3
$\begingroup$

I'm stuck on the following problem: let $S$ be a compact orientable hypersurface in the symplectic manifold $(M,\omega)$. Prove that there exists a smooth function $H: M \to \mathbb R$ such that $0$ is a regular value of $H$ and $S \subset H^{-1}(0)$.

Since $S$ and $M$ are orientable, I can find a tubular neighborhood $N\simeq S \times (-\epsilon, \epsilon)$, which is open in $M$ and with $S$ corresponding to $S \times \{0\}$. Then $S$ is the inverse image of the regular value 0 under the projection onto the second factor $N \to (-\epsilon,\epsilon)$. Is there a clean way to see that I can extend this map to all of $M$ such that 0 remains a regular value?

  • 2
    ah, I'm on leave of absence from my mind :) But one more attempt: take a smooth function on $\mathbb{R}$ which is $1$ away from $(-\epsilon,\epsilon)$ and has two regular zeros, one at $0$ and the other at (say) $\epsilon/2$. This gives you a function on $S\times(-\epsilon,\epsilon)$ where $0$ is a regular value and $H^{-1}(0)$ are two copies of $S$. Extend this function by $1$ to the resto of $M$.2011-04-21

1 Answers 1

1

Perhaps my idea is silly, but could not you take $H=f\circ pr_2$? where $pr_2$ is your projection on the second factor and $f$ is a real-valued smooth function on $]-\epsilon,+\epsilon[$ such that: its support is compact and $f(0)\neq 0$ is one of its regular values. For example you could take f(x)=exp(1/(x−2ϵ/3))exp(−1/(x+ϵ/3)) for x∈]−ϵ/3,2ϵ/3[ and f(x)=0 otherwise.

The proposed $H$, defined on $N$, has compact support $S\times\mathrm{supp}(f)$, so it prolongs to the smooth function on $M$ vanishing outside $N$. In such a way, you get a smooth function $H$ on $M$ with a constant value $f(0)$ on $S$ which is not one of its singular values.

  • 0
    Sure $H^{-1}(c)$ could contain more than $S$. It is properly $pr_2^{-1}(f^{-1}(f(0)))$ by constuction, and does not contain singular points because $pr_2$ is non singular and because I requested $f(0)$ is not a singular value for $f$( so in particular $c=f(0)\neq 0$.2011-04-21