Let $B$ be the open unit ball in $\mathbb C^n$. Consider the space $\mathcal F$ of holomorphic embeddings of $B$ in $\mathbb C^n$ equipped with the compact-open topology. (A holomorphic embedding of $B$ in $\mathbb C^n$ is a holomorphic map $f: B\to \mathbb C^n$ such that $f(B)$ is open and there is a holomorphic inverse $f^{-1}: f(B)\to B$). Is it known whether the space $\mathcal F$ is connected?



Space of holomorphic embeddings of open unit ball in ${\mathbb C}^n$
The space of holomorphic embeddings with Jacobi matrix = 1 at zero is contractible due to the standard construction $f\_t=(1/t)f(zt)$, and $f\_0$ defined as a limit will be equal to the identity map ($t$ changes from 1 to 0). The similar proof also works in the smooth category.


