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Let $C$ be a round circle in $R^2$ centered at $p$. Let $g$ be a rotation fixing $p$; then, as a Moebius transformation of $R^2\cup \infty$, $g$ has two fixed points $p, \infty$. Let $g^\sigma$ be the …

Schlicht domain over ${\mathbb C}^n$ is the same as a domain in ${\mathbb C}^n$. The point is that one also defines domains over ${\mathbb C}^n$ as connected complex manifolds $M^n$ equipped with a lo …

There are essentially two cases when the limit is not a covering map: (a) the limit is constant, (b) the images $S_n:=f_n(D)$ are annuli which collapse to a circle. There could be a more elementary ex …

In Theorem 3 of this paper, for every $\alpha>1$, Chris Bishop constructs examples of Cantor sets $E\subset {\mathbb C}$ whose Hausdorff dimension is in the interval $(1, \alpha)$ and homeomorphisms …

For the 2nd part one can construct an example by imitating the Sierpinski Carpet construction. Or, you use Kleinian groups. Take two Fuchsian subgroups $F_1, F_2$ of $PSL(2,C)$ with disjoint limit cir …

Not only $f$ admits an analytic continuation across boundary, in fact, $f$ is the restriction of a linear transformation. Indeed, the interior of $Q\times Q$ is the 2-dimensional polydisk (more precis …

Proposition. If $D$ is a bounded connected domain in ${\mathbb C}^n$ and $\Gamma$ is a nilpotent group of biholomorphic transformations of $D$ acting freely and properly discontinuously on $D$, then …

Two relevant references: W. Abikoff, Some remarks on Kleinian groups. 1971 Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969) pp. 1–5. Ann. of Math. Studies, No. 66. P …

No, we cannot. Let $S$ be a connected noncompact Riemann surface whose group of conformal automorphisms is nondiscrete. Then (this follows from the uniformization theorem) $S$ is either conformal to …

A more general result is proven in Gunning, R. C., Narasimhan, R., Immersion of open Riemann surfaces. Math. Ann. 174, 103–108 (1967). As for compact surfaces with boundary, it is essentially a par …

Isolated points are removable singularities for quasiconformal maps, see for instance Theorem 17.3 in Vaisala's book (where higher-dimensional case is proven too). J. Vaisala, Lectures on n-dimensio …

It follows from the proof in Sasane's paper that Krull dimension of a (connected) complex manifold $M$ is infinite iff $M$ admits a nonconstant holomorphic function $F: M\to {\mathbb C}$. Namely, usin …

Take product of a hyperbolic manifold and a Riemann surface.

The cocompact case in this setting is due to Donaldson: S.Donaldson, Twisted harmonic maps and the self-duality equations. Proc. London Math. Soc. (3) 55 (1987), no. 1, 127–131. Simpson proved a gen …

An example exists already for $M={\mathbb C}P^2$, furthermore, there exists an injective holomorphic map $f: {\mathbb C}^2\to {\mathbb C}^2\subset {\mathbb C}P^2$ whose image is open but not dense. R …