Let $V$ be an analytic subvariety of a domain $\Omega$ in $\C^n$. When does $V$ have the Isometric Extension Property (IEP), i.e. when does every bounded holomorphic function $f$ on $V$ has an extension to a bounded holomorphic function on $\Omega$ with the same norm?
If $V$ is a retract, i.e. if there exists a holomorphic $r: \Omega \to V$ so that $r|_V = {\rm id}$, then there is an obvious isometric extension, namely $f \circ r$. If $\Omega$ is very nice, for example the ball, this condition is also necessary. We shall discuss why convexity assumptions lead to theorems that say only retracts have the IEP.
If the convexity assumption is dropped, functional analysis can be used to show that every $V$ has the isometric extension property for some $\Omega$. We shall discuss the proof of this theorem. This is joint work with Jim Agler and Lukasz Kosinski. | 
