Is it true that every bounded holomorphic functions on a smooth analytic hypersurface $X$ of $\Bbb C^n$ is constant?


* Remark that if $X$ is algebraic, the answer is yes.


Otherwise can you provide counterexamples?



Bounded holomorphic functions on hypersurfaces of $\Bbb C^n$
This is not true: there are smooth hypersurfaces (even curves in $C^2$)
which are holomorphically equivalent to the unit disk in the plane.


MR1359950 
Globevnik, Josip, Stensønes, Berit,
Holomorphic embeddings of planar domains into $C^2$. 
Math. Ann. 303 (1995), no. 4, 579–597.


