Let $f$ be a given Morse-Smale function $f$ on $\mathbb R^n$ with finite many critical points and sufficient growth at $\infty$ like $\langle x, f(x)\rangle \geq |x|$ (cp. MO120858).

Is there a way to find a perturbation $\tilde f$ of $f$ such that:

  1. the eigenvalues of the Hessian are controlled in a quantitative way without changing the index
  2. and the stable manifolds of $\tilde f$ and $f$ are the same?

One naive way is to use the local stable manifold theorem to linearize the coordinates of the Morse function $f$ around a critical point $p$ and then do a quadratic perturbation. But this is only sufficient, if there are no other stable manifolds in the neighborhood of the critical point $p$, i.e. only if $W^u_{\mathrm{loc}}(p)\cap W^s(q) =\emptyset$ for all $q\in\mathrm{Crit(f)}$.

On the other hand the Morse-Smale property ensures that $W^u(p)\cap W^s(q)$ is again a manifold and one can use the fact in the construction. From this point of view such a construction looks possible but technical involved. Is there a more simple approach?



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