Path of Diffeomorphisms Fixing the Boundary

Question

Could you please let me know if the following is true. The problem came up while constructing a solution of a PDE. I have browsed through the net for an answer. While I came across some articles regarding the identity component of the diffeomorphism group, with my poor geometry and topology I could not really figure out what really is happening. Any help in this direction is appreciated.

QUESTION:

Let $n\geqslant 2$, $k\geqslant 1$, $\Omega\subset\mathbb{R}^n$ be open, bounded, smooth, simply connected with $\partial\Omega$ connected. Let $u:\overline{\Omega}\to\overline{\Omega}$ be a diffeomorphism of class $C^k$ satisfying $$\det(\nabla u)>0\text{ in }\overline{\Omega}\text{ and }u(x)=x\text{ on }\partial\Omega.$$ Does there exist $H\in C^k\left([0,1]\times\overline{\Omega};\mathbb{R}^n\right)$ satisfying

1. $H(0,\cdot)=\text{Id}$ in $\overline{\Omega}$.
2. $H(1,\cdot)=u$ in $\overline{\Omega}$.
3. $\det(\nabla H(t,\cdot))>0\text{ in }\overline{\Omega}$, for all $t$.
4. $H(t,x)=x\text{ on }\partial\Omega$, for all $t$.

Note that, 3 and 4 imply that $H(t,\cdot)$ is a diffeomorphism of $\overline{\Omega}$.

If the result is negative in general, it will be great to have an explicit counterexample. It will also be good to know some cases, if any, when the result is positive.

Answer 1

A diffeomorphism for which there exists such an H is called an isotopy (relative to the boundary).

This has been much studied in the case where Omega is the unit ball in R^n. Your question is well-known as: does pi_0(D^n,\partial D^n)=0?

The answer is positive for the unit balls in R^2 (Smale 1958) and R^3 (Cerf's famous "Gamma4=0" in 1969). This answers positively tour question for n=2 and 3, since in these dimensions, every simply connected compact domain with smooth connected boundary is diffeomorphic to the n-ball.

For the compact unit ball in R^4, your question is a big open question.

The answer to your question is widely negative in large dimensions.

The answer to your question is negative for the unit ball in R^6, as discovered by John Milnor in 1959. It is linked to the existence of "exotic spheres". There are many more counterexamples in large dimensions; see for example Hatcher's review "A 50 year-view on diffeomorphism group". He writes "\pi_0 of Diff(D^n, \partial D^n) is not zero for most n ≥ 5. However it is zero for n = 5, 11, 60." It seems that no other exception than 2,3,5,11,60 is known.