Topologies of level sets of nearby functions

Question

Suppose we have two smooth, real valued functions $\Phi$ and $\hat{\Phi}$ on a manifold $M$. Suppose $\Phi$ and $\Phi$ are close under some function space topology like $L^2$ or $L^\infty$. I am intentionally leaving some details ambiguous.

Let $c$ be a regular value of both functions. What conditions will allow me to conclude $\Phi^{-1}(c)$ and $\hat{\Phi}^{-1}(c)$ are diffeomorphic?

For context, I am studying equipotential surfaces of knotted electric charge distributions. At large distances, the electric potential $\Phi$ can be accurately approximated by the so-called "far-field potential" $\hat{\Phi}$. which treats the knotted charge distribution as a single charged particle. The equipotential surfaces of a single charged particle are spheres, and I want to verify that this approximation is so slight, that it does not affect the topology of a low-voltage (close to $0$) equipotential surface for the actual potential $\Phi$. This result is accepted as true within physics, but I am trying to find a mathematical justification. If it helps, we can assume both functions are harmonic.

Answer 1

I will restrict attention to compact level sets $\hat{\Phi}^{-1}(c)$ of $\hat{\Phi}:M\to \mathbb{R}$. Below I sketch a proof of what is essentially a global implicit function theorem in your setting. For more details, see Lemma 20.3 in "Transversal Mappings and Flows" by Abraham and Robbin.

Fix a regular value $c$ of $\hat{\Phi}$ so that $\hat{N}:= \hat{\Phi}^{-1}(c)$ is a closed embedded submanifold of $M$. Let $\rho:E\hookrightarrow M$ be a smooth tubular neighborhood, where $\pi:E\to \hat{N}$ is a vector bundle. Equip $E$ with a continuous fiberwise norm $\|\cdot\|$ and set $E^r:=\{e\in E: \|e\|\leq r\}$ for $r > 0$. Given $U\subset N$ and $r>0$, define $U^r:=\pi^{-1}(U)\cap E^r$.

For each $n\in \hat N$, the implicit function theorem (Banach space version) yields $r_n > 0$ and neighborhoods $U_n\subset \hat{N}$ of $n$ and $V_n\subset C^\infty(M,\mathbb{R})$ of $\hat \Phi$ in the weak $C^1$ topology such that, for all $\Phi\in V_n$, $U_n^{r_n}\cap (\Phi\circ \rho)^{-1}(c)$ is the image of a smooth local section of $E$.

Since $\hat{N}$ is compact, there are $n_1,\ldots, n_k\in \hat N$ such that $\hat{N}\subset U:= \bigcup_{i=1}^k U_{n_i}$. Define $r:=\min\{r_1,\ldots, r_k\}$ and $V:=\bigcap_{i=1}^k V_{n_k}$. Then for any $\Phi\in V$, we have that $U^r\cap (\Phi\circ \rho)^{-1}(c)$ intersects each fiber of $E$ transversely and thus is the image of a smooth global section $\sigma:\hat N\to E^r$. Defining $N:= \Phi^{-1}(c)\cap \rho(E^r)$, it follows that $\rho\circ \sigma:\hat{N}\to N$ is a diffeomorphism as desired.

Note that, under my assumptions above, if $M$ is noncompact then there may well be other connected components of $\Phi^{-1}(c)$ outside of the neighborhood $\rho(E^r)$ of $\hat{N}$ that cannot be eliminated by approximating $\hat \Phi$ closely in the weak $C^1$ topology by $\Phi$. These can be eliminated if you are able to approximate sufficiently closely in the strong $C^1$ topology instead.