}(0) \cap \partial M$. Suppose that such a metric was constructed. Then the flow $\phi_s$ of $X$ preserves $F_s^{-1}(0) \cap \partial M$, and therefore it induces a homotopy equivalence of pairs, as wanted.
Below I'll try to define such a metric. I'll be happy to get feedback regarding the possible problems in my argument.
- Choose an open cover $U_{\alpha}$ of $M$, such that $U_0 \simeq \partial M \times [0,1)$, that is an open neighbourhood of $\partial M$ in $M$, and the other $U_{\alpha}$ are charts. Let also $\rho_{\alpha}$ be a partition of unity subordinate to this cover.
- Let $g_0$ be a metric on $\partial M$. Extend it to $U_0$ in the following way: $$ g_{\alpha}(\partial_t, \partial_t) = 1 \quad \text{and} \quad g_{\alpha}(\partial_t, X) = 0, \quad \text{for all} \quad X \in T \partial M, $$ where $\partial_t$ is the vector tangent to the coordinate in $[0,1)$.
- The form $g := \sum_{\alpha} \rho_{\alpha} g_{\alpha}$ is then a metric on $M$.
The idea now is to modify the open set $U_0$ so that the inward pointing vector $\partial_t$ at each point $x \in F_s^{-1}(0) \cap \partial M$ is tangent to $F_s^{-1}(0)$. I am not sure that such a modification is possible. If it is, then I think the metric $g$ should do the job.
I will be very grateful to get feedback on this.
Regarding the other case, where we deal with a family of sub manifolds $\Gamma_s$ and a function $F: M \to \mathbb{R}$ such that $0$ is a regular value of $F_{|\Gamma_s}$ for any $s \in [0,1]$, I honestly have no idea how to build the vector field, even to obtain a homotopy $$ \lbrace F_{\Gamma_0} \leq 0 \rbrace \simeq \lbrace F_{\Gamma_1} \leq 0 \rbrace, $$ without trying to preserve the boundary.
Thanks a lot in advance.