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Liouville's theorem states that all bounded holomorphic functions on $\mathbb{C}^n$ are constant.

I'm wondering which connected complex manifolds have this property ?

Connected compact complex manifolds have it since all holomorphic functions there are constant.

There are no simply connected proper open subsets of $\mathbb{C}$ satisfying this because of the Riemann Mapping Theorem. But are there some other open subsets satisfying it ?

In higher dimension there are some, such as the Fatou–Bieberbach domains (open subsets of $\mathbb{C}^n$ biholomorphic to it).

I would be interested in references on this property on complex manifolds.

Incidentally, is it true that any complex manifold where all holomorphic functions are bounded is compact ?

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3 Answers 3

up vote 12 down vote accepted

We have that there are no non-constant bounded functions on $\mathbb C^*=\mathbb C\setminus\{0\}$. The easiest way to see that is to notice that such a function has a removable singularity at the origin and hence comes from a bounded function on $\mathbb C$ (which incidentally has a removable singularity at $\infty$ and hence extends to the Riemann sphere and therefore is constant).

As for the higher-dimensional problem it is no doubt hopeless to get anything like a complete description: There are as you say domains in $\mathbb C^n$ (and there are many of those), the compact manifolds but also compact manifolds minus a codimension $2$ closed analytic subspace, the blowing up of some space that has the property, any product of two manifolds with the property and so on. Complex manifolds in higher dimension are simply too varied.

Finally, there are non-compact manifolds with only constant holomorphic functions. If $L\rightarrow X$ is an analytic line bundle over a complex manifold $X$ and $f\colon L\rightarrow\mathbb C$ is a holomorphic function, then we may Taylorexpand $f$ along the zero section of $L$: First we just look at the restriction of $f$ to the zero section which gives a function on $X$. Then we may take any local section of $L$, think of that as a tangent vector at the zero section and take the derivative of $f$ along this tangent vector. This glues together to give the first derivative as a global section of $L^{-1}$ and similarly the $n$'th derivative of $f$ along the zero section will be a section of $L^{-n}$. If $X$ is compact and $L^{-n}$ for $n>0$ only has the zero section as global section, then $f$ is constant along the zero section and all its higher derivatives along it are zero so that $f$ is constant in a neighbourhood of the zero section and hence constant. There are lots of such examples $(X,L)$.

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As it was already mentioned in the previous answers, although it is not really a problem to give an example of a non-compact complex manifold with the Liouville property (absence of non-constant bounded holomorphic functions), it is hopeless to describe all such manifolds. This question becomes more approachable if one restricts it to the class of covering manifolds with a reasonable (e.g., compact) quotient. Any nilpotent cover of a compact manifold (more generally, of any manifold with no non-constant bounded plurisubharmonic functions) has the Liouville property.

It is interesting to compare this result with what is known about the Liouville property for harmonic functions with respect to a Hermitian metric on a complex manifold, or just with respect to a Riemannian metric on a real manifold. If the quotient manifold has no Green function (in particular, compact), then any nilpotent cover is Liouville. Moreover, any polycyclic cover of a compact manifold also has the Liouville property.

An example of compact complex manifolds with a polycyclic fundamental group is provided by so-called Inoue surfaces, for which the universal covering manifold is the product $\mathbf{H}\times\mathbb{C}$, and therefore has lots of bounded holomorphic functions. For any Kähler and, more generally, semi-Kähler metric (one for which the Kähler form is $\delta$-closed) any holomorphic function is harmonic. Therefore, Inoue surfaces do not admit such metrics.

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The reference for complex manifolds in dimention 1 (Riemann surfaces) is the subject known under the title "Classification theory of (open) Riemann surfaces". It used to be very popular in the 1950-s. Some books are:

M. Tsuji, Potential theory in modern function theory,

L. Ahlfors and Sario, Riemann surfaces.

Look also at the nice paper by P. Doyle, On deciding whether a surface is parabolic or hyperbolic.

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