Let $M^n$ be a smooth closed embedded hupersurface in $\mathbb R^{n+1}$. Denote by $D$ the bounded connected component of $\mathbb R^{n+1}\backslash M$. We assume that $\mathbb R^{n+1}\backslash D$ is simply connected. Let $B=B_R^n$ be the ball centered at the origin with radius $R>>1$ whose boundary is denoted by $S$.

Question: Does there exist a smooth map $$F:M\times [0,1]\rightarrow \bar B\backslash D$$ such that:

1) $F(x,0)=x,$, $f(x, 1)\in S, \forall x\in M$,

2) For any fixed $x\in M$, $F(x,\cdot):[0,1]\rightarrow \bar B\backslash D$ is injective.

The motivation of this question comes from several complex variables, where $M$ is taken to be the boundary of a bounded domain.